Cassels, Froehlich (Eds) - Algebraic Number Theory (378s)

378
Algebraic Number Theory PROCEEDINGS OF AN INSTRUCTIONAL CONFERENCE ORGANIZED BY THE LONDON MATHEMATICAL SOCIETY (A NATO ADVANCED STUDY INSTITUTE) WITH THE SUPPORT OF THE INTERNATIONAL MATHEMATICAL UNION Edited by J. W. S. CASSELS Trinity College, University of Cambridge, U.K. and A. FRiiHLICH King’s College, University of London, U.K. ACADEMIC PRESS Harcourt Brace Jovanovich, Publishers London San Diego . New York . Berkeley Boston . Sydney . Tokyo . Toronto

Transcript of Cassels, Froehlich (Eds) - Algebraic Number Theory (378s)

Page 1: Cassels, Froehlich (Eds) - Algebraic Number Theory (378s)

Algebraic Number Theory

PROCEEDINGS OF AN INSTRUCTIONAL CONFERENCE ORGANIZED BY THE LONDON MATHEMATICAL SOCIETY

(A NATO ADVANCED STUDY INSTITUTE) WITH THE SUPPORT OF THE

INTERNATIONAL MATHEMATICAL UNION

Edited by

J. W. S. CASSELS

Trinity College, University of Cambridge, U.K.

and

A. FRiiHLICH

King’s College, University of London, U.K.

ACADEMIC PRESS

Harcourt Brace Jovanovich, Publishers London San Diego . New York . Berkeley

Boston . Sydney . Tokyo . Toronto

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CONTRIBUTORS

J. V. ARMITAGE, Department of Mathematics, University of Durham, U.K. M. F. ATIYAH, Mathematical Institute, University of Oxford, U.K. B. J. BIRCH, Mathematical Institute, University of Oxford, U.K. D. A. BURGESS, Department of Mathematics, University of Nottingham, U.K. J. W. S. CASSELS, Trinity College, University of Cambridge, U.K. A. FR&XLICH, Department of Mathematics, Kings College, University of London,

(I. K. K. GRUENBERG, Department of Mathematics, Queen Mary College, University of

London, U.K. H. HALBERSTAM, Department of Mathematics, University of Nottingham, U.K. H. HASSE, Mathematisches Seminar der Universitat, Hamburg, Germany H. A. HEILBRONN, Department of Mathematics, University of Toronto, Canada K. HOECHSMANN, Mathematisches Institut der Universitat, Tiibingen, Germany M. KNESER, Mathematisches Institut der Universitiit, Giittingen, Germany R. R. LAXTON, Department of Mathematics, University of Nottingham, U.K. A. LUE, Department of Mathematics, Kingg’s College, University of London, (1. K. I. G. MACDONALD, Mathematical Institute, University of Oxford, U.K. J. NEGGERS, Department of Mathematics, University of Puerto Rico, Puerto Rico,

U.S.A. P. ROQUETTE, Mathematisches Institut der Universitiit, Tiibingen, Germany J-P. SERRE, Chaire d’Algtbre et Gtomttrie, College de France, Paris, France H. P. F. SWINNERTON-DYER, Trinity College, University of Cambridge, U.K. J. T. TATE, Department of Mathematics, Harvard University, Cambridge, Mass.,

U.S.A. C. T. C. WALL, Department of Mathematics, University of Liverpool, U.K.

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PREFACE

This volume is the fruit of an instructional conference on algebraic number theory, held from September 1st to September 17th, l-965, in the University of Sussex, Brighton. It was organized by the London Mathematical Society under the auspices and with the generous financial aid of the International Mathematical Union and of the Advanced Study Programme of NATO. The organizers of the conference owe a great deal to the constant support which they received from the authorities of the host University.

All the lectures held during the conference are recorded here with the exception of a few informal seminars. The drafts for publication were either supplied by the speakers themselves, or were prepared by members of the conference in collaboration with the lecturers. We wish to express our deep gratitude to the lecturers both for ensuring the success of the meeting itself, as well as for enabling us subsequently to publish this volume. We are no less grateful to the “note-takers” for their co-operation. Both they and the lecturers also assisted in the proof correction. The editors must emphasize, however, that neither the lecturers nor the note-takers have any responsibility for any inaccuracies which may remain: they are an act of God.

Apart from accounts of the lectures this volume also contains exercises compiled by Tate with Serre’s help, and above all Tate’s doctoral thesis, which is for the first time published here after it had over many years had a deep influence on the subject as a piece of clandestine literature.

Finally we wish to express our appreciation for the co-operation which we received from our publishers.

January 1967 J. W. S. Cassels A. Frohlich

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CONTENTS

CONTRIBUTORS . . . . . . . .

PREFACE . . . . . . . . . .

1NTR00ucT10~ . . . . . . . .

CHAPTER I

Local Fields

A. FR~HLICH

1. Discrete Valuation Rings . . . . 2. Dedekind Domains . . . . . . 3. Modules and Bilinear Forms . . 4. Extensions . . . . . . . . 5. Ramification . . . . . . . . 6. Totally Ramified Extensions. . . . 7. Non-ramified Extensions . . . . 8. Tamely Ramified Extensions . . 9. The Ramification Groups . . . .

10. Decomposition . . . . . . Bibliography . . . . . . . .

1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12.

CHAPTER II

Global Fields

J. W. S. CASSELS

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Valuations . . . . . . . . . . Types of Valuation . . . . . . . . Examples of Valuations . . . . . . Topology . . . . . . ’ . . . . Completeness . . . . . . . . . Independence . . . . . , . . . . Finite Residue Field Case . . . . . . Normed Spaces . . . . . . . . Tensor Product . . . . . . . . Extension of Valuations . . . . . . Extensions of Normalized Valuations . . Global Fields . . . . . . . . . .

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V

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42 44 45 46 47 48 49 52 53 56 58 60

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X CONTJZNTS

13. Restricted Topological Product ........ 14. Adele Ring .............. 15. Strong Approximation Theorem ........ 16. Idele Group .............. 17. Ideals and Divisors ...... ...... 18. Units ................ 19. Inclusion and Norm Maps for Adeles, Ideles and Ideals. .

APPENDIX A

Norms and Traces ............

APPENDIX B

Separability ..............

APPENDIX C

Hensel’s Lemma ............

CHAPTER III

Cyclotomic Fields and Kummer Extensions

B. J. BIRCH

1. Cyclotomic Fields ............ 2. Kummer Extensions ............

APPENDIX

Kummer’s Theorem ............

CHAPTER IV

Cohomology of Groups

M. F. ATIYAH AND C. T. C. WALL

. . 62

. . 63

. . 67 . . 68 . . 70 . . 71 . . 73

. . 76

. . 80

. . 83

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. . 92

1. 2. 3. 4. 5. 6. 7. 8. 9.

10.

(Prepared by I. G. Macdonztylh;he basis of a manuscript of

Definition of Cohomology . . . . . . . . . . . . 94 The Standard Complex . . . . . . . . . . . . 96 Homology . . . . . . . . . . . . . . . . 97 Change of Groups . . . . . . . . . . . . . . 98 The Restriction-Inflation Sequence . . . . . . . . 100

The Tate Groups . . . . . . . . . . . . . . 101 Cup-products . . . . . . . . . . . . . . . . 105

Cyclic Groups: Herbrand Quotient . . . . . . 108 Cohomological Triviality . . . . . . . . . . :: 111

Tate’s Theorem . . . . . . . . . . . . . . 113

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CONTENTS

CHAPTER V

Profinite Groups

K. GRUENBERG

1. The Groups . . . . . . . . . . . . . . 1.1. Introduction . . . . . . . . . . . . 1.2. Inverse Systems . . . . . . . . . . . . 1.3. Inverse Limits . . . . . . . . . . . . 1.4. Topological Characterization of Profinite Groups . . 1.5. Construction of Profinite Groups from Abstract Groups 1.6. Profinite Groups in Field Theory . . . . . .

2. The Cohomology Theory . . . . . . . . . . 2.1. Introduction . . . . . . . . . . . . 2.2. Direct Systems and Direct Limits . . . . . . 2.3. Discrete Modules . . . . . . . . . . 2.4. Cohomology of Profinite Groups . . . . . . 2.5. An Example: Generators of pro-p-Groups . . . . 2.6. Galois Cohomology I: Additive Theory . . . . 2.7. Galois Cohomology II: “Hilbert 90” . . . . . . 2.8. Galois Cohomology III: Brauer Groups . . . .

References . . . . . . . . . . . . . . . .

xi

. . 116 . . 116 . . 116 . . 117 . . 117 . . 119 . . 119 . . 121 . . 121 . . 121 . . 122 . . 122 . . 123 . . 123 . . 124 . . 125 . . 127

CHAPTER VI

Local Class Field Theory

J-P. SERRE

(Prepared by J. V. Armitage and J. Neggers)

Introduction . . . . . . . . . . . . . . . . 129 1. The Brauer Group of a Local Field . . . . . . . . 129

1.1. Statements of Theorems . . . . . . . . . . 129 1.2. Computation of H*(K,,/K) . . . . . . . . . . 13 1 1.3. Some Diagrams . . . . . . . . . . . . 133 1.4. Construction of a Subgroup with Trivial Cohomology . . 134 1.5. An Ugly Lemma. . . . . . . . . . . . . . 135 1.6. End of Proofs . . . . . . . . . . . . . . 136 1.7. An Auxiliary Result . . . . . . . . . . . . 136

APPENDIX Division Algebras Over a Local Field . . . . . . 137

2. Abelian Extensions of Local Fields . . . . . . . . . . 139 2.1. Cohomological Properties . . . . . . . . . . 139 2.2. The Reciprocity Map . . . . . . . . . . . . 139

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xii CONTENTS

2.3. Characterization of (01, L/K) by Characters . . 2.4. Variations with the Fields Involved . . . . 2.5. Unramified Extensions . . . . . . . . 2.6. Norm Subgroups . . . . . . . . 2.7. Statement of the Existence Theorem . . . . 2.8. Some Characterizations of (a, L/K) . . . . 2.9. The Archimedean Case . . . . . . . .

3. Formal Multiplication in Local Fields . . . . 3.1. The Case K= Q, . . . . . . . . 3.2. Formal Groups . . . . . . . . . . 3.3. Lubin-Tate Formal Group Laws . . . , 3.4. Statements . . . . . . . . . . 3.5. Construction of FJ and [a]/ . . . . . . 3.6. First Properties of the Extension K, of K . . 3.7. The Reciprocity Map . . . . . . . . 3.8. The Existence Theorem . . . . . .

4. Ramification Subgroups and Conductors . . . . 4.1. Ramification Groups . . . . . . . . 4.2. Abelian Conductors . . . . . . . . 4.3. Artin’s Conductors . . . . . . . . 4.4. Global Conductors . . . . . . . . 4.5. Artin’s Representation . . . . . . . .

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1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12.

CHAPTER VII

Global Class Field Theory

J. T. TATE (Prepared by B. J. Birch and R. R. Laxton)

Action of the Galois Group on Primes and Completions Frobenius Automorphisms . . . . . . . . . . Artin’s Reciprocity Law . . . . . . . . . . Chevalley’s Interpretation by Ideles . . . . . . Statement of the Main Theorems on Abelian Extensions Relation between Global and Local Artin Maps . . . . Cohomology of Ideles . . . . . . . . . . Cohomology of Idele Classes (I): The First Inequality . . Cohomology of Idele Classes (II): The Second Inequality Proof of the Reciprocity Law . . . . . . . . Cohomology of Idtle Classes (III): The Fundamental Class Proof of the Existence Theorem . . . . . . . .

List of Symbols . . . . . . . . . . . . . .

. . 140

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. . 147 . . 147 . . 148 . . 149 . . 151 . . 152 . . 154 . . 155 . . 155 . . 157 . . 158 . . 159 . . 160

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. . 174

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. . 193 . . 201 . . 203

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CONTENTS

CHAPTER VIII

Zeta-Functions and L-Functions

H. HEILBRONN

(Prepared by D. A. Burgess and H. Halberstam)

1. Characters . . . . . . . . . . 2. Dirichlet L-series and Density Theorems . . 3. L-functions for Non-abelian Extensions . . References . . . . . . . . . . . .

CHAPTER IX

On Class Field Towers

PETER RCQUE-M’E

1. Introduction . . . . . . . . . . 2. ProofofTheorem3 . . . . . . . . 3. Proof of Theorem 5 for Galois Extensions References . . . . . . . . . . . .

CHAPTER X

. . . . . . . . . . . . . . . .

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Semi-Simple Algebraic Groups

M. KNESFJR (Prepared by I. G. Macdonald)

Introduction . . . . . . . . . . . . . .

1. Algebraic Theory . . . . . . . . . . . . 1. Algebraic Groups over an Algebraically Closed Field 2. Semi-Simple Groups over an Algebraically Closed Field 3. Semi-Simple Groups over Perfect Fields . . . .

2. Galois Cohomology . . . . . . . . . . . . 1. Non-Commutative Cohomology . . . . . . 2. K-forms 3. Fields of Dimension < 1 ’ * : :

. . . . . .

. . . . . . 4. p-adic Fields . . . . . . . . . . . . 5. Number Fields . . . . . . . . . . . .

3. Tamagawa Numbers . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . 1. The Tamagawa Measure . . . . . . . . . . 2. The Tamagawa Number . . . . . . . . . . 3. The Theorem of Minkowski and Siegel . . . .

References . . . . . . . . . . . . . . . .

. . . xni

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. . 209

. . 218

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. . 231 . . 234 . . 241 . . 248

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xiv CONTENTS

CHAPTER XI

History of Class Field Theory

HELMUT HASSE

(Prepared by A. Lue on the basis of a preliminary manuscript of the lecturer) . . . . . . 266

References . . . . . . . . . . . . . . . . . . 277

CHAPTER XII

An Application of Computing to Class Field Theory

H. P. F. Swinnerton-Dyer . . . . . . 280

CHAPTER XIII

Complex Multiplication

J-P. SERRE

(Prepared by B. J. Birch)

Introduction . . . . . . . . . . . . . . . . . . 292 1. The Theorems . . . . . . . . . . . . . . 292 2. The Proofs . . . . . . . . . . . . . . : : 293 3. Maximal Abelian Extension . . . . . . . . . . . . 295 References . . . . . . . . . . . . . . . . . . 296

CHAPTER XIV

l-Extensions

K. HOECHSMANN

Introduction . . . . . . . . . . . . . . . . . . 297 1. TwoLemmas . . ,. . . . . . . . . . . . . 297 2. Local Fields . . . . . . . . . . . . . . , . 298 3. Global Fields . . . . . . . . . . . . . . . . 300 APPENDIX

Restricted Ramification . . . . . . . . . . . . 302

References . . . . . . . . . . . . . . . . . . 303

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CONTENTS

CHAPTER XV

Fourier Analysis in Number Fields and He&e’s Zeta-Functions

J. T. TATE (Thesis, 1950)

ABSTRACT . . . . . . . . . . . . . .

1. Introduction . . . . . . . . . . . . 1.1. Relevant History . . . . . . . . 1.2. This Thesis . . . . . . . . . . 1.3. “Prerequisites” . . . . . . . . . .

2. The Local Theory . . . . . . . . . . 2.1. Introduction . . . . . . . . . . 2.2. Additive Characters and Measure . . . . 2.3. Multiplicative Character and Measure . . 2.4. The Local C-Functions; Functional Equation 2.5. Computation of p(c) by Special [-Functions . .

3. Abstract Restricted Direct Product . . . . . . 3.1. Introduction . . . . . . . . . . 3.2. Characters . . . . . . . . . . 3.3. Measure . . . . . . . . . . . .

4. The Theory in the Large , . . . . . . . 4.1. Additive Theory . . . . . . . . . . 4.2. Riemann-Roth Theorem . . . . . . 4.3. Multiplicative Theory . . . . . . . . 4.4. The i-Functions; Functional Equation . . 4.5. Comparison with the Classical Theory . .

A few Comments on Recent Related Literature . .

References . , . . . . . . . . . . . .

Exercises

Exercise 1: The Power Residue Symbol . . . . . . Exercise 2: The Norm Residue Symbol . . . . . . Exercise 3: The Hilbert Class Field . . . . . . Exercise 4: Numbers Represented by Quadratic Forms Exercise 5: Local Norms not Global Norms . . . . Exercise 6: On Decomposition of Primes . . . . Exercise 7: A Lemma on Admissible Maps . . . . Exercise 8: Norms from Non-Abelian Extensions . .

AUTHOR INDEX . . . . . . . . . . . .

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xv

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. . 306

. . 306

. . 306

. . 307 . . 308 . . 308 . . 308 . . 311 . . 313 . . 316 . . 322 . . 322 . . 324 . . 325 . . 327 . . 327 . . 331 . . 334 . . 338 . . 341 . . 346 . . 347

. . 348

. . 351

. . 355

. . 357

. . 360

. . 361

. . 363

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INTRODUCTION

The chapters of this book, with the exception of the final one, are edited texts of lectures and lecture courses, delivered at the Brighton Conference. The topics and the general course programme were chosen with the principal purpose of the Conference in mind. This was to give the non-specialist mathematician (i.e. the mathematician who specializes in some other field) an introduction to algebraic number theory, starting from the more elemen- tary aspects and going on to class field theory, and to acquaint him with some of the recent developments in the subject. The individual contributions thus fit into an overall plan.

The first three chapters provide a broad introduction into algebraic number fields, containing in particular all the more elementary theory needed later on. The subject matters of Chapters I and II are closely inter- woven and the respective chapter headings were chosen to indicate-none too accurately-the demarcation line. An alternative choice of titles would have been “Algebraic Theory of Dedekind Domains” for Chapter I and “Topological Arithmetic” for Chapter II.

Chapters IV and V are frankly utilitarian, preparing the tools that are needed for class field theory.

The backbone of the book consists of the two chapters by Serre and by Tate on local and global class field theory, of which the second depends on the first. A feature of special interest in Serre’s contribution is that it in- cludes for the fist time as an integral part of local class field theory the des- cription of the maximal Abelian extension in terms of formal groups, due to Lubin-Tate. This yields a new approach to the existence theorem and to the refinements of the local reciprocity law dealing with the filtrations of the Galois group and the group of units.

While the first seven chapters were designed as a coherent whole, the rest are more loosely articulated and deal with various aspects and applications of the theory. They presuppose a knowledge of some of the material in the first seven chapters but are substantially independent of one another. The exercises at the end of the book, which were compiled by Serre and Tate after the Conference, are intended to indicate lines of thought for which there was no time at the Conference.

The final chapter does not represent a lecture at the Conference itself. It is the unchanged text of Tate’s thesis (1950). There is therefore a certain amount of overlap with the material in earlier chapters.

It was impossible (even if it had been desirable) to impose a completely xvii

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. . . xv111 INTRODUCTION

uniform system of notation on all the contributors. But, in addition to usages which have become too classical for mention, the following more recent conventions are adopted as standard (except in Tate’s thesis):

Q, Z, R, C are the rational numbers, the rational integers, the reals and the complex numbers respectively, and F, unless otherwise defined, is a finite field.

The special arrow H denotes the effect of a map on a typical element of a set. Thus the map R -+ R consisting of squaring can be described by: r H r2(r E R).

In general, bibliographies are at the ends of chapters and are referred to by the author’s name and the date of publication, with a further italic letter to differentiate works by the same author in the same year [e.g. Artin (1927), Hilbert (1900 b)]. Titles of periodicals are usually abbreviated as in “The World List of Scientific Periodicals” (Butter-worth).

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CHAPTER I

Local Fieldsi

A. FR~~HLICH

1. Discrete Valuation Rings ............ 1 2. Dedekind Domains .............. 6 3. Modules and Bilinear Forms ........... 9 4. Extensions ................ 13 5. Ramification ................ 18 6. Totally Ramified Extensions ............ 22 7. Non-ramified Extensions ............. 25 8. Tamely Ramified Extensions ............ 29 9. The Ramification Groups ............ 33

10. Decomposition ............... 39 Bibliography ................. 41

1. Discrete Valuation Rings Preliminaries on Fractional Ideals: Let R be an integral domain (i.e. a commu- tative ring with 1 + 0 and no zero divisors) and K its quotient field. For R-submodules Ii, Ia of K we define in the usual way the operations

II + I, (sum of submodules)

I1nl2

ZlZ2 (submodule generated by products). In addition we have, for an R-submodule I of K to consider the R-sub- modules

I-’ = [x E KlxZ c R]

R(Z) = [x E K[xZ c Z].

LEMMA 1. (i) Addition, multiplication and intersection are commutative and associative.

(ii) Z(Zr +Z2) = ZZ, +ZZ2.

(iii) R(Z) 3 R 3 ZZ- ‘.

(iv) Zf Z c R, then I-’ 3 R.

An R-submodule I of K is a fractional ideal of R if it is non-zero and if

t Back references. Propositions, Lemmas, etc., within the same section are referred to by their numbers, for other sections this is preceded by the number of the relevant section.

1

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2 A. FRijHLICH

there is a non-zero element a of K so that aZ c R. Here a may always be chosen to lie in R.

LEMMA 2. Zf Z, II, Zz are fractional ideals, then so are ZI +Zz, ZI n I,, ZIZz, Z-l and R(Z).

Proof. For the first three operations this is obvious. For the last two we prove more generally that if Z,, I, are fractional ideals then so is

J = [x E zqxzz c z1-J.

Let a, b be non-zero elements with al, c R, b E Zi n R. Then ab is a non-zero element of J. Let c, d be non-zero elements with cZ1 c R, d E Zz. Then cdJ c R.

LEMMA 3. Suppose R is Noetherian. A non-zero R-submodule Z of K is a fractional ideal if and only if it is finitely generated.

Proof. Necessity: Z g aZ c R.-Sufficiency: Multiply up by the denomi- nator product of the generators.

Let K* be the multiplicative group of a field K, and let Z be the integers under addition. A map

v:K-+Zuca

is a discrete valuation of K, if

(i) v defines a surjective homomorphism

K*+Z (again denoted by v);

(ii) v(0) = co ;

(iii) v(x+ y) 2 inf v(x), v(y) (usual conventions for the symbol co).

We now give the translation into the language of “multiplicative” valua- tions (see Chap. II): If v is a discrete valuation of K and p is real 0 < p < 1, then lx]” = p”@) is a discrete (non-Archimedean) valuation as defined in the other place. Every discrete (multiplicative) valuation is of this form, and equivalent valuations will correspond to the same v with varying p. The results proved in Chap. II can now be translated back. In particular :

(a) If V(X) # v(y), then V(X+ y) = inf v(x), v(y). (b) The set R, = [x E Klv(x) 2 0] is an integral domain with quotient

field K, the valuation ring of v, and the set pu = [x E Z+(x) > 0] is a maximal ideal of R,, the valuation ideal.

See the other course for valuation-topology and completion.

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LOCAL FIELDS 3

PROPOSITION 1. A discrete valuation v of afield K can uniquely be extended to a discrete valuation on the completion $ of K with respect to the valuation topology.

Proof. See Chap. II, $10: A discrete valuation, in the multiplicative sense, extends uniquely to the completion, and with the same set of values.

Example. Let F be a field and let K be the field of formal series

f a,f, with a, E F for all n E Z ng-co

(i.e. 3 m E Z with a, = 0 for all n 5 m). Then we have a “standard” discrete valuation v of K, given by

v ($yntn) =a25n.

K is complete in the valuation topology. We now turn to a ring-theoretic description of the pair K, v where v is a

discrete valuation of the field K. The elements u with V(U) = 0 form a subgroup U = U, of K*, the group of units (invertible elements) of R,. We now choose an element K with v(n) = 1. Then every a E K* has a unique representation

a 4 n”u, n E Z, u E u,

namely with n = v(a).

Turning to the fractional ideals Z of R, we define

v(Z) = inf v(x). x=x

A priori v(Z)EZU 00 v -CO. But Z = aJ, where J is a non-zero ideal of R,, and aE K*. Hence v(Z) = v(J) + v(a) E Z. Choose bsZ with v(b) = v(Z). Then

x”(~)R, = bR, c I.

But z c [x E z+(x) 2 v(l)J.

If t.(x) 2 v(Z) then x = rc”(‘)y, with y E R,. Thus Z c ,+)R = ,fV’JK

” “9

hence Z = (nR,)“(‘).

In particular P” = ~4,

and so z = p;“‘.

The last equation shows that R, has one and only one non-zero prime

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4 A. FRijHLICH

ideal, namely p”, and the preceding one that R, is a principal ideal domain. We now make the following definition: A discrete valuation ring (d.v. ring) R is a principal ideal domain with one and only one non-zero prime ideal. We have then proved one half of

PROPOSITION 2. The valuation ring R, of a discrete valuation v is a d.v. ring.

Conversely a d.v. ring R is the valuation ring R, for a unique discrete valua- tion v of its quotientJield K.

Proof of converse. Let p = nR be the non-zero prime ideal of R. R is a unique factorization domain and hence each non-zero element x of R has a unique representation

x = 7c”u, u a unit, n 2 0.

Allowing n to vary over Z we get the corresponding statement for x E K*. But then

v(x) = n defines a discrete valuation of K with R = R,. Uniqueness is obvious.

PROPOSITION 3. An integral domain R is a d.v. ring if and only if it is Noetherian, integrally closed and possesses one and only one non-zero prime ideal.

(An element x of an extension ring of R is integral over R, if it is the root of a manic polynomial with coefficients in R, i.e. if the ring R[x] is a finitely generated R-module. R is integrally closed if every element of the quotient field of R, which is integral over R, will already lie in R.)

Proof. (Sufficiency of the conditions.) Let Z be a fractional ideal. R(Z) is a ring (see the definition preceding Lemma 1). Hence for all x E R(Z), R[x] is a submodule of R(Z). By Lemmas 2 and 3, R(Z) is a finitely generated R-module, hence so is R[x]. Therefore x is integral over R, i.e. x E R. Thus

(1) R(Z) = R.

Let p be the non-zero prime ideal of R. We next show that

(2) p-’ #R.

There are non-zero ideals Z of R with I-’ # R, e.g. all principal ideals aR with a E p, a # 0. Let then J be a non-zero ideal, maximal with respect to this property. We have to show that J is prime.

Let x, y E R, x $ J, xy E J, z 4 R, z E J-‘. Then zy(xR+J) c R, hence zy E R and thus z(yR+ J) c R. Thus (yR+J)-’ # R, but yR+J I J. Therefore y E J. We have thus established (2).

By Lemma 1, (iii), (iv) R 3 pp-1 =) pR = p.

But pp-l = p would imply p - ’ c R(p), which contradicts (1) and (2).

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LOCAL FIELDS

Hence

(3) R = pp-?

Clearly p-l c R(n p”). By (l), (2)

(4) r-l p” = 0.

5

Hence we can choose an element a E p, with aR + p2. Then ap-’ c R, but by (3) ap-’ $ p. Hence ap-’ is an ideal of R not contained in any maximal ideal, i.e. ap-’ = R, and so

p = aR.

By (4) every non-zero element of R has then a unique representation a%, n 2 0, u a unit of R. Thus R is a d.v. ring.

We shall finally give a description of some groups associated with a given discrete valuation z) of a field K, in terms of the residue class field k = R/p, where R is the valuation ring of v and p the valuation ideal.

The additive group of K is the union of open (and hence closed) sub- groups p” (n E Z), whose intersection is zero. The quotient groups p”/p”+l have the structure of k-modules and we have

LEMMA 4. There is an isomorphism

k E pn/pn+ ’

of k-modules. Proof. If p = Rx, then multiplication by rc” induces such an isomorphism

Turning to the multiplicative group K* of non-zero elements of K, we first note that the valuation gives rise to an exact sequence

” (5) O+U+K*-+Z+O,

where U is the group of units of R. For each n 2 1 the set

(6) U” = 1+p”

is an open subgroup of U, and n U, = 1. The associated subgroup

topology of U coincides with that ;nduced on the subset U of K by the valuation. The quotient groups can again be described in terms of the residue class field.

PROPOSITION 4. (i) The residue class map R + k gives rise to an iso- morphism

U/U1 z k* (multiplicative group).

(ii) For each n 2 1 the map u H u- 1 gives rise to an isomorphism

UJJ,,, = p”/jP’.

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6 A. FRtkLICH

Proof. Straightforward. Observe that for ul, u2 E U,

(ulup-1)-(q-l)-(uz-1) = (q-l)(uz-1) E p.

COROLLARY. For n 2 1

UJUn+1 g k (additive group).

PROPOSITION 5. (i) If k is of prime characteristic p, then for n 2 1

u,P = u,+,.

(ii) If K is complete and tf the natural number m is not a multiple of the residue class field characteristic, then for each n 2 1, the map u H urn is an automorphism of U,,.

Proof. (i) follows from the preceding Corollary. For (ii) we frrst note that by the same Corollary the endomorphism of

v&Jg+1, induced by the endomorphism f : u H u”’ of U,,, is bijective for each q 2 n. Hence in the first place f is injective. In the second place we can find for each element u of U,, elements v0 E U,,, wi E U,,, r so that u = VZWl. Again we can find elements v1 E U,,+l, w2’o U,,,, so that wl = vyw2, i.e. 2.4 = (vOvJmw2. And so on. The sequence wl, w2,. . . will tend to 1, and as U, is now complete the infkite product vOvl. . . will converge to an element v of U,. But then u = vm E U:. We have thus shown that f is surjective, hence bijective.

2. Dedekind Domains

Throughout R is an integral domain, K its quotient field. If p is a prime ideal of R, we define the local ring of fractions by

R,=[xY-‘EK~~,YER,Y~P].

pR, is the only maximal ideal of Rp. Clearly p c pR, n R. If x E R, x # p then x-l E R,, and so x 4 PR,. Thus :

LEMMAI. p=@+,nR. Next we show

LEMMA 2. If J is an ideal of R, then

J = (J n R)R,.

Proof. Let x,y~R, y#p and xyB1eJ. Then XEJ~R, whence xy-’ E (J n R)R,. Thus (J n R)R, 3 J. The opposite relation is even more trivial.

PROPOSITION 1. Each of the following conditions on an integral domain R implies the others:

(i) R is Noetherian, integrally closed and its non-zero prime ideals are maximal.

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LOCAL FIELDS 7

(ii) R is Noetherian and, for every non-zero prime ideal p, R,, is a d.v. ring. (iii) AN fractional ideals of R are invertible.

(A fractional ideal I is invertible if II-’ = R.) A domain R satisfying the conditions of the Proposition is a Dedekind

domain. E.g. a principal ideal domain is a Dedekind domain. Proof. (a) (i) implies (ii). We shall use 1. Proposition 3. By Lemma 2 every ideal of R, is of form

IR,, where Z is an ideal of R. A finite generating set of Z over R is also a finite generating set of ZR, over Rp. Thus R, is Noetherian.

If x is integral over R,, i.e. if x”+ b-la,-lx”-l + . . . + b-la, = 0,

with b, a, E R, b # p then bx is integral over R. Hence if x lies in the quotient field K of R then bx E R, whence x E R,.

Let J be a non-zero prime ideal of R,. J n R is certainly a prime ideal of R, and is non-zero by Lemma 2. But J c PR,, this being the only maximal ideal of R,, whence J n R c p, by Lemma 1. Hence J n R = p and so, by Lemma 2, J = pR,.

(b) (ii) impZies (iii).

Let Z be a fractional ideal of R with generators a,, . . . , a,. Then for some i, say i = 1, v,(ai) = inf vp(x), vp being the valuation with valuation

xeI ring R,. But then IR, = a,R,. Hence

aila, = XiyiT’, withx,,yioR,yi#p (i=l,..., n).

Let y = lj y,. Then ya,-‘a, E R, hence ya,-’ EZ-’ and so y E II-‘. But

y 4 p. Thus II-’ C$ p. This is true for all maximal ideals p of R. Hence II-’ = R.

(c) (iii) impZies (i).

Let Z be a fractional ideal of R. Then 1 al,. . . , a,, E Z, bl,. . . , b, E I-’ with c a,bl = 1. If x EZ then x = c a,(b,x), b,x E R. Hence a,, . . ., a, generate I. Thus R is Noetherian.

Let x E K be integral over R. By 1. Lemma 3, S = R[x] is a fractional ideal. It is also a ring, i.e. S2 = S. Hence

S = SR = SSS-’ = SS-’ = R.

Thus R is integrally closed. Let Z be a non-zero prime ideal, p a maximal ideal containing it. Then

Zp-’ is an ideal of R and (Zp-‘)p = I. Hence either Zp-’ t Z or p c Z, i.e. p = I. The first relation however would imply that

P -%I-‘Ip-%I--‘I=R,

i.e. p-l = R and so p = R, which is nonsense.

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8 A. FRijHLICH

If p is a non-zero prime ideal of a Dedekind domain R we shall denote by up the valuation of its quotient field K with valuation ring R,.

COROLLARY 1. Let 1x1 be a non-trivial multiplicative valuation of K, with IRI s 1. Then 1x1 = p’p@) for some p, 0 < p < 1, and some non-zero prime ideal p of R.

Proof. The inequality 1x1 < 1 defines a non-zero prime ideal p of R. Hence R, is characterized in K by the inequality 1x1 I 1. This yields the result.

If I is a non-empty subset of K, define

v,(Z) = inf v,(x) XSI

(possibly v,(Z) = - co).

PROPOSITION 2. The fractional ideals of a Dedekind domain R form an Abelian group 9(R) under multiplication. This group is free on the non-zero prime ideals p. The representation of a fractional ideal I in terms of these generators is given by

1 = I-I p%v) P

Also then

IR, = (pR,)“P(?

Proof: The first assertion follows from 1. Lemma 1 and from Proposition 1. To show that the prime ideals p generate Y(R) it suffices to show that

every integral ideal I (i.e. I c R), different from R, is product of prime ideals. Such an ideal I is contained in a maximal ideal p. Hence I = p(lp-‘) with I c Ip-’ c R. The ascending chain condition now yields the required result.

By 1. Lemma 1, (IR,)(.IR& = (IJ)R,. We thus obtain a surjective homomorphism

(1) fp : J(R) --, -V,)

which acts injectively on the subgroup generated by p. If pi # p, then plR, = R,. For if aopl, a $ p then already aR, = R,. Thus the p1 other than p lie in Kerf,. As a first consequence we see that the non-zero prime ideals form a free generating set of 9(R). Secondly we see that if

I = n p

then

IR, = (pR,)‘P.

Hence

rp = vp(IRp) = v,,(I) + v,(R,) = v+,(I).

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COROLLARY 1. If a E K* then v+,(a) = 0 for almost all p. COROLLARY 2.

v&u1 12) = vpw + VpUA

vpu- 9 = - vpm,

vvUl + 1,) = infv,Ud, v,Ud,

V,Vl n 12) = SUP ~,VI)> VpUJ.

COROLLARY 3. The maps fp induce an isomorphism

9(R) ru JqR,).

(The symbol u stands for the restiicted direct product (direct sum) of groups.)

Let &, be the valuation ring of the completion of K at vp. By 1. Proposition 1,

9(R,) z Y(Rv) (z Z!).

Hence :

COROLLARY 4.

Y(R) rJp(ir,). v

3. Modules and Bilinear Forms

In this section we introduce some concepts, which will subsequently be used in the discussion of ideal norms, of differents and of discriminants for extensions of Dedekind domains. Throughout this chapter R is a Dedekind domain, K its quotient field, U a finite dimensional vector space over K of dimension n > 0. The symbol T always stands for R-submodules of U and the symbols L, M, N for finitely generated R-submodules which span U, i.e. contain a basis of U. If p is a non-zero prime ideal of R, then Tp = TR, is the R,-module generated by T.

LEMMAI. nT,=T.

Proof. This” is true for any integral domain R, not necessarily Dedekind, provided only that p runs through a set of prime ideals containing the maximal ideals.

Clearly T c n Tp. For the converse we consider an element u of n T,,

and show that tie ideal V

J, = [x E Rlxu E T]

is the whole of R, i.e. is not contained in any given maximal ideal p. In fact u = x-lw with WET, XER, but x#p. As xczJ,, we now see that Ju Q: p.

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10 A. l?RthLICH

LEMMA 2. Given M and N, there is a non-zero element a of K with aMcN.

Proof. Let {Us} be a basis of U, contained in N. For a given finite generating set (wi) of M, choose the element a as a “common denominator” of the coefficients of the wI with respect to the basis (Us}.

LEMMA 3. For almost all p, Mp = Np. Proof. By the previous Lemma we can find non-zero elements a, b of K

with aM c N c bM. Thus Mp = N, whenever v,(a) = 0 = v,(b). By 2. Proposition 2, Corollary 1, this is the case for almost all p.

Now suppose for the moment that M and N are free R-modules. They are both of rank n, hence isomorphic. Therefore there is a non-singular linear transformation G of U with Ml = N. The determinant det (e) is non-zero and, apart from a unit in R, solely depends on M and N. Hence the fractional ideal

(1) R det (6’) = [M : NJ,

solely depends on M and N. If the restriction on M and N to be free is removed we still get, for each p,

a fractional ideal [M, : Np] of Rp. Also whenever Mp = ND then [Mp : NJ = R,. By Lemma 3, and by 2. Proposition 2, Corollary 3, there is a unique fractional ideal

[M:N]=[M:N],

of R, the module index, such that for all p

(2) [M: N]R, = [M,: NJ

One verifies that definitions (1) and (2) agree when M and N are free. One also sees that when R = Z and M =I N then [M: N] is just the ordinary group index, viewed as a Z-ideal.

PROPOSITION 1. (i) k: : J$[NRI,] = [M : L]

. . .

(ii) Suppose iat M 3 N. Then [M: N] is an integral ideal, and [M:N] = R implies M = N.

Proof. Locally (i.e. for the 4) this is obvious. For the global form one applies Lemma 1, and 2. Proposition 2.

PROPOSITION 2. If t is a non-singular linear transformation of U, then

[Mt : Nt] = [M : N].

Proof. Localize and apply definition (1).

One can also show that M z N, if and only if [M: N] is a principal fractional ideal.

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LOCAL FIELDS 11

Now let B(u, u) be a non-degenerate, symmetric, K-bilinear form on U. If {uI} is a basis of U, its dual basis {u,} is defined by

B(u,, vi) = 6,, (Kronecker symbol).

The dual module of T is

(3) D(T) = D,(T) = [u E UlB(w, T) c R].

LEMMA 4. If M is the free R-module on {ur}, then D(M) is the free R-module on the dual basis {II,}, and

D(D(M)) = M.

Proox Obvious.

In what follows the symbol D stands for duals with respect to R and in place of DHp we shall write D,.

PROPOSITION 3.

(i) D(M) is ajinitely generated R-module, spanning U,

(ii) D(M), = ~,(M,h (iii) D(M) = II DJM,),

P (iv) D@(M)) = M,

(v) [D(M): D(N)] = [N : M].

ProoJ (i) M contains a free R-module N, and by Lemma 2 is contained n a free R-module L = bN, both L and N spanning (1. Hence

D(N) 3 D(M) 3 D(L).

By Lemma 4, D(L) and D(N) are free and span (1. This gives (i). (ii) Let (Wi} be a finite generating set of M. Suppose D E D&M,,). Then

for all i, B(u, WJ = b-la,, with ai, b E R, b 4 p. Thus o E D(M)b-’ c D(M),. We have shown that

400, = WW,.

To get the opposite inclusion, observe that

W&W,)> Mp) = W,(M), WR, = R,-

(iii) follows from (ii) and Lemma 1, (iv) from (ii) and Lemma 4. The proof of(v) reduces by (ii) to the case when M and N are free. We

then only have to recall that if {u,} and {vi} are dual bases and {uiC) and {o,d*} are dual bases then

det (C) det (E*) = 1.

We define the discriminant of M by

(4) b(M) = b(M/R) = [D,(M) : M-ja.

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12 A. FRijHLICH

PROPOSITION 4.

(i) b(N) = b(M)[M : N]‘.

(ii) b(M,/R,) = b(M/R)R,.

(iii) If M is the free R- module on {ui> then b(M) is the fractional ideal generated by

det B(u,, uj).

Proof. For (i):

[D(N) : N] = [D(N) : D(M)] [D(M) : M] [M : N] (Prop. l(i))

= [D(M) : M][M : N]* (Prop. 3(v)).

(ii) follows from Proposition 3(h).

For (iii) let (a[} be the dual basis of {uI} and let uI = uid. By Lemma 4,

[D(M) : M] = Rdet(C).

On the other hand

det B(u,, 0~6) = det (6’) det B(Ui, u/) = det (d).

COROLLARY 1. Suppose M =I N. Then b(M) diuides b(N), and b(M) = b(N) implies M = N.

The last proposition shows that when R = Z, our discriminant is the same as the classical discriminant over Z (to within a sign).

Let now U = U,+ CJ, (direct sum of vector spaces). Suppose that MI and Ni span U, and write M = MI + M2, N = N1 +N,. For (ii) and (iii) in the following proposition also assume that B(U,, U,) = 0, so that B gives rise to non-degenerate bilinear forms on U1 and U2.

PROPOSITION 5.

(i) [M : N] = [MI : N,][M, : N2].

(ii) D(M) = D(M,)+D(M,).

(iii) b(M) = b(M,)b(M,).

Proof. Obvious.

To make life easy, we shall for the next and final proposition consider only free R-modules M and N, although it would remain true generally. Let R be a Dedekind domain containing R, with quotient field R. We shall view U as embedded in the vector space U = U @x R over R. The bilinear form B can uniquely be extended to a R-bilinear form R, which is again symmetric and non-degenerate. The R-module MR generated by M is free and spans 8.

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PROPOSITION 6.

(i) [Ma : N& = [M : NIRa,

(ii) DR(MR) = D,(M)a,

(iii) b(MR/@ = b(M/R)R.

,Proof Obvious.

13

4. Extensions

Throughout R is a Dedekind domain, K is its quotient field and L is a finite separable algebraic extension field of K. The condition of separability is not needed for part of Proposition 1, and for Proposition 2 (see Z.S., Ch. V, Th. 19 and Serre, Ch. II, Prop. 3).t But we shall not here consider the inseparable case.

The elements of L which are integral over R form a domain S, the integral closure of R in L and S is integrally closed (in L) (see Z.S., Ch. V, $ 1).

LEMMA 1. If p is a prime ideal of R then SR, is the integral closure of Rp in L.

Proof. The elements of SR, are obviously integral over Rp. Conversely, if x is integral over R,, i.e.

x”+(b-Ia,-Jx”-‘+ .. . +b-‘a, = 0 with a, E R, b E R, b # p, then bx E S, hence x E SR,.

A prime ideal q of S is said to lie over the prime ideal p of R if

!JJnR=p.

We then write Sp[p.

PROPOSITION 1. S is a finitely generated R-module which spans L over K and is a Dedekind domain.

Every non-zero prime ideal $3 of S lies over a non-zero prime ideal of R, and there is a prime ideal of S lying over every non-zero prime ideal p of R.

Proof Applying Lemma 1 to p = (0) we see that S spans L over K. The trace tL,= : L + K defines a non-degenerate, symmetric K-bilinear

form N4 4 = t&Jv)

on L. As SK = L, S contains a free R-module N spanning L. But then, in the notation of $3, D(N) is free and spans L. Also

W) = N9. The traces of integral elements lie in R, and hence

D(S) = s,

t For the references “Z.S.” and “Serre” see the literature list at the end of Chapter I.

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14

i.e.

A. FRijHLICH

D(N) 3 s.

Thus S is a finitely generated R-module. It follows that S is Noetherian, and we have already noted that S is integrally closed.

Let ‘$ be a non-zero prime ideal of S and let

b”+qb”-‘+...+a,=0

be the minimal equation of the non-zero element b of Sp. Then a, E R, for all i, hence a, E p = ‘$3 n R. Thus p is non-zero. Moreover !J3 3 pS, i.e. $3/pS is a prime ideal of the commutative algebra S/pS over the field R/p. As S is finitely generated over R, S/pS is a finite dimensional algebra, and the same is then true for (S/pS)/(p/pS) = Sl’p. Thus S/p is a field, i.e. ‘$ is a maximal ideal. We have now shown that S is a Dedekind domain.

Let p be a non-zero prime ideal of R. pS = S would imply p-‘5 = p-‘(pS) = S, i.e. p-l c S n K = R, which is false. If now the prime ideal ‘$3 of S is a factor of pS then ‘$3 n R 3 p, i.e. ‘$3 n R = p.

COROLLARY 1. (Used in Chap. II.) Every discrete (multiplicative) valuation of a field K can be extended to a

finite, separable extension field L. Proof. Take R as the valuation ring in K. A suitably normalized valuation

of L of form P”O(~) will do. COROLLARY 2. The map II-+ IS is an injective homomorphism 3(R) + #(S). Proof. By the proposition, together with the observation that if Ii and I2

are integral ideals of R with Ii + Iz = R then also I1 S+ I, S = S. Combining Proposition 1 with the theorems proved in Chap. II (see 9 10)

on the uniqueness and the completeness, for extensions of valuations with complete base fields, we have

PROPOSITION 2. If R is a d.v. ring and K is complete, then S is a d.v. ring and L is complete.

In the remainder of the present section and the early part of the next one we shall study a number of concepts, associated with the embedding of R in S. In each instance our aim is to obtain a reduction to the complete local case.

A fractional ideal J of S is iinitely generated over S, hence over R. Also if 0 # a E J then J =) as, hence J spans L over K. We can thus define the ideal norm by

(1) NL/K(~ = I? : JIR. The connection to element norms is given by

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LoCAL HELIX 15

PROPOSMTON 3. Zf u EL* then

&,K(W = h,KWR. Proof: N,,,(a) is the determinant of the linear transformation x H ux of L!

Note that when R = Z, and when J is an ideal of S then N&J) is just the number of residue classes of S mod J, viewed as a Z-ideal. This follows from our remark (cf. Q 3) on the interpretation of the module index as a group index in the case R = Z.

The ideal norm commutes-in a sense to be made precise-with the isomorphism exhibited in 2. Proposition 2, Corollary 4. We first recall a theorem, proved in Chap. II (see 9 10). Let R be a d.v. ring, p its maximal ideal, R the completion of K at up. Let ‘p, run through the non-zero prime ideals of S and denote by E, the corresponding completions. Then we can identify (as algebras and topological vector spaces over R)

(2) L @xE = c f;, (direct sum of fields).

View L, L, and R as embedded in this algebra. Denote by R the valuation ring of R and by S, that of L,. Then we have

LEMMA 2. RS = x3,. Proof. C S, is the integral closure of R in the algebra C LP Hence

RS c c 3,. RS is complete, hence closed. It will thus suffice to show that S is dense in C SI. As we have not stated or proved the general approxi- mation theorem which would yield this result, we shall give an ad hoc proof.

From Chap. II we know that L is dense in CL,, and so C S, is the closure of (c 3,) n L. We shall show that the latter module is contained in S. The minimal polynomial of an element x of (c 33 A L over R has coefficients in R. But it coincides with the minimal polynomial over K, i.e. its coefficients lie in K n i? = R. Thus x E S.

Now return to the case of an arbitrary Dedekind domain R. If p is a non-zero prime ideal of R write Kp for the associated completion of K and i?,, for its valuation ring. For a non-zero prime ideal Sp of S denote the corresponding objects by & and S,.

PROPOSITION 4. Zf J is a fractional ideal of S then

%,dJ.& = n N 8/p WdJV Proof. In view of the definition of the module index and by Lemma 1

we may assume that R = R, is a d.v. ring. But then the Proposition follows from Lemma 2 and from 3. Propositions 5 and 6.

COROLLARY 1. NL,x is a homomorphism Y(S) + S(R) of groups. Proof. By 2, Proposition 2, Corollary 4 and by the above Proposition 4

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16 A. l&kLICH

the proof reduces to the case when R is a d.v. ring and K is complete. But then every fractional ideal of S is principal, and the Corollary follows from Proposition 3 and the multiplicativity of element norms.

By similar reasoning we get COROLLARY 2. ForZE$(R)

N&IS) = I",

where n = (L : K) is the degree.

COROLLARY 3. Zf L 3 Fx Kthen

We can now return to the bilinear form defined by the trace. The dual DR(S) of S is clearly an S-module. By Proposition 1 and by 3, Proposition 3 it is finitely generated over R, hence over S. As D&S) 2 S we have

(3) DR(S) = a-'

where 3 = %&!3/R) is an integral ideal of S, the different. We write

b = b(S/R)

for the discriminant defined in 3(4). As DR(S) z) S, b is an integral ideal of R. Moreover

(4) b = NL,~). In fact

b=[D,(S):S]=[S:D,(S)]-’

= NL,K(D-l)-l = N&D).

PROPOSITION 5. In the notation of Proposition 4:

0) W/R)30, = W&J.

(3 ‘pip = J-J w%l/~,).

Proof. By Lemma 2 and 3, Proposition 4-6.

The next proposition establishes a connection between the discriminant b(S/R) and the discriminants of an integral generator of L. This proposition is usually established via the theory of the “Noether conductor” of a ring, but the concept of a module index will enable us to do without this.

Let x be an element of S so that L = K[x], and let g(X) be the minimal polynomial of x over K. The ring R[x] then spans L and is the free R- module on 1, x,. . ., x”-‘. In the next Proposition g’(X) is the derivative of g(X).

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LOCAL FIELDS

PROPOSITION 6.

17

69 WCxl) = WLIK~WR- (iii) R[x] = S if and only ijB(S/R) = g’(x)S.

Proof. By Euler’s formulae,

t&i/dW) E K fori=O,...,n-1.

Hence

(5)

By 3, Proposition 4

b(R[x]) = det tLIR(x’+‘)R,

but (any old-fashioned textbook on algebra)

det t,&‘+j) = f N,&‘(x)), and so

CWCxl) : RCxll = JL,r&W)R = [g.&CXI : Rbl] 3

whence, by (5), in fact

NRCxl) = &) RCxl-

We have now established (i) and (ii), and the necessity of the condition in (iii) is a trivial consequence.

Suppose that D(S/R) = g’(x)S, then

NR[x]) = WI = -& S = &W = Wbl).

Now take duals again, and apply 3, Proposition 3, to get S = R[x].

Finally we prove the tower formula:

PROPOSITION 7. If L 3 F 2 K and if T is the integral closure of R in F then

(0 W/R> = W/TP(T/R>,

(ii) W/R) = WIR)“(NF&WT)),

where m = (L : F).

Proof. We shall show that

D(S/R)-’ = a(S/T)-‘CD(T/R)--‘.

then (ii) will follow from (4) and the Corollaries to Proposition 4.

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18 A. FR&lLICH

By the transivity of the trace we have

fL,Kcw = b,KC h,FW TI . Hence, writing b, = D(T/R),

fL,&x) c R o tLIF(Sx) c 3, ’

est,,p&(sxD,) c TexlDo c ID(s/T)-’

-x E a,‘a(s/T)-‘.

5. Ramification

We first consider a pair of Dedekind domains R, and R,, with quotient fields Kl and K,, and with RI c Rz. Let pZ be a non-zero prime ideal of R2 and suppose that the prime ideal

PI = PZ A RI is also non-zero. Then the residue class field k = RI/p, is embedded naturally in the residue class field k, = R,/p,. The degree

(1) 6% : W =~(Pz/PI) is the residue class degree (possibly f = CO). The ramification index e(p,/p,) is defined by the equation

(2) ~JP~RJ = e(pA), i.e. by

(3) Restriction of up2 to KT = e(p,/p,)u,,. In the obvious notation we have

PROPOSITION 1. fWP2lf(P2IPl) =f(PJPlL 4~3/pJ&2/pi) = ehh).

Proof Obvious.

PROPOSITION 2. Let jj be the valuation ideal In the completion o/‘K = K, with respect to the valuation vp = 9,. Then

f@h) = 1 = eGVp>. Proof. Let R, be the valuation ring of up in K. By 2, Proposition 2,

e(pR,lp) = 1 and by 1, Proposition 1, e@/pR,) = 1. By Proposition 1, e(W) = 1.

Every element of RJpR, is of form xy-’ with x, y E R/p. Hence the two fields coincide, i.e. f(pR,/F) = 1. Also Rp is dense in the valuation ring R, of the completion. Hence the image of R&R, is dense in the discrete group &,l$i, i.e. f(ji/pR,) = 1. Thus f(fi/p) = 1.

COROLLARY.

JWPl) =f@z/~1)

&2/P,) = &%m

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LOCAL FIELDS 19

The results of 5 4, together with the last Corollary, show that differents, discriminants, residue class degrees, ramification indices and ideal norms, for extensions L/K, can be described locally in terms of the completions. From now on, and up to 5 10, we shall always assume R to be a d.v. ring, and K to be complete in the valuation topology. This situation is inherited by finite separable extensions (4, Proposition 2). The reformulation of our results in global terms will mostly be left to the reader.

We shah now make some changes in the notation, which was introduced in $4, and shall use for “relative” objects the symbols D(L/K), b(L/K), fW/K), 4W). V, is the valuation of L, k= the residue class field. For k, we also write k. p is the maximal ideal of R, ‘p the maximal ideal of the integral closure S of R in L.

Except for Q 9, all our results will be established without assuming a priori that the residue class field extensions are separable.

PROPOSITION 3.

4mMLIK) = CL : K). Proof. The vector-space S/pS over k has the sequence of quotient spaces

WP, %w=, * * * 9 ‘p’-‘I~= w = Ps), and by 1, Lemma 4 these are isomorphic. The dimension of S/!@ over k is f = f(L/K), hence the dimension of SlpS is ef. On the other hand, as S is a free R-module of rank (L: K), the dimension of S/pS is (L: K).

Let UK and U, be the groups of units of R, and of S respectively. We already know that the embedding j: K* +L* yields a commutative diagram (with exact rows) (cf. l(5))

UK

&-+UK-+K*-,z--+0

1 1,) 0-4J,-+L*--,z-0

Now we get in addition

COROLLARY. The element norm yields a commutative diagram

i.e.

and

O-+UL-+ L*-+z+o

1 1 lf ~--+~~-+~*--+~--+~

f&) = vK(NL/Kx)

NL/Kcw = P’.

proof. NL,KK(UL) c U, and NLIx 0 j = (L: K).

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20 A. FRijHtICH

Interlude on traces: Let A be a finite dimensional commutative algebra over a field k with the following properties: (i) If N is its radical then A/N = B is a field. (ii) N’ = 0, N’-’ # 0 and for i < e, B r N’/N’+’ (isomorphism of B-modules). Denote by G the image in B of an element a of A. Then we have

LEMMA 1. t,,,(a) = et,,,(Z). Sketch of proof: tA,,(a) is the trace of the linear transformation x I-+ xa

of the k-space A. Let B = B, = A/N,. . ., B, = N’/N’+‘,. . . be the suc- cessive quotient modules of the A-module A and let a, be the linear trans- formation of Bi induced by a. Then

tAlk(a) = C trace (a,). i

But the Bi are isomorphic A-modules and hence

trace (ai) = trace (ao) = ttllk(ii).

Therefore the Lemma.

Now write a H a for the residue class map S + S/‘Q

LEMMA 2. t&a) = et,,,,(a).

Proof. By Lemma 1, with A = S/pS, N = $3/p%

Remark: Analogously one can show that

(4) N&a) = Nk,&V.

PROPOSITION 4. uL(z)) 2 e-l. Proof. Write again A = SJpS, N = p/pS and denote by I the image

in A of an element x of S, Choose a k-basis (al) of A, so that for 1 5 i I (e- 1)f the a, form a k-basis of N. We can lift (ai} back to an R-basis {xl} of S, so that jY, = a,. For 1 5 i I (e-1)fand for allj, gi.Zzi will lie in N, i.e. Xixi = 0, whence by Lemma 2

tL/K(xixj) E Pm

Hence each of the first (e- I)frows of the matrix

(tL/K(XiXj))

lies in p, and hence by 3, Proposition 4

db) 2 (e - l>f, i.e.

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LOCAL FIELDS 21

L is said to be non-ramified over K, if

(i) e(L/K) = 1. (ii) kL is separable over k.

THEOREM 1. L is non-ram$ed over K if and only if b(L/K) = R. Proof. By Proposition 4, b = R, i.e. 2, = S implies e = 1. Now suppose

that e = 1. Let {x1} be a free generating set of S over R and let d = det t,,,(xlxj).

By 3, Proposition 4, b = R if and only if the residue class a is non-zero, But by Lemma 2,

ii! = det fk& %*. 3,)

is the discriminant of the basis {a,} of kL over k, and this does not vanish if and only if kL is separable over k.

Let in the sequel x be the characteristic of k. L is said to be tamely ramified over K if

(9 xt@IK). (ii) kL is separable over k.

Thus if x = 0, L is always tamely ramified. THEOREM 2. The following conditions are equivalent:

(i) L is tamely ramified over K. (ii) tLIK(S) = R.

(iii) v,(n) = e - 1. Note: tLIK(S) is always a non-zero ideal of R. Proof. The equivalence of (i) and (ii) is immediate from Lemma 2. For the equivalence of (ii) and (iii) first note that when a E K, then

b#a) = h.lK@k. Hence

tL,K(S)-1 = a-’ n K,

or writing v = v,(D), r = vK(tLIK(S))

rlU<r+l. e

Thus if v = e- 1 then r = 0. If r = 0, then v < e, and so by Proposition 4. v=e-1.

Remark: If L is normal over K one can deduce from condition (ii) in the last theorem a further criterion. For this purpose denote by R(T) the group ring over R of the Galois group r. One then has the

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22 A. FR&ILICH

Normal basis theorem: The R(r)-module S is isomorphic with R(IJ if and only ifL is tamely ramified over K.t

The proof is left as an exercise. Hints: Interpret the existence of an element of trace 1 in terms of endomorphism rings to deduce that when L is tamely ramified over K then S is projective. Then use Swan’s theorem.:

Global AppIication We shall explicitly restate Theorem 1 in terms of arbitrary Dedekind domains. Let then just for the moment R be a Dedekind domain, not necessarily a d.v. ring, K its quotient field, and S the integral closure of R in a finite separable extension field L of K. A non-zero prime ideal ‘!$I of S is non-ramified (over K) if its ramification index over v n R has value 1 and if S/p is separable over R/q A R. A non-zero prime ideal p of R is non-ramified (in L) if all prime ideals !$ in S, .above p, are non-ramified over K.

COROLLARY 1 TO THEOREM 1. p is non-ramified in L if and only if p does not divide the discriminant b of SIR.

Proof. On the one hand we know (cf. Proposition 2) that both the ramification index and the actual residue class field extension remain unchanged on transition to the completions. Thus p is non-ramified in L if and only if the completions &, for all ‘$ above p, are non-ramified over the completion Kp of K.

On the other hand it follows from 4, Proposition 5, and 2, Corollary 4 to Proposition 2, that p does not divide b if and only if the product of the local discriminants is trivial. As each factor is an integral ideal, this is the same as saying that each of the local discriminants is trivial.

Now apply Theorem 1. (By replacing the discriminant by the different one can obtain a sharper

criterion for the individual !$ to be non-ramified.) COROLLARY 2 TO THEOREM 1. Almost all non-zero prime ideals of R are

non-ramified in S. In fact all those which do not divide the discriminant !

6. Totally Ramified Extensions

The notation here, and in $ 7 to 9, is that introduced in 5 5. R is always a d.v. ring, K is complete and L is a finite, separable extension field of K.

A polynomial g(X) in K[X] is separable if (g(X), g’(X)) = 1. An Eisen- stein polynomial in K[X] is a separable polynomial

(1) E(X)=Xm+b,-IXm-‘+...+b,X+bO,

t See E. Noether, Normalbasis bei Kiirpern ohne hiihere Verzweigung, Creole 1931. $ See R. S. Swan, Induced Representations and Projective Modules, Ann. of Math.

1960, Corollary 6.4.

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LOCAL FIELDS 23

with

(2) trg(bJ 2 1 for i = 1,. . . , m- 1, and u&J = 1.

(The condition of separability on either L or E(x) is not really necessary for the following theorem. But as we are only considering separable exten- sion fields, we had to impose the corresponding restriction on E(X)).

L is totally ramified over K if e(L/K) = (L: K), i.e. f(L/K) = 1.

THEOREM 1. (i) An Eisenstein polynomial E(X) is irreducible. If ll is u root ofE(x), then L = K[ll] is totally ramified and u,(n) = 1.

(ii) If L is totally ramified over K and u&II) = 1, then the minimal poly- nomial of II over K is Eisenstein and

s = Nm, L = K[II].

For the proof we shall need a proposition on the representation of elements of a complete field K by convergent series. If, for all n E 2, v(II,) = n, u(a,) 2 p (constant), then the series

“z,a”l& (i.e. a, = 0 for all n near - co)

converges and thus has a sum in K. Suppose now that we are given maps II : Z + K*, r : k + R, so that r(0) = 0, and that

Z-LK*-L=z

kr- R “2-k

are the identity maps. Writing II, for the image of n under II, and ‘8 = Im r, we have

PROPOSITION 1. Every element of K has a unique representation of form

a =n$,aJL a, E 93.

The value of a is then given by o(a) = inf n.

#a”#0 Proof. Standard. Proof of Theorem 1. First suppose that the polynomial E(X) in (1) is

Eisenstein and that L = K[ll], where E(ll) = 0. Write n = (L: K), e = e(L/K).

Clearly u,(lI) 2 0. Hence

uL(IIm) = ut(bmJIm-l+. . . +bo) r 1, i.e. u,(II) 2 1. Let s be the integer with

e - > s-l.

s r %(n)

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24 A. FRhLICH

Then

(3) m>n>e>s. If m > s then ZJ#YI~) > e. Also

uL(bj) = eu,(bJ 2 e. and so

u~(~~-~II~-~+. . . +blII) > e. Hence v&J > e, i.e. ZJ,&,) > 1 contrary to (2). We have shown that m I s, and so by (3)

m=n=e=s.

Therefore

u#I) = 1. All the assertions of(i) have now been established.

For (ii) we apply Proposition 1 to L. Asf(L/K) = 1 we may choose % to be in K. Taking c to be an element of K, with ox(c) = 1, we put

l-r qe+r = CqI’ (q E Z, 0 2 r -c e). A rearrangement of the sum c 4IT, now shows that in fact L = K[II] and S = R[II].

Let now the polynomial E(X) in (1) be the minimal polynomial of II over K. Then it is also the characteristic polynomial. Therefore, in the first place, b0 = &N&II) and so, by 5, Proposition 3, Corollary 1,

u&b()) = u&-I) = 1.

In the second place, E(X) reduced mod p is the characteristic polynomial of the nilpotent element II mod pS of the algebra S/pS over k. Hence E E X”(mod p), i.e. UK(bi) > 0 for all i.

COROLLARY. K has a totally ramified extension of prescribed degree e. Proof. Let uK(c) = 1 and

E(X) = xp--cx-c.

It is worth mentioning some results, which follow on from Proposition 1, but which we shall not be able to give in this course. (See Serre, Ch. II.)

If K is the field of formal series c a, t” over a field F (see example in 5 1) ll*-W

then one can clearly take II, = t” and % = F (i.e. F z k). Here the charac- teristics of k and K coincide. Conversely if this is the case then K is (to within value isomorphism) the field of formal power series over k.

There remains the case when x = p # 0 but the characteristic of K is zero. (Typical example the rational p-adic field Q,.) If one assumes also that k is perfect (e.g. k is finite) then one can at least choose % to be multiplica- tively closed, and uniquely so. As for the existence problem, one can show

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LOCAL FIELDS 25

that if k is perfect of characteristic p # 0 then there is one and (essentially) only one discretely valued, complete field K of characteristic zero which has k as its residue class field and for which ux(p) = 1.

7. Non-ramified Extensions

A separable extension L of K determines by transition to the residue class fields an algebraic extension of k. If L is non-ramified over K then the residue class field extension is separable. One of our aims in the present section is to show conversely that each separable extension of k can uniquely be lifted to a non-ramified extension of K (in a proper functorial sense). We first established an analogue to the theorem of the last chapter, which gives a description of non-ramified extensions as root fields of polynomials. Notation: The image in kL of an element a of S will always be denoted by a, and analogously the image in the polynomial ring k,[X] of a polynomial g(X) E SIXI ‘v BP3

PROPOSITION 1. (i) Suppose L to be non-ram$ed over K. Then there exists an element x of S with kL = k[a]. If x is such an element and g(X) is its minimal polynomial over K, then S = R[x], L = K[x] and g(X) is irreducible in k[X] and separable.

(ii) Suppose g(X) is a manic poIynomia1 in R[X], such that g(X) is irreducible in k[X] and separable. If x is a root of g(X) then L = K[x] is non-ram$ed over K and kL = k[i?].

Proof. (i) As k, is separable over k it is of form k[n] with x E S. For every such x the minimal polynomial G(X) of R over k is separable. Also

(L : K) 2 degree g(X) 2 degree G(X) = (kL : k) = (L : K).

Thus in fact G(X) = g(X), i.e. g(X) is irreducible, and L = K[x]. The equation S = R[x] can now be deduced either from 6, Proposition 1, as applied to L, or from 4, Proposition 6.

(ii) We have

(L : K) = degreeg(X) = (k[?] : k) I (k, : k) I (L : K).

Hence in the first place (L: K) = f(L/K), i.e. e(L/K) = 1 and in the second place kL = k[jz], i.e. kL is separable over k.

Consider now a class of algebraic extension fields E, E,, . . . of a given field F. A homomorphism CJ : E --, E, over F is a homomorphism of fields, leaving F elementwise fixed. Example: The identity map E -+ E. If (r : E + E, and r : E, -+ E2 are homomorphisms over F then so is aoz:E+ E2. In other words, we have a “category” (for those familiar with the language). We denote by HomF (E, E,) the set of homomorphisms

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r: :.” .‘t ;+ +<,; ,:% 26 .i A. PRijHLICH

E 4 El over F. If E is normal over F then Horn’ (E, E) is just the Galois group.

Let E” be the separable closure of F in E, i.e. the maximal subfield of E which is separable over F. We then obtain maps

(1) Horn (E, E,) + Horn’ (ES, Ei)

which preserve the compositions o 0 r and also the identity maps of the fields E, i.e. we have a functor. The maps (1) are moreover injective and if E = E* also bijective.

Now we apply this formalism to the class of finite, separable, algebraic extensions L of K. As L will vary, we shall now use the symbol RL for its valuation ring and pL for its valuation ideal.

PROPOSITION 2. Let o : L + L’ be a homomorphism over K. Then, for all XEL

vLs(xo) = e(L’/La)v,(x).

Proof. The function v on L defined by

e(L!/La)v(x) = vL,(xo)

is a discrete valuation of L, which on K coincides with v,. By uniqueness (see Chap. II) v = v,.

COROLLARY 1. Application to normal extensions L of K and their Galois groups.

COROLLARY 2. A homomorphism Q : L + L’ over K induces, by restriction to the valuation rings and reduction module the valuation ideals, a homomor- phism 5 : kL -+ kLn over k. The resulting maps

HomK (L, L’) + Horn& (k,, kL,)

preserve the identity maps offields and map composition. Thus the residue class fields kL define a functor, and so do the separable

closures k; of k in kti Our next theorem asserts, in the language of category theory, that the functor k: has an adjoint. This will then in particular yield an isomorphism between the category of separable extensions of k and the category of non-ramified extensions of K.

THEOREM 1. Let E be a finite, separable, algebraic extension of k. Then there exists a finite, separable, algebraic extension L = L(k) of K, such that

(i) E z kL (over k), (ii) L is non-ramified over K, (iii) the maps

HomK (L, L’) + Homk (k,, kL,)

are bijective for all L’. Properties (i), (ii) or (i), (iii) determine L uniquely, to within isomorphism

over K.

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u)CAL FlEL.DS 27

Note: In (iii) one may of course replace kL and kL, by VL and kL,. For the proof we need the LEMMA 1. Let g(X) be a manic polynomial in R[X] such that g(X) is

separable, and suppose that the element a of k is a root of g(X). Then there is one and only one element of R, such that

g(x) = 0 and Is = a.

Proof. See Chap. II, Appendix C. Proof of Theorem 1. We know that li = k[a], where the minimal poly-

nomial G(X) of a over k is separable. Choose any manic polynomial g(X) in R[X] with g(X) = G(X) and let L = K[x], where x is a root of g(X). By Proposition 1, L has properties (i) and (ii).

To show that L has property (iii) consider a homomorphism w : kL + kL, over k. By the Lemma, L’ will contain a unique element y, so that g(y) = 0 and J = fw. But then there exists a unique homomorphism Q : L + L’ over K with xt7 = y. Clearly i = w. If also Z = w then xr = y and so z = 0.

Now suppose that L’ is non-ram&d over K and that w is an isomorphism kL E kLo over k. Then (L’ : K) = (L : K) and therefore the lifted homo- morphism e : L + L’ over K must be an isomorphism. We have thus seen that properties (i) and (ii) determine L uniquely, to within isomorphism over K. The corresponding uniqueness result in terms of (i) and (iii) is standard.

COROLLARY. L@) is normal over K if and only if li is normal over k, and if so then the Galois groups are isomorphic.

In the sequel a “subfield” of a finite separable, algebraic extension L of K is always one which contains K.

THEOREM 2. L has a subfield L,,, such that the subfields L’ of L which are non-ramified over K are precisely the subfields qf L,. Also kLo = ki.

If L is normal over K with Galois group r then L, is normal over K and is the fixed field of

ro=[~Erlv,(xy-x)>OforaIZxERJ

rO is called the inertia group of L over K. Proof. The existence of a subfield LO which is non-ramified over K,

with kLO = VL, follows from Theorem 1. All subfields of L, are then also non-ramified over K. In fact, from the definition of the term “non-ramified” we see that, for any tower E 3 F 3 K of fields, E/K is non-ramified if and only if E/F and F/K are non-ramified.

Conversely, let L’ be a subfield of L, non-ramified over K. Then k L’ c k”= = kb. By Theorem 1, as applied to li = kL, we obtain a homo- morphism Q : L’ + L, over K so that d is the inclusion map. Now let

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28 A. FRihILICH

k L, = k[X], with x EL’. Then x and X(T are elements of L with the same residue class and so, by Lemma 1, x = XQ. But, by Proposition 1, L = K[x], and so L’ c LO.

Now suppose L is normal. The conjugate fields of L, in L are all non- ramified over K, hence coincide with L,,, i.e. L, is normal. The inertia group I?,, by its definition is the kernel of the homomorphism

I? --, Homk (kL, kJ.

As the maps (1) are injective, To is also the kernel of the homomorphism from r into the Galois group Horn’ (ki, ka of kLO. If Q is the Galois group of L,/K, it follows from Theorem 1 that

r. = Ker (r -+ Q),

which is the required result. COROLLARY 1. The composite field of non-ramified extensions L and L

in a given separable closure of K is non-ramified. The union K, of all non-ramified extensions L of K in a given separable

closure of K is called the maximal non-ramified extension of K. COROLLARY 2. Every finite extension of K in K,,, is non-ramiJied. The

Galois group I’(K,,JK) is isomorphic (as a topological group) with the Galois group I’(P/k) of the separable closure 7i” of k.

Application (see Chapters III and V). We suppose now that k is a finite field of characteristicp with q = pm elements. Denote by Z the completion of Z with respect to the topology defined by the subgroups nZ (n > 0). Then r(li”/k) is an isomorphic copy of 9: under the map

VHWqY

where awq = aq.

Hence :

I. There is a unique element oq in T(K,,,/K) with the following property: If L is a subfield of K,,JK, then for all a E R=

abq = aq (mod t-4. The map v H CT: is an isomorphism Z z lY(K,,/K) of topological groups.

(oq is the Frobenius substitution)

This result implies that for each integer n > 0 the field K has one and (to within isomorphism over K) only one non-ramified extension L of degree n. Further L is normal over K with cyclic Galois group.

From the theory of finite fields and by Proposition 1 we also have II. K, is the union of thefields of m-th roots of unity (in a given separable

closure of K) for all m prime to p.

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LOCAL FIELDS 29

In conclusion we consider the effect of the norm map on the groups of units. (See the Corollary to 5, Proposition 3.) The subgroups defined in 4 l(6) for the fields K and L will be denoted by UK,” and U,,“.

PROPOSITION 3. If L is non-ramified over K then for all n 2 1

NL,KK(UL,“) = UK,“. Proof, Choose an element II in K with v&I) = 1, i.e. vL(II) = 1. Then

U,,” consists of the elements

u = l-ITX, XERL.

If m = (L : K), the characteristic polynomial h(X) of II? over K is of form

h(X) = X”-II”t~,&)X”‘-l+l-I”+‘h,(X), h,(X) E WI. Therefore

NLIK(u) = h(1) = 1 +IInt,,K(x) (mod p”+‘).

This implies in the first place that

NL,K(UL.n) = UK,.. By 5, Theorem 2, tLIK(RL) = R and hence also

NL,K(LWK,~+I = uK,n-

Now one uses completeness, proceeding as for 1, Proposition 5, to finish the proof.

COROLLARY. Suppose that L is non-ramified over K. Then a unit in U, is norm of U, if and only ifits residue class mod p is norm of kp In particular if k is finite then

NL,K(%.) = uK-

Proof. By 5, (4).

8. Tamely Ramified Extensions

The notation is the same as in $7. x is the characteristic of k. The term “subfield” is used as in 7, Theorem 2. TO is always the inertia group de&red there.

THEOREM 1. (i) L has a subfield Li such that the subjields L’ of L which are tamely ramified over K are preciseIy the subfieIds of L1. If x = p # 0 then (L : L,) is a power of p.

(ii) Suppose that L is normal over K with Galois group I?. Then Ll is normal over K and is thejixedfield of

rl = [y E rlv,(xy-x) 2 t+(x)+1 for all x E RL].

!IfVL(W = 1, then the map

Y z-b mm

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30 A. FRGHLICH

defines a homomorphism 8,, of rO into kz,, which is independent of the choice of ll and whose kernel is rl. To/r1 is a cyclic group. If x = p # 0 then rl is the (unique) p-Sylow group of r,,.

Note: If x = 0 then L is tamely ramified anyway. In this case, the theorem just asserts the existence of the homomorphism &, and the resulting fact that r. is cyclic.

Proof. We first consider the normal case, and begin by establishing the properties of the homomorphism 8,. I1 is clearly a subgroup of I0 and a normal subgroup of I?.

If y E r,, and if u is a unit of R, then uy/u E 1 (mod pa. Hence the

residue class O,(y) = IIy/II in kz is independent of the particular choice of II (within the condition v,(H) = 1). As IYO acts trivially on kE one sees that &(Y~ YJ = &,(y,)&,(y,). As I’,, is a finite group, the 8,(y) are roots of unity, and so lie in (FL)* = k&. Moreover, it follows that T,/Ker 8, r Im 8, is cyclic, and that its order is not divisible by x. Finally, as uy/u = 1 for -- all units U, we have, for a EL*, the equation ay/a = OO(y)vL(0). Hence in fact rl = Ker 8,,.

Let now Li be the fixed field of rl. We have seen that if x = p # 0 then the degree (L, : L,) is prime to p. As kL 3 kL, =3 k,, and as (kL : kLO) is a power of p it follows that f(L,/L,) = 1. Also e(L1/LO) is prime to p. Thus Li is tamely ramified over L,.

In the sequel we shall use repeatedly the fact that, for any tower E 3 F 13 K of fields, EJK is tamely ramified if and only if E/F and F/K are tamely ramified. This follows from the definition of tame ramification. As a first consequence we conclude the L, and its subfields are tamely ramified over K.

Let L’ be a subfield of L, tamely ramified over K. We shall now show that L’ c Li. Let L’L, = E, L’ n L, = F. Then L’/F is tamely ramified and hence, by 5, Theorem 2, there exists an element a E RL, with tlelp(a) = 1. As r,, is normal, we then also have t,,,,(a) = 1, and so, again by 5, Theorem 2, E is tamely ramified over L,, i.e. over K. Hence k, c k& i.e. kE = kLO and so E is also totally ramified over L,.

Let Q(C) = 1 and let g(X) be the minimal polynomial of c over L,. By 6, Theorem 1 and 4, Proposition 6, we have

W/W = R&3, and hence by 5, Theorem 2,

&f(4) = e - 1, e = 4Wo), i.e.

(1) d?‘(c)) = (e - lb(c). Now let E be the fixed field of the subgroup A of IO. Choose in each

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LOCAL FIELDS 31

right coset of r,, mod A, other than A itself, an element y. Then

g’(c) = n cc - CY). Y

For each of the e- 1 factors of the product we have

%(C - 4 2 %(4 and so by (1)

UL(C - 4 = %(C). Hence y 4 rl. This however implies that A =) rl, i.e. that E c L1, and so L’ C L,.

Now suppose that x = p # 0 and that L” is a proper extension of L1 in L. Then L” is not tamely ramified over L,, i.e. either ple(L”/L,) or pIf(L”/L,) and so certainly pl(L” : L,). It follows that ri coincides with its p-Sylow group.

The results for non-normal L now follow by embedding L in a normal extension of K. The details are left to the reader.

COROLLARY 1. The inertia group To is always soluble. More precisely, 1yx = 0, then r0 is cyclic, and if x = p # 0, then r0 is the extension of a p-group by a cyclic group.

If k is finite then the Galois group of a normal extension is soluble. COROLLARY 2. The composite Jield of tamely ramified extensions L and L’

in a separable closure of K is again tamely ramified. The maximal tamely ramified extension K,, of K is the union of all tamely

ramified extensions in a separable closure of K. COROLLARY 3. AN finite extensions of K in K,, are tamely ramified.

Kt, contains K,. If X” 0 then I’(KJKM) z Z, if x = p # 0 then Wtr/L~ = gf’)pZ,-

Remark: The group rr has an interesting module theoretic characteriza- tion. R= has the structure of a module over the group ring RQ. For a subgroup A of the Galois group l? one can show that RL will be relative projective with respect R(A) if and only if A 3 rl.

From Theorem 1, one sees that in order to “catch” the tamely ramified extensions one has to proceed in two steps. First construct a non-ramified extension (dealt with in § 7) and then a totally and tamely ramified normal extension. This last step can also be described explicitly.

We first recall a bit of Kummer theory (see Chap. III). Suppose that K contains the primitive e-th roots of unity and that the element c of K* has true order e modulo K*‘. Then the field K(II), II’ = c, is normal over K of degree e and the equations

$A9 = mm

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32 A. FRiiHLICH

define an injective homomorphism $, of the Galois group into K*. We shall then write

&,,<Y> = tic(r)- PROPOSITION 1. (i) Let L be a normal totally and tamely ramified extension

of degree e. Then K contains the primitive e-th roots of unity and there is an element c E K* with Q(C) = 1 and L = K(c”~).

Moreover FC coincides with the homomorphism 4, of Theorem 1. Also L = K(b”‘) with UK(b) = 1 if and only if

bc-1 E k*=.

(ii) If x ,j’ e and af K contains the primitive e-th roots of unity, and if vl&) = 1, then the field L = K(c”‘) is normal, totally and tamely ramiJied over K, of degree e.

Proof. (i) By Theorem 1, Im 0, is precisely the group of primitive e-th roots of unity of k*. As the polynomial X’- 1 is separable over k it follows from 7, Lemma 1, that K contains the primitive e-th roots of unity and that these are mapped injectively into k*. The Galois group I? of L/K being cyclic of order e it follows from Kummer theory that L = K(c”‘), where e is the order of c mod K*‘. Moreover, both & and 8e are now injective homomorphisms r + k*, with the same image. Therefore

(2) & = or,, (r,e) = 1.

But as seen in the proof of Theorem 1,

(3) r = v#‘e) = Q(C).

Thus (Q(C), e) = 1. We may now replace c by c”a’ with (s, e) = 1 and a E K*, and we can thus ensure that Q(C) = 1, and then by (2) (3) E = 8,.

If now also L = K(b”‘) then, by Kummer theory, b = c’a’ with a E K*, (r, e) = 1 and 0 < r < e. If vi[(b) = 1 then we must have r = 1, and so the element a is a unit. The final form of our criterion will now follow from 1, Proposition 5.

(ii) follows from Kummer theory and by 6, Theorem 1. COROLLARY 1. Let Q(C) = 1. Then K,, is the union of fields K,,,(c”‘),

for all e not divisible by x.

We finally turn again to unit norms. Using the notation of 7, Proposi- tion 3, we have

PROPOSITION 2. If L is tamely ramified over K then

NL,RWL, 1) = UK, 1’

Proof. By 5(4) we know that always iV&Y,,,) c U,,,. In view of 7, Proposition 3 and the transitivity of the norm, we may also assume that

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LOCAL FIELDS 33

L is totally ramified over K, say of degree e. But then we have by 1, Pro- position 5,

u K, 1 = U&,1 = %,KWL. 1).

9. The Ramification Groups

Throughout this chapter L is normal over K with Galois group l? = T(L/K). The series of subgroups beginning with To, l?r (cf. § 7,8) can be continued. Apart from a brief indication of the general case, we shall from now on always assume that kL is separable over k, i.e. that kL = kLo. This is certainly the case whenever k is a finite field. Under this hypothesis we have (irrespective of whether L is normal or not)

PROPOSITION 1. R, = R[a].

Proof. In the notation of 7, Theorem 2, and by 7, Proposition 1, there exists an element b of L,, so that kL = k[6] and so that the residue class polynomial g(X) of the minimal polynomial g(X) of b over K is separable. Thus g’(h) # 0.

Let now a = b + h, with uL(h) = 1. Then we get from the Taylor expansion of g(X) the equation

g(u) = hg’(b) + O(h’). Hence

G7(a)) = 1.

Now we apply 6, Proposition 1 to L. As kL = k[a] we can choose an 83 consisting of “polynomials” in a with coefficients in R. If n = me +r, e = e(L/K), 0 I r < e, m E 2 we take II, = g(u)lcm, where c E K, Q(C) = 1. Re-arranging the series expansion for an element of R, one sees that R, = R[u].

With a as in the proposition we now define a function i = iLlg : I? --) Z v CO

by

(1) iLIK(Y) = i(y) = day-4

We furthermore define (for i z - 1)

(2) rl = [r E rlv,(xy-x) 2 i+l,for all x E RL].

Thus r-r = I?. For i = 0 this was precisely the definition of the inertia group in 9 7. For i = 1 we shall subsequently see that this definition agrees with that in 5 8.

The following proposition connects the Pi with the function &, and shows incidentally that this function is independent of the choice of the generator a.

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34 A. PR&ILICH

PROPOSITION 2.

(i) yErlei(y)2 i+l.

(ii) i(yS) L inf (i(y), i(6)).

(iii) i(6yX’) = i(y). Proof. Obvious. COROLLARY 1. The r, are normal subgroups of r with ri+ 1 c Fe and

rm = 1 for large m. In fact

m = ;z~ i(Y)

will do. COROLLARY 2. If i(y) # i(6) then

i(yS) = inf (i(y), i(6)).

The rl are the ramification groups (Hilbert).

If A is a subgroup of l?, then it is the Galois group of L over the fixed field of A. We have clearly:

PROPOSITION 3. A, = rI n A.

In the sequel U is the group of units of RL and the U, are the subgroups defined in 3 l(6) (with K replaced by L).

THEOREM 1. Let i 2 1. Then y E rr if and only v for all x EL*, xylx lies in U,.

Choose an element II of L with u,(lS) = 1. Then the map

Y I+ HY/~ mod Ut+ I

is a homomorphfim

ei: r,-, we,

which is independent of the particular choice of II, and whose kernel is r,+ i. Note: We now know that lY,(§ 8) = I’,(§ 10). Proof. If xy/x E U, for all x EL*, then y acts triviall. on k,, hence y E r,,.

As kL = kLO the elements of R, are of form y +z, y E pL, z E R,. But then

uJy+z)y-(y+z)) = t&y-y) = u, y - 1 ( >

+ cL(y)2 i+l.

Thus ydl. Suppose conversely that y E rl. If y E U then

~YY/Y-~) = &Y-Y) 2 i+L hence

(3) YYlY E vi+,.

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LocALFIELDS 35

Now let &‘I) = 1. Then

&Iy/II- 1) = aL(IIy-IQ-1 2 i,

i.e. IIy/II o U,. Write

e,(y) = IIy/II mod U,, i.

By (3) we see firstly that 8,(y) is independent of the choice of II, and secondly that if x EL* then

ei(y)“L(x) = z&mod U1+l.

Hence in the first place

xrlx E u,, and in the second place-applying what we have proved already-

tl,(y)=lifandonlyify~T,+,.

Finally for y, 6 E TI we have

rIys/n = (IIy/n)s(rIs/II).

As IIy/II E U, we know from (3) that

(nr/W = HY/~ (mod ul+J and thus we have

ei(Ys> = G94(~). We know that if x = 0, then already I?r = 1. On the other hand, if

x = p # 0, then by 1, Proposition 5, U/’ c U,, 1. Hence: COROLLARY. If x = p # 0 then, for i 2 1, r,/r,+ 1 is an elementary

Abelian p-group. For the commutator properties of the ramification groups see Serre,

Ch. IV.

Note: If kL is not assumed to be separable over k, one has to define two sequences of subgroups. One is given by (2) (for i 2 0), and we shall denote these groups for the moment by rl*. On the other hand, the criterion of Theorem 1: “xy/x E U, for all x E L*” defines a second sequence of sub- groups I: (i r 1). Corollary 1 to Proposition 2 and Proposition 3 remain valid for each of the two sequences, and so does the Corollary to Theorem 1. In other words, for i 2 1 the quotients IYr*/IYr+ i* and r:/IY: + 1 are elementary Abelian p-groups.

The two sequences are interwoven, i.e. for i 2 0, lYl* 3 r:+, 3 rl+l*. If k, is separable over k then, by Theorem 1, I*+i = rr+i*. On the other hand if e(L/L,) = 1 then Pi* = I:+ i (for i 2 1).

From now on kL is again assumed to be separable over k. The ramification groups yield an explicit determination of the different, which generalizes the

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36 A. IX6HLICH

formula (cf. 5, Theorem 2) holding in the tamely ramified case. We shall write

(4) gi = order Tr.

Note that L/L, is now totally ramified, i.e.

90 = 4Wo) = 4Wh

PROPOSITION 4.

Proof. Let RL = R[a] and let g(X) be the minimal polynomial of a over K. By 4, Proposition 6,

3 = gWR,, and hence

dQ = &?‘(~N = % (

II (a - 4 Y+l )

COROLLARY. Let A be a subgroup of I?, F its jixedfield. Then

4W’hF(W/~)> =y&iLj~(~).

Proof. By the tower formula (cf. 4, Proposition 7) we have

4W9uF(WKN = dW/W - ULWLIF)).

Now apply the last proposition to evaluate the right-hand side, noting that for 6 E A, iLjK(8) = i&j) (by Proposition 3).

From now on we shall be concerned with the following situation: A is a normal subgroup of r with fixed field F. F is then normal over K with Galois group r/A. Our first aim is to determine the function iFIR.

PROPOSITION 5. For OE T/A

4LIFh~d~) = c i&9. Y-m

Proof. For w = 1 both sides are infinite. Assume now that o # 1. Let R, = R[u] and let g(X) be the minimal polynomial of a over F. Acting with o on g(X) coefficientwise, we obtain a polynomial (gw)(X). Then

iFjK(m) - PF

_ p;WWF/d~)

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LOCAL FIELDS 37

divides the coefficients of (gw) (X) - g(X), and so divides

(g~)(a)-da) = (w>(a) =yQpaY).

In other words

(5) 4Wli&4 5 C i&9. Y-m

Now evaluate e(L/F)u#(F/K)) first via Proposition 4 (with F in place of L) and then via the corollary to that proposition. Comparing the results we obtain the equation

But this implies that we must have equality in (5).

We now turn to the ramification groups of r/A. Our results in 8 7 and $8 imply that r/A)i = rlA\/A for i = 0, 1. But the same is no longer generally true for i > 1. To obtain an analogue to Proposition 3 for quotient groups one has, following Herbrand, to introduce a new enumeration of the ramification groups.

In the sequel x is a real variable 2 - 1. We write

(6) l?, = rl, where i is the least integer 2 x.

We then define a function 4 = +=I= by x,if -lSx_<O,

(7) 444 =

i

-$I+. . . +9m+(X-mkLn+II

if x 2 0 and m is the integral part of x

(i.e. m the integer with m I; x < m+ 1). 4(x) is a continuous, strictly increasing function and thus possesses a

continuous, strictly increasing inverse function t,Q) (- 1 5 JJ). The new, “upper” enumeration of the ramification groups is then given by

(8) r-y = r, if x = I&), i.e. y = 4(x).

We shall need some formal properties of the function 4. LEMMA 1. The function r#~ is characterized by the following properties:

(i) 4(O) = 0. (ii) r$(x) is continuous.

(iii) If m is un integer 2 - 1, then &c) is linear in the closed inter& [m, m + l] and has derivative

4’(x) = order r&@/K)

in the open interval (m, m + 1). Proof. Obvious.

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38 A. FR&IL.ICH

LEMMA 2. If 4(x) is integral, then so is x. Proof. For x E [- 1, 0] this is obvious. If x E [m, m + I] (m 2 0) and

y = 4(x), then

x = g$l Cg0y+w,+i-(gi+. . . +s,N. The lemma now follows by observing that g,,,+ 1 divides go,. . . , g,,,.

LEMMA 3.

#J(X) + 1 = $ rTrinf(i(r), x + 1).

Proof. We shall give the proof for x > 0 only. Let m be the integer with m < x 9 m+l. Then

x+1 = inf(i(y),x+l)*y EI,+l. Therefore

L C WW9x+l) = k( Qo7or Y J ,W+g,+~(x+l))

= ~~~~,-,-8,)i+g.,,(x~l))

l2 =-

( 90 1st) gf-gm+i(m-l-l)+g,+,(x+l)

>

= (b(x)+ 1. The theorem we wish to prove is:

THEOREM 2.

(0 h&) = ~F&h&)~. (ii) For all y 2 - 1

(I/A)’ = I”A/A. Proof. For o B P/A define

(9) A4 = sup i&9. 7--J

Our first step is to prove that

(10) b&4 - 1 = djLlp(A4 - 1). Choose y0 E I’ so that y. + o and that iLlr(yo) = j(m). Then the equation

in Proposition 5 can be written in the form

(11) 4W9bIRW =aJAit&O 6).

If i&5) <j(o), then by Proposition 2, Corollary 2, iLIX(yOb) = iLIx(6). If iLIR(b) S j(o), then by Proposition 2,

A4 2 iLI&o 8 2 inf(iL&h i(4) = ib-4, i.e.

iLdyo 8) = iW.

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LOCAL FIELDS 39

Thus in all cases

(12) iLI&O 6) = WiLIF@M4),

(recalling that iLlF(@ = iLIK(8) by Proposition 3). Now substitute (12) into (11) and apply Lemma 3 (for L/F), to obtain (10).

Using (10) we now have:

0 E r, A/A * x S it4 - 1* ~L,F(x) 5 &,~(1’(4 - 1)

* 4L&) S b&4 - 1 * 0 E WV, with y = &,F(~). In other words we have shown that

(13) r,A/A = (WQ, for Y = 4LIF(-4. We now establish Theorem 2(i). Write 0(x) = r/+,K(&,F(x)). We shall

use the characterization of &,R gr ‘ven in Lemma 1. 0(x) clearly has pro- perties (i) and (ii). Moreover, if x varies over an open interval (m, m+ 1) then, by Lemma 2, the interval (r&(m), &(m + 1)) does not contain integral values. Hence 4&y) is linear in the closed interval [~#&r), &,r(11? + l)] and so e(x) is linear in [m, m+ 11.

It remains to compare the derivatives for non-integral x. We have

(14) W4 = 4&O. d&Ax), with Y = h&). By Lemma 1 (for F/K)

4&W = order UP),/eV’/K) and so by (13)

(1% 4~~&~ = (r,A : 4)leWO By Lemma 1 (for L/F) and by Proposition 3

4&(x) = order (r, n A)/e(L/fl.

Hence by (14), (15) and by Lemma 1 (for L/K)

e’(x) = 4;xw, and so finally

But this equation, in conjunction with (13) now implies that

I=A/A = (r/A)

(z = &,&) = &,x(x)). We have thus established the theorem. For further properties of the Herbrand function see Serre, Ch. IV and V.

10. Decomposition

We now return to the global case. R is an arbitrary Dedekind domain, K its quotient field, and S the integral closure of R in a finite separable extension L of K. p is a non-zero prime ideal of R and p a non-zero prime ideal of S. The associated completions will be denoted by K,,, and by J$. For the

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40 A. FRijHLICH

ramification index of Cp over ‘$3 n R we write eW and for the residue class degree f@. Thus

(0 PS = @ll W’-

(2) &,KW) = PfV (P = rp n a. PROPOSITION 1.

(L : K) = eqfv. VP 7

Proof. By 4(2) CL : K) =v&‘Lca : K&J

and by 5, Proposition 3,

t&l: qJ= e,fa. From now on suppose that L is normal over K with Galois group r.

As Sy = S, the conjugate ‘@py (y E I) of the prime ideal ‘p will again be a prime ideal of S, and if ‘$?lp then clearly also ‘$y]p. Conversely one has

PROPOSITION 2. All prime ideals in S lying above p = !J.J n R are conjugates VPr of Ip.

Proof. The proof involves a special case of the Chinese remainder theorem, which we have not even stated earlier on. Let I be the product of the prime ideals of S, above p and different from Ip. Then ‘@ +I = S. Thus there exist elements a E ‘$ b E I with a + b = 1. But then

a E % a 4 % if YI~P and ‘4% ZV. More generally

ay E IPY, ay 4 % of RIP and ‘4% Z YY. Taking the product over y E l? we see that

n<ay> 4 ‘% if %IP and % z ‘$Y for all Y. But

n<ay> = ~&4 E Sg n R = P. Thus !&lp does in fact imply !I$ = !Qy.

COROLLARY. ecp = e and fv = f solely depend on p = Ip n R. If g is the number of prime ideals of S, lying above p, then

(3) efg = (L : IQ.

The elements y of r with !J3y = !@ form a subgroup IYe, the decomposition group. They are characterized by the equation

v&r) = v&) for all x EL (or for all x E S)). If y E l? then clearly

(4) r,, = prwy.

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MCAL FIELDS 41

PROPOSITION 3. Suppose ‘$I lies above p. Then 4 is normal over Kp. Denote the Galois group by I&. The restriction to L of the automorphisms d E 2+ then gives rise to an isomorphism

c, g r,.

Proof. Define in the first place C, as the group of automorphisms of & leaving K, elementwise fixed. An element (T of Xc, maps L into the extension field ,$ of K and leaves the elements of K fixed. As L is normal it follows that La = L. In other words, there is a unique element B of r whose action on L coincides with that of a. Moreover, by 7, Proposition 2, v,(xa) = v~(x) for all x EL, i.e. 8 E I-,. We thus obtain a homomorphism

(5) h:C,-+l-,, h(a) = a.

Let y E r,. y acts continuously on L with respect to the v+opology. It then follows from the universal embedding property of completions that there is an element y’ = t(y) of C,, making the diagram

Y L --+L

1 Y' I L,-+L,

commutative. We obtain a homomorphism

t:rv4,, t(r) = Y’s so that both h o t and t o h are identity maps, and thus h is in fact an isomorphism.

We now only have to show that the order of &, i.e. of Te coincides with the degree (4 : K,) = ess f,, = ef = (L:K)/g (cf. Corollary to Proposi- tion 2), i.e. that the group index (r : r,) is g. This, however, is immediate from the definition of rs, and by Proposition 2.

Now we can get a definition for the ram$kation groups of L/K. Identify r, with the Galois group of L,/K, and define the rcq,* in the first place locally as in 3 9. As S is dense in each of its completions it follows then that in fact

rg,*= [~d-&cq(xy-x)2 i+l, forallxES].

For i 2 0, one can moreover show that

r~,r=[~~rlvc4(XY-X)ri+1, forallxES].

BIBLIOGRAPHY

J-P. Serre, “Corps locaux”, Hermann, Paris 1962. (Quoted as “Serre”.) 0. Zariski and P. Samuel, “Commutative Algebra”, Vol. 1, Van Nostrand 1958.

(Quoted as “Z.S.“.)

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CHAPTER II

Global Fields

J. w. s. CAsSELS

;: 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

if: 18. 19.

Valuations ............. Types of Valuation ........... Examples of Valuations .......... Topology .............. Completeness. ............ Independence. ...... Finite Residue Field ‘Cask : : : : ..... Normed Spaces ............ Tensor Product ............ Extension of Valuations .......... Extension of Normalized Valuations ...... Global Fields. ............ Restricted Topological Product. ........ Adele Ring ............. Strong Approximation Theorem ....... Idele Group ............. Ideals and Divisors ........... units . . Inclusion and Norm Maps for kdeies,‘Ideles ‘and Ideals’

. . . 42

. . . 44

. . . 45

. . . 46

. . . 47

. . . 48

. . . 49

. . . 52

. . . 53

. . . 56

. . . 58

. . . 60

. . . 62

. . . 63

. . . 67

. . . 68

. . . 70

. . . 71

. . . 73

Appendix A. Norms and Traces . . . . . . . . . . . 76 Appendix B. Separability . . . . . . . . . . . . . 80 Appendix C. Hensel’s Lemma . . . . . . . . . . . . 83

1. Valuations

We shall be concerned only with rank 1 valuations, so for brevity, valuation will mean “rank 1 valuation”.

DEFINITION. A valuation 1 1 on a field k is a function defined on k with values in the non-negative real numbers satisfying the foIlowing axioms.

(1) Ial = 0 ifand onZy ifa = 0.

(2) IaS1 = Ial ISI. (3) There is a constant C such that 11 +orl 5 C whenever ICC] 5 1.

DEFINITION. The trivial valuation of k is that for which Ial = 1 for all a # 0.

Note: This will often be tacitly excluded from consideration. 42

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GLGBAL. FIEZDS 43

From (2) we have

Ill = 1’1-111~ so 111 = 1 by (1). If now some power of cc E k is 1, say o” = 1 we have

I4 = 1 by (2). In particular the only valuation of the finite fields is the trivial one.

The same argument shows that I - 11 = 1 and so

I-al = lcll allask.

DEFINITION. TWO vuluutions I II, I I2 on the same field k are equivalent if there is a c > 0 such that

I& = I#. (l-1)

Note: If lall is a valuation then lalz defined by (1.1) is one also. Equivalence is clearly an equivalence relation.

Trivially every valuation is equivalent to one with C = 2. For such a valuation it can be shown? that

(31 Is+4 5 Pl+bl (The “triangle inequality”.) Conversely (l), (2) and (3’) trivially imply (3) with C = 2. We shall at first be almost entirely concerned with properties of valuations unaffected by equivalence and so will often use (3’) instead of (3).

t We shall actually be concerned only with valuations with C = 1. for which (3’) is trivial (see next section), or with valuations equivalent to the ordinary absolute value of the real or complex numbers, for which (3’) is well known to hold: and we use (3) instead of (3’) (following Artin) only for the technical reason that we will want to call the square of the absolute value of the complex numbers a valuation. For completeness, however, we give the deduction of (3’) from (3) with C = 2. First, Ia1 + a,15 2 max [all, [a& on putting a1 = aai if, say, Ia,1 2 laal. Then, by induction,

w I B aA 5 2’max M. j-1

and so for any n > 0, we have n

where 2’-l < n 4 2’. on inserting 2’ - n zero summands. In particular

InI I2nlll - 2n (?I > 0). But now

IS + YI” = IX 0 /w-‘I

5 & + 1) max I(;)1 Islrlyl+j -z 40~ + 1) max $1 I/V Irl”-’ IeJ + 1w1 + Irk

and (3’) follows on extracting nth roots and making n + ~0.

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44 J. W. S. CASSELS

For later use we note the formal consequence

IIPI-lrll SIP-rl of (3’) where the outside 1 1 are the ordinary absolute value. For one need only apply the triangle inequality to the identity

P = r+P-r, Y = B+(r-B>*

2. Types of Valuation

We define two important properties of a valuation, both of which apply to whole equivalence classes of valuation.

DEFINITION. The uuluution 1 1 is discrete if there is a 6 > 0 such that

l-6 < Ial < 1+6

impZies Ial = 1. This is the same as saying that the set of log 1~11, a E k, a # 0 form a

discrete subgroup of the reals under addition. Such a group is necessarily free on one generator, i.e. there is a c < 1 such that Ial, a # 0 runs through precisely the set of cm, m E Z. If Ial = cm we call m = m(a) the order of a. Axiom 2 implies

ord (a/3) = ord a + ord j?.

DEFINITION. The v&&ion 1 I is non-archimedean if one can take C = 1 in Axiom 3, i.e. if

ID+rl s max {IPI9 Irll. (2.1)

If it is not non-archimedean, then it is archimedean. We note at once the consequence

IP+rl = IPi if Irl < IPl of (2.1). For

IPl = I@+Y>-rl 5 max{IP+rl, 1~11. For non-arch. I I the a with la) 5 1 clearly form a ring, the ring o of

integers. Two non-archimedean valuations are equivalent if and only if they give the same o: for [/?I < 171 if and only if /?y-’ E o, j?-ly 4 o (cf. $4).

The set of a with Ial < 1 form an ideal p in o, clearly maximal. It consists precisely of the a E o with a- ’ $ o.

The notation o and p will be standard. The reader will easily prove the LEMMA. Let 1 I b e non-archimedean. A necessary and suJicient condition

for it to be discrete is that p is a principal ideal. We need later the LEMMA. A necessary and su$icient condition that I I be non-archimedean

is that InI 5 1 f or all n in the ring generated by 1 in k. Note: We cannot identify this ring with Z if k has a characteristic.

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Proof. Necessity is the triangle inequality

GLOBALFIELDS 45

obvious. For sufficiency let Ial < 1, and then by

jl+aj” = ](l+a>“j

r;l+l+...+l=n,

so (n + co), p+a1 5 1. COROLLARY. If Char k = p # 0 then any valuation of k is non-

archimedean. For the ring generated by 1 in k is the field F of p elements. If b E F,

then bps’ = 1 and so lb] = 1.

3. Examples of Valuations

The archetypal example of an arch. valuation is the absolute value on the field C of complex numbers. It is essentially the only one:

THEOREM (Gelfand-Tornheim). Any field k with an arch. valuation is isomorphic to a subfield of C, the valuation being equivalent to that induced by the absolute valuation on C.

We do not prove this as we do not need it. See e.g. E. Artin, “Theory of Algebraic Numbers” (Striker, Gottingen), pp. 45 and 67.

The non-arch. valuations are legion. On the rationals Q there is one for every prime p > 0, the p-adic valuation defined by

IP”uIvIp = P-’ for a, 24, v E Z, p $ u, p y 0.

THEOREM (Ostrowski). The only non-trivial valuations on Q are those equivalent to the I Jp or the ordinary absoIute value I Im.

Proof. Let I 1 b e a non-trivial valuation on Q which (without loss of generality) satisfies the triangle inequality.

Let a E Z be greater than 1. Every b E Z can be put in the shape

b=b,a”+b,-,a”-‘+...+b,

where

OIbj<a (O<jIm)

and log b

mI- log a’

By the triangle inequality

II- ( b <.&I + 1) max (1, [al’s}

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46 J. W. S. CASSELS

where

Jr = lyJl~

On putting b = c” and letting n + 03, we nave

(3.1)

Firs? Case. 3 c > 1 in Z with ICI > 1. Then Ial > 1 for every a > 1 in Z and (3.1) gives

I,gc = l,p$.

Hence I I is equivalent to the ordinary absolute value. Second Case. ICI s 1 for all c E Z so by a previous lemma I I is non-arch.

Since I I is non-trivial the set a of a E Z with Ial < 1 is non-empty and is clearly a Z-ideal. Since lbcj = lb1 ICI th e 1 ea a is prime, say belonging to ‘d 1 p > 0 and then clearly I I is equivalent to I IP.

Now let kO be any field and let k = k,,(r), where t is transcendental. Ifp = p(r) is an irreducible polynomial in the ring k,[t] we define a valuation by

I(Po>>“~o>l~wlp = c-’ (3.2)

where c < 1 is Exed, a E Z and u(t), u(t) E k,[t], p(t) 1 u(t), p(t) ,j’ u(t). In addition there is the non-arch. valuation I Im defined by

(3.3)

Note the analogy between k,,(t) and Q, which is however not perfect. If S= t-‘, so k,(t) = k,,(s), the valuation I Im is seen to be of the type (3.2) belonging to the irreducible polynomial p(s) = s.

The reader will easily prove the LEMMA. The only non-trivial valuations on k,,(t) which are trivial on k,,

ure equivalent to the valuation (3.2) or (3.3). COROLLARY. If F is a jinite field the only non-trivial valuations on F(t)

are equivalent to (3.2) or (3.3).

4. Topology

A valuation I 1 on a field k induces a topology in which a basis for the neighbourhoods of u are the *‘open spheres”

&A4 = {tl It-al < d}

for d > 0. Equivalent valuations induce the same topology. A valuation satisfying the triangle inequality gives a metric for the topology on defining the distance from a to /I to be la-/$

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GLOBAL FIELsDS 47

LEMMA. A field with the topology induced by a valuation is a topological field, i.e. the operations sum, product, reciprocal are continuous.

Proof. For example (product) the triangle inequality implies that

[(a+e)(B+&-aBI 5 l~lI4I+IaIlO~+~~~~~I is small when 161, 141 are small (a, /I fixed).

LEMMA. Iftwo valuations I Ii, I I2 on the samefield induce the same topology then they are equivalent in the sense defined above.

Proof. Iall < 1 if and only if a” -+ 0 (n --, + co) in the topology and so lall < 1 if and only if lalz < 1. On taking reciprocals we see that 1al1 > 1 if and only if lalz > 1 so finally lall = 1 if and only if lalz = 1.

Let now p, y E k and not 0. On applying the foregoing to

a = P”y” (m,nEZ) we see that

m lwISIl +nlwl~l&O according as

mlogIB12+nlog[~12S0 and so

1% IPll _ log IYll 1% lslz log Irlz’

5. Completeness

A field k is complete with respect to a valuation I 1 if it is complete as a metric space with respect to the metric [a-pi (a, /3 E k) i.e. if given any sequence a, (n = 1, 2,. . .) with

Iam-cI,I -+ 0 (m,n+ aha) (a fundamental sequence), there is an a* E k such that

cl,+a* w.r.t. II

(i.e. Ian-a*1 + 0). THEOREM. Every Jield k with valuation I I can be embedded in a complete

field R with a valuation I I extending the original one in such a way that E is the closure of k with respect to I I. Further, li is unique (up to isomorphism).

Proof (sketch). We define l as a metric space to be the completion of k as a metric space with respect to I I. Since the field operations + , x and inverse are continuous on k they are well-defined on ff. Q.E.D.

COROLLARY 1. I I is non-arch. on & if and only if it is so on k. If that is so, the set of values taken by I I on k and It are the same.

Proof. Use second lemma of 5 2. Alternatively, if k is non-arch., the functional inequality

IS+YI I max <IPI, Irl>

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48 J. W. S. CASSELS

holds also in rE by continuity. If now jl E i?, b # 0 there is a y E k such that Ij?-71 < ISI and then 1~1 = 171. Converse trivial.

COROLLARY 2. Any valuation-preserving embedding of k in a complete field K can be uniquely continued to an embedding of k.

6. Independence

The following theorem asserts that inequivalent valuations are in fact almost totally independent. For our purposes it will be superseded by the result of 5 15.

LEMMA (“weak approximation theorem”). Let I 1. (1 5 n I N) be inequivalent non-trivial valuations of a field k. For each n let k,, be the topo- logical space consisting of the set of elements of k with the topology induced by 1 In. Let A be the image of k in the topological product n =1 SF< Nk,,

(with the product topology). Then A is everywhere dense in fl. The conclusion of the lemma may be expressed in a less topological

manner: given any u. E k (1 I n 5 N) and real E > 0 there is a r E k such that simultaneously :

Ian-$ <e (1 r; n 5 N).

Note. If k = Q and the 1 I are p-adic valuations this is related to the “Chinese Remainder Theorem”, but the strong approximation theorem is the real generalization.

Proof. We note first that it will be enough to find 0, E k such that

p”l” > 1, p”lm < 1 (n Z m) (6-l) wherel<nSN,lSm<N. Forthenasr++cowehave

0: 1 1 1 w.r.t. 1 In -c--p i+e; i+e;r 0 w.r.t. 1 jm, m # n

and it is enough to take

g =“il l&p ”

with sufficiently large r. By symmetry it is enough to show the existence of 8 = e1 with

Ieli> lel.4 (25nnN) and we use induction on N.

N=2. Since III and II2 are unequivalent there is an 01 such that

Ia11 < 1, Ial2 2 1 and similarly a p such that

and then 8 = /la-’ will do.

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GLOBAL FIELDS

N 2 3. By the case N- 1 there is a 4 E k such that

l&>l, I&<1 (21n~N-1)

and by the case N = 2 there is a $ E k such that

p11 > 19 MN < 1. Then put

P q#JIN< 1

where r E Z is sufficiently large.

7. Finite Residue Field Case

Let k be a field with non-archimedean valuation I I. Then the set of a E k with Ial 5 1 form a ring o, the ring of integers for 1 I. The e E k with IsI = 1 are a group under multiplication, the group of units. Finally, the set of a with Ial < 1 is a maximal ideal p, so the quotient ring o/p is a field. We consider the case when o/p has a finite number P of elements.

Suppose further, that 1 I is discrete. Then p is a principal ideal (n), say, and every a is of the form a = x”e, where E is a unit. We call u the order of a. If also p = (n’) then n/n is a unit and conversely, so the order of a is independent of the choice of a.

Let ii, p be defined with respect to the completion k of k. Then clearly a/p = o/p and fi = (n) as an a-ideal.

LEMMA. Suppose, further, that k is complete with respect to I I then o is precisely the set of

a= ~~j,l

J=O (7.1)

where the aJ run independently through some set c of representatives in 0 of oh.

By (7.1) is meant of course the limit of the fundamental sequence J

Cain’ as J+ CO. For there is a uniquely defined a0 E c such that Ia --a01 c 1. Then

a1 = IC -‘(a-a,) E o. Now define a, EC by Ial-a,l < 1. And so on. THEOREM. Under the conditions of the preceding lemma o is compact

with respect to the I I-topology. Proof. Let 0, (A E A) be some family of open sets covering o. We must

show there is a finite subcover. We suppose not. Let c be a set of representatives of o/p. Then o is the union of the finite

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50 J. W. S. CASSELS

number of sets a + no (a E C). Hence for at least one a, E C the set a0 + ZD is not covered by finitely many of the O1. Then similarly there is an al E c such that a, +a1 x+zzo is not finitely covered. And so on. Let a = u,+ula+.... Then aEOA, for some L,EA. Since OLO is open, a+n’o c OAO for some J. Contradiction.

COROLLARY. k is Iocully compact. [The converse is also true. If k is locally compact with respect to a non-arch. valuation 11

then

(1) k is complete; (2) the residue field is tinite; (3) the valuation is discrete.

For there is a compact neighbourhood c of 0. Then lrVo c c for sufficiently large u so zoo is compact, being closed. Hence o is compact. Since 1 1 is a metric, D is sequentially compact, i.e. every fundamental sequence in o has a limit, which implies (1). Let 01 (A E II) be a set of representatives in 0 of o/p. Then 01 : II - all < 1 is an open covering of o. Thus (2) holds since o is compact. Finally, p is compact being a closed subset of o. Let S. be the set of Q E k with ltcl < 1 - l/n. Then S,, (1 5 n < 03) is an open cover of 0, so 4 = S, for some n, i.e. (3) is true.

If we allow I 1 to be archimedean the only further possibilities are k = R and k = C with 1 I equivalent to the absolute value.]

We denote by k+ the commutative topological group whose points are the elements of k, whose law is addition and whose topology is that induced by 1 I. General theory tells us that there is an invariant measure (Haar measure) defined on k+ and that this measure is unique up to a multiplicative constant. We can easily deduce what that measure p is.

Since p is invariant

p(a+7c”o) = p”

is independent of a. Further

a+n”o = u (a+7c”uj+x”+‘o) l<j<P

where ui (1 5 j < P) is a set of representatives of o/p. Hence

P” = PP”, 1. If we normalize p by putting

P(O) = 1, (7.2)

we have

p” = P-“.

Conversely, without the theory of Haar measure, it is easy to see that there is a unique invariant measure on k+ subject to (7.2).

Everything so far in this section has depended not on the valuation 1 1 but only on its equivalence class. The above considerations now single out one valuation as particularly important.

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GLOBAL FIELDS 51

DEFINITION. Let k be a field with discrete valuation 1 1 and residue class fieId with P -C 00 elements. We say that 1 1 is normalized if

17cl = P-l, where p = (7~).

THEOREM. Suppose, further, that k is complete with respect to the nor- malized valuation I I. Then

Aa+Po> = IPI where ,a is the Haar measure on kf normalized by ,u(o) = 1.

We can express the result of the theorem in a more suggestive way. Let /? E k, p # 0 and let p be a Haar measure on k+ (not necessarily normalized as in the theorem). Then we can define a new Haar measure pfl on k+ by putting CL,@) = p(PE) (E c k+). But Haar measure is unique up to a multiplicative constant and so &(E) = pL(PE) = fp(E) for all measurable sets E, where the factor f depends only on p. The theorem states that f is just I/II in the normalized valuation.

[The theory of locally compact topological groups leads to the consideration of the dual (character) group of k+ . It turns out that it is isomorphic to k+. We do not need this fact for class field theory so do not prove it here. For a proof and applications see Tate’s thesis (Chapter XV of this book) or Lang: “Algebraic Numbers” (Addison Wesley), and, for generalizations; Weil: “Adeles and Algebraic Groups” (Princeton lecture notes) and Godement: Bourbaki seminars 171 and 176. The determination of the character group of k” is local class-field theory.]

The set of non-zero elements of k form a group k” under multiplication. Clearly multiplication and taking the reciprocal are continuous with respect to the topology induced in k” as a subset of k, so k” is a topological group with this topology.? We have

k” DEEE~

where E is the group of units of k and where E, is the group of einseinheiten, i.e. the E E k with Is- 1 I < 1. Clearly E and E, are both open and closed ink”.

Obviously k”/E is isomorphic to the additive group Zf of integers with the discrete topology, the map being

YC’E+V (v E Z).

Further, E/E, is isomorphic to the multiplicative group rcx of the non-zero elements of the residue class field, where the finite group JC’ has the discrete topo1ogy.S Further, E is compact, so k” is locally compact, Clearly the

t We shall later have to consider the situation for topological rings R, where R” in general is given a different topology from the subset topology.

$ xX is cyclic of order P - 1. It can be shown that k always contains a primitive P - 1-th root of unity p and so the elements of k” are just the Yp%, E E El, i.e. k” is the direct product of Z, Z/(P - l)Z and El.

In fact let f(X) = Xpel - 1 and let M: E o be such that 0: mod p generates K’. Then 1 f(a) 1 < 1, 1 f’(a)1 = 1. Then by Hensel’s Lemma (App. C) there is p E k such that

f(p) = 0, p = a (mod p).

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52 J. W. S. CASSELS

additive Haar measure on Ei is also invariant under multiplication so gives a Haar measure on E,: and this gives the Haar measure on k” in an obvious way.

Finally we note the

LEMMA. k+ and k” are totally disconnected (the only connected sets are points).

Beweis. Klar.

[It is perhaps worth mentioning that k” and k+ are locally isomorphic if k has charac- teristic 0. We have the exponential map

01+expo: = X1$

valid for all sut?kiently small a with its inverse

,og a = z (-I+% - 1)” n

valid for all a sufficiently near to 1.1

8. Normed Spaces

DEFINITION. Let k be a field with valuation 1 1 and let V be a vector space over k. A real-valuedfunction 11 11 on V is called a norm tf

(1) llall > 0 for a cz V, a # 0.

(2) lla+fJII 5 lbll+ II% (3) /aall = Ial llall @Ek, a EW. DEFINITION. Two norms II II 1, 11 II2 on the same space are equivalent tf

there exist constants cl, c2 such that

II4 22 4lQllz IMI s 44 , This is clearly an equivalence relation. LEMMA. Suppose that k is complete with respect to I I and that V isfinite-

dimensional. Then any two norms on V are equivalent. Note. As we shall see, completeness is essential. Proof. Let a1, . . . , a, be any basis for V. We define a norm II IlO by

IIC r”a”llo = my ILI. It is enough to show that any norm II 11 is equivalent to 1 I,,. Clearly

IIC e.41 22 c Ia IMI s clIIC tmanllo

with

cl = C ,pnl,.

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GLGBAL FIELDS

Suppose that there is no cz such thatt

ho 5 4lQII~ Then for any E > 0 there exist Ti,. . ., & such that

0 < IIC Ca,ll 5 Emax It.1. By symmetry we may suppose that

max IL1 = ltNl and then by homogeneity that

&= 1.

Form= 1,2,...,wethushave&,(l <n<N-1)with

ll~~~t...a. + aNIl --f 0 (m + a>;

SO

N-l

53

The lemma being trivial for N = 1, we may suppose by induction that it is true for the (N- I)-dimensional space spanned by al,. . . , a,- I and hence

~5n,d-~“,ml+o V,rrf-+%~) for 1 5 n I N- 1. Since k is complete there are c.* E k with

Ic,n-t*p (m-,aJ). Then

IIN~‘t.*an+aNll li; IIN~ltn,nan+%ll +NilIt?-tn.mj llanll -+O cm + co) . in contradiction to (1).

9. Tensor Product We need only a special case. Let A, B be commutative rings containing a field k and suppose that B is of finite dimension N over k, say with basis

1 =@1,02 ,..., 0,.

Then B is determined up to isomorphism by the multiplication table

% %I = c c~nrn~n c,, E k. We can define a new ring C containing k whose elements are expressions of

t When k is not merely complete with respect to 1 1 but locally compact, which will be the case of primary interest, one can argue more simply as follows. By what has been shown already, the function Iloll is continuous in the II /lo-topology, and so attains its lower bound 6 on Ilull = 1. Then 6 > 0 by condition (i), and then llallo I S-lllajl by homo- geneity for all a.

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54 J. W. S. CASSELS

the type

c amwm a,EA

where the w,,, have the same multiplication rule

QJcPm = x GAmPJn as the q,,. There are ring isomorphisms

i:a+aw, and

j:~L*m+~L*II, of A and B respectively into C. It is clear that C is defined up to isomor- phism by A and B and is independent of the particular choice of basis w,. We write

C=A@,B

since it is, in fact, a special case of the ring tensor-product.

[The reader will have no difficulty in checking that C together with the maps i, j possesses the defining Universal Mapping Property.]

Let us now suppose, further, that A is a topological ring, i.e. has a topology with respect to which addition and multiplication are continuous. The map

Cash -‘(al,. . .,aJ is a l- 1 correspondence between C and N copies of A (considered as sets). We give C the product topology. It is readily verified (i) that this topology is independent of the choice of basis q,. . . cc, and (ii) that multiplication and addition in C are continuous with respect to it; i.e. C is now a topological ring.

We shall speak of this topology on C as the tensor product topology. Now let us drop our supposition that A has a topology but suppose

that A, B are not merely rings but fields. LEMMA. Let A, B be jields containing the field k and suppose that B is a

separable extension of degree [B : k] = N < co. Then C = A & B is the direct sum of a finite number of fields Kj, each containing an isomorphic image of A and an isomorphic image of B.

Proof. By a well-known theorem (appendix B) we have B = k(B) where fvlh;nO,l f; someNseyble f(X) E k[X] of degree N irreducible in k[X].

. . , /l - is a basis for B/k and so A I& B = Au] where bit. . . , b-” are linearly independent over A and f(J) = 0.

Although f(X) is irreducible in k[X] it need not be in A[X], say

fwl = l Jp,cx,

where gj(X) E A[X] is irreducible. The gj(X) are distinct because f(X) is

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GLOBALFIELDS 55

separable. Let K, = A(Bj) where Sjclsi> = 0. Clearly the map

AC&B%,

given by

wb 11: W) h(x) E AX

is a ring homomorphism. We thus have a ring homomorphism

Pl@...@PJ ARABS 8 Kj.

t5jrJ (9-l)

Let h(B), h(X) E A[X] be in the kernel. Then h(X) is divisible by every g,(X), so also byf(X), i.e. h(p) = 0. Thus (9.1) is an injection. Since both sides of (9.1) have the same dimension as vector spaces over A it must be an isomorphism, as required.

It remains to show that the ring homomorphisms

QB+A&B:K,

are injections. If Aj2j(B) # 0 for any /3 E B then Aj(81) # 0 for all PI # 0 because Aj@) = njcs,)nj~~-‘). H ence all we have to show is that A, does not map the whole of B onto 0: and this is trivial.

COROLLARY. Let a E B and let F(X) E k[X], Gj(X) E A[X] (1 I i < J) be the characteristic polynomial of a over k and of the image of a under

B+A&B-+Kj

over A respectively. Then

F(X) = n Gj(X)* (9.2) ISjSJ

Proof: We show that both sides of (9.2) are the characteristic polynomial T(X) of the image of a in A @‘k B over A. That E’(X) = T(X) follows at once by computing the characteristic polynomial in terms of a basis m,,*-* WN, where ol,. . ., o, is a basis for B/k. That T(X) = IIGj(X) follows similarly by using a base of

A&B= QK,

composed of bases of the individual KjIA. COROLLARY. For a E B we have

Norm,,,a = 1 jp-%L4 a

Trace,,, a = i *paCeK,,A a.

ProoJ For the norm and trace are just the second and the last coefficient in the characteristic equation.

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56 J. W. S. CASSELS

10. Extension of Valuations Let k c K be fields and 1 I, )I b

II

e valuations on k, K respectively. We say that 11 11 extends I I if lb1 = IJb for all b E k.

THEOREM. Let k be complete with respect to the valuation k and let K be an extension of k with [K: k] = N < 00. Then there is precisely one extension of 1 1 to K namely

llall = INorm,,l,crll’N. (10.1)

Proof. Uniqueness. K may be regarded as a vector space over k and then I\ 11 is a norm in the sense defined earlier. Hence any two extensions I( ((r and 11 II2 of I 1 are equivalent as norms and so induce the same topology in K. But as we have seen two valuations which induce the same topology are equivalent valuations, i.e. 11 I! 1 = )I 11; for some c. Finally c = 1 because llblll = /lb/l2 for all b E k.

Existence. For a proof of existence in the general case see e.g. E. Artin: “Theory of Algebraic Numbers” (Striker, Gottingen) and for a proof valid for separable non-arch. discrete valuations see Chapter I, 5 4, Prop. 1, Corollary. Here we give a proof (suggested by Dr. Geyer at the conference) valid when k is locally compact, the only case which will be used. In any case it is easy to see that the definition (10.1) satisfies the conditions (i) that llal\ 2 0 with equality only for a = 0 and (ii) 11aflll = [[a11 l//?/l: the difficulty is to show that there is a constant C such that /all 5 1 implies 111+a\] I C. Let 11 Ilo be any norm on K considered as a vector space over k. Then /alI defined by (10.1) is a continuous non-zero function on the compact set (Ial(O = 1, so A 2 llall 2 6 > 0 for some constants A, 6. Hence by homogeneity

ArM II40

26>0. (alla#O).

Suppose, now, that IIan 5 1. Then IIan 5 6-r and so

lP+4 s Al11+4J 22 ~Wllo+ 11410~ 5 A(~~~~)o+~-‘) = C (say),

as required. Formula. Geyer’s existence proof also gives (10.1). But it is perhaps worth

noting that in any case (10.1) is a consequence of unique existence, as follows. Let L 1 K be a finite normal extension of k. Then by the above there is a unique extension of I I to L which we shall denote also by 11 11. If r~ is an automorphism of L/K then

114~ = lbll

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GLOBALFIEZDS

is also an extension of 11 to L, so 1 us = (I 11, i.e.

lluall = llall (all c1 E L).

But now

Norm,,,a = ulao2a.. . ona

foraoK,whereol,..., oN are automorphisms of L/k. Hence

INormK,k aI = IINo~Ic,G~~

57

= llQllN~ as required.

COROLLARY. tit O1,..., an be a basis for KJk. Then there are constants ct, c2 such that

c * IC bJ@A< c ’ max lb.1 ’

for b l,...,bNEk(notaNO). Proof. For 12 b, “1 d o an max lb,,1 are two norms on K considered as a

vector space over k. COROLLARY 2. A finite extension of a completely valued$eld k is complete

with respect to the extended valuation. For by the preceding corollary it has the topology of a finite-dimension

vector space over k. When k is no longer complete under I I the position is more complicated: THEOREM. Let K be a separable extension of k of degree [K : k] = N < CO.

Then there are at most N extensions of a valuation I 1 of k to K, say II 11, (1 I j s J). Let I%, K, be the completion of k req. K with respect to I I, resp. 11 11,. Then

k@kK= @ Kj (10.2) 1SjS.l

algebraically and topologically, where the R.H.S. is given the product topology. Proof. We know already that R @I K is of the shape (10.2) where the Kj

are finite extensions of R. Hence there is a unique extension I 17 of I 1 to the K, and the K, are complete with respect to the extended valuation. Further, by a previous proof, the ring homomorphisms

+K--&&K+K,

are injections. Hence we get an extension II 11, of I I to K by putting

IlPllj = Inj<P>li*-

Further, K GZ ;li(K) is dense in KI with respect to 11 11, because K = k & K is dense in k Bpk K. Hence K, is exactly the completion of K.

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58 J. W. S. CASSELS

It remains to show that the 11 Ilj are distinct and that they are the only extensions of I I to K.

Let 11 ]I be any valuation of K extending I I. Then II 11 extends by con- tinuity to a real-valued function of R Ok K, a function also to be denoted by 11 II. By continuity we have

Ib+~II 5 max II4 Ml ll@ll = lbll IPII >

c1 p E k(g) K ’ *

We consider the restriction of II I/ to one of the Kj. If lltlll # 0 for some CI E Kj the ll~lll = ll~ll IIuP-‘II for every P + 0 in Kj SO l/~/l # 0. Hence either I] II is identically 0 on Kj or it induces a valuation on Kj.

Further, II 11 cannot induce a valuation on two of the KP For

(a,@O@. . . @O).(O@a,OO.. .@O) = (000. . .@O)

and so

II%II ll%II = 0 %EKlY a2 E&*

Hence II II induces a valuation in precisely one of the Kj and it clearly extends the given valuation I I of li. Hence II II = II II j for precisely one j.

It remains only to show that (10.2) is also a topological homomorphism. For ul,. . .,pJ~K~@...@K,put

11& . * *,PJ>llO =l~~~JllPjllj*

Clearly, II Ijo is a norm on the R.H.S. of (10.2), considered as a vector space over k and it induces the product topology. On the other hand, any two norms are equivalent, since R is complete, and so II II0 induces the tensor product topology on the left-hand side of (10.2).

COROLLARY. Let K = k(P) and let f(x) E k[X] be the irreducible equation for j7. Suppose that

in k[X], where the gj are irreducible. Then Kj = &?j) where gj<pj) = 0.

11. Extensions of Normalized Valuations

Let k be a field with valuation 1 I. We consider the three cases:

(1) 1 1 d is iscrete non-arch. and the residue class field is finite. (2(i)) The completion of k with respect to is R. (2(ii)) The completion of k with respect to II is C.

[In virtue of the remarks in 3 7, these cases can be subsumed in one: the completion k is locally compact.]

In case (1) we have already defined a normalized valuation (9 7). In case (2(i)) we say I I is normalized if it is the ordinary absolute value and in

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GLOBAL FELDS 59

case (2(ii)) if it is the square of the absolute value. Thus in every case the map

a: <-+a< <EL+ (aEE)

of the additive group R+ of the completion of k multiplies the Haar measure on li’ by la[: and this characterizes the normalized valuation among equivalent ones.

LEMMA. Let k be complete with respect to the normalized valuation 1 1 and let K be an extension of k of degree [K: k] = N < co. Then the normalized valuation 11 II of K h’ h w IC is equivalent to the unique extension of I I to K is given by the formula

llall = INor+ka( (a E K).

Proof. By the preceding section we have

llall = INormK,kalc (a E K> (11.1)

for some real c > 0 and all we have to do is to prove that c = 1. This is trivial in case 2 and follows from the structure theorems of Chapter I in case 1. Alternatively one can argue in a unified way as follows. Let ml,. . *, ON be a basis for K/k. Then the map

~=~bw-G,...,5~) (tf1,--.,LEk)

gives an isomorphism between the additive group K+ and the direct sum ONkf of N copies of kf, and this is a homomorphism if the R.H.S. is given the product topology. In particular, the Haar measures on Kt and CBNkf are the same up to a multiplicative constant. Let b E k. Then the map

b:E+bS

of Kt is the same as the map

(t 1,...,5N)-,(b~l,...,br~)

of eNkt and so multiplies the Haar measure by IbIN, since I I is normalized. Hence

llbll = IbIN* But NormKltb = bN and so c = 1 in (11.1).

In the incomplete case we have THEOREM. Let I I b e a normalized valuation of a field k and let K be a

finite extension of k. Then

where the (1 II j are the normalized valuations equivalent to the extensions of

II to K.

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60

Proof. Let

J. W. S. CASSELS

E@kK= @ Kj, 15lZZ.I

where li is the completion of K. Then (I 9)

Norm,,, a = n lSj<J

(Norm,,,x a).

The theorem now follows from the preceding lemma and the results of 5 10.

12. Global Fields

By a global field k we shall mean either a finite extension of the rational field Q or a finite separable? extension of F(t), where F is a finite field and t is transcendental over F. We shall focus attention in the exposition on the extensions of Q (algebraic number case) leaving the extension of F(t) (func- tion field case) to the reader.

LEMMA. Let a # 0 be in the global field k. Then there are only finitely many unequivalent valuations 1 1 of k for which

[aI > 1.

Proof. We know this already for Q and F(t). Let k be a finite extension of Q, so

a”+a,a”-‘+. . . +a, = 0

forsomenanda,,...,a,,. If[ 1 is a non-arch. valuation of k we have

Iaj”=j-aala”-l-...-aa,i

and so

5 max(l,Iar-‘)max(jarj,. . .,\a$

Ial 5 maxU,lall,. . .,la$. Since every valuation of Q has finitely many extensions to k and since there are only finitely many arch. valuations altogether, the theorem for k follows from that for Q.

All the valuations of a global field k are of the type described in $ 11, since this’is true of Q and F(f). Hence it makes sense to talk of normalized valuations.

THEOREM. Let a E k, where k is a globalfieId and a # 0. Let I IV run through all the normalized valuations of k. Then IaIV = 1 for all except finitely many v and

Note. We shall later give a less computational proof of this.

t Thii condition is not redly necessary. If k is any Ikite extension of F(t) there is a “separating element” s, i.e. an s E k such that k is a finite separable extension of F(S).

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GLOBAL FIELDS 61

Proof. By the lemma Ial0 < 1 for almost all 1) (i.e. all except finitely many). Similarly la-‘lV I 1 for almost all u, so lalD = 1 for almost all 0.

Let V run through all the normalized valuations of Q [or F(t)] and write VI V to mean that the restriction of u to Q is equivalent to K Then

by the preceding section. This reduces the theorem to the case k = Q. But if now

b=kn~+Q, P

where p runs through all the primes and BP E Z, we have

lblp = p+p

for the p-adic valuation 1 IP and

lblo3 =,pz’*

for the absolute value I I m. P

Q.E.D. Let K be a finite separable extension of the global field k. Then for every

valuation v of k we have an isomorphism

where k, is the completion of k with respect to u and K,, . . . , K,, are the completions of K with respect to the extensions V,, . . . , V, of u to K (5 lo), the number J = J(u) depending on u. We shall later need the

LEMMA. Let cut,. . ., a+,? be a basis for K/k. Then for almost all normalized u we have

~,o~~~o~...~~No=~l~...~~J (12.1)

where N = [K: k], o = o, is the ring of integers of k for I I,, and 0, c K, is the ring of integers for I IV, (1 5 j 5 J). Here we have identified a E K with its canonical image in k, @ K.

Proof. The L.H.S. of (12.1) is included in the R.H.S. provided that lo.&, 5 1 (1 I n 5 N, 1 5 j < J). Since IaIr 5 1 for almost all Y it follows that L.H.S. c R.H.S. for almost all u.

To get an inclusion the other way we use the discriminant

NYl,. * .,YN> = det,, dtraCeK/k Y~Y,),

where yr,. . ., yN E k, Bk K. If y. E R.H.S. (1 < n I N) we have (5 9)

traceKlk YntYn = c traceKj/Eymyn E o = Ov l&jsiJ

and so %b. . * > YN) E 0,.

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62 J. W. S. CASSELS

Now suppose that a E R.H.S. and that

,!I=$b,,w,ER.H.S. (b,Ek,). 1

(12.2)

Then for any m, 1 I m I N we have

I.@, ,..., w,,-~,P,w,,+~ ,..., d=b:W’, ,..., R>, and so

dbi E o, (1 I m 5 N)

where

d=D(o,,...,w,)Ek.

But (Appendix B) we have d # 0, and so ldlv = 1 for almost all u. For almost all u the condition (12.2) thus implies

b, E o, (15 m 5 N),

i.e.

R.H.S. c L.H.S. This proves the lemma.

[COROLLARY. Almost all v are unramijied in the extension K/k. For by the results of Chapter I a necessary and sutlicient condition for v to be un-

ramified is that there are yl,. . ., yN E R.H.S. with ID(y,,. . ., yN)lv = 1. And for almost all v we can put yn = ~f”-~.]

13. Restricted Topological Product

We describe here a topological tool which will be needed later: DEFINITION. Let sZA (A E A) be a family of topological spaces and for almost

alIt L let On c Q, be an open subset of a,. Consider the space n whose points are sets u = h&h where aA E a, for every A and cr, E On for almost all 1. We give Q a topology by taking as a basis of open sets the sets

l-m where rl c G12, is open for all rZ and rA = O1 for almost all 1. With this topology R is the restricted topologicalproduct of the R, with respect to the O1.

COROLLARY. Let S be a finite subset of A and let Rs be the set of a E fi with aA E O1 (2 q! S), i.e.

(13.1)

Then G!, is open in Sz and the topology induced in as as a subset of R is the same as the product topology.

Beweis. Klar. The restricted topological product depends on the totality of the 0,

but not on the individual 0,:

t i.e. all except possibly finitely many.

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GLOBAL FIELDS 63

LEMMA. Let 0; c Jz, be open sets defined for almost all II and suppose that OA = 0: for almost all 2. Then the restricted product of the f2, with respect to the 0; is the same ast the restrictedproduct with respect to the 0,.

Beweis. Klar. LEMMA. Suppose that the RA are locally compact and that the 0, are

compact. Then R is locally compact. Proof. The as are locally compact by (13.1) since S is finite. Since

n = u C& and the S& are open in 0, the result follows. DEFINITION. Suppose that measures ,uI are defined on the RA with

~~(0,) = 1 when OA is defined. We define the product measure p on R to be that for which a basis of measurable sets is the

where MA c Sz, has finite PA-measure and MA = On for aImost all 1 and where

COROLLARY. The restriction of ,u to Q, is just the ordinary product measure.

14. Adele Ring (or Ring of Valuation Vectors)

Let k be a global field. For each normalized valuation 1 1” of k denote by k, the completion of k. If 1 1” is non-archimedean denote by D, the ring of integers of k,. The adele ring V, of k is the topological ring whose under- lying topological space is the restricted product of the k, with respect to the D, and where addition and multiplication are defined componentwise:

W% = aA (a+/% = av+/L a,BE 6. (14.1)

It is readily verified (i) that this definition makes sense, i.e. if a, fi E V, then an, a+ fl whose components are given by (14.1) are also in V, and (ii) that addition and multiplication are continuous in the V,-topology, so V, is a topological ring, as asserted.

V, is locally compact because the k, are locally compact and the o, are compact (5 7).

There is a natural mapping of k into V’ which maps cc E k into the adele every one of whose components is a: this is an adele because a E o, for almost all O. The map is an injection, because the map of k into any k, is an injection. The image of k under this injection is the ring of principal adeles. It will cause no trouble to identify k with the principal adeles, so we shall speak of k as a subring of Yk.

LEMMA. Let K be afinite (separable) extension of the globaljield k. Then

&c&K = v, (14.2)

t A purist would say “canonically isomorphic to”.

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64 I. W. S. CASSELS

aIgebraicaIIy and topologically. In this correspondence k &K = K c V’ Blk K, where k c V,, is mapped identically on to K c V,.

Proof. We first established an isomorphism of the two sides of (14.2) as topological spaces. Let ml,. . . , oN be a basis for K/k and let u run through the normalized valuations of k. It is easy to see that the L.H.S. of (14.2), with the tensor product topology, is just the restricted product of the

k,C&K=k,o,@...@kvcoN (14.3)

with respect to the

o,,t& @ . . . @ o,wN. (14.4)

But now (cf. $lO), (14.3) is just

G, CB . . .c?3 Kv,, <VllU,. * *, v,lu> (14.5)

where V,,. . ., V,, J = J(u) are the normalized extensions of u to K. Further (5 12) the identification of (14.3) with (14.5) identifies (14.4) with

D,,a3...cEmJ (14.6) for almost all? u. Hence the L.H.S. of (14.2) is the restricted product of (14.3) with respect to (14.4), which is clearly the same thing as the restricted product of the K, with respect to the DV, where V runs through all the normalized valuations of K. This is just the R.H.S. of (14.2). This establishes an isomorphism between the two sides of (14.2) as topological spaces. A moment’s consideration shows that it is also an algebraic isomorphism.

Q.E.D. COROLLARY. Let Vc denote the topological group obtained from V, by

forgetting the multiplicatiue structure. Then

Vi+ =V;‘@...@V;’ (N=[K:k]). N

In this isomorphism the additive group K+ c Vz of the principal adeles is mapped into k’ @ . . . @k+, in an obvious notation.

ProoJ: co&’ c V$, for any non-zero o E K, is clearly isomorphic to Vk’ as a topological group. Hence we have the isomorphisms

Vi+ =Vk+C&K=~iVk+@ . [email protected]+ = v;‘@...@&+.

THEOREM. k is discrete,+ in V’ and V’+/k + is compact in the quotient topology.

Proof. The preceding corollary (with k for K and Q or F(t) for k) shows that it is enough to verify the theorem for Q or F(t) and we shall do it for Q.

To show that Q + is discrete in VG it is enough because of the group

t This was proved there only when CO,, = h-l, where K = k(a). We should therefore take this choice of CD”.

$ It is impossible to conceive of any other uniquely defined topology in k. This meta- mathematical reason is more persuasive than the argument that follows!

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structure to find a neighbourhood U of 0 which contains no other elements of k+. We take for U the set of a = (GC,} E Vo! with

Iaalg, < 1

Ia& 5 1 W ~1, where 1 IP, 1 I- are respectively the p-adic and the absolute values on Q.

If b E Q n U then in the first place b E Z (because lblp 5 1 for all p) and then b = 0 because lblm < 1.

Now let W c V$ consists of the a = {a”} with

la,la S 3, l~plp 5 1 611 10. We show that every adele /I is of the shape

j = b+a, beQ, aEW. (14.7)

For each p we can find an

such that rP = zpp (Z,EZ, XpEZ, x, 2 0)

I&-4 5 1 and since a is an adele we may take

rP = 0 (almost all p).

Hence r = c rp is well defined and

Nowchoos:soZsuchthatlh-r”l (allp)S

Ifi, -r-s1 S $.

Then b = r +s, fl = a - b do what is required. Hence the continuous map W + V,‘/Q’ induced by the quotient map

V$ + V,‘/Q’ is surjective. But W is compact (topological product of la,la 2 3 and the D& and hence so is V,‘/Q’.

As already remarked, Vz is a locally compact group and so it has an invariant (Haar) measure. It is easy to see that in fact this Haar measure is the product of the Haar measures on the k, in the sense described in the previous section.

COROLLARY 1. There is a subset W of Vk defined by inequalities of the type I<& 5 a,, where 6, = 1 for almost all v, such that every cp E V, can be put in the form

(P = e+y, OEW, yak

Proof. For the W constructed in the proof is clearly contained in some W of the type described above.

COROLLARY 2. V,‘/k’ has finite measure in the quotient measure induced by the Haar measure on V,’ .

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66 J. W. S. CASSELS

Note. This statement is, of course, independent of the particular choice of the multiplicative constant in the Haar measure on Vk+. We do not here go into the question of finding the measure of V,+/k+ in terms of our explicitly given Haar measure. (See Tate’s thesis, Chapter XV of this book.)

Proof. This can be reduced similarly to the case of Q or F(t), which is almost immediate: thus W defined above has measure 1 for our Haar measure.

Alternatively finite measure follows from compactness. For cover Vz /k’ with the translates of F, where F is an open set of finite measure. The existence of a finite subcover implies finite measure.

[We give an alternative proof of the product formula lI l<jv = 1 for 6 E k, 6 # 0. We have seen that if b,, E kv then multiplication by /3. magnifies the Haar measure in kc by the factor I& Hence if B = {jV} E Vk, multiplication by B magnifies Haar measure in I’,,? by I2 I& In particular multiplication by the principal adele c magnifies Haar measure by l7 I&. But now multiplication by < takes k+ c V,+ into k+ and so gives a well-defined 1 - 1 map of V,+/k+ onto V,+/k+ which magnifies the measure by the factor I7 I<[.. Hence Ii’ I& = 1 by the Corollary.]

In the next section we shall need the LEMMA. There is a constant C > 0 depending only on the global field k

with the following property: Let dl = {cI”> c V, be such that

lJ IU”l” ’ c- (14.8)

Then there is a principal adele B E k c V,, /I # 0 such that

lPlu 5 ‘l~l, (all 9.

Proof. This is modelled on Blichfeldt’s proof of Minkowski’s Theorem in the Geometry of Numbers and works in quite general circumstances.

Note that (14.8) implies [a,l” = 1 for almost all u because l~,l, 5 1 for almost all v.

Let co be the Haar measure of V,+/k+ and let c1 be that of the set of y = (yJ c Vc with

IY”l” s ik if v is arch.

lY”l” 2 1 if u is if u is non-arch.

Then 0 < co < co and 0 < ci < co because the number of arch. u’s is finite. We show that

c = CO/Cl

will do. The set T of z = (tU) c V> with

Iz,I, I &$x,1, if u is arch.

j~“l” 5 I@“\” if u is non-arch.

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has measure

Hence in the quotient map F’k’ + V,+/k+ there must be a pair of distinct points of T which have the same image in Vz/k+, say

z’ = (2;) E T, Z” = {TV”} E T

and z’-r” = p (say) E k’.

Then

IPI” = I+cl” 2 I%I” for all u, as required.

COROLLARY. Let v0 be a normalized valuation and let 6, > 0 be given for all v # I,+, with 6, = 1 for almost all v. Then there is a /? E k, p # 0 with

I& 5 6, (all u # ud.

Proof. This is just a degenerate case. Choose a, E k, with 0 < aolv I 6, and ]a,/, = 1 if 6, = 1. We can then choose avO E kUO so that

all “K ” I avIu ’ =. 0 Then the lemma does what is required.

[The character group of the locally compact group V,+ is isomorphic to V,+ and k+ plays a special role. See Chapter XV (Tate’s thesis), Lang: “Algebraic Numbers” (Addison- Wesley), Weil : “Adeles and Algebraic Groups” (Princeton lecture notes) and Godement : Bourbaki seminars 171 and 176. This duality lies behind the functional equation of 5 and L-functions. Iwasawa has shown (Annuls of Math., 57 (1953), 331-356) that the rings of adeles are characterized by certain general topologico-algebraic properties. ]

15. Strong Approximation Theorem

The results of the previous section, in particular the discreteness of k in V, depend critically on the fact that all normalized valuations are used in the definition of V,:

THEOREM. (Strong approximation theorem.) Let v,, be any valuation of the global field k. Define Y to be the restricted topological product of the k, with respect to the Q~, where v runs through all normalized v # vO. Then k is everywhere dense in Y.

Pro0f.t It is easy to see that the theorem is equivalent to the following statement. Suppose we are given (i) a finite set S of valuations u # ue, (ii) elements a, E k, for all u E S and (iii) E > 0. Then there is a /I E k such that Ip-a”l “< E for all u E S and l/31u 5 1 for all v 4 S, v # ve.

By Corollary 1 to the Theorem of $14 there is a W c V, defined by inequalities of the type Ir,lu 5 6, (6, = 1 for almost all U) such that every

t Suggested by Prof. Kneser at the Conference.

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68 J. W. S. CASSELS

q E V, is of the form cp=e+y, OEW, yEk. (15.1)

By the corollary to the last lemma of § 14, there is a rZ E k, A # 0 such that

I$ < 43 (u E 8,

I& s 6,’ (1) 4 s,u # uo). (15.2)

Hence, on putting cp = A-la in (15.1) and multiplying by L we see that every a E V, is of the shape

a=$+P, $EAW, BEk, (15.3)

where AW is the set of AC, C E IV. If now we let a have components the given a, at u e S and (say) 0 elsewhere, it is easy to see that fi has the properties required.

[The proof clearly gives a quantitative form of the theorem (i.e. with a bound for I&,). For an alternative approach, see K. Mahler: Inequalities for ideal basea, J. Australian Math. Sot. 4 (1964), 425A48.1

16. Idele Group

The set of invertible elements of any commutative topological ring R form a group R” under multiplication. In general, R” is not a topological group if it is endowed with the subset topology because inversion need not be continuous. It is usual therefore to give R” the following topology. There is an injection

x-+,x-1) (16.0)

of R” into the topological product Rx R. We give to R” the corresponding subset topology. Clearly R” with this topology is a topological group and the inclusion map RX + R is continuous.

DEFINITION. The idele group Jk of k is the group V,” of invertible elements of the adele ring V, with the topology just defined.

We shall usually speak of Jk as a subset of V, and will have to distinguish between the Jk- and V’-topo1ogies.t

We have seen that k is naturally embedded in V, and so k” is naturally embedded in Jk. We shall call kx considered as a subgroup of Jk the principal ideles.

LEMMA. k” is a discrete subgroup of Jk. ProojI For k is discrete in V, and so k” is injected into V, x V, by (16.0)

as a discrete subset. LEMMA. Jk is just the restricted topoIogica1 product of the k,” with respect

to the units U,, c k. (with the restricted product topology). Beweis. Klar.

t Let a@) for a rational prime 4 be the element of JQ with components a?) = q, a:” = 1 (u # q). Then a@) + 1 (q -+ 00) in the Y,-topology, but not in the J,-topology.

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GLGBAL FIELDS 69

DEFINITION. For a = (Q~> c Jk we define c(a) = fl IQ& lo be the con- all ”

tent of a. LEMMA. The map a + c(a) is a continuous homomorphism of the topological

group Jk into the multiplicative group of the (strictly) positive real numbers. Beweis. Klar.

[Lemma. Let a E .&. Then the map 5 + a6 of V,+ onto itself multiplies Haar measure on V,+ by a factor c(a).

Beweis. Klar. Note also that the &topology is that appropriate to a group of operators on V,+ : a

basis of open sets is the S(C, 0) where C, 0 C V,+ are respectively K-compact and Vk-open and S consists of the a l Jk such that (1 - a)C c 0, (1 - a-l)C c 0.1

Let Ji be the kernel of the map a + c(a) with the topology as a subset of Jk. We shall need the

LEMMA. Jl considered as a subset of V, is closed and the Vk-subset topology on Jj coincides with the J,-topology.

Proof. Let aE Vk, a#Ji. We must find a Vk-neighbourhood W of a which does not meet Ji.

1st Case. n [a& < 1 (possibly = 0). Then there is a finite set S of v such that

(i) S contains all the v with lcc,l, > 1 and (ii) n Ic(& < 1. Then the set W can be defined by

ves ls”-~“l” <& VES

IC”lS 1 v#S for sufficiently small E.

2nd Case. n Ia& = C (say) > 1. Then there is a finite set S of v such

that (i) S con:ains all the v with Ia&, > 1 and (ii) if v 4 S an inequality It,]” < 1 implies? l&l, < *C. We can choose E so small that I&-a&, < E (v E S) implies 1 < ,?[, l&,1 < 2C. Then W may be defined by

IL-slo < 8 (0 Es)

jr”1 5 1 (0 4 s). We must now show that the Jk- and &topologies on Ji are the same.

If a E Ji we must show that every J,-neighbourhood of a contains a I’,-neigh- bourhood and vice-versa.

Let: W c J,,! be a V,-neighbourhood of a. Then it contains a V,-neigh-

t I f k =) Q and u is a normalized extension of the p-adic valuation then the value group of II consists of (some of the) powers of p. Hence it is enough for (ii) to include in S all the arch. u and all the extensions of p-adic valuations with p 5 2C. Similarly if k = F(r).

$ This half of the proof of the equality of the topologies makes no use of the special properties of ideles. It is only an expression of the fact noted above that the inclusion R” -+ R is continuous for any topological ring R.

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70 J. W. S. CASSELS

bourhood of the type IL-a,l, < 8 CUE s)

[rvlv 5 1 (v # s) > (16.1)

where S is a finite set of v. This contains the J,-neighbourhood in which 5 in (16.1) is replaced by = .

Now let H c Ji be a J,-neighbourhood. Then it contains a J,-neigh- bourhood of the type

15U--C1,1,<~ C-S) Ir”l” = 1 (u # 9 >

(16.2)

where the finite set S contains at least all arch. Y and all v with ICI,], # 1. Since n Ia& = 1 we may also suppose that E is so small that (16.2) implies

rJ Ir”l” < 2.

Then the intersection of (16.2) with Jk is the same? as that of (16.1) with Ji, i.e. (16.2) defines a V,-neighbourhood.

By the product formula we have k” c Jk . r The following result is of vital importance in class-field theory.

THEOREM. JiJkx with the quotient topology is compact. Proof. After the preceding lemma it is enough to find a V,-compact set

W c V, such that the map W n J: + J,‘/kX

is surjective. We take for W the set of c = {&,} with

It& 5 Ia& where a = (a”} is any idele of content greater than the C of the last lemma of § 14.

Let fl = {B,} E J,f. Then by the lemma just quoted there is a g E k” such that

I$ I IP;‘a,l, (all 0). Then qfi E W, as required.

[&l/c” is totally disconnected in the function field case. For the structure of its connected component in the number theory case see papers of Artin and Weil in the “Proceedings of the Tokyo Symposium on Algebraic Number Theory, 1955” (Science Council of Japan) or Artin-Tate: “Class Field Theory”, 1951/Z (Harvard, 1960(?)). The determination of the character group of Jk/kx is global class field theory.]

17. Ideals and Divisors

Suppose that k is a finite extension of Q. We define the ideal group Ik of k to be the free abelian group on a set of symbols in 1 - 1 correspondence

t See previous footnote.

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GLOBAL FIELDS

with the non-arch. valuations v of k, i.e. formal sums

c n,.v “non-arch.

71

(17.1)

where n, E Z and n, = 0 for almost all v, addition being defined component- wise. We call (17.1) an ideal and call it integral if n, 2 0 for all v. This language is justified by the existence of a l-l correspondence between integral ideals and the ideals (in the ordinary sense) in the Dedekind ring

o= n 0,: non-arch.

cf. Chapter I, $2, Prop. 2. There is a natural continuous map

Jk--+& of the idele group on to the ideal group? given by

a = (a,> --f C (ord, a). v.

The image of k” c Jk is the group of principal ideals. THEOREM. The group of ideal classes, i.e. Ik module principal ideals, is

jinite. Proof. For the map Ji + I, is surjective and so the group of ideal classes

is the continuous image of the compact group Jl/k” and hence compact. But a compact discrete group is finite.

When k is a finite separable extension of F(t) we define the divisor group Dk of k to be the free group on all the v. For each v the number of elements in the residue class field of v is a power, say qdy of the number q of elements in F. We call d, the degree of v and similarly define C n,d, to be the degree of c n, . v. The divisors of degree 0 form a group Dz. One defines the prin- cipal divisors similarly to principal ideals and then one has the

THEOREM. 0,” module principal divisors is a jinite group. For the quotient group is the continuous image of the compact group

J,‘/k”.

18. Units

In this section we deduce the structure theorem for units from our results about idele classes.

Let S be any finite non-empty set of normalized valuations and suppose that 5’ contains all the archimedean valuations. The set of q E k with

[?I” = 1 (v#s) (18.1)

are a group under multiplication, the group Hs of S-units. When k 3 Q and S is just the archimedean valuations, then Hs is the group of units tout court.

t Zk being given the discrete topology.

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72 J. W. S. CASSELS

LEMMA 1. Let 0 < c -c C c co. Then the set of Sunits 4 with

c~[lrlltsC CUES) (18.2)

is finite. Proof. The set W of ideles at = (CL,} with

I~“10 = 1 (u6s)s c 2 lct”l” s c (u E S) (18.3)

is compact (product of compact sets with the product topology). The required set of units is just the intersection of W with the discrete subset k of Jk and so is both discrete and compact, hence finite.

LEMMA 2. There are only finitely many E E k such that l&Iv = 1 for every v. They are precisely the roots of unity in k.

Proof. Ifs is a root of unity it is clear that IsI0 = 1 for every v. Conversely, by the previous lemma (with any S and c = C = 1) there are only finitely many E E k with Islo = 1 for all u. They form a group under multiplication and so are all roots of 1.

THEOREM. (Unit theorem.) Hs is the direct sum of a finite cyclic group and a free abelian group of rank s- 1.

Proof. To avoid petty notational troubles we treat only the case when Q c k and S is the set of arch. valuations.

Let J, consist of the ideals at = {a”} with lcl&, = 1 (t, 4 S) and put Ji = J, n Ji.

Clearly Jl is open in Jk and so

Jfi/H, = Ji/(Ji n k”) (18.4)

is open in Jl/k” . Since it is a subgroup, it is also closed, and so compact (§ 16).

Consider the map

L:J,-+R+@R+&?&R+, \ ,

s times

where R+ is the additive group of reals, given by

a-,(logI~,ll,logla,l,,. . .Joglcl,l,), where 1,2,. . ., s are the valuations in S. Clearly A is both continuous and surjective.

The kernel of L restricted to Hs consists just of the E with Is],, = 1 for every t), so is a tite cyclic group by Lemma 2. By Lemma 1 there are only Iinitely many q E H, with

341~1.~2 VES. (18.5)

Hence the group A (say) = A(H,) is discrete. Further, T = A(J,‘) is just the set of (x,, . . . , x,) with

x,+x2+. . . +x, = 0,

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GLOBAL FIELDS 73

i.e. an S- 1 dimensional real vector space. Finally, T/A is compact, being the continuous image of the compact set (18.4). Hence A is free on S- 1 generators, as asserted.

Of course this structure-theorem (Dirichlet) and the finiteness of the class- number (Minkowski) are older than ideles. It is more usual to deduce the compactness of J,‘/k” from these theorems instead of vita versa.

19. Inclusion and Norm Maps for Adeles, Ideles and Ideals Let K be a finite extension of the global field k. We have already seen

($ 14, Lemma) that there is a natural isomorphism

Kc&K = VK (19.1) algebraically and topologically. Hence V, = V, Bk k can naturally be regarded as a subring of V, which is closed in the topology of I’,. This injection of V, into I’, is called the injection map or the conorm map and is written

con : a + con a = con,,, a E V, (a o V,).

Explicitly if A = con a, then the components satisfy

Ay=uvEk,c K, (19.2)

where V runs through the normalized valuations of K and v is the normalized valuation of k which extends to V. If k c L c K it follows that

conKlk a = conLlk (conKIL a). (19.3) Finally, for principal adeles the conorm map is just the usual injection of k into K.

It is customary, and usually leads to no confusion, to identify conKlk a with a.

One can also define norm and trace maps from V, to vk by imitating the usual procedure (cf. Appendix A). Let or,. . . , w, be a basis for K/k. Then by (19.1) every A E I’, is uniquely of the shape

A=Cajmj ait? vk (19.4)

and the map A + aj of V, into vk is continuous by the very definition of the tensor product topology (5 9). Hence if we define

aij = aij(A) E Vk

(19.5)

the n x n matrices (Q) give a a continuous representation of the ring VR over vk. In particular, the

Sxc,kA = 1 aii (19.6)

NK,kA = det (aij) (19.7)

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74 J. W. S. CASSELS

are continuous functions of A and have the usual formal properties

SK,,& + AZ) = SK,& + SK,,& (19.8)

SKIkconK,ka = na (19.9)

NK,PCWJ = NK,A NK,L& (19.10)

NKIR con,,, a = a”. (19.11)

Further, the norm and trace operations are compatible with the embedding of k, V in V,, V, respectively, i.e. if A E K c V, we get the same answer whether we compute NKIkA, SKIK A in K or in V,, so there is no ambiguity in the notation.

Finally if K 2 L 2 k we have V’ c V, c V, (on regarding conorm as an identification), and so the usual relations (cf. Appendix A)

S,,,(SK,A = S,,,A (19.12) and

NL,~NKILA = = N,,,A- (19.13)

We can express the maps (19.6), (19.7) componentwise if we like. Let VI,. . *, V, be the extensions of any given valuation u of k to K. Then (§ 9)

K, (say) = 0 Ki = k, Blk K = 0 kvq (19.14) 1sijS.J 15is;n

where k,, Ki are the completions of k, K with respect to v, Vj respectively. Any A E V, can be regarded as having components

A,,O...OA,,=A, (19.15)

in the K, and then the components in the matrix representation (19.5) of A are just the representations of the A,. In particular

SK,IM = {SK,,~, A,) (19.16) and

NKIJ = {NK,,v%)- (19.17)

Finally, making use of the final remarks of $9, we deduce that

SK/IA = c SK~,,JAV) { 1

(19.18) Vlv ”

and

NK,& = f-/

NK.p,Av 9 >

(19.19) VU ”

where VIv means “V is a continuation of v”. We now consider the consequences for ideles. If et is an idele, it is clear

from the definition (19.2) that con,,, a is an idele, so we have an injection

conKlk : Jk --) JR

which-is clearly a homomorphism of Jk with a closed subset of V,. Further,

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GLOBAL FIELDS 75

if A E JR c V,, so A is invertible, it follows from (19.9) that NKIk A is invertible, i.e. is an element of Jn. Hence we have a map

N K/k - -JK+Jk

which is continuous by the definition of the idele topology ($16) and which clearly satisfies (19.10), (19.11), (19.13) and (19.19). On the other hand, the definition of trace does not go over to ideles.

FinalIy, we consider the conorm and norm maps for ideals, where k is a finite extension of Q. The kernel of the map ($17)

Jk -+ Ik

of the idele group into the ideal group is just the group u,, (say) of ideles a = a, which have ICI& = 1 for every non-archimedean V. If K is a finite extension of k, it is clear that

ConKlk uk = UK

and from the Lemma of $11 and (19.17) we have

NK,kUK = uk*

Hence on passing to the quotient from Jk we have the induced maps

ConKlk . - I, --) I,

N K/k * - I, --, Ik

with the usual properties (19.10), (19.11) and (19.13); and these maps are compatible with the norm and conorm maps for elements of K and k on taking principal ideals. By definition (19.2) we have

conKlkv = c ev v (19.20) Vlv

where the positive integers ey are defined by

b”I” = pblevv, (19.21)

rr, and lT, being prime elements of k,, Kr, respectively. Similarly, it follows from (19.19) that

NK,k v = fv v, (19.22)

where fV is the degree of the residue class field of V over that of v. We note in passing that (19.11), (19.20) and (19.22) imply that

zvevfv = h

as it should since +fv = [Kv: bl.

Similarly, when k is a finite extension of F(t) one defines conorm and norm of divisors, with the appropriate properties.

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76 I. W. S. CASSELS

APPENDIX A

Norms and Traces

Let R be a commutative ring with 1. By a vector space V over R of dimen- sion n we shall mean a free R-module on n generators, say q,. . . , o, (a basis). If o;, . . ., c$, is another basis, there are z+,, uu E R such that

01 =zy utp;, 0; = c %j@f (A.1)

and

7 %J vjl = C "lj"jZ = 6il (A-2)

(Kronecker 8).

The set of all R-linear endomorphisms of V is a ring, which we denote by End, V. The ring R is injected into End, V if we identify b E R with the module action of b on V, and we shall do this. The ring End, V is isomor- phic but not canonically, to the ring of all n x n matrices with elements in R. The isomorphism becomes canonical if Y is endowed with a fixed choice of basis. In fact if /I E End, (V) and

PA = T hf./w,, (bij E R) (A-3)

the I- 1 correspondence between /I and the transposed matrix (b,J is a ring isomorphism.

For p E End, we denote by

FB(x) = det (x6,1 - b,) (A-4) the characteristic polynomial of R. On using (A.2) it is easy to see that I;B(x) is independent of the choice of bases of V. The Cayley-Hamilton theorem? states that

F&3) = 0. (A-5) We define further the truce

&pdPl= WI = C b,

= - coefhient of 9-r in FP(x) 64.6)

and the norm

&I~(B) = NP) = det (hi) (A-7) = (-)” constant term in F@(x),

t Proof. Write (A.3) in the form

T (4,8 - wwi = 0.

Working in the commutative ring R[Bl multiply the equations (*) by the cofactors of the “coefficients” of wl, and add. Then war. . . , w, are “eliminated” and one obtains F&?)w~ = 0. Similarly F$./!QD, = 0 (2 5 j I n) and so F&9) = 0.

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GLOBAL FIELDS 77

which are independent of the choice of basis because Q(x) is. Clearly

WI +BJ = WI) + WJ (A.9 S(b) = nb (b E R) (A.9

N(P, PJ = NmwM (A.10)

N(b) = b” (b E R), (A.ll)

because the correspondence (A.3) between /? E End, (V) and the matrix bji is a ring isomorphism.

LEMMA A.1. Let t be transcendental over R. Then

N(t-p) = F,(t). (A.12)

Pedantically, what is meant is, of course, that we consider a vector space V with basis ol, . . ., w, defined over R[t] and a p given by (A.3).

Proof. We have (t-P)oi = C (tsij-bij)wj

j and so

N(t-p) = det(t$-b,J

= F,(t) by (A.4) and (A.7).

COROLLARY. (A.12) holds for any t E R. LEMMA A.2. Let /II,. . . , j3, E End, V and let t be transcendental over R.

Then

N(t’+j& t'-'+ . . . +/3,) = t”‘+g1 P-l+ . . . +g”l (A.13)

where gl,. . . , gnl E R and in particular

91 = WI); 9.1 = NW (A.14)

Proof. Similar to that of Lemma A.1 and left to the reader. Now let R and P c R be commutative rings with 1 and suppose that R

regarded as a P-module is free on a finite-number, say, m of generators a,,. - *, 0, (i.e. an m-dimensional P-vector space). Let V be an n-dimensional R-vector space with basis or,. . . , 0,. Then V can also be regarded as an mn-dimensional P-vector space with basis

n,Oj (l<i<m, lljln)

and there is an obvious natural injection of End, (V) into End,, (V). We have now the key

THEOREM A.l. Let

p E End, (V) c End, (V). (A.15) Then

&,I4 = &4f&J9, (A.16)

NvpB = &,P(&,R P>- (A.17)

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78 J. W. S. CASSELS

Further

w3 = NR,PWh (A.18)

where a,(x) E P[x], F(x) E R[x] are the characteristic polynomials of p in Endp (V) and End, (V) respectively.

Proof. If j? is given by (A.3) let y E End, (V) be given by

Y%=% -jFl bijwj

ywi = b,lo, (i > 1). (A.19)

Then for cc = yp we have

ami = bllw, awi = Bill + C (bll bij-bil blj)wj

= bilw, +ji’aijw, (say). (A.20) j>l

Hence

Nv,Ra = bllNwlRa* (A.21)

where W is the n- 1 dimensional R-vector space spanned by 02,. . . , w, and a* is the R-linear map

COi +jFIU*jWj (i > 1).

Consequently

NRIPWvIR a> = NRIp 4 I . NRIP(NWIRa*). (A.22)

We now use induction on the dimension n, since the Theorem is trivial for n = 1. Since W has dimension n - 1 we have by the induction hypothesis

(A.23)

(A.24)

%,PWV,R a*) = N,,,a*.

On the other hand, it follows directly from (A.20) that

hIpa= NR~~blN~,~a*

and so

NV/G = NRIPNvIpa.

Further, clearly

Nv,,Y = NR,,Nv,PY = WR,PW’-~-

Since a = j?y and both Nv,r and NRIPNYIR are multiplicative (by (A. lo)), it follows from (A.24) that

W~,dd”-%,J = (NR,Pbll)n-lNR,p(Nv,RP). (A.25)

If NRIPbll were invertible, this would give (A.17) at once. In general, however, this is not the case and we must use a common trick.

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GLOBAL FIELDS 19

Let t be a transcendental over R and let /?, be the transformation obtained from /? by replacing b,, by bil +t but leaving the remaining b,j unchanged. Then (A.25) applied to /?, gives

(&p&l + 0)“-%,~Bt = NR,Y@II + O”-lN~,~Wv,dr)~ WW

All the norms occurring in (A.26) are polynomials in t. On comparing coefficients of powers of t in (A.26), starting at the top, we deduce that

NvpPt = NR,PWY,R PJ (A.27)

because the coefficient of the highest power of i in iVRIP(bl 1 + t) is 1. Then (A.17) follows on putting t = 0.

We now prove (A.18). By Lemma 1 we have

W4 = NV,& -PI, F(x) = NV,& -B)

and so (A.18) is just (A.17) with x-p for p. Finally, (A.16) follows from (A.18) on using (A.6) and the first half

of (A.14). When R = k is a field there is some simplification, since every finitely-

generated module V over k is free, i.e. is a vector space. Further each /? E End, (V) has a minimum polynomial, i.e. a non-zero polynomial f(x) of lowest degree, with highest coefficient 1, such that f(p) = 0. Then g(p) = 0 for g(x) E k[x] if and only iff(x) divides g(x) in k[x]. In particular the Cayley-Hamilton theorem (A.5) now states thatf(x) divides the charac- teristic polynomial FP(x).

Finally we have THEOREM A.2. Let K be a field of finite degree n over the field k and let

p E K. Then the degree m (say) of the minimum polynomial f(x) of p over k divides n and

J-64 = (fW>“‘“,

where F(x) is the characteristic polynomial of p. In particular

&,k(P) = ;GB,+...+P,h

&,&V = (PI 9 IL - . -9 AJ”‘m,

where PI,. . . , p,,, are the roots of f(x) in any splitting field. Proof. Suppose first that K = k(B). Then the minimum polynomial f(x)

and the characteristic polynomial F(x) of p have the same degree and highest coefficient, so F(x) = f(x) by the remarks preceding the enunciation of the Lemma.

The general case now follows from Theorem A. 1, with V = K, R = k(P), P = k on using (A.ll).

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80 J. W. S. CASSELS

Separability

APPENDIX R

In this book we are primarily interested in separable algebraic field exten- sions. Here we recall their most important elementary properties.

LEMMA B. 1. Let K, M be extensions of finite degree of the field k. Then there are at most [K: k] injections of K into M which leave k elementwise fixed.

Proof. Trivial when K = k(a) for some a on considering the minimal polynomial for a. For general K we have a chain

k=K,,CK,cK,... cKJ=K (B.1) where Kj = Kj_,(ai_,) and use induction on J.

DEFINITION. The finite field extension K/k is separable if there is some finite extension M/k such that there are [K: k] distinct injections of K into k which leave k elementwise fixed. If K/k is not separable then it is said to be inseparable.

COROLLARY 1. Let K 2 L I k. If K/k is separable then so are K/L and L/k.

Proof. By Lemma 1 there are at most [L: k] distinct injections of L into M and by Lemma 1 again each of these can be extended in at most [K: L] ways into injections of K into M. By definition, there are

[K:k]=[K:L][L:k]

injections of K into M, and so there must be equality both times. COROLLARY 2. Let a E K where K/k is separable and let CQ,. . ., a,,, be

the roots in M of the irreducible poIynomiaI f(x) for a over k. Then the oia (1 I i I n = [K: k]) are just the ul,. . . , LX,,, each taken n/m times, where cl,. . . , a,,, are the injections of K into some M.

Proof. For put L = k(a) in the preceding argument. COROLLARY 3.

Proof. Follows from Theorem A.2 and the preceding Corollary. LEMMA B.2. Let K/k be aJinitejield extension and let o be an injection of k

into some field M. Then there is a finite extension M, of M and an injection al of K into Ml which reduces to o on k.

Proof. Trivial if K = k(a), and then follows for general K on using a chain (B. 1).

THEOREM B.1. Let K/L and L/k be separable extensions. Then K/k is a separable extension.

Proof. Let U/L be a finite extension and

pi : K ~ U (1 I i I [K : L])

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GLOBAL FIELDS 81

be injections extending the identity on L and similarly let V/k be a finite extension and

Cj:L+‘v (lljl[L:k])

extend the identity on k. By repeated application of Lemma 2 there is a finite field extension M/V and [L : k] injections

oi:U+M (lljl[L:k])

which extend the aj. Then the ojri give

[K:L][L:k]=[K:k]

distinct injections of K into it4 extending the identity on k. COROLLARY. In characteristic zero livery finite Jield extension is separable. Proof. For a simple extension k(cr)/k clearly is, and then apply the theorem

to a tower of simple extensions. THEOREM B.2. Let K/k be a separable extension. Then it is simple, i.e.

K = k(y) for some y. Note. The converse is, of course, false. Proof: If k is a finite field then so is K and so indeed K = U(a) for

some u E K, where II is the prime field, by the structure theory of finite fields. Hence we need consider only the case when k has infinitely many elements. Suppose first that K = k(cr, /I) and let pi,. . . , gn where n = [K: k] be the distinct injections of K into M (say). If i # j, distinctness implies that

either oia# aja or o,Q# oJp

(or both). Hence we may find a, b E k to satisfy the finitely many inequalities

a(oior-ajcr)+ b(a;P-oj~) # O(i # j). Put

y = act+ b/3, SO

Oiy # CjY (i #j).

The a,y are all roots of the irreducible equation fcr y over k and so

[k(y) : k] 2 n.

But k(y) c K, so K = k(y). For the genera1 case when K = k(cr,, CQ,. . . , aJ) with J > 2 one uses

induction on J. We have k(cc,, . . ., crJ) = k(B) for some /I and then W,, l-9 = k(y).

THEOREM B.3. Let K/k be a separable extension. Then

0, PI = &&dO is a non-degenerate symmetric bilinear form on K considered as a vector space over k.

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82 J. W. S. CASSELS

Proof. Only the non-degeneracy needs proof. Let oi, . . . , w, be a base of K/k. The statement of the Theorem is equivalent to

D (say) = det {S~/d~i”‘));;~~; Z 0.

Let or,. . ., a,, be distinct injections of K into some M. By Lemma B.l, Corollary 3 we have

D = A2 where

A = det(aiwj)lSiS.. l<j<n

By Theorem B.2 we have K = k(y) and so can take Oj = yj-‘. Then

as required.?

We now consider when a simple extension k(z)/k is separable. Let f(x) be an irreducible polynomial in k[x] and let f’(x) be its derivative. If f’(x) # 0 it must be coprime tof(x), since it is of lower degree, and so there are a(x), b(x) E k[x] such that

a(x)f(x> + b(x)f’(x) = 1.

Hence f(p) = 0 for /I in any extension of k, implies that f’(/3) # 0, and so B is a simple root. Hence the number of roots off(x) in a splitting field is equal to the degree. On the other hand, iff’(x) = 0, every root off(x) is multiple, and so the total number of roots is less than the degree. In the first case we say thatf(x) is separable, in the second inseparable. The second case occurs if and only iff(x) = g(x”> for some g(x) E k[x], where p is the characteristic.

LEMMA B.3. A necessary and suficient condition for k(or)/k to be separable is that the irreducible polynomial f(x) E k[x] for cc be separable.

Proof. Clear. COROLLARY 1. Let K I k, and suppose that k(cr)/k is separable. Then

K(or)/K is separable. Proof. For the irreducible polynomial F(x) E K[x] over K divides f(x). COROLLARY 2. A necessary and suficient condition that K/k be separable

is that every element of K be separable /k. Proof. Suppose that every element of K is separable and that K is given

by a chain k=KOCK1c . . . cK,=K

t Instead of using the fact that K = k(y) we could have used Artin’s theorem that any set of injections of one field into another is linearly independent. See Artin: “Galois Theory” (Notre Dame) or Adamson: “Introduction to Field Theory” (Oliver and Boyd).

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GLOBAL FIELDS 83

where Kj = K~_,(Orj_ 1). Then KJIK,- 1 is separable by the previous corollary and so K/k is separable by Theorem 1.

The converse follows from Lemma 1 Corollary. In striking contrast to Theorem B.3 we have THEOREM B.4. Let K/k be inseparable. Then the trace S,,,(/?) vanishes

for all /I E K. Proof. Suppose, first, that K = k(a) where up E k, cc $ k and p is the

characteristic. Then

o~~=l,o~=a,...,o~=cI~-~

is a basis for K/k. If /3 = b,+b,a+ . . . +bpaP-’ with bj E k, then

PWi = C b,jOj bij E k where clearly

bii = b, (1 < i I p).

Hence SK,LB = C bii = pb, = 0.

L Now let K/k be any inseparable extension. By the latest Corollary there

is an inseparable CI E K. Put L = k(x), M = k(uP), so L/M is an extension of the kind just discussed. The general result now follows because of the transitivity of the trace:

&,I4 = &&L,M(SX,LP)l.

Hensel’s Lemma APPENDIX C

In the literature a variety of results go under this name. Their common feature is that the existence of an approximate solution of an equation or system of equations in a complete valued field implies the existence of an exact solution to which it is an approximation, subject to conditions to the general effect that the approximate solution is “good enough”. These results are essentially just examples of the process of solution by successive approximation, which goes back to Newton (at least). In this appendix we give a typical specimen.

LEMMA. Let k be a field complete ,l,ith respect to the non-archimedean valuation 1 1 and let

J-(X) E 4IXlY cc. 1) where o c k is the ring of integers for 1 I. Let a0 E o be such that

If( < I~‘<GJI~~ (C.2)

w,here f'(X) is the (formal) derivative of f(X). Then there is a solution of

.I@> = 0, I--~01 5 lf<4l/lf’<41. (C-3)

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84 J. W. S. CASSELS

Proof. (Sketch.) Let f,(X) E o[X] be defined by the identity

j-(X+ Y) =f(X)+f,(X)Y + . . . +fj(x>Yj+ . . ., (C.4) where X, Y are independent variables, so fi(X) = f’(X). Define PO by

f&J + Pof1(4 = 0. (C.5)

Then by (C.4) and sincefj(a,) E D we have

lfh +/Ml 5 max Ifj<%>P',I

On using the analogue of (C.4) forfi(X), it is easy to verify that

Ifi(ao+Po>-fi(ao>l < Jfi(ao)l. Thus on putting a1 = aO+po, we have

lf<adl 2 If<a0>12/lfl<ao>l’~

If&dl = Ifh.Jl and

Ial -a01 5 If(aO>l/lfi<ao>l. On repeating the process with al, etc., we get a sequence aO, ar, al,. . ., which is easily seen to be a fundamental sequence. By the completeness of k there is an a = lim a, E k, which clearly does what is required.

“*co In fact, the solution of (C.3) not merely exists, but is unique. For if

a+p, j? # 0 is another solution one readily gets a contradiction by putting X = a Y = p in (C.4).

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CHAPTER III

Cyclotomic Fields and Kummer Extensions

B. J. BIRCH

1. Cyclotomic Fields ............... 85 2. Kummer Extensions .............. 89 Appendix. Kummer’s Theorem ............ 92

1. Cyclotomic Fields Let K be any field of characteristic zero, and m > 1 be an integer. Then there is a minimal extension L/K such that x” - 1 splits completely in L. The zeros of x”‘- 1 form a subgroup of the multiplicative group of L; this subgroup is cyclic (since every finite subgroup of the multiplicative group of a field is). The generators of this subgroup are called the primitive mth roots of unity. If [ is a primitive mth root of unity then every zero of (x” - 1) is a power of i, and L = K(c). Clearly, L is a normal extension of K; we write L = K(ql).

If u is an element of the Galois group G(L/K), then oi must be another primitive mth root of unity, so a( = ck for some integer k, (k, m) = 1. If [ is another primitive root of unity then a[” = pk; accordingly, (TH k is a canonical map of G(L/K) into the multiplicative group G(m) of residues modulo m prime to m. In particular, [L : K] I 4(m).

If m = rs where (r, s) = 1 then there exist integers a, b with ar + bs = 1, I = (rywb, so K(l) = K(5’, r”) ; one obtains the extension K(c) by com- posing K(c) and K(r). So to some extent it is enough to consider K(mJ1) when m is a prime power. If p is odd then the group G(p”) is cyclic, so if m =pn, L = K(vl), then G(L/K) is cyclic; on the other hand G(2”) is generated by - 1 and 5, so if we write q = [+c-’ where c2” = 1 then

I K(c) = K(i, q) and G[K(q)/K] is cyclic.

We are particularly interested in the extensions Q(r;l/l) and Q&‘/l); by Chapter I (Section 4 and start of Section 5) the study of the factorization of the prime p in the extension Q(vl)/Q is essentially the same as the study of the extension Q,(vl)/Q,. A s one of my jobs is to supply explicit examples for abstract theorems, I will prove things several times over by different routes. Good accounts of cyclotomic extensions are given by Weyl: “Algebraic Theory of Numbers” (Princeton U.P., Annals of Math. Studies

85

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86 B. J. BIRCH

No. l), pp. 129-140, and by Weiss: “Algebraic Number Theory” (McGraw Hill), Chapter 7.

LEMMA 1. Q(rC/l) is a normal extension of Q of degree 4(m); its Galois group is naturally isomorphic to G(m).

Proof (after van der Waerden). Let 5 be a primitive mth root of unity; after what has been said already, it is enough to show that an equation of minimal degree for [ over Q has degree d(m) so we need to show that if

j”(x) E Z[x] and f(c) = 0 then f(p) = 0 whenever (a, m) = 1. For this, it is enough to show thatf(cP) = 0 whenever p is a prime not dividing m.

Consider the field kp with p elements; if f(x) E Z[x], denote its natural image in k,[x] byf*(x). Let L*, a finite extension of kp, be a splitting field for (xl- 1) ; (xm - 1) is prime to its derivative mxmwl, so has distinct roots.

Suppose that the factorization in Z[x] of (xm- 1) isfi(x).fZ(x). . .L(x), so that the A(x) are irreducible. Then in k&l, (x”- 1) = nfr(x), and all the zeros in L* of all the f *, being roots of x’” = 1, are distinct. Choose the numl&ing so that [ is a root of fi(x), and suppose that CP is a root of fJ(x). Then tiI(x) divides fJ(xp), so f T(x) divides f J(xp). Let [* be a zero off T(x); then fT([*p) = 0, and on the other hand f :(c*3 = ~~(~*)]p = 0. So f, is the same as fi.

COROLLARY 2. If p ,/’ m there is a unique element op of the Galois group G[Q(vl)/Q] such that opu s or(p) for all integers a E Q(r;/l). In fact, zf[ is a primitive mth root of unity then op is given by o,(c air’) = xaicip for a, E Q. (op is the Frobenius automorphism.)

Proof. The field basis 1, 5, . . . , [@(“‘)-l has discriminant prime to p, so by Chapter I, Section 3, certainly every integer can be expressed as c ail ‘lb with al,. . ., ap-2, b E Z and (b,p) = 1; so cp defined above does what is wanted. We need to show that it is unique. The effect of an element c of the Galois group is clearly determined by its effect on 5; and clearly every CJ takes c to another primitive root, p say, with (a,m) = 1. We need to show that r” I cb(p) only if r” = 5’; that is, 1 - cb E O(p) only if lb = 1.

First method. Suppose cb =j= 1. We know

x”-1 = fi (x-q), i=l

so mxmml = T i~(x-,I),

and m-1

m = ,I-! (1 -C)-

So (1 -lb) divides m, so p does not divrie (1 - cb).

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CYCLOTOMIC FIELDS AND KUMMER EXTENSIONS 87

Second method. Take the completion Q, of Q, and form the extension Q&‘/l). Let L* be the residue class field of Q,(r;/l). Then L* is a finite extension of k, and is a splitting field for x”- 1. Q(‘;/l) c Q,(r;l/l), so the residue class map makes a root of xm- 1 in L* correspond to each root of x”- 1 in Q(vl). Conversely, the roots of x” = 1 in L* are distinct, and each of them is the residue class of an element of Q,(r;l/l), by Hensel’s Lemma (Chapter I, Section 7, Lemma 1, or Chapter II, Appendix C). So we have a one-one correspondence between the mth roots of unity in L* and in Q(ql), and everything follows.

LEMMA 3. If q = pf is a prime power and c is a primitive qth root of unity, then the prime p is totally ramified and in fact (p) = (1 - c)9(q).

First Proof. Write Iz = 1 - [. Then cq = 1 but cqtp # 1, so I is a root of the polynomial F(x) = [(l +x)“- l]/[(l +x)~‘~- I]. f has leading coefficient 1 and constant term p, and it is readily verified that all the other coefficients are divisible by p. So F is an Eisenstein equation, and by Chapter I, Section 6, Theorem 1, Q,(ql) is a totally ramified extension of Q, of degree 4(q) and (p) = (A)4(q). Going back to Q(ql), we see that Q(ql) is an extension of Q of degree 4(q) (so we get another proof of Lemma 1 in this case) and p is totally ramified in this extension.

Second Proof. We can see all this more explicitly. If (a,p) = (b,p) = I, then we can solve a I bs(q), so

(l-r”)/(l-[b) = (l-[b”)/(l--56) = 1+[b+. . . +rb(s-l)

is an integer, and similarly so is (1 -[‘)/l -tJ; so (1 - y)/(l -lb) is a unit whenever (a,~) = (b,p) = 1. Also,

p=lim xq-1 - =f.. (Jp-i”) = (l-i)“(q) n ls x-1 xq’p- 1

O&<q

so (p) is simply the +(q)th power of (A) = (1 - 5).

LEMMA 4. Let c be a primitive mth root of 1. If p is a prime not dividing m then it is unramifed in Q(r;/l) and its residue class degree f, (see Chapter I, Section 5) is the least integer f 2 1 such that p* E l(m).

Proof [after Serre: “Corps Locaux” (Hermann, Paris)]. Consider the extension Q,(c) of Q,; the residue class field k, has p elements. The poly- nomial x’“- 1 splits in k,, if and only if ml(pf- 1). Take the least f with p/ = l(m), and construct the unramified extension L of Qp with residue class field k,, (see Chapter I, Section 7, Theorem 1). There is a multiplicative map k$ + Lx as in the second proof of Corollary 2, so xrn - 1 splits in L, and clearly L is minimal with this property. So L = Q,(c).

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88 B. J. BIRCH

Going back to Q(c), we see that p is unramified in Q(c) and has residue class degree f, as described.

(In fact, the lemma is an immediate consequence of the properties of the Frobenius automorphism obtained in Corollary 2.)

COROLLARY. If p,j’ m, then p splits completely if and only ifp s l(m). LEMMA 5. If q = p’ is a prime power and [ is a primitive qth root of unity,

the discriminant of the extension Q(c) of Q is q4(4)/pp’p; a basis for the Z-module of integers @“Q(c) is 1, [, 5’, . . . , [9(q)-1.

Proof. Consider first the case t = 1. By Lemmas 3 and 4, the only prime ramified in Q(5) is p, and the ramification ofp is tame with e = p- 1. Hence by Chapter I, Section 5, Theorem 2, the discriminant is ppm2. Hence by Chapter I, Section 4, Prop. 6, l,c, c2,. . . , cpm2 is a basis as asserted.

In case t 2 2, the ramification is wild, so all that is obvious from Chapter I (until the final section) is that the discriminant is a power of p, at least p4(q). We give a direct proof that the powers of 5 form a basis for the integers, which works for all t. Consider the Z-module Z[[]. By Chapter I, Section 4, Prop. 6, Z[[] has discriminant N,,I,,,[g’(c)], where

g(x) =(kgcl(x-Ik)i

after dull computation this turns out to be q”(q)/pq’p. We want to show that Z[l] is the whole ring of integers of Q(c); after

Chapter I, Section 3, Prop. 4 it remains to check that

c airi OSiS$(q)-1

with ai E Z is divisible by p if and only if all the ai are divisible by p; that is, that

C b,(l-5)’ OSij+(q)-1

is divisible by p if and only if all the bi are. To check this, suppose that PIE b,(l-[‘); recollect that (p) = (1 -#‘(q), and suppose that plbi for i= l,..., s- 1. Then

(l-~)+‘q’lb,(l-~)“+b,+,(l-~)“+‘+. . ., so (1-Ojb,, so plb,;

so by induction we get what we want. The conclusions of the Lemma follow.

LEMMA 6. If C is a primitive mth root of unity, then Q(C) is an extension of Q of degree 4(m); the discriminant of Q(c) over Q is

m9(m) /n

pmol(P- 1) . ,

PM

a basis for the Z-module of integers of Q(C) is 1, [, r2, . . , C4Cm)- ’ ; p is ramified if and only ifplm.

Most of this has been proved already (Lemmas 1, 3 and 4); the extra assertions about the discriminant and the basis for integers are equivalent to

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CYCLOTOMIC FIELDS AND KUMMER EXTENSIONS 89

each other, and may in fact be proved directly (see Weiss, Section 7.5). One may also pick up the assertions about Q(r;/l) by fitting together the extensions Q(q1) where q runs through the prime power factors of m.

First, remark that if q,q’ are powers of different primes,

Q(ql> n Q($) = Q. Now work out the discriminant. Suppose phlIm. By Lemma 5 the dis- criminant of Q,(phJ1) over Q, isphph-(h+l)ph-‘. There is no further ramifica- tion of p in Q,(Ij/l), so by Chapter I, Section 4, Prop. 7, the discriminant of Q,(r;/l) Over Q, is pW(~)-WO/(p-l)~ By Chapter I, Section 4, Prop 6, the discriminant of Q(c) over Q is

mWO In

p~(W(p - 1). Ph

By computation this is the discriminant of Z[(], so by Chapter I, Section 3, Prop. 4, 1, t;, c2,. . . , [0cm)- l really is a basis for the Z-module of integers.

Remark. All assertions about factorization of primes in Q(y1) may be regained from Lemma 6, by using Kummer’s theorem (see for instance Weiss, Section 4.9; or the Appendix). “If Z[[] is the complete ring of integers of Q(c), and g(x) = 0 is the characteristic equation of [ over Q, then a prime p factorizes in Q(c) in the same way as g(x) factorizes modulo p, and in fact if g(x) E n gi(x) (mod P> then (PI = fl [P, sLO1.”

It is, of course, perfectly possible to work out the ramification groups explicitly in this case-for this we refer to Serre : “Corps Locaux” (Hermann, Paris), pp. 84-87.

To conclude. We have shown that every cyclotomic extension, and so every subfield of a cyclotomic extension of Q has abelian Galois group. The converse is true-every abelian extension of Q is a subfield of a cyclotomic extension (Kronecker’s Theorem); but that is Class Field Theory (Chapter VII, § 5.7.)

2. Kummer Extensions

Throughout this lecture, K is a field of characteristic prime to n in which x”- 1 splits; and [ will be a primitive nth root of unity. It will be shown that the cyclic extensions of K of exponent dividing n are the same as the so-called “Kummer” extensions K(ya).

Let a be a non-zero element of K. If L is an extension of K such that x”=a has a root CI in L, then all the roots ct,[tL,[2a,...,~-1a of x”=a are in L, and any automorph of L over K permutes them. We denote the minimal splitting field for x” - a by K(qa). If r~ is an element of the Galois group G[K(!$‘a)/K], then, once we have chosen a root of x” = a, d is deter- mined completely by the image ba = cba. In particular, if a is of order n in the multiplicative group K “/(K “>“, then d is a nth power only if nlr, so

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90 B. J. BIRCH

X” - a is irreducible; in this case, the map B I+ lb gives an isomorphism of the Galois group on to a cyclic group of order n. Let us state all this as a lemma.

LEMMA 1. If a is a non-zero element of K, there is a well-defined normal extension K(va), the splitting field of x” - a. If a is a root of x” = a, there is a map of G[K(va)/K] into K” given by ok au/a; in particular, if a is of order n in K “/(K “>“, the Galois group is cyclic and can be generated by u with OCR = ccc.

LEMMA 2. If L is a cyclic extension of K (i.e. G(L/K) is cyclic) then L = K(yb) for some b E K. (b must generate K “/(K”)“.)

The proof is by direct construction, which is providentially possible. Let CJ be a generator of G(L/K); then L = K(y) for some y, since all separable extensions are simple, and we can choose y so that y, ay,. . . , rr”-l y is a basis for L over K. Form the sum

n-1 P =,zo rse

Then o/3 = C-‘p, and p # 0 since y,. . . , a”-‘~ are linearly independent over K; so p” E K and p’ # K for 0 < r < n. Thus p” = b is of order n in Kx /(Kx >“; by Lemma 1, K(lb) is a cyclic extension of degree n contained in L, so L = K(;lb).

LEMMA 3. Two cyclic extensions K(qa), K(gb) of K of the same degree are the same if and only if a = b’ c” for some c E K and r E 2 with (r, n) = 1.

In fact, “if” is obvious; we have to prove the “only if” part. This is easy by elementary Galois theory. Suppose that K(a) = K(p) with ci’ = a, j3” = b. Let (T be a generator of the Galois group with aa = &x, then a/? = [‘/3 for some i. Suppose

n-l

p = T cjc2,

then ab = 1 cjjicli,

so we can only have /3 = cic2.

COROLLARY. Let L be a jinite extension of K with abelian Galois group G of exponent dividing n. Then G is the direct product of cyclic subgroups

GI,. . ., Cr. For each i, let Li be the$xedfield of G, x . . . x Gi+l x . . . x G,; then G(Li/K) = Gt, Lt = K(ai) with UT= a, E K, and L = K(a,, . . . , a,).

We may approach the last couple of lemmas via Galois cohomology (see Serre, p. 163). We quote a generalization of Lemma 2.

LEMMA 4. If L is a normal algebraic extension of K with Galois group G then

H1(G, Lx) = 0.

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CYCLOTOMIC FIELDS AND KUMMER EXTENSIONS 91

[A I-cocycle is a “continuous” map G -+ Lx in which a -+u, with u(aJ = -1 a0 a,l; “continuous” means that the map may be factored through a finite

quotient G’ of G. Form the sum /I = X a, O(Y); by the linear independence OEG’

of automorphisms [Artin: “Galois Theory”, (Notre Dame), Theorem 12 and Corollary and Theorem 211, we may choose y so that p # 0; and then a, = ,9/u fl.1

Apply Lemma 4 taking L as the (separable) algebraic closure of K. There is an exact sequence 0 + E,, .Y+ L x --, Lx --) 0 where v is the map XHX” and E,, is the nth roots of unity, Taking homology, we get

H’(G,E,) + H’(G,L”) ” H’(G, Lx) + H’(G,E,) + H’(G, Lx) +. . .

Since G acts trivially on E,, H’(G, E,) is Hom(G, E,), and of course H’(G, L “) is the part of L” fixed under G, so this is

E,-+K”&KX + Hom(G, EJ --) 0, i.e.

K ‘/(K ’ >” ES Hom(G, E,).

One can make the isomorphism explicit. Recollect that if 0 + A + B + C + 0 then an element c E Cc gives a map G --, A by taking b with b + c, and then cr I+ blab is a map G + A. In our case, we start with c E Kx ; take y so that y” = c; and then 6~ y/oy is a map G + E.. Conversely, if 4 E Hom(G, E,), let G’ be the kernel of 4. Then G/G’ is the Galois group of a cyclic extension K’ of K of exponent dividing n, and if u E G is such that 4(o) = 5, there is an element y E K’ such that oy = [y; then y” E K ‘, so determines a class of K”IK”“.

The elements of G whose exponent divides n are preserved in Hom(G, EJ. Let K, be the union of the abelian extensions of K whose Galois groups have exponent dividing n; then Hom[G(KJK), E,] E K “/(K ‘>,. In particular the finite abelian extensions of G whose exponents divide n correspond to finite subgroups of Kx /(Kx >“.

Finally we want to look at the factorization of primes p of K in the extension K(J’/a); by Chapter I, as before, this is the same as the study of the extensions K+,(qu) of the local field KP.

LEMMA 5. The discriminant of K(va) over K divides n” a”-’ ; p is unram$ed if p J’na. If af is the least power of a such that af = x”(p) is soluble, then f is the residue class degree.

Suppose a” = a; then, if o is the ring of integers of K, o[tl] is a submodule of the ring of integers of K(a), and by Chapter I, Section 4, Prop 6, its discriminant is @a”-‘)” = n” u”- l; so the discriminant of K(U) over K divides .“a”-I. In particular, by Chapter I, Section 5, p is unramified if p$na, and indeed by Kummer’s theorem (or by Chapter, II Section 16 and Hensel’s Lemma) the factorization of p then mimics that of x”- a modulo p. Hence, if a/ is the least power of a such that x” = a/(p) is soluble,

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92 B. J. BIRCH

then p factors as the product of n/fprime ideals in K(a), and the residue class degree isf. Alternatively, this last bit can be proved like Lemma 4 of the cyclotomic lecture, by forming the unramified extension of KP in which x” - a splits.

LEMMA 6. Zf pla, p,j’n and p”,f a then p is tamely ramified in K(qa); if pla and p2,fa then p is totally ramijied in K(qa).

Let u be a root of x” = a. The second part of the lemma is easy; if pla but pz #a then, quite explicitly, p = (p, cc)“. The result can also be obtained from Chapter I, Section 6, Theorem 1, since in this case x”- a is an Eisenstein polynomial.

If p,t’n, so that p does not divide the degree of the extension then any ramification must certainly be tame (see Chapter I, Section 5). On the other hand, if pla but +~“,/‘a then p is certainly ramified, since (p, or)lpl(p, a)“, but a 4p.

There remains the case where flu, p’+ ’ )(a, p ,j’n with 2 I r I n - 1. If (r,n) = 1, we have k,m so that rk+nm = 1, and chooseqGK with p/q, p’,j’q. Then K[;i@q-“‘“)I = K($’ a , and we have got back to the case r = 1. If ) (r,n) = s > 1, the ramification is no longer total, and there may or may not be splitting as well. The ramification index is n/s : p is unramified in the extension K(la) of K, and then the factors of p are totally ramified in the extension K(;/u) of K(qa).

APPENDIX

Kummer’s Theorem

Throughout this appendix, K is an algebraic number field with ring of integers o, and L = K(O) is an extension of degree n; we suppose that 13 is an integer and that f(x) 5 o[x] is the characteristic polynomial of 8. p is a prime ideal of o, v is the associated valuation, Kg is the completion of K at p, oP is the ring of integers of K,, and K,* is the residue class field. The prime ideal factorization of p in L is p = n q:J; the associated valuations, completions, rings of integers, and residue class fields are Vj, L,, oj, LT. If f, g are poly- nomials in o[x], o,[x] their images under the residue class map are denoted by f *, g*.

In Chapter II, Section 10, it was shown that, if the irreducible factorization off(x) in o[x] is

Ax> = 1 %jgJ 9 jCx>

then Lj = Kp(Oj) where gj(0j) = 0. There is an injection map pj : L + L,, with p, 0 = tlj; and a residue class map *j : oj -+ LT.

We wish to relate the factorization of p in L to the factorization off;(x) in K~[x]. First we need a lemma.

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CYCLOTOMIC FIELDS AND KUMMER EXTENSIONS 93

LEMMA. With the above notations, g;(x) is a power of an irreducible polynomial of K,*[x], say g;(x) = [GT(x)rJ, and $j 61j is a zero of G:(x).

This will be clear enough if we show that the modulo p reduction of every irreducible polynomial of op[x] is a power of an irreducible polynomial of K:[x]. Put another way, we wish to show that if h(x) E op[x] and

h*(x) = h:(x). h;(x) with (h:(x), h;(x)) = 1, then we can find h,(x), h,(x) from op[x] such that h(x) = h,(x). h,(x). This isjust a version of Hensel’s Lemma (see Chapter II, Appendix C; Weiss 2.2.1.).

KUMMER’S THEOREM. Suppose that p, 8 have the property that n-l

T a,&K[B]

is an integer only if v(ai) 2 0 for i = 0,. . . , n - 1. Suppose that the irreducible factorization off*(x) in K,*[x] is f*(x) = n [G,*(x)]~‘, and for each j let Gj(x) E o[x] be a manic polynomial whose image by the residue class map is G;(X). Then the prime idealfactorization of p is p = fl q;l, with qj = (p, Gj(0)).

[cf. Chapter II, 19.20; p = IT q;J is equivalent to saying that the valuations ofL extending v are V, for j = 1, . . . . J, the corresponding ramification indices are the e,, and the set of integers a of L with V,(a) > 0 is q,. In our notation, V,(a) > 0 if v,a = 0.1

We have to check that the ramification indices are correct and that q, = (p, Gj(0)); also, that different polynomials GT come from different polynomials gj.

We know already that Lj E KP[x]/(gj(x)). By the hypothesis of the theorem, every integer of L is of form c a, 0’ with u(ai) 2 0, so every element of LT is of form c a*(tij8)‘; so Lj* z K,*[x]/(GT(x)), and ej = [Lj : K,]/[LT; K,*] is the correct ramification index.

If a E (p, G,(e)) then clearly wjcr = 0, SO, CI E qj. Conversely, suppose aeqi; then

n-1 a = 1 a&3*,

0

and by the hypothesis of the theorem v(aJ 2 0 for i = 0,. . ., n- 1. Write h(x) = 1 ai.xi so that h(x) E oP[x]. Since u E qj, tjj CI = 0 so h*(ll/j 0) = 0. We can express h(x) = Gj(x)qj(x)+ri(x), with qj, rj E o~[x] and

de&j) < deg(Gj) ; and then rj*(t,bjO) = 0, so r;(x) is identically zero, and h(B) E (p, Gj(@). Hence qj = (p, Gj(B)),‘as required.

Finally, if GT = GTlwith i #.j, then qi = qj, so g,(x) would be the same as gj(x),Ia contradiction.

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CHAPTER IV

Cohomology of Groups

M. F. ATIYAH and C. T. C. WALL

1. Definition of Cohomology . . . 2. The Standard Complex . . . . 3. Homology. . . . . . . . 4. Change of Groups . . . . . 5. The Restriction-Inflation Sequence 6. TheTate Groups. . . . . . 7. Cup-products . . . . . . . 8. Cyclic Groups; Herbrand Quotient 9. Cohomological Triviality . . .

10. Tate’s Theorem . . . . . .

......... 94

......... 96

......... 97

......... g8 .

......... 100

......... 101

......... 105

......... 108

......... 111

......... 113

1. Definition of Cohomology

Let G be a group, A = Z[G] its integral group ring. A (left) G-module is the same thing as a (left) A-module. By G-module we shall always mean left G-module. Note that if A is a left G-module we can define a right G-module structure on A by putting a. g = g - ’ . a.

If A, B are G-modules, the group of all abelian group homomorphisms A + B is denoted by Horn (A, B), and the group of all G-module homo- morphisms by Horn, (A, B). Horn (A, B) has a G-module structure defined as follows: if cp E Horn (A, B), g. cp is the mapping a H g. cp(g-‘a) (a E A).

For any G-module A, the subset of elements of A invariant under the action of G is denoted by AG. AG is a n abelian group which depends func- torially on A. It is the largest submodule of A on which G acts trivially. If A, B are G-modules then

Horn, (A, B) = (Horn (A, B))G; u.0 in particular,

Horn, (Z, A) = (Horn (Z, A))G r AG,

regarding Z as a G-module on which G acts trivially. Since the functor Horn is left-exact, it follows that AG is a left-exact covariant fun&or of A, i.e. if

O+A-+B+C+O (1.2) is an exact sequence of G-modules, then

O+AG~BG+CG

is an exact sequence of abelian groups. 94

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COHOMOLOGY OF GROUPS 95

If X is any abelian group we can form the G-module Horn (A, X). A G-module of this type is said to be co-induced.

By a cohomological extension of the functor AG we mean a sequence of functors H4(G, A) (q = 0, 1,. . .), with H’(G, A) = A’, together with connecting (or boundary) homomorphisms

6 : Hq(G, C) --) H”+‘(G, A)

defined functorially for exact sequences (1.2), such that

(i) the sequence

. . . --f Hq(G, A) + Hq(G, B) + Hq(G, C): Hq+l(G, A) --f . . . (1.3) is exact; and

(ii) Hq(G, A) = 0 for all q 2 1 if A is co-induced.

THEOREM 1. There exists one and, up to canonical equivalence, only one cohomological extension of the functor AG.

The groups Hq(G, A), uniquely determined by Theorem 1, are called the cohomoIogy groups of the G-module A.

The existence part of Theorem 1 is established by the following con- struction. Choose a resolution P of the G-module Z (G acting trivially on Z) by free G-modules:

. . . +P1-+Po+Z+O

and form the complex K = Horn, (P, A), i.e.

O+Hom,(P,,A)+Hom,(P,,A)+.... Let Hq(K) (q > 0) denote the qth cohomology group of this complex. Then Hq(G, A) = Hq(K) satisfies the conditions for a cohomological extension of the functor AG. For by a basic theorem of homological algebra, the Hq(G, A) so defined satisfy the exactness property (1.3); also H’(G, A) = Ho(K) = Horn, (Z, A) = AG; finally, if A is co-induced, say A = Horn (A, X) where X is an abelian group, then for any G-module B we have

Horn, (B, A) z Horn (B, X) (the isomorphism being as follows: if cp : B + A is a G-homomorphism, then cp corresponds to the map B + X defined by b I-+ e@)(l), where 1 is the identity element of G). Hence the complex K is now

O-,Hom(P,,X)+Hom(P,,X)+ . . .

which is exact at every place after the first, because the Pi are free as abelian groups; and therefore Hq(G, A) = 0 for all q > 1.

To prove the uniqueness of the cohomology groups we consider, for each G-module A, the G-module A* = Horn (A, A). There is a natural injection

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96 M. F. ATIYAH AND C. T. C. WALL

A + A* which maps a E A to cp,,, where cp,, is defined by p,(g) = ga. Hence we have an exact sequence of G-modules

O+A+A*+A’-,O (1.4) where A’ = A*/A; since A* is co-induced, it follows from (1.3) that

6 : Hq(G, A’) -+ H4+ ‘(G, A) (1.5)

is an isomorphism for all q > 1, and that H’(G, A) g Coker (H’(G, A*) + H’(G, A’)). (1.6)

The Hq(G, A) can therefore be constructed inductively from Ho, and so are unique up to canonical equivalence. This procedure could also be used as an inductive definition of the Hq.

Remark. It follows from the uniqueness that the Hq(G, A) are independent of the resolution P of Z used to construct them. So we may take any con- venient choice of P.

2. The Standard Complex

As a particular choice for the resolution P we can take Pi = Z[G’+‘], i.e. Pi is the free Z-module with basis G x . . . x G ((i+ 1) factors), G acting on each basis element as follows:

s(go, 91,. * . , Si> = (sS09sSl,. . *,sC7i)*

The homomorphism d: Pi + Pi-l is given by the well-known formula

Go,. . ~.83=j~o(-1)j(80~~~~~9j-l,Bj+l~~ . .sgi)7 (2.1)

and the mapping E : PO + Z is that which sends each generator (go) to 1 E Z. (To show that the resulting sequence

d d

. ..+P.+P,:z+o (2.2)

is exact, choose an element s E G and define h : Pi + Pi+ 1 by the formula

Mg 0,...,9i)=(s,B0,91,...,91).

It is immediately checked that dh +hd = 1 and that dd = 0, from which exactness follows.)

An element of K’ = Horn, (Pi, A) is then a function f: G’+ ’ -+ A such that

mo, ?I13 * * *, W3 = s*fh709f?l,+ * *,Si>*

Such a function is determined by its values at elements of G’+ ’ of the form t1,gl,glg,,...,glg,....g3:ifweput

(Pkll,. ..,gj)=f(l,gl,g,Y,,...,Yl...Yi~

the boundary is given by the formula

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COHOMOLOGY OF GROUPS 97

tdv)(g,, . . -,9*+1)

= 91 .(Pt92, . . ..Bi+l)+j~l(-1Ym(gl....,8j8j+l.. * *,Si+l)+

+(-iY+w,,. .4x (2.3)

This shows that a I-cocycle is a crossed homomorphism, i.e. a map G -+ A satisfying

and cp is a coboundary if there exists a E A such that cp(g) = gu-a. In particular, if G acts trivially on A then

H’(G, A) = Horn (G, A). (2.4) From (2.3) we see also that a 2-cocycle is a function 40 : G x G + A such

that

91~t92,93)-4p(9192,93)+(Pt91~9293)-(Pts1,92) = 0. Such functions (called factor systems) arise in the problem of group exten- sions, and H’(G, A) describes the possible extensions E of G by A, i.e. exact sequences 1 + A + E + G + 1, where A is an abelian normal subgroup of E, and G operates on A by inner automorphisms. If E is such an extension, choose a section o : G + E (a system of coset representatives). Then we have

4g1). h72) = (P(s19 g2M3192) for some (p(gr, gJ E A. The function cp is a 2-cocycle of G with values in A; if we change the section (r, we alter cp by a coboundary, so that the class of cp in H’(G, A) depends only on the extension. Conversely, every element of H2(G, A) arises from an extension of G by A in this way.

For later use, we give an explicit description of the connecting homo- morphism 6 : H’(G, C) + H’(G, A) in the exact sequence (1.3). Let c E H’(G, C) = CG, and lift c up to b E B. Then db is the function s H sb - b; the image of sb- b in C is zero, hence sb- b E A and therefore db is a I-cocycle of G with values in A. If we change b by the addition of an element of A, we change db by a coboundary, hence the class of db in H’(G, A) depends only on c, and is the image of c under 6.

3. Homology

If A, B are G-modules, A @I B denotes their tensor product over Z, and A @o B their tensor product over A. A @ B has a natural G-module structure, defined by g(a 8 b) = (gu) @I (gb).

Let I, be the kernel of the homomorphism A + Z which maps each s E G to 1 E Z. IG is an ideal of A, generated by all s- 1 (s E G). From the exact sequence

o-+I,+A+z+o (3.1)

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98 M. F. ATIYAH AND C. T. C. WALL

and the right-exactness of @I it follows that, for any G-module A,

Z BG A z A/Z,A.

The G-module A/&A is denoted by A,. It is the largest quotient module of A on which G acts trivially. Clearly A, is a right-exact functor of A. For any two G-modules A, B we have

A&B z (A@&. (3.2)

A G-module of the form A @ X, where X is any abelian group, is said to be induced. By interchanging right and left, induced and co-induced, we define a homological extension of the functor A,.

THEOREM 2. There exists a unique homological extension of the functor A o

The homology group H&G, A) given by Theorem 2 may be constructed from the standard complex P of $2 by taking

H&G, A) = f&U’ 63~ A).

Uniqueness follows by using the exact sequence

O-+A’-+A*-+A-+O (3.3)

where A, = A @ A. The details are exactly similar to those of the proof of Theorem 1.

The connecting homomorphism 6 : H,(G, C) + &(G, A) may be described explicitly as follows. A l-cycle of G with values in C is a functionf: G --, C such thatf(s) = 0 for almost all s E‘ G and such that df = C (s-l - l)f(s) =O.

SEG

For each s E G lift f(s) to j(s) E B (if j(s) = 0, choose j(s) = 0). Then dj has zero image in C, hence is an element of A. The class of djin H,(G, A) is then the image under 6 of the class off.

PROPOSITION 1. H,(G, Z) g G/G’, where G’ is the commutator subgroup of G.

Proof. From the exact sequence (3.1) and the fact that A is an induced G-module, the connecting homomorphism

6 : H1(G, z) --) H,(G, zo) = I&

is an isomorphism. On the other hand, the map s H s- 1 induces an isomorphism of G/G’ onto &/I$

4. Change of Groups

Let G’ be a subgroup of G. If A’ is a G’-module, we can form the G-module A = HomGP (A, A’): A is really a right G-module, but we turn it into a left G-module as described in $ 1 (if cp E A, then g . cp is the homomorphism g’ H (p(g’g-I)). Then we have

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COHOMOLOGY OF GROUPS 99

PROPOSITION 2 (Shapiro’s Lemma).

Hq(G, A) = Hq(G’, A’) for all q Z 0.

Proof. If P is a free A-resolution of Z it is also a free A’-resolution, and Horn, (P, A) z Horn, (P, A’).

The analogous result holds for homology, with Horn replaced by 63. Note that Prop. 2 may be regarded as a generalization of property (ii) of the cohomology groups (9 1) : if G’ = (l), then A’ = Z and A is a co-induced module, and the Hq(G’, A’) are zero for q 2 1.

Iff: G’ + G is a homomorphism of groups, it induces a homomorphism P’ + P of the standard complexes, hence a homomorphism

f * : Hq(G, A) + Hq(G’, A)

for any G-module A. (We regard A as a G-module via J) In particular, taking G’ = H to be a subgroup of G, and f to be the embedding H --) G, we have restriction homomorphisms

Res : Hq(G, A) --+ Hq(H, A).

If H is a normal subgroup of G we considerf: G + G/H. For any G-module A we have the G/H-module AH and hence a homomorphism Hq(G/H, AH) + Hq(G, AH). Composing this with the homomorphism induced by AH + A we obtain the inflation homomorphisms

Inf : Hq(G/H, Ai + Hq(G, A).

Similarly, for homology, a homomorphism f: G’ y G gives rise to a homomorphism

f* : K#X A) + H,G A) ;

in particular, taking G’ = H to be a subgroup of G, and f: H + G the embedding, we have the corestriction homomorphisms

Cor : H,(H, A) + H&G, A).

Consider the inner automorphism s H tst -’ of G. This turns A into a new G-module, denoted by A’, and gives a homomorphism

Hq(G, A) + Hq(G, A’). (4.1) Now a H t -Ia defines an isomorphism A’ + A and hence induces

Hq(G, A) + Hq(G, A). (4.2)

PROPOSITION 3. The composition of (4.1) and (4.2) is the identity map of H’(G, A).

The proof employs a standard technique, that of dimension-shifting: we verify the result for q = 0 and then proceed by induction on q, using (1 S) to shift the dimension downwards.

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100 M. F. ATIYAH AND C. T. C. WALL

For q = 0, we have H’(G, A’) = (A’)G = t. AG, and (4.1) is just multi- plication by t. Since (4.2) is multiplication by t-r, the composition is the identity.

Now assume that q > 0 and that the result is true for q - 1. Corresponding to the exact sequence (1.4) we have an exact sequence

0 --f A’ + (A*)’ + (A’)’ --* 0.

Since (A*)’ is G-isomorphic to A*, it is a co-induced module, hence we have functorial isomorphisms

Hq(G, A’) z Hq-‘(G, (A’)‘) (q Z 2)

and H’(G, A’) z Coker (H’(G, (A*)‘) --t H’(G, (A’)‘)).

Now apply the inductive hypothesis.

5. The Restriction-Inflation Sequence

PROPOSITION 4. Let H be a normal subgroup of G, and let A be a G-module. Then rhe sequence

Inf RCS

0 --) H’(G/H,AH) -+ H’(G, A) + H’(H, A)

is exact.

The proof is by direct verification on cocycles. (1) Exactness at H’(G/H, AH). Let f: G/H + AH be a 1-cocycle, then

f induces f : G + G/H --f AH + A, which is a I-cocycle, and the class off is the inflation of the class off. Hence if f is a coboundary, there exists aEA such that f(s) = sa-a (sEG). But f is constant on the cosets of H in G, hence sa-a = sta- a for all t E H, i.e. ?a = a for all t E H. Hence a E AH and thereforef is a coboundary.

(2) Res 0 Inf = 0. If rp : G + A is a I-cocycle, then the class of cplH: H + A is the restriction of the class of cp. But if q = j, it is clear thatf/H is constant and equal to f(l) = 0.

(3) Exactness at H’(G, A). Let cp : G + A be a 1-cocycle whose restriction to H is a coboundary; then there exists a E A such that cp(t) = ta-a for all t E H. Subtracting from cp the coboundary s H sa -a, we are reduced to the case where rp\H = 0. The formula

then shows (taking t E H) that cp is constant on the cosets of H in G, and then (taking s E H, t E G) that the image of cp is contained in AH. Hence rp is the inflation of a 1-cocycle G/H + AH, and the proof is complete.

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COHOMOLOGY OF GROUPS 101

PROPOXTION 5. Let q > 1, and suppose that H’(H, A) = Ofor 1 < i <q- 1. Then the sequence

Inf Res 0 + Hq(G/H, AH) + Hq(G, A) -+ Hq(H, A)

is exact. This is another example of dimension-shifting: we reduce to the case

q = 1, which is Proposition 4. Suppose then that q > 1 and that the result is true for q- 1. In the exact sequence (1.4), the G-module A* is co-induced as an H-module (since A = Z[G] is a free Z[H]-module), hence

Hi(H,A’)zHi+‘(H,A)=O forl<i<q-2.

Also, since H’(H, A) = 0, the sequence 0 + AH + (A*)H + (A’)H -+ 0

is exact, and (A*)a is co-induced as a G/H-module (because (A*)H E Horn (Z[G/H], A)). Hence in the diagram

0 + H4-‘(G/H, (A’)H) + Hq-‘(G, A’) + Hq- ‘(H, A’)

16 16 ld 0 + Hq(G/H, AH) + Hq(G, A) + Hq(H,A)

the three vertical arrows are isomorphisms, the diagram is commutative, and by the inductive assumption applied to A’, the top line is exact. Hence so is the bottom line.

COROLLARY. Under the hypotheses of Prop. 5,

H’(G/H, Aa) g H’(G, A), 1 < i < q - 1.

6. The Tate Groups

From now on we assume that G is finite, and we denote by N the element 1 s of A. For any G-module A, multiplication by N defines an endomor-

S6G phism N: A + A, and clearly

I,A 6 Ker (N), Im(N) 5 AG.

Hence N induces a homomorphism

N* : H,(G, A) + H’(G, A)

and we define

fi,(G, A) = Ker (N*), fi’(G, A) = Coker (N*) = AGIN(

Since G is finite, we can define a mapping Horn (A, X) + A @ X (where X is any abelian group) by the rule

and it is immediately verified that this is a G-module isomorphism. Hence for a finite group the notions of induced and co-induced modules coincide.

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102 M. F. ATIYAH AND C. T. C. WALL

PROPOSITION 6. If A is an induced G-module, then &(G, A) = Z%‘(G, A) = 0.

Proof. Let A = A @ X, X an abelian group. Since A is Z-free, every element of A is uniquely of the form c s @ x,. If this element is G-invariant,

SGG then c gs @ x, = C s @ X, for all g E G, from which it follows that all

the x,‘are equal. hence such an element is of the form N.(l 8 X) and therefore lies in N(A). Hence fi’(G, A) = 0.

Similarly, if N.C s @ X, = 0, we find that c X, = 0, and therefore

1 s @I x, = C (s- lj(l @I x,) E Io A. Hence fj,(G, A) = 0.

Now we define the Tate cohomology groups I?*(G, A) for all integers q by

fiq(G, A) = Hq(G, A) forq 2 1

8- ‘(G, A) = fl,(G, A) P-Q(G, A) = Hq- ,(G, A) for 4 2 2.

THEOREM 3. For every exact sequence of G-modules O+A*B-+C-+O

we have an exact sequence

. . . + Aq(G, A) + A~(G, B) -+ A~(G, c) L f++ ‘(G, A) .+ . . .

Proof. We have to splice together the homology and cohomology sequences. Consider the diagram

. . . + H,(G, C) : H,(G, A) --) H,(G, B) + H,(G, C) + 0 1 &G 1% PE. d 1 0 + H’(G, A) + H’(G, B) + H’(G, C) + H’(G, A) + . . .

where N: is the homomorphism N* relative to A, and so on. It is clear that the inner two squares are commutative, and for the outer two squares commutativity follows immediately from the explicit descriptions of the connecting homomorphism 6 given in $I 2 and 3.

We define 8 : fj,(G, C) + I?‘(G, A) as follows. If c E fio(G, C) = Ker (NE), lift c to b E H,(G, B), then N;(b) E H’(G, B) has zero image in H’(G, C) and therefore comes from an element a E H’(G, A), whose image in A’(G, A) is independent of the choice of b; this element of fi’(G, A) we define to be (c). The definitions of the other maps in the sequence

H,(G, C> + fio(G, -4 --) fio(G, B> --) fio(G C)

: AO(G, A) + fi’(G, B) -+ BO(G, C) --t H’(G, A)

are the obvious ones, and the verification that the whole sequence is exact is a straightforward piece of diagram-chasing.

The Tate groups can be considered as the cohomology groups of a

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COHOMOLOGY OF GROUPS 103

complex constructed out of a complete resolution of G. Let P denote a G-resolution of Z by finitely-generated free G-modules (for example, the standard resolution of $2), and let P * = Horn (P, Z) be its dual, so that we have exact sequences

. ..-tP.-tP,-:Z+O z*

O+Z-+P;r-+PT+ . . .

(the dual sequence is exact because each P, is Z-free). Putting P-, = Pzdl and splicing the two sequences together we get a doubly-infinite exact sequence

L: . . . +P1-+Po--+P-1+P-2+ . . . .

The Tate groups are then the cohomology groups Hq(Hom, (L, A)) for any G-module A. This assertion is clear if q > 1. If q < -2 we use the following fact: if C is a finitely generated free G-module, let C* = Hom(C, Z) be its dual; then the mapping B: C 8 A + Horn (C*, A) defined as follows :

o(c 8 a) mapsfe C* tof(c).a

is a G-module isomorphism. Hence the composition

T : C gG A = (C @ A), z (C @ A)G : (Horn (C*, A))’ = Horn, (C*, A)

is an isomorphism (N* is an isomorphism because C 0 A is an induced G-module). From this it follows that Horn, (P-,, A) z Pndl @o A, and hence that Hbq(HomG (L, A)) = H,-,(G, A) for q 3 2.

Finally, we have to consider the cases q = 0, 1. The mapping

HOIll, (P- 1, A) + HOm, (PO, A) (6.1) .?*

is induced by the composition P, 2 Z + Pml. If we identify Horn, (P- 1, A) with P, OG A by means of the isomorphism r, the mapping (6.1) becomes a mapping from P,, @o A to Horn, (PO, A), and from the definition of r it is not difficult to see that this mapping factorizes into

P,~,A~A,~AG_tHomG(P,,A) (6.2)

where the extreme arrows are the mappings induced by E. From this it follows that Hq(HomG (L, A)) = flq(G, A) for q = 0, 1.

Remark. Since any G-module can be expressed either as a sub-module or as a quotient module of an induced module, it follows from Prop. 6 and Theorem 3 that the Tate groups A4 can be “shifted” both up and down.

If H is a subgroup of G, the restriction homomorphism

Res : Hq(G, A) + Hq(H, A)

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104 M. F. ATIYAH AND C. T. C. WALL

has been defined for all q > 0. It is therefore defined for the Tate groups fig, q 2 1, and commutes with the connecting homomorphism 6. By dimension-shifting it then gets extended to all fiq (use the exact sequence (3.3) and the fact that A* is induced as an H-module). Similarly, the corestriction, which was defined in the first place for H, (i.e. keqel, q 2 1) gets extended by dimension-shifting to all fig (use (1.4) likewise).

PROPOSITION 7. Let H be a subgroup of G, and let A be a G-module. Then (i) Res : I?,,(G, A) + fjo(H, A) is induced by N&/n : A, + AR, where

N;,,(a) = c s; Ia

and (sJ is a system of coset representatives of G/H;

(ii) Cor : fi’(H, A) + fi’(G, A) is induced by No,n : A“ + AG, where

NGIH(a) = 7 sia*

We shall prove (i), and leave (ii) to the reader. First of all, since 8 : &‘(G, A) + H’(G, A’) is induced by 6 : H’(G, A) + H’(G, A), and since Res : H’(G, A) + H”(H, A) is the embedding AG + AH and is compatible with 6, it follows that Res : fi’(G, A) -+ fi’(H, A) is induced by AG + A”. Now let v: fio(G, A) + l?,(H, A) be the map induced by NAIH. We have to check that the diagram

fio(G, A) : 8’(G, A’)

fi,;fr, A) : A’(&‘)

is commutative. Let a E A be a representative of Z E fj,(G, A), so that N,(a) = 0. Lift a to b E A,, then No(b) has zero image in A and is G-invariant, hence belongs to (A’)G E (A’)H. The class of N,(b) mod. N,(A’) is Res 0 8(E). On the other hand, v(Z) is the class mod. I,A of N&,,(a), which lifts to N;,,,(b), and 8 0 v(Z) is represented by NH 0 N;,,(b) = N,(b).

Note: For q = -2 and A = Z we have Z?-‘(G, Z) = H,(G, Z) z G/G’; Res : G/G’ + H/H’ is classically called the transfer and can be defined as follows. G/G’ is dual to Horn (G, C*), hence the transfer will be dual to a homomorphism

Horn (H, C*) + Horn (G, C*).

This homomorphism is given by

P I+ det (i*p)ldet td),

where i,p is the representation of G induced by p, and det denotes the corresponding one-dimensional representation obtained by taking deter- minants (Horn (G, C*) is here written multiplicatively).

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COHOMOLOGY OF GROUPS 105

PROPOSITION 8. If (G : H) = n, then

Cor 0 Res = n.

Prooj For I?’ this follows from Prop. 7(ii): Res is induced by the embed- ding AC -+ AH, and Cor by NC/n :AH+AG; and N,,,(a) = na for all a E A’. The general case then follows by dimension-shifting.

COROLLARY 1. If G has order n, all the groups @(G, A) are annihilated by n.

ProoJ Take H = (1) in Prop. 8, and use the fact that fiq(H, A) = 0 for all q.

COROLLARY 2. If A is a jinitely-generated G-module, all the groups fiq(G, A) are finite.

ProoJ The calculation of the fiq(G, A) from the standard complete resolution L shows that they are finitely generated abelian groups; since by Cor. 1 they are killed by n = Card (G), they are therefore finite.

COROLLARY 3. Let S be a SyIow p-subgroup of G. Then

Res : Bq(G, A) + fiq(S, A)

is a monomorphism on the p-primary component of Aq(G, A).

Proof. Let Card (G) = pa. m where m is prime to p. Let x belong to the p-primary component of Aq(G, A), and suppose that Res (x) = 0. Then

mx = Cor 0 Res(x) = 0

by Prop. 8, since m = (G: S). On the other hand, we have pax = 0 by Cor. 1; since (p”, m) = 1, it follosvs that x = 0.

COROLLARY 4. If an element x of fiq(G, A) restricts to zero in Aq(S, A) for all Sylow subgroups S of G, then x = 0.

7. Cup-products

THEOREM 4. Let G be a jinite group. Then there exists one and only one family of homomorphisms

@(G, A) 6 Aq(G, B) + fip+q(G, A @I B)

(denoted by (a @ b) + a. b), defined for all integers p, q and all G-modules A, B, such that:

(i) These homomorphisms are functorial in A and B; (ii) For p = q = 0 they are induced by the natural product

AG@BG-+(A@B)G;

(iii) If 0 + A + A’ --) A” + 0 is an exact sequence of G-modules, and if O+ A@ B+ A’@ Bd A”@ B-0 is exact, thenfora”EpP(G, A”) and b E fiq(G, B) we have

@a”).b = &Y’.b) (E~?~+~+‘(G,A@B));

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106 M. F. ATIYAH AND C. T. C. WALL

(iv) If 0 ‘+ B + B’ + B” + 0 is an exact sequence of G-modules, and if O+A@B+A@B’+A@B” -+ 0 is exact, then for a E fip(G, A) and b” E aq(G, B”) we have

a. (66”) = (- 1jP6(a. b”) (E fiP+n+l(G, A @ B)).

Let (PnL be a complete resolution for G, as in 5 6. The proof of existence depends on constructing G-module homomorphisms

qpP*q * - P,+, -+Pp@Pq

for all pairs of integers p, q, satisfying the following two conditions:

qp.q o d = (d 8 1) o ‘pp+l,q+(-ljp(l @ 4 0 qp,q+i; (7.1)

(6 63 4 0 90.0 = 8, (7.2) where E : PO + Z is defined by s(g) = 1 for all g E G.

Once the (ppSq have been defined, we proceed as follows. Let f E Horn, (P,, A), g E Horn, (P,, B) be cochains, and define the product cochain f 0 g E Horn, (P,,,, A @ B) by

f * 9 = UC3 s> 0 ‘pp.q.

Then it follows immediately from (7.1) that

d(f. s> = WI. g+(-ljPf. (dgj. (7.3)

Hence iff, g are cocycles, so is f. g, and the cohomology class off . g depends only on the classes off and g: in other words, we have a homomorphism

fiP(G, A) @ fiq(G, B) -i fiP+q(G, A 0 B).

Clearly condition (i) is satisfied, and (ii) is a consequence of (7.2). Consider (iii). We have an exact sequence

0 + Horn, (P,, A) + Horn, (P,, A’)+ Horn, (P,, A”) + 0.

Let tl” E Horn, (P,, A”) be a representative cocycle of the class a”, and lift a” back to tl’ E Horn, (P,, A’); da’ has zero image in Horn, (P,, r, A”) and therefore lies in Horn, (P,,,, A). The class of da’ in fiP+‘(G, A) is &a”). Hence if p E Horn, (P,, B) is a cocycle in the class b, then a”./3 represents the class a”. b; d(a’. j?) represents 6(a”. b); and (da’). j? represents (da”). b. But (since d/II = 0) we have d(a’./.?) = (da’) ./? from (7.3); hence s(a”. b) = (da”). b. The proof of (iv) is similar.

Thus it remains to define the (P~,~, which we shall do for the standard com- plete resolution (Pq = Z[Gq+‘] if q > 0; P-, = dual of PqSl if q > 1).

Ifq a l,P-, = P,*_, has a basis (as Z-module) consisting of all (gr, . . . , g,*), where (gy,. . ., g,*) maps (gl,. . ., gq) E PqSl to 1 E Z, and every other basis element of Pq _ 1 to 0. In terms of this basis of P-,, d: P-, -+ PWqml is given by

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COHOMOLOGY OF GROUPS

andd:P,+P- I by GoI =s;G’s*‘-

We define qpsq : Pp+4 + Pp @ P4 as follows: (1) ifp>Oandq>O,

(Pp,qk70,. * * 2 gp+q) = (903 * * * > sp) Q (gp, * * * 7 gp+q); (2) if p 2 1 and 4 > 1,

107

cp-,.-q(& * *d7f+ql = <gT,. * 4l;m<s;+1,* - 4lp*+q>; (3) if p > 0 and 4 > 1,

l ‘pp. --p-&:2 . . ..gq*)=C(gl.Sl,...,Sp)Q(Sp*1...,ST,g:,...,gq*);

(P-p-q.p(d” - .,s:> = CG* * *d?:JT,.~ d;) QDsp, . - .,s1,gq);

Pp+q, -q(go,. * .,$I,> = C(909. * .,gp,s1,. * .Jq)QJSp*,. - .,a;

(P-q.p+q ( go,* * -9 sp) = c (ST, * * . , $1 Q (sq, * * *, Sl, go, - * *, gp).

(In the sums on the right-hand side, the si run independently through G.) The verification that the (pp.4 satisfy (7.1) is tedious, but entirely straight- forward.

This completes the existence part of the proof of Theorem 4. The unique- ness is proved by starting with (ii) and shifting dimensions by (iii) and (iv): the point is that the exact sequence (3.3), namely

O-+A’-+A*-+A-+O,

splits over Z, as the Z-homomorphism A + A, = A 0 A defined by a I--+ 1 @ a shows; hence the result of tensoring it with any G-module B is still exact, and A, @ B = A 6 A 0 B = (A @ B),. Similarly for the exact sequence (I .4).

Note the following properties of the cup-product, which are easily proved by dimension-shifting :

PROPOSITION 9.

(i) (a.b).c = a.(b.c) (identvying (A @ B) @ C with A @ (B @ C)). (ii) a.b = (-l)dim@. dimb b.a (identifying A @I B with B Q A). (iii) Res (a. b) = Res (a). Res (b). (iv) Cor (a. Res (b)) = Cor (a). b.

As an example, let us prove (iv). Here His a subgroup of G, a E Ap(H, A), b E Aq(G, B), so that both sides of (iv) are elements of fjp+q(G, A @ B). If p = q = 0, a is represented by say CI E AH, and by Prop. 7(ii) Cor (a) is represented by N,,,(a) = c Sitl E AG; b is represented by /I E BG, hence

I Cor (a). b is represented by

NG/H(Oo @ p = <c sia> @ b = c sita @ 8 = NG/H(a @ p)*

On the other hand, a. Res (b) is represented by tl @ p E (A 8 B)H, hence

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108 M. F. ATIYAH AND C. T. C. WALL

Cor (a. Res (b)) by N,,,(a @ j?). This establishes (iv) for p = q = 0. Now use dimension-shifting as in the proof of the uniqueness of the cup-products and the fact that both Cor and Res commute with the connecting homo- morphisms relative to exact sequences of the types (3.3) and (1.4).

We shall later have to consider cup-products of a slightly more general type. Let A, B, C be G-modules, cp : A @ B + C a G-homomorphism. If we compose the cup-product with the cohomology homomorphism cp* induced by q, we have mappings

fiP(G, A) Q fiq(G, B) + l?P+q(G, C); 4

explicitly, a Q b I-+ cp*(u. b). cp*(a. b) is the cup-product of a, b relative to cp.

8. Cyclic Groups; Herbrand Quotient

If G is a cyclic group of order n, and s is a generator of G, we can define a particularly simple complete resolution K for G. Each K, is isomorphic to A, and d: Ki+l + K, is multiplication by T = s- 1 if i is even (resp. by N if i is odd). The kernel of T is AG = N. A = image of N, and the image of T is IG = kernel of N. Hence for any G-module A the complex HOIll, (K, A) is

N T N T

and therefore . . . +AtA+AtA+...

tt’q(G, A) = l?‘(G, A) = AGINA,

A2q+l(G, A) = fi,(G, A) = .A/Io A,

where NA is the kernel of N: A + A.

In particular, H’(G, Z) = ZG/NZ = ZlnZ is cyclic of order n.

THEOREM 5. Cup-product by a generator of H’(G, Z) induces an iso- morphism

fiq(G, A) -+ fiq+2(G, A)

for all integers q and all G-modules A. Proof. The exact sequences

give rise to isomorphisms

O-tIG-+h-+Z-+O, N T

o+z-+I\+I,+o,

(8.1)

(8.2)

8O(G, Z) : H’(G, lo) : H2(G, Z).

Since both (8.1), (8.2) split over Z, they remain exact when tensored with A, and we are therefore reduced to showing that cup-product by a generator of fi’(G, Z) induces an automorphism of Aq(G, A). By dimension-shifting

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COHOMOLOGYOFGROIJPS 109

again, we reduce to the case q = 0. Since &‘(G, Z) = Z/nZ, a generator h of fi”(G, Z) is represented by an integer fi prime to n, and cup-product with b is multiplication by /j’. Now /I is prime to n, hence there is an integer y such that j?y E 1 (mod. n); fi’(G, A) is killed by n, hence multiplication by /? is an automorphism of fi’(G, A).

Let h,(d denote the order of fiq(G, A) (q = 0, 1) whenever this is finite. If both are finite we define the Herbrand quotient

PROPOSITION 10. Let 0 + A + B --t C + 0 be an exact sequence of G-modules (G a cyclic group). Then if two of the three Herbrand quotients h(A), h(B), h(C) are defined, so is the third and we have

h(B) = h(A). h(C).

Proof. In view of the periodicity of the fiq, the cohomology exact sequence is an exact hexagon:

Ho(A) + Ho(B)

H’(C) L H’(C)

s J H’(B) t H’(A)

where Ho(A) means &‘(G, A), and so on. Suppose for example that H’(A), H’(A), Ho(B), H’(B) are finite. Let M, be the image of Ho(A) in H’(B), and so on in clockwise order round the hexagon. Then the sequence 0 + M2 -+ Ho(C) -+ M3 + 0 is exact, and M2, M3 are finite groups (M, because it is a homomorphic image of H’(B), M3 because it is a subgroup of H’(A)). Hence Ho(C) is finite, and similarly H’(C) is finite. The orders of the groups H’(A), . . . , H’(C) are respectively m6m1, mlm,, . . ., m5m6 (mi = order of M,), hence h(B) = h(A). h(C).

PROPOSITION Il. If A is a jnite G-module, then h(A) = 1.

Proof. Consider the exact sequences T

O+AG+A--+A+Ao--+O,

. 0 + H’(A) + Ao 5 AC + Ho(A) + 0.

The first one shows that AC and Ao have the same order, and then the second one shows that Ho(A) and H’(A) have the same order.

COROLLARY. Let A, B be G-modules, f: A + B a G-homomorphism with finite kernel and cokernel. Then if either of h(A), h(B) is dejined, so is the other, and they are equal.

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110 M. F. ATIYAH AND C. T. C. WALL

Proof. Suppose for example that h(A) is defined. From the exact sequences

O+Ker(J)+A+f(A)-+tJ

0 +f(A) + B + Coker (f) --) 0

it follows from Prop. 10 and 11 that h(f(A)) is defined and equal to h(A), then that h(B) is defined and equal to h(f(A)).

PROPOSITION 12. Let E be a finite-dimensional real representation space of G, and let L, L’ be two lattices of E which span E and are invariant under C Then if either of h(L), h(L’) is defined, so is the other, and they are equal.

For the proof of Prop. 12 we need the following lemma: LEMMA. Let G be a finite group and let M, M’ be two finite-dimensional

Q[G]-modules such that MR = M @Q R and ML = M’ Q. R are isomorphic as R[G]-modules. Then M, M’ are isomorphic as Q[G]-modules.

Proof. Let K be any field, L any extension field of K, A a K-algebra. If Yis any K-vector space let V, denote the L-vector space V BK L. Let M, M’ be A-modules which are finite-dimensional as K-vector spaces. An A-homo- morphism cp: M + M’ induces an A,-homomorphism cp @ 1 : ML + ML, and rp H 40 0 1 gives rise to an isomorphism (of vector spaces over L)

(HomA CM, M’)), zz HomAr. (ML, ML). (8.3) In the case in point, take K = Q, L = R, A = Q[G], so that AL = R[G]. The hypotheses of the lemma imply that M and M’ have the same dimension over Q, hence by choosing bases of M and M’ we can speak of the determinant of an element of Homorcl (M, M’), or of Hom,rc, (MR, ML). (It will of course depend on the bases chosen.)

From (8.3) it follows that if Ti are a Q-basis of Homot,, (M, M’), they are also an R-basis of Hom,tc, (MB, Mi). Since MR, Mi are R[G]-isomorphic, there exist ai E R such that det (c airi) # 0. Hence the polynomial

F(t) = det(z ti5J E Q[tl,. . *, ~1, where ti are independent indeterminates over Q, is not identically zero, since F(a) # 0. Since Q is infinite, there exist bi E Q such that F(b) # 0, and then 2 bi5; is a Q[G]-isomorphism of M onto M’.

For the proof of Prop. 12, let M = L 0 Q, M’ = L’ @ Q. Then MR and ML are both R[G]-isomorphic to E. Hence by the lemma there is a Q[G]-isomorphism cp : L @ Q + L’ 0 Q. L is mapped injectively by cp to a lattice contained in (l/N)L’ for some positive integer N. Hence f = N. rp maps L injectively into L’; since L, L’ are both free abelian groups of the same (finite) rank, Coker (f) is finite. The result now follows from the

Corollary to Prop. 11.

.

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COHOMOLOGY OF GROUPS 111

9. Cohomological Triviality A G-module A is cohomologically triviaI if, for every subgroup H of G, fiq(H, A) = 0 for all integers q. For example, an induced module is cohomologically trivial.

LEMMA 1. Let p be a prime number, G a p-group and A a G-module such that pA = 0. Then the foIlowing three conditions are equivalent:

(i) A = 0; (ii) H’(G, A) = 0;

(iii) H,(G, A) = 0.

Proof. Clearly (i) implies (ii) and (iii).

(ii) => (i): Suppose A # 0, let x be a non-zero element of A. Then the submodule B generated by x is finite, of order a power of p. Consider the G-orbits of the elements of B; they are all of p-power order (since the order of G is a power of p), and there is at least one fixed point, namely 0. Hence there are at least p fixed points, so that H’(G, A) = AG # 0.

(iii) + (i). Let A’ = Horn (A, FP) be the dual of A, considered as a vector-space over the field FP of p elements. Then

H’(G, A’) = (A’)G = HOIll, (A, F,)

is the dual of H,(G, A). Hence H’(G, A’) = 0, so that A’ = 0 and therefore A = 0.

LEMMA 2. With the same hypotheses as in Lemma 1, suppose that H,(G, A) = 0. Then A is a free module over F,[G] = AlpA.

Proof. Since pA = 0, we have p. H,(G, A) = 0 and therefore H,(G, A) is a vector space over FP. Take a basis e, of this space and lift each e, to a, E A. Let A’ be the submodule of A generated by the a,, and let A” = A/A’. Then we have an exact sequence

Ho (G, A’) : Ho (G, A) --, Ho (G, A”) --) 0

in which by construction tl is an isomorphism. Hence H,(G, A”) = 0 and therefore A” = 0 by Lemma 1, so that the a, generate A as a G-module. Hence they define a G-epimorphism cp : L -+ A, where L is a free F,[G]-module. By construction, q induces an isomorphism

B : ffo(G L> --, Ho(G A).

Let R = Ker (9). Then since H,(G, A) = 0, the sequence

0 + H,(G, R) + H,(G, L) : H,(G, A) + 0

is exact; since j? is an isomorphism, H,(G, R) = 0 and therefore R = 0 by Lemma 1. Hence cp is an isomorphism.

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112 M. F. ATIYAH AND C. T. C. WALL

THEOREM 6. Let G be a p-group and let A be a G-module such that pA = 0. Then the following conditions are equivalent:

(i) A is a free F,[G]-module; (ii) A is an induced module; (iii) A is cohomologically trivial; (iv) kg(G, A) = 0 for some integer q.

Proof. Clearly (i) * (ii) * (iii) * (iv). (iv) => (i). By dimension-shifting we construct a module B such that

pB = 0 and fiq+‘(G, A) = A’-‘(G, B) for all r. Hence H,(G, B) = 0 and therefore (Lemma 2) B is free over FJG]; hence

I?-‘(G, A) = fi-g-4(G, B) = 0

and therefore (Lemma 2 again) A is free over FJG].

THEOREM 7. Let G be a p-group and A a G-module without p-torsion. Then the following conditions are equivalent:

(i) A is cohomologically trivial; (ii) fig(G, A) = Ag+l(G, A) = 0 for some integer q;

(iii) A/pA is a free F,[G]-module.

Proof. (i) => (ii) is clear.

(ii) + (iii): From the exact sequence

O+A:A+A/pA-,O

we have an exact sequence fig(G, A) + Ag(G, A/PA) --, fi4+‘(G, A), hence fig(G, A/PA) = 0. Hence, by Theorem 6, A/pA is free over F,,[G].

(iii) =z= (i): From the same exact sequence it follows that

fig@, A) i: fig(H, A)

is an isomorphism for all integers q and all subgroups H of G. But fig(H, A) is a p-group (Prop. 8, Cor. I), hence fig(H, A) = 0.

COROLLARY. Let A be a G-module which is Z-free andsatisfies the equivalent conditions of Theorem 7. Then, for any torsion-free G-module B, the G-module N = Horn (A, B) is cohomologically trivial.

Proof. Since A is Z-free, the exact sequence

O+B:B+B/pB+O

gives an exact sequence

O+N:N+Hom(A,B/pB)-tO,

so that N has no p-torsion and N/pN z Horn (A/PA, B/pB). Since A/pA is a free F,[G]-module, it is induced, hence is the direct sum of the

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COHOMOLOGYOFGROUPS 113

s. A’ (s E G), where A’ is a subgroup of A/PA. Hence N/pN is the direct sum of the subgroup s. Horn (A’, B/pB) and is therefore induced. Therefore N is cohomologically trivial by Theorems 6 and 7.

A G-module A is projective if Horn, (A, ) is an exact functor, or equiva- lently if A is a direct summand of a free G-module. A projective G-module is cohomologically trivial.

THEOREM 8. Let G be a finite group, A a G-module which is Z-free, Gp a Sylow p-subgroup of G. Then the following are equivalent:

(i) For each prime p, the G,-module A satisfies the equivalent conditions of Theorem 7;

(ii) A is a projective G-module.

Proof. (ii) * (i) is clear. (i) =- (ii): Choose an exact sequence 0 + Q --) F + A --) 0, where F is

a free G-module. Since A is Z-free, this gives an exact sequence

0 + Horn (A, Q) + Horn (A, F) + Horn (A, A) + 0

By the Corollary to Theorem 7, Horn (A, Q) is cohomologically trivial as a G,-module for each p, hence H’(G, Horn (A, Q)) = 0 by Prop. 8, Cor. 4. Bearing in mind that H’(G, Horn (A, Q)) = (Horn (A, Q))G = Horn, (A, Q), and so on, it follows that Horn, (A, F) + Horn, (A, A) is surjective, hence the identity map of A extends to a G-homomorphism A + F. Consequently A is a direct summand of F and is therefore projective.

THEOREM 9. Let A be any G-module. Then the following are equivalent:

(i) For each prime p, fiq(Gp, A) = 0 for two consecutive values of q (which may depend on p);

(ii) A is cohomologically trivial; (iii) There is an exact sequence 0 + B, --f B, + A + 0 in which B. and

B, are projective G-modules.

Proof. (ii) * (i) is clear; so is (iii) => (ii), since a projective G-module is cohomologically trivial.

(i) * (iii): Choose an exact sequence of G-modules

O-+B1-+Bo+A+O,

with B, a free G-module. Then fiq(Gp, B,) z kq-r(Gp, A) for all q and all p, hence fiq(Gp, B,) = 0 for two consecutive values of q. Also B, is Z-free (because B, is); hence, by Theorem 8, B, is projective.

10. Tate’s Theorem

THEOREM 10. Let G be afinite group, B and C two G-modules and f: B + C a G-homomorphism. For each prime p, let G, be a Sylow p-subgroup of G,

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114 M. F. ATNAH AND C. T. C. WALL

and suppose that there exists an integer n,, such that

f; : flq(Gp, B) + fiq(G,, C)

is surjective for q = np, bijective for q = n,+ 1 and injective for q = n,+ 2. Then for any subgroup H of G and any integer q,

f$ : &(H, B) --f f&H, C)

is an isomorphism.

Proof. Let B* = Horn (A, B) and let i: B -+ B* be the injection (defined by i(b)(g) = g. b). Then (f, i) : B + C @ B* is injective, so that we have an exact sequence

O+B+C@B*-+D-+O.

Since B* is cohomologically trivial, the cohomology of C @ B* is the same as that of C. Hence the cohomology exact sequence and the hypotheses of the theorem imply that fiq(Gp, 0) = 0 for q = np and q = n,+ 1. It follows from Theorem 9 that D is cohomologically trivial, whence the result.

THEOREM 11. Let A, B, C be three G-modules and cp : A @I B + C a G-homomorphism. Let q be a fixed integer and a a given element of Aq(G, A). Assume that for each prime p there exists an integer np such that the map B”(G,, B) + fin+q(G,, C) induced by cup-product with Res,,,, (a) (relative to cp) is surjective for n = np, bijective for n = n,+ 1 and injective for n = n,+2. Then, for all subgroups H of G and aN integers n, the cup-product with Res,,, (a) induces an isomorphism

A”(H, B) + B”+q(H, C).

(Explicitly, this mapping is b I-+ cpz+,(ResclR (a). b).)

Proof The case q = 0 is essentially Theorem 10. We have a E fi’(G, A): choose a E AC representing u (then a also represents Res,,, (a) for every subgroup H of G). Define f: B -+ C by fcB> = q(a @I /3); f is a G-homo- morphism, since a is G-invariant. We claim that, for every b E A”(H, B),

P*(R%,n (a>. b) = f*(b). (10.1)

Indeed, this is clear for n = 0 (from the definition off), and the general case then follows by dimension-shifting. To shift downwards, for example, assume (10.1) true for n + 1, and consider the commutative diagram

O+B’-+B*+B-+O YJ l@fJ 5-f (10.2)

o+c’+c*-+c+o

where B, = A @ B, C, = A 8 C, and the rows are exact. B,, C, are induced modules and therefore cohomologically trivial, hence the con- necting homomorphisms 8 are isomorphisms, and the diagram

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COHOMOLOGY OF GROUPS

fjt"(H, B) : AR+l(H, B')

lf* 8 Jr* A"(H, C) + Bn+yH, C')

115

is commutative. Moreover, the rows of (10.2) split over Z, hence (10.2) remains exact (and commutative) when tensored with A (over Z). Let cp”: A @ B’ + C’ be the homomorphism induced by rp : A &I B + C. Then using the inductive hypothesis and the compatibility of cup-products with connecting homomorphisms, we have

8 o f*(b) = f '* O 8(b) = #‘*(Res,,, (a). 8(b)) = cp ,,* o &Res,,, (a). b) = 8 0 ‘p*(ResclH (a). b).

Since 8 is an isomorphism, (10.1) is proved. Nowfsatisfies the hypotheses of Theorem 10, hencef,* is an isomorphism.

This establishes Theorem 11 for the case q = 0. The general case now follows by another piece of dimension-shifting. To

shift downwards from q+ 1 to q, for example, consider the exact sequence O+A’+A*-+A-+O

where A, = A @ A; this gives rise to isomorphisms 8 : fiq(H, A) + Aq+ ‘(H, A’). Let u = Res,,, (a) E Aq(H, A); then U’ = 8(u) = Res,,, (&a)). Also 40 : A 8 B -+ C induces cp’ : A’ @J B -+ C’. Consider the diagram

A”(H, B) : A”+q(H, A 0 B) Z @+q(H, c)

A.$ B)“‘. A 9 -+ n+‘l+l(H, A’@&)jn+9+&, Cl);

it is commutative, because 8 0 (3*(u. b) = gf* 0 S(u . b) = rp’*(&u). b) = q’*(u’. b);

by the inductive hypothesis, the bottom line is an isomorphism, and b is an isomorphism; hence the top line is an isomorphism.

THEOREM 12 (fate). Let A be a G-module, a E H2(G, A). For each prime p let G, be a Sylow p-subgroup of G, and assume that

(i) H’(G,, A) = 0; (ii) H’(G,,, A) is generated by Res,,cP (a) and has order equal to that of G,.

Then for all subgroups H of G and all integers n, cup-product with Res,,, (a) induces an isomorphism

&(H, Z) + A” + ‘(H, A).

Proof: TakeB=Z,C=A,q=2,n,=-linTheorem11. Fern=-1 the surjectivity follows from (i). For n = 0, fi’(G,, Z) is cyclic of order equal to the order of GP, so the bijectivity follows from (iii). For n = 1, the injectivity follows from the fact that H’(G, Z) = Horn (G,,, Z) = 0. Thus all the hypotheses of Theorem 11 are satisfied.

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CHAPTER V

Profinite Groups

K. GRUENBERG

1. The Groups ............... 1.1. Introduction ............. 1.2. Inverse Systems ............ 1.3. Inverse Limits ............. 1.4. Topological Characterization of Protinite Groups . . 1.5. Construction of Prokite Groups from Abstract Groups 1.6. Profinite Groups in Field Theory .......

2. The Cohomology Theory ........... 2.1. Introduction ............. 2.2. Direct Systems and Direct Limits ....... 2.3. Discrete Modules ............ 2.4. Cohomology of Profinite Groups ....... 2.5. An Example: Generators of pro-p-Groups. .... 2.6. Galois Cohomology I: Additive Theory ..... 2.7. Galois Cohomology II: “Hilhert 90” ...... 2.8. Galois Cohomology III: Brauer Groups .....

References ................

. . 116 . . 116 . . 116 . . 117 . . 117 . . 119 . . 119 . . 121 . . 121

. 121 . 122 . . 122 . . 123

123 . 124

. . 125 . . 127

1. The Groups

1.1. zntroduction

A profinite group is an inverse limit of finite groups. We begin by explaining this definition in some detail. [ For all basic facts concerning inverse limits (and direct limits-these will be needed later) we refer to Chapter VIII in Eilenberg-Steenrod : Foundations of Algebraic Topology (E-S).]

All our topological groups are assumed to have the Hausdorff separation axiom. We recall that a morphism of topological groups means a continuous homomorphism.

1.2. Inverse Systems

Let Z be a directed set with respect to a relation I. This means that 5 is reflexive, transitive and to every il, iz in I, there exists i in Z such that i 2 il and i 2 i,.

An inverse system of topological groups over Z is an object (I; G’; xi), where, for each i in Z, Gi is a topological group and, for each i I j in I,

116

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PROFINITE GROUPS 117

n{ is a morphism: Gj + Gi. Moreover, zf is the identity on GI; and if i _< j I k, then xi rrj” = nf (read mappings from right to left). We shall often simply write our inverse system as (Gi).

Suppose (G!,) (over Z’) is a second inverse system. Let 4: I’ + Z be an order preserving mapping and assume that to each i’ in I’ we are given a morphism +i,: G4u,, + G;. such that, whenever i’ < j’ in I’,

commutes. Then we call (4; 4Y, i’ E I’) = @ a morphism of (Gi) to (Gi.).

1.3. Inverse Limits

We shall view finite groups as topological groups with the discrete topology. Let (Gi) be an inverse system offinite groups and form IIG, the Cartesian product, made into a topological group in the usual way by letting the kernels of the projections IIGi --f Gi form a sub-basis of 1. Let L be the subset of all (xi) in IIGi with the property that whenever i I j, ni(xj) = Xi* Then L is a subgroup of IIGi and we give it the induced topology. We say L is the inverse (or projective) limit of the system (Gi) (or, more loosely, of the groups Gi, i E Z). If all the groups Gi are finite p-groups we call L a pro-p- group. The notation will be L = lim Gi.

Clearly, L is closed in IIGi: for if x 4 L, there exists i I j so that niXj # xr and the set of all elements in IIGi with i-term xi and j-term xi is open and contains x but contains no element of L.

If CD: (Gi) + (Gi,) is a morphism of inverse systems, we may define a mapping $ of IIGi into IIG;, as follows: given x in IIGi and any i’ in I’, $(x) is to be the element in IIG$ whose i’-term is $Y (x+(~,,). Obviously, @ is a group homomorphism and is continuous because, for each i’, the kernel of IIGi + Gi, is open. By restriction, we obtain a continuous homo- morphism of $IJ Gi into lim G!,.

1.4. Topological Characterization of Projinite Groups

The main result of this section will not be needed later in these two lectures (but the corollaries will be : the reader may take them on trust). Nevertheless, we give the proof in some detail because it is not too easy to extract from the existing literature.

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118 K. GRUENBERG

THEOREM 1. A topological group is profinite if, and only if, it is compact and totaI[v disconnected.

We recall two facts for which we refer to Montgomery-Zippin : Topological Transformation Groups (M-Z).

(i) To say that a compact group is totally disconnected is equivalent to saying that 1 is the meet of all compact open neighbourhoods of 1: M-Z, p. 38.

(ii) In a compact, totally disconnected group every neighbourhood of 1 contains an open normal subgroup (and hence 1 is the meet of all open normal subgroups): M-Z, p. 56.

Proof. Let (GJ be an inverse system of finite groups and write L = &IJ Gi, C = IIGi. Then C is compact (Tychonoff’s theorem) and hence L is also compact because L is closed in C.

We must show next that L is totally disconnected. Consider first C. If xf linC,thenxi# lforsomei. PutUi=li,Uj=G,forj#iand U = IIU,. Then U is compact and open and contains 1 but not x. Hence C is totally disconnected (by (i), above). Now in a compact group, a subset is compact if, and only if, it is closed. Thus in C, 1 is the meet of all open and closed subsets containing 1. It follows that the same is true in L (induced topology) and hence L is totally disconnected.

Next assume, conversely, that G is a compact, totally disconnected group. Let (Hi) be the family of all open, normal subgroups of G. Then (G/H,) forms an inverse system (we put i I i whenever Hi 2 Hj and use the natural epimorphism G/Hj --f G/H,). Let L = lim G/H,. Since each G/H, is finite, L is profinite. The mapping 8: g + (gH,) is clearly a continuous homomorphism of G into L. Since 1 = n H, (by (ii), above), 0 is one-one. On the other hand, if a = (ai Hi) E L and S = n aI Hi, then S is not empty (by compactness) and so, if g E S, B(g) = a: thus 0 is surjective. Hence 8-l is also a mapping; and it is continuous. We conclude that 0 is an isomornhism and thus G is profinite.

The last part of our argument also yields COROLLARY 1. For any profnite group G,

G g & G/U,

where U runs through all open, normal subgroups of G.

COROLLARY 2. If H is a closed subgroup of G,

Hr&H/HnU.

ProoJ If V is an open normal subgroup of H, V = 0 n H for some neigh- bourhood 0 of 1 in G. Hence 0 contains some open normal subgroup Uof G ((ii) above) and so U n H E V. Thus the family (U n H), for varying U,

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PROFINITE GROUPS 119

is cofinal in the family of all open normal subgroups of H. The result follows now by Corollary 1 and E-S, p. 220, Corollary 3.16.

COROLLARY 3. If H is a closed normal subgroup of G,

G/H “= l@t~ G{UH.

Proof. We have only to show that G/H is profinite. Clearly, G/H is compact. It remains to check that G/H is totally disconnected. Take any x 4 H. For each h in H we may choose an open and compact neighbourhood Oh of h not containing x (because G is totally disconnected). Then H s U 0, and so, by the compactness of H, H c O,,, u . . u O,,, = S, say. Then S is open and compact and contains H but not x.

1.5. Construction of Profinite Groups from Abstract Groups

Let G be an abstract group and suppose (Hi; i E 1) is a family of normal subgroups such that given any Hi,, Hi>, there exists H, E Hi, n Hi,. If we partially order I by defining i I j whenever H, 2 Hi, then I becomes a directed set and (G/H,) is an inverse system of groups ($ being the natural homomorphism G/HI -+ G/Hi). The mapping 8: g + (gHJ is a homomor- phism of G into L = Jinx G/H,.

If Go = (I H,, then G/G0 becomes a topological group if we use as basis for open sets at 1 the groups Hi/G, (M-Z, p. 25). The homomorphism 8 induces a continuous homomorphism: G/G, -+ L and L is the completion of G/G,, with respect to (Hi/G,).

There are two important cases.

(i) (HJ is the family of all normal subgroups of finite index in G. We denote the profinite group lim G/Hi by G. For example, 2 is the inverse limit of all finite cyclic groups.

(ii) (HJ is the family of all normal subgroups of index a power of p, where p is a given prime number. Here lim G/H, = G,, is a pro-p-group.

For example, 2, (usually written Z$ is the inverse limit of all finite cyclic p-groups. This is the group ofp-adic integers.

Exercise: 2 = HZ,. P

1.6. Profinite Groups in Field Theory

Let E/F be a Galois extension of fields. This means that E is algebraic over F and the group G = G(E/F) of all F-automorphisms of E has no fixed points outside F. [ For simple facts about field theory (including finite Galois theory) we refer to Bourbaki: AZgSbre, Chapter V (B).]

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120 K. GRUENBERG

Let (K,, i E Z) be the family of all finite Galois extensions of F contained in E Then E = U Ki (B, Section 10.1).

Now (i) if K, G Kj, then we have a natural homomorphism ni: G(Kj/F) + G(K,/F) (B, Section 10.2);

IO OK,

I I

and (ii) the F-composite of K,, and K,, is some Kj. Hence the finite Galois

1

groups (G(K,/F)) form an inverse G(Ki/F) G(KJKi) 7 7”’

system and we may construct its inverse G(Kj/F) 0 OF 3

limit call it L .

PROPOSITION 1. G(E/F) g L.

Proof For each i in I, we have a homomorphism G(E/F) + G(KJF). These together yield a homomorphism 8: G(E/F) --t nG(K,/F). The image of 8 is obviously contained in L. We assert 0 is an isomorphism onto L.

If g # 1 in G, there exists x in E such that g(x) # x; and then there exists K, containing x. Now the image of g in G(K,/F) maps x to g(x) and thus is not the identity. Hence i3 is one-one.

Take (gJ in L. If x E E and we set g(x) = gi(x), where x E K,, then this is an unambiguous definition of a mapping g of E into E. It is easy to check that g is, in fact, an F-automorphism of E. Since B(g) = (gi), L is the image 0f 8.

We use the isomorphism 8 to transfer the topology on L to G. Thus G = G(E/F) is now a profnite group and if Vi = G(E/K,), (Vi) is a defining system of neighbourhoods of 1 (M-Z, pp. 25-26).

Example. If E is an algebraic closure of the field F, of p elements, then G(E/F,) r 2 (cf. Section 1.5).

THEOREM 2 (Fundamental theorem of Galois theory).

Let E/F be a Galois extension with group G, 9 the set of all closed subgroups of G and 9 the set of aN fields between E and F. Then K --f G(E/K) is a one-one mapping of 9 onto 9. The inverse of this is the mapping S + ES, where ES denotes the $eld of fixed points for the closed subgroup S.

Proof: First Step. We show that if K E F, then G(E/K) E 9’. Let (Lj; j E .Z) be the family of all finite extensions of F contained in K.

Then K = U Lj and thus G(E/K) = n G(E/Lj). Each Lj is contained in some finite Galois extension Kt and hence

G(E/Lj) 2 G(E/Ki). By Proposition 1, G(E/K,) is open and therefore G(E/Lj) is also open in G(E/F). Hence G(EIL,) is closed and thus n G(E/LJ = G(E/K) is closed. Second step. If K E % then K = EC@“) (B, Section 10.2).

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PROFINITE GROUPS 121

Third step. Let S be a closed subgroup of G, K = ES and T = G(E/K). Clearly S c T. But by the second step, ES = ET.

10 OE

’ I

Hence for every open normal subgroup Vi of T, if Li = BY’ LT”‘ = K = Lfvftvl. By finite Galois theory,

SO we conclkde T/V, = SV,/Vi, i.e., T = SV,. Hence S is

YK dense in T. But S is closed by the first step and so S =I T.

TO

I I 10 OE Go OF

vi 0 lo OL, OL,

SV‘ 0

To T/V,0 OK OK

2. The Cohomology Theory

2.1. Introduction

In order to define the cohomology of profinite groups we make use of two facts : (i) a profinite group is put together from finite groups ; and (ii) we know how to define the cohomology of finite groups.

2.2. Direct Systems and Direct Limits

We shall only be concerned here with the category of all discrete abelian groups (written additively).

We consider a family (AJ of abelian groups, indexed by a directed set I. Assume that for each i 5 j in Z we are given a homomorphism ri : A, + A, and that (i) zf is the identity on Ai, (ii) if i < j pi k, then r; 7; = 7:. We call (I; Ai ; 7:‘) a direct system of abelian groups over I. Frequently we just write (A,).

If (A;,) (over Z’) is a second direct system, let $: I + I’ be an order preserving mapping and to each i in 1, let et be a homomorphism: A, + A;,,,. If i I j always implies that

Al Jlr

‘A;,‘)

commutes, then we call ($; pi) = Y a morphism of direct systems: (4) -, (A;+

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122 K. GRUENBERG

Given (A J, let S be the disjoint union of the groups A *, i E I. If x E A I, Y E A,, write x - y to mean that there exists k 2 i, k 2 j such that rfx = 7;;~. Then - is an equivalence relation on S. We write lim AI for the set of equivalence classes dnd X for the class containing x.

Now A = & Ai is made into an abelian group (the direct limit of the groups A i, i E Z) as follows. If 2, J E A, where x E A i, y E Aj, then find k so that k 2 i and k 2 j and define x” + J to be the class containing T:X + zty; also - 2 is to be % . Clearly A is now an abelian group.

If Y: (A,) + (A;,) is a morphism of direct systems, we obtain a homorph- ism of&A i into lint Af. by defining the image of X, where x E A,, to be the class Of It/i(X).

2.3. Discrete Modules

Let G be a profinite group and A a (left) G-module. If Uis an open subgroup of G, we shall denote, as usual, the set of all fixed elements in A under U by AU. We shall consider only G-modules A satisfying the condition

A = UA”,

where the union is taken over all open normal subgroups of G. Such modules are called discrete G-modules.

It is not difficult to see that the following three conditions on the G-module A are equivalent:

(i) A is a discrete G-module; (ii) the stabilizer in G of every module element is an open subgroup of G;

(iii) the pairing G x A + A is continuous, where A is viewed as a discrete space and G has its usual topology as a profinite group.

2.4. Cohomology of Projinite Groups

Let A be a discrete G-module and let (Ui ; i E Z) be the family of all open normal subgroups of G. Then G g lim G/U, (Corollary 1 to Theorem 1) and A E lim A”’ (because A = lJ A”* and U A”’ is naturally isomorphic to )im A”?.

Let q be a fixed non-negative integer. For every i _< j, we obtain a homo- morphism (inflation)

Ai: H4(G/Ui, A”‘) --f Hq(G/Uj, A”J)

in the standard way (cf. Chapter IV, Section 4). Clearly, we now have a direct system of abelian groups

(I; Hq(G/U,, Aul); Ai).

DEFINITION. lim Hq(G/Ui , Au’) is called the q-th cohomology group of G in ;4 and written Hq(G,A).

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PROFINITE GROUPS 123

There is another way of defining these cohomology groups. We consider the additive group C” = C”(G,A) of all continuous mappings of G” into A and define a coboundary d: C” --, Cn+’ by the standard formula:

WXg,, . ..> &+1) = g,- f(g2, ***3 gn+l)+ i$l (- 1Yfk19 **-3 gigi+ ***9 gn+l)

+ (-lY”fkl9 . . ..gJ* This yields a complex C(G,A) and the homology groups of it are precisely Hq(G,A).

Of course, this must be proved. The argument is not difficult but (when written out in full) is both long and tedious. The crucial point is this. A mapping 4 of a profinite group H into a discrete space S is continuous if, and only if, there exists an open normal subgroup K of H and a mapping 1+4 of the finite group H/K into S such that 4 is the product of the natural projection H -+ H/K and $. It follows that if f E C”(G,A), then there is an open normal subgroup U, of G such thatf is

G” + (G/U,)” + A.

Sincef is finitely valued, the image offlies in AU2, for some open normal U,. If U = U, n U,, then f is

G”+(G/U)“+Av-+A

andf’: (G/U)” --f AU is an element of C”(G/U,AU). This makes it more than plausible that C”(G,A) = lim C”(G/U,A’).

2.5. An Example: Generators of pro-p-Groups

Let G be a pro-p-group and FP the field ofp elements, viewed as a discrete G-module with trivial action. The formula above for df shows immediately that

H’(G,F,) = Horn (G,F,). If G* = GP[G,G], then the right-hand side is Horn (G/G*, F,).

Assume G is finitely generated as a topological group. Then G/G* is finite and its dimension d, as a vector space over Fp, is the dimension of H’(G,F,). Let xlG*, . ., x,,G* be a basis of G/G*. If U is any open normal subgroup of G contained in G*, then xi, . . , x,, generate G module U (by the Burnside basis theorem for finite p-groups) and hence x1, . . , x, generate G topologically. Thus dim p, H’(G,F,) is the minimum number of generators of G.

2.6. Galois CohomoIogy I: Additive Theory

[ Our reference for elementary Galois cohomology is Chapter X of Serre: Corps Locaux (S).]

As in Section 1.6, let E/F be a Galois extension of fields with group G = G(E/F). Denote by (Ki; i E Z) the family of all finite Galois extensions of F contained in E and set Ui = G(E/KJ. Then G = bG/U,.

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124 K. GRUENBERG

The action of G on E turns the additive group of E into a G-module. Now E”’ = K, and E = u Ki so that E is a discrete G-module. Moreover, K, is a W,IF) - module and G(Ki/F) z G/U,. Thus we have

Hq(G,E) g lint Hq(G(K,/F), K,). (1)

PROPOSITION 2. Hq(G,E) = Ofir all q r 1.

By equation (1) this is an immediate consequence of LEMMA 1. Let E/F be a finite Galois extension and G = G(E/F). Then

Hq(G,E) = 0 for all q 2 1. Proof. The normal basis theorem states that E, as FG-module, is free on

one generator (B, Section 10.8), i.e., is isomorphic to the induced module on F. Hence the result (cf. Chapter IV, Section 6).

Note that this argument yields the following stronger fact:

COROLLARY. If E/F is a finite Galois extension, then the Tare cohomology groups fiq(G(E/F), E) = 0 for all integers q.

(Recall that A4 = Hq for q 2 1.)

2.7. Galois Cohomology II: “Hilbert 90”

We continue with the notation of Section 2.6. We saw that E as a G-module turned out to be cohomologically uninteresting. The situation is very different when we look at E* (the multiplicative group of E) as a G-module.

Again, since (E*)” = KF and E* = U K:, E* is a discrete G-module; and

Hq(G,E*) z lim Hq(G(K,/F), K;). (2:

PROPOSITION 3. H’(G,E*) = 0. Proof. By equation (2), we need only prove this when E/F is finite. Let f be a I-cocycle of G in E*. By the theorem on the independence of

automorphisms (B, Section 7.5), there exists c such that

b = 1 f(x) . x(c) # 0. xeo

Apply y in G to this:

y(b) = X;oIyf WllvxWl = X;of - 1 (y)f (yx) . yx(c)

(since f Cvx) = f(y) . yf (x))

= f -‘(Y)&f (4 * z(c)

= f-‘(y)b. /

Thus f (y) = b.y (b)-‘, i.e., f is a coboundary.

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PROFINITE GROUPS 125

COROLLARY (Hilbert Theorem 90). If G = G(E/F) is finite cyclic with generator g and a E E* is such that N&a) = 1, then there exists b in E* such that a = b/g(b).

Proof. Since G is cyclic, H’(G, A) = .A/(1 - g)A (cf. Chapter IV, Section 8). Here A = E* and so, in multiplicative notation, (1 - g)A is (b/g(b), b E E*}. Since Hl(G,E*) = 0, the result follows.

2.8. Galois Cohomology III: Brauer Groups Let EJF, EJF be two Galois extensions and write Gi = G(E,/F). Suppose j is an F-homomorphism: E1 -+ Ez. Then j(E,)/F is Galois

and hence the restriction g + glj(E,) yields a morphism j: G, + G,. Let U = G(EJj (El) ). Now

inf

Hq(G1, E:) S Hq(GJU, E;‘) --, Hq(G,,E,*)

and we shall write j* for the product of these two homomorphisms:

j*: Hq(G,,E:) -+ Hq(G,,E,*).

We assert j* is independent of j.

-7” P’ E1 O& j(E,) 0 U

I I ’ 1

E G F o OF oG,

Let j’ be another F-homomorphism: E1 --) E,, yielding j’ : G, --) G1. Since j’(E,) = j(Ei) (B, Section 6.3), j’ = jg, for some g in G1. Hence (j’)* = j*g*.

Now g: E1 --f E1 will yield, by the above procedure, a morphism 3: G1 + G1. Clearly g is the inner antomorphism x + g-‘xg of G1. Therefore g* is the identity on Hq(G,,EI*). (Cf. Chapter IV, Proposition 3.)

Thus (j’)* = j* and hence j* is indeed independent of j. Now suppose E,, Ez are two separable closures of F. Then there always

exist F-isomorphisms: E1 + E, and all these yield the same isomorphism Hq(G,,E,*) + Hq(G,,E,*). From the cohomological point of view, it is therefore immaterial which separable closure one uses and we shall simply write Hq(F) for the group Hq(G(E/F),E*), where E is any given separable closure of F. The groups Hq(F) depend functorially on F.

DEFINITION. The Brauer group of the field F is the group H’(F).

THEOREM 3. If EJF is a Galois extension containing a Galois extension K/F, then we have the exact sequence

0 + H’(G(K/F),K*) -+ H’(G(E/F), E*) -+ H’(G(E/K),E*).

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126 K. GRUENRERG

COROLLARY I. If K/F is a Galois extension, the following sequence is exact:

0 + H’(G(K/F),K*) + H’(F) + H’(K).

Proof. Take E to be any separable closure of F containing K and apply Theorem 3.

An immediate consequence is the following corollary.

COROLLARY 2. If (KJ is the family of all$nite Galois extensions of F in a separable closure of F, then H’(F) = U H2(G(Kt/F),Kt*).

To prove the theorem, we first translate it into a result about abstract profinite groups.

Let G = G(E/F), H = G(E/K) and A = E”. Then, by Galois theory, G/H z G(K/F) and AH = K*. The mapping H’(G,A) + H2(II,A) is restriction and the mapping H2(G/H,AH) + H’(G,A) is inflation. (Restriction and inflation for profinite groups are defined precisely as for abstract groups.) Theorem 3 is now seen as a consequence of the following result, together with Proposition 3.

PROPOSITION 4. Let H be a closed normal subgroup of the profinite group G and A be a discrete G-module such that H’(H,A) = 0. Then the following sequence is exact:

inf res 0 + H’(G/H,An) + H’(G,A) --) H’(H,A).

Outline of proof. The condition H’(H,A) = 0 is equivalent to

H’(HU~lUi, A”) = 0

for all open normal subgroups Vi of G. Hence, for each Vi,

0 --, H’(G/HUI , AHUt) --) H2(G/Ui, A”) --* H2(HUJUi, A”)

is exact (Chapter IV, Proposition 5 with q = 1). If i 2 j, we have an exact row for i, one for j and the inflation mappings connecting the two and producing a commutative diagram. All these diagrams together constitute an “exact sequence of direct systems”. The result we want now follows by taking the direct limit and using two facts:

(1) lint is an exact functor on the category of direct systems over a fixed indexing set (E-S, p. 225);

(2) lint H/H n U, r Hand @ G/Hut z G/H (Corollaries 2 and 3 to Theorem 1, above).

Alternatively, one may prove the result directly by a method almost identical to that used for establishing the abstract case.

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PROFINITE GROUPS

REFERENCES

127

The basic reference for profinite groups is Serre, J-P., “Cohomologie Galoisienne”. Springer Vet-lag, Berlin (1965).

Books referred to in this paper: Bourbaki, N., “Algkbre,” Chapter V Hermann, Paris (B) (1950). Eilenberg, S. and Steenrod, N., “Foundations of Algebraic Topology” (E-S).

Princeton University Press, Princeton, New Jersey, U.S.A. (1952). Montgomery, D. and Zippin, L., “Topological Transformation Groups” (M-

Z). Interscience, New York and London (1955). Serre, J-P., “Corps Locaux”. Hermann, Paris (S) (1962).

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CHAPTER VI

Local Class Field Theory

J-P. SERRE

Introduction ................

1. The Brauer Group of a Local Field .......... 1 .l Statements of Theorems. ........... 1.2 Computation of H2(K,,/K) ........... 1.3 Some Diagrams .............. 1.4 Construction of a Subgroup with Trivial Cohomology . . . 1.5 AnUglyLemma. ............. 1.6 End of Proofs .............. 1.7 An Auxiliary Result ............. Appendix. Division Algebras Over a Local Field ......

2. Abelian Extensions of Local Fields .......... 2.1 Cohomological Properties ........... 2.2 The Reciprocity Map ............ 2.3 Characterization of (a, L/K) by Characters ...... 2.4 Variations with the Fields Involved ........ 2.5 Unramified Extensions ............ 2.6 Norm Subgroups ............. 2.7 Statement of the Existence Theorem ........ 2.8 Some Characterizations of (a, L/K) ........ 2.9 The Archimedean Case ............

3. Formal Multiplication in Local Fields ......... 3.1 TheCaseK=Q, ............. 3.2 Formal Groups .............. 3.3 Lubin-Tate Formal Group Laws ......... 3.4 Statements 3.5 Construction of i, and ia], . : : : : : : : : : : 3.6 First Properties of the Extension K, of K. ...... 3.7 The Reciprocity Map ............ 3.8 The Existence Theorem ............

4. Ramification Subgroups and Conductors ........ 4.1 Ramification Groups ............ 4.2 Abelian Conductors ............. 4.3 Artin’s Conductors ............. 4.4 Global Conductors ............. 4.5 Artin’s Representation ............

128

129 129 129 131 133 134 135 136 136 137 139 139 139 140 141 141 142 143 144 145 146 146 147 147 148 149 151 152 , 154 155 155 157 158 159 160

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Introduction

LOCAL CLASS FIELD THEORY 129

We call a field K a local field if it is complete with respect to the topology defined by a discrete valuation v and if its residue field k is finite. We write q = pf = Card (k) and we always assume that the valuation o is normalized; that is, that the homomorphism a: K* -+ Z is surjective. The structure of such fields is known:

1. If K has characteristic 0, then K is a finite extension of the p-adic field Qp, the completion of Q with respect to the topology defined by the p-adic valuation. If [K: Q,] = n then n = ef where f is the residue degree (that is, f = [k : F,,] and e is the ramification index u(p).

2. If K has characteristic p (“the equal characteristic case”), then K is isomorphic to a field k((T)) of formal power series, where T is a uniformiz- ing parameter.

The first case is the one which arises in completions of a number field relative to a prime number p.

We shall study the Galois groups of extensions of K and would of course like to know the structure of the Galois group G(K,/K) of the separable closure KS of K, since this contains the information about all such extensions. (In the case of characteristic 0, KS = R). We shall content ourselves with the following :

1. The cohomological properties of all galois extensions, whether abelian or not.

2. The determination of the abelian extensions of K, that is, the determina- tion of G modulo its derived group G’.

Throughout this Chapter, we shall adhere to the notation already intro- duced above, together with the following. We denote the ring of integers of K by OR, the multiplicative group of K by K* and the group of units by U,. A similar notation will be used for extensions L of K, and if L is a galois extension, then we denote the Galois group by G(L/K) or GLIR or even by G. If s E G and a EL, then we denote the action of s on a by ‘a or by s(a).

In addition to the preceding Chapters, the reader is referred to “Corps Locaux” (Actualites scientifiques et industrielles, 1296; Hermann, Paris, 1962) for some elided details. In what follows theorems etc. in the four sections are numbered independently.

1. The Brauer Group of a Local Field

1.1. Statements of Theorems

In this first section, we shall state the main results; the proofs of the theorems will extend over $5 1.2-1.6.

We begin by recalling the definition of the Brauer group, Br (K), of K.

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130 J-P. SERRE

(See Chapter V, 5 2.7.) Let L be a finite galois extension of K with Galois group G(L/K). We write H’(L/K) instead of H2(GyK, L*) and we con- sider the family (LJ,,, of all such finite galois extensions of K. The induc- tive (direct) limit lii H’(L,/K) is by definition the Brauer group, Br(K), of K.

It follows from the definition that Br (K) = H’(K,/K). In order to compute Br (K) we look first at the intermediate field K,,, K c K,,, c KS, where K,, denotes the maximal unramified extension of K. The reader is referred to Chapter I, 5 7 for the properties of K,,,. We recall in par- ticular, that the residue field of K,, is 2, the algebraic closure of k, and that G(K,,/K) = G(k/k). We denote by F the Frobenius element in G(KJK); the effect of F on the residue field k is given by 1 H Aq. The map u H F” is an isomorphism 2 + G(K,,/K) of topological groups. From Chapter V 3 2.5, we recall that 2 is the projective (inverse) limit, lim Z/nZ, of the cyclic groups Z/nZ.

-

Since K,, is a subfield of KS, H2(K,,/K) is a subgroup of Br(K) = H’(K,/K). In fact:

THEOREM 1. H’(K,,/K) = Br (K). We have already noted above that H’(K,,/K) = Hz@, K,‘). THEOREM 2. The valuation map v : K,‘r + Z defines an isomorphism

H’(K,,/K) --) HZ@, Z). We have to compute Hz@, Z). More generally let G be a profinite

group and consider the exact sequence

O-+Z-+Q--+Q/Z-+O

of G-modules with trivial action. The module Q has trivial cohomology, since it is uniquely divisible (that is, Z-injective) and so the coboundary 6 : H’(Q/Z) + H’(Z) yields an isomorphism H’(Q/Z) --, H’(G, Z). Now H’(Q/Z) = Horn (G, Q/Z) and so Horn (G, Q/Z) E Hz (G, Z).

We turn now to Horn (2, Q/Z). Let 4 E Horn (2, Q/Z) and define a n$p y : Horn (2, Q/Z) + Q/Z by 4 H 4(l) E Q/Z. It follows from Theorem 2 that we have isomorphisms

H’(K,,/K) : Hz@, Z)‘: Horn (2, Q/Z) L Q/Z.

The map inv, : H’(K,,/K) + Q/Z is now defined by

invK=yo6-loo.

For future reference, we state our conclusions in:

COROLLARY. The map inv, = y 0 6-l Q v defines an isomorphism between the groups H’(K,,/K) and Q/Z.

Since, by Theorem 1, H’(K,,/K) = Br (K), we see that Iye have defined an isomorphism inv, : Br (K) + Q/Z.

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LOCAL CLASSFIELD THEORY 131

If L is a finite extension of K, the corresponding map will be denoted by inv,.

THEOREM 3. Let L/K be a jinite extension of degree n. Then

inv, 0 Res,,, = n . inv,.

In other words, the following diagram is commutative

Br(K) ResK/L, Br(L) ilwR

! invr.

!

QIZ n f QIZ

. . (For the definition of Res,,,, the reader is referred to Chapter IV $4

and to Chapter V 3 2.7.)

COROLLARY 1. An element CI E Br (K) gives 0 in Br (L) if and only if ncc = 0.

COROLLARY 2. Let LJK be an extension of degree n. Then H’(L/K) is cyclic of order n. More precisely, H’(L/K) is generated by the element uMK E Br (K), the invariant of which is I/n E Q/Z.

Proof. This follows from the fact that H’(L/K) is the kernel of Res.

1.2 Computation of H’(K,,/K)

In this section we prove Theorem 2. We have to prove that the homo- morphism H’(K,,/K) + Hz@, Z) is an isomorphism.

PROPOSITION 1. Let K,, be an unramified extension of K of degree n and let G = G(KJK). Then for all q E Z we have:

(1). Hq(G, U,) = 0, where U, = U,“; (2). the map v : Hq(G, K,*) + Hq(G, Z) is an isomorphism.

(Theorem 2 is evidently a consequence of (2) of Proposition 1, since H2K,/K) = Hz@, KJ.)

Proof. The fact that (1) implies (2) follows from the cohomology sequence

Hq(G, U,,) --) Hq(G, K,*) -+ Hq(G, Z) -+ Hq+‘(G, U,)

It remains to prove (1). Consider the decreasing sequence of open sub- groups u, 1 u, =J u,’ 2 . . . defined as follows: x E Ui if and only if v(x- 1) > i. Now let rc E K be a uniformizing element; so that Vi = 1 +x*0,,, where 0, = 0,. Then U, = lim U,,/Uj. The proof will now be built up from the three following lemma:

LEMMA 1. Let k,, be the residue field of K,,. Then there are galois isomor- phisms lJ,JU, z k,* and, for i > 1, U#Uj’ ’ z k.’ .

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132 J-P. SERRE

(By a galois isomorphism we mean an isomorphism which is compatible with the action of the Galois group on either side.)

Proof. Take a E U,, and map a H E where B is the reduction of a into k.. By definition, Uj = 1 +x0,; so if a E U,’ then I = 1, and the first part of the lemma is proved.

To prove the second part, take a E Ui and write a = 1 +&I where /3 E 0,. Now map a H 8. We have to show that in this map a product au’ corresponds to the sum p + j?. By definition, au’ = l+n’(/I+jI’)+. . ., whence au’ H p + j’.

Finally, the isomorphisms are galois since “a = 1 +n’.“j?.

LEMMA 2. For alI integers q andfor all integers i L 0, Hq(G, Vi/ Ujt ‘) = 0.

Proof. For i = 0, LJt = U,, and the first part of Lemma 1 gives

Hq(G, UJ U.‘) = Hq(G, k:) = Hq(GknIk, k:).

Now for q = 1, H’(Gk,,k, k,*) = 0 (“Hilbert Theorem 90”, cf. Chapter V, $2.6). For q = 2, observe that G is cyclic. Since k.* is finite, the Herbrand quotient h(k,*) = 1 (cf. Chapter IV, $ 8, Prop. 11); hence the result for q = 2. For other values of q the result follows by periodicity.

For i > 1, the lemma follows from Lemma 1 and the fact that k.’ has trivial cohomology.

The proof of Theorem 2 will be complete if we can go from the groups u;/uy to the group U, itself and the following lemma enables us to do this.

LEMMA 3. Let G be a finite group and let M be a G-module. Let M’, i > 0 and Ma = M, be a decreasing sequence of G-submodules and assume that M = lim M/M’; (more precisely, the map from M to the Iimit is a btjection).

Then, gfor some q E Z, Hq(G, M’JM” ‘) = 0 for all i, we have Hq(G, M) = 0.

Proof. Let f be a q-cocycle with values in M. Since Hq(G, M/M’) = 0, there exists a (q- I)-cochain $i of G with values in M such that f = S$, +fi, where fi is a q-cocycle in M’. Similarly, there exists lclz such that fl = S$, +f2, fi E M2, and so on. We construct in this way a sequence ($., f,) where tj. is a (q- I)-cochain with values in M”-l and f, is a q-cocycle with values in M”, andf, = 6.1,4.+ i +f, + i. Set IJ? = $I + e2 + . . . . In view of the hypotheses on M, this series converges and defines a (q-l)- cochain of G with values in M. On summing the equations fn = 6$,,+ 1 +f.+ i, we obtain f = SJ/, and this proves the lemma.

We return now to the proof of Proposition 1. Take M in Lemma 3 to be U,. It follows from Lemma 3 and from Lemma 2 that the cohomology of U, is trivial and this completes the proof of Proposition 1 and so also of Theorem 2.

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LOCALCLASSFIELD THEORY 133

1.3 Some Diagrams

PROPOSITION 2. Let L/K be ajinite extension of degree n and let L,,, (resp. K,,) be the maximal unramljied extension of L (resp. K); so that K,, c L,,. Then the folIowing diagram is commutative.

fwL/w 2% fw”,/L) invK I invr. 1 Q/Z -2-4 Q/Z

Proof. Let lYK = G(K,,/K) and let FK be the Frobemus element of lYK; let rL and FL be defined similarly. We have FL = (FK)f where f = [I: k] is the residue field degree of L/K.

Let e be the ramification index of L/K, and consider the diagram:

H2G K,*,) 22 H2(G, Z) 6-’ Horn U-m Q/Z) -EL Q/Z I

RCS l

(1) e.Res I

(2) .?.R.ZS I

(3) “1

fmL9 L,*,) 2L, H2(L, Z> --!.I?+ Horn (r,, Q/Z) yr. Q/Z, where Res is induced by the inclusion I?, + JYx, and yK (resp. y,) is given by rp H cp(F,) (resp. cp H cp(F,)). The three squares (I), (2), (3) extracted from that diagram are commutative; for (l), this follows from the fact that vL is equal to e.vK on K,*,; for (3), it follows from FL = FL, and n = efi for (2), it is obvious.

On the other hand, the definition of inv, : H’(f’,, K,*,) + Q/Z is equiva- lent to:

inv~=y,~6-‘~v,, and similarly :

inv, = yL 0 6-l 0 VL.

Proposition 2 is now clear.

COROLLARY 1. Let H’(L/K),, be the subgroup of H2(K,,/K) consisting of those a E H’(K,,/K) which are “killed by L.” (that is, which give 0 in Br (L)). Then H2(L/K),, is cyclic of order n and is generated by the element uLIK in H’(K,,/K) such that inv, (uLlg) = l/n.

Proof. Note that a less violent definition of H2(L/K),I is provided by H’(L/K),, = H’(L/K) n H2(K,,/K).

Consider the exact sequence

0 4 H2(L/K),, 4 H2(K,,/K) Relr, H2(L”,/L).

The kernel of the map H’(K,,/K) -+ H’(L,,/L) is H2(L/K),, and this goes to 0 under inv L : H’(L,,,/L) + Q/Z. On the other hand, it follows from Proposition 2 that inv, 0 Res = n. inv,. The kernel of the latter is (l/n)Z/Z and so H’(L/K),, is cyclic of order n, and is generated by u,,, E H’(K,,/K) with inv, (uL,x) = I/n.

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134 J-P. SERRE

COROLLARY 2. The order of H’(L/K) is a multiple of n.

Proof. H2(L/K) contains a cyclic subgroup of order n by Corollary 1.

1.4 Construction of a Subgroup with Trivial Cohomology

Let L/K be a finite galois extension with Galois group G, where L and K are local fields. According to the discussion in Proposition 1, the G-module U, has trivial cohomology when L is unramified.

PROPOSITION 3. There exists an open subgroup, V, of U, with trivial cohomology. That is, Hq(G, V) = 0 for all q.

Proof. We shall give two proofs; the first one works only in characteristic 0, the second works generally.

Method 1. The idea is to compare the multiplicative and the additive groups of L. We know that L+ is a free module over the algebra K[G]. That is, there exists a EL such that [SalssG is a basis for L considered as a vector space over K.

Now take the ring 0, of integers of K and define A = cssc OK. “a. This is free over G and so has trivial cohomology. Moreover, by multiplying a by a sufficiently high power of the local uniform&r zx, we may take such an A to be contained in any given neighbourhood of 0.

It is a consequence of Lie theory that the additive group of L is locally isomorphic to the multiplicative group. More precisely, the power series ex = 1+x+. . . +x”/n!+. . ., converges for u(x) > V(P)&- 1). Thus in the neighbourhood v(x) > v(p)l(p - 1) of 0, L* is locally isomorphic to L+ under the map XI+ eX. (Note that, in the same neighbourhood, the inverse mapping is given by log (1 +x) = x-x2/2+x3/3-. . . .)

Now define V = eA; it is clear that V has trivial cohomology. The foregoing argument breaks down in characteristic p; namely at the

local isomorphism of L+ and L*. Method 2. We start from an A constructed as above: A = CSEo 0,. “a.

We may assume that A c 0,. Since A is open in OL, $OL c A for a suitable N. Set M = z;A. Then M.M c n,M if i > N+l. For M. M = 7ti?q. A c niiOL and if i > N+l then

~~‘0 c X,.&A c n,M. KL

Now let V = 1+ M. Then Y is an open subgroup of U,. It remains to be proved that V has trivial cohomology. We define a filtration of V by means of subgroups V’ = 1 +nkM, i Z 0. (Note that V’ is a subgroup since (1 +&x)(1 +&y) = 1 +&(x+y+&xy), etc.) This yields a decreasing filtration V = V” 3 v’z v2 3 . . . . As in 5 1.2, Lemma 2, we are reduced to proving that Hq(G, Vi/Vi+l) = 0 for all q. Take x = 1+&j?, /IEM and associate with this its image fl E M/n,M. This is a group isomorphism

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LOCAL CLASS FIELD THEORY 135

of Yi/v’+’ and M/n,M and we know that the latter has trivial cohomology, since it is free over G.

This completes our proofs of Proposition 3. We recall the definition of the Herbrand quotient h(M). Namely,

h(M) = Card (fiO(M))/Card (H’(M)), when both sides are finite. (See Chapter IV, 5 8.)

COROLLARY 1. Let L/K be a cyclic extension of degree n. Then we have h(U,) = 1 and h(L*) = n.

Proof. Let V be an open subgroup of U, with trivial cohomology (cf. Prop. 3). Since h is multiplicative, h(U,) = h(V) . h( UJV) = 1.

Again, L*JU, z Z. So h(L*) = h(Z). h(UJ. Now h(U,) = 1 and h(Z) = n, since fi’(G, Z) = n and H’(G, Z) is trivial. Hence h(L*) = n.

COROLLARY 2. Let L/K be a cyclic extension of degree n. Then H’(L/K) is oforder n = [L: K].

Proof: We have

h(L*) = Card (iY’(G, L*)) Card (H’(G, L*))’

Now Corollary 1 gives h(L*) = n. Moreover, H’(G, L*) = 0 (Hilbert Theorem 90). Hence Card (H’(G, L*)) = n. But H’(G, L*) is H’(L/K), whence the result.

1.5 An Ugly Lemma LEMMA 4. Let G be a finite group and let M be a G-module and suppose

that p, q are integers with p 2 0, q > 0. Assume that:

(a) H’(H, M) = 0 for all 0 < i < q and all subgroups H of G; (b) if H c K c G, with H invariant in K and K/H cyclic of prime order,

then the order of Hq(H, M) (resp. I?‘(H, M) if q = 0) divides (K: H)P. Then the same is true of G. That is, Hq(G, M) (resp. I?‘(G, M) is of order

dividing (G : l)p. Proof. Since the restriction map Res : fiq(G, M) + fiq(Gp, M) is injective

on the p-primary components of fiq(G, M), where Gp denotes a Sylow p-subgroup of G, we may confine our attention to the case in which G is a p-group. We now argue by induction on the order of G.

Assume that G has order greater than 1. Choose a subgroup H of G which is invariant and of index p. We apply the induction hypothesis to G/H. We know from (b) that, for q > 0, the order of Hq(G/H, MH) divides (G : H)P = pp and by the induction hypothesis Hq(H, M) divides (H: 1)“. Now it follows from (a) that we have an exact sequence (Chapter IV, 5 5).

0 --, Hq(G/H, MH) Inf , Hq(G, M) Res --f Hq(H, M).

Thus Hq(G, M) has order dividing pp. (H : l)p = (G : 1)“.

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136 J-P. SERRE

For g = 0, we recall (see Chapter IV, 8 6) that

fiO(G, M) = MG/NGM.

Then we have the exact sequence

MH/NHM NG’n, MGINo M --+ (Ma)G’HjNo,H MH

where NGIH denotes the norm map and the second map is induced by the identity. The remainder of the argument now runs as before.

1.6 End of Proofs

PROPOSITION 4. Let L/K be a finite galois extension with Galois group G of order n = [L : K]. Then H’(L/K) is cyclic of order n and has a generator u,,~ E H’(K,,JK) such that inv, (u,& = l/n.

Proof. In Lemma 4, take M = L*, p = 1 and g = 2. Condition (a) is satisfied by “Theorem 90” and (b) is true by Prop. 3, Cor. 2. Hence H’(G, L*) has order dividing (G: 1) = n. But by Prop. 2, Cor. 1, H’(L/K) contains a cyclic subgroup of order n, generated by u,,, E H’(K,,,/K) and such that inv, (u& = l/n. Whence Proposition 4.

It follows from this proposition that H2(L/K) is contained in H’(K,,/K). We turn now to the proof of Theorem 1. The theorem asserts that the

inclusion Br (K) 2 H’(K,,/K) is actually equality. Now by definition, Br (K) = u H’(L/K), where L runs through the set of finite galois extensions of K. But as remarked above, H’(L/K) c H’(K,,/K). Hence Br (K) c H’(K,,/K), as was to be proved.

Evidently, Theorem 3 follows from Theorem 1 and Proposition 2.

1.7 An Auxiliary Result

We have now proved all the statements in 5 1.1 and we conclude the present chapter with a result which has applications to global fields.

Let A be an abelian group and let n be an integer > 1. Consider the cyclic group Z/nZ with trivial action on A. We shall denote the correspond- ing Herbrand quotient by h,(A), whenever it is defined. We have

hnG4 = Order (A/nA)

Order .A -

where .A is the set of tl E A such that ncr = 0. (Alternatively, we could begin with the map A 5 A and take h,(A) to be:

order (Coker (n))/order (Ker (n)).)

Now let K be a local field. Then for u E K there is a normalized abso- lute value, denoted by jalx (see Chapter II, § 11). If a E OK, then lc& = l/Card (O&O,).

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LOCALCLASSFIELDTHEORY 137

PROPOSITION 5. Let K be a localfield and let n 2 1 be an integer prime to the characteristic of K. Then h,(K*) = n/l&

Proof. Suppose that K has characteristic 0. We have h,(K*) = h,(Z). h,( U,>. Now h,(Z) = n; so we must compute h,(U,). As in Proposition 3, we con- sider a subgroup V of U which is open and isomorphic to the additive group of 0,. We have h,(U,) = h,(V). h,(U,/V) and since U,/V is finite, UWV) = 1. We have

h,(V) = MO,) and

Whence h,(O,) = Card (O&OK) = I/[&.

h,(K*) = n. (~/l&J = dl& Suppose now that K has characteristic p. We take the same steps as

before. First, h,(K*) = n. h,( U,). Now consider the exact sequence

0 -* U; --) U, + k* + 0

where Vi is a pro-p-group (cf. Lemma 1). Since n is prime to p it follows that h,(Ui) = 1 and that h,(k*) = 1. So n, h,(U,) = n. Whence the result.

We note that the statement of the proposition is also correct for R or C. In these cases we have lnlR = Inl, I& = lnl2 and one can check directly that, for R, h,(R*) = n/In1 = 1 and, for C, h,(C*) = n/l& = l/n.

APPENDIX

Division Algebras Over a Local Field

It is known that elements of Brauer groups correspond to skew fields (cf., for instance, “Stminaire Cartan”, 1950/51, Exposes 6/7), and we are going to use this correspondence to give a description of skew fields and the corresponding invariants. Most results will be stated without proof.

Let K be a local field and let D be a division algebra over K, with centre K and [D : K] = n2. The valuation v of K extends in a unique way from K to D (for example, by extending first to K(a), c1 E D, and then fitting the resulting extensions together). The field D is complete with respect to this valuation and, in an obvious notation, 0, is of degree n2 over OK. Let d be the residue field of D; we have n2 = ef where e is the ramification index andf = [d: k].

Now e < n; for there exists a E D such that Q(U) = e- ’ and a belongs to a commutative subfield of degree at most n over K, The residue field d is commutative, since k is a finite field, and d = k(z) for some a E D. Hence f < n. Together with n2 = ef, the inequalities e < n and f < n yield e = n andf = n.

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138 J-P. SERRE

Since [d: k] = n, we can find B E d such that k(E) = d. Now choose a corresponding a E OD and let L = K(a). Evidently [L : K] < n, since L is a commutative subfield of D. On the other hand, cii is an element of I (the residue field of L) and I = d; hence [I: k] = n, It follows that [L : K] = n and L is unramified. We state this last conclusion as: D contains a maximal commutative subfield L which is unramijied over K.

The element 6 E Br (K) corresponding to D splits in L, that is 6 E H’(L/K). So any element in Br (K) is split by an unramified extension and we have obtained a new proof of Theorem 1.

Description of the Invariant

The extension L of K constructed above is not unique, but the Skolem- Noether theorem (Bourbaki, “Algebre”, Chap. 8, $ 10) shows that all such extensions are conjugate. The same theorem shows that any automorphism of L is induced by an inner automorphism of D. Hence there exists y E: D such that yLy -I = L and the inner automorphism x H yxy-’ on L is the Frobenius F. Moreover y is determined, up to multiplication by an element of L”.

Let vL be the valuation uL : L* + Z of L; so that a, : D* --t (I/n)Z extends uL on D. The image i(D) of u,(y) in (I/n)Z/Z c Q/Z is independent of the choice of y. One can prove that i(D) = inv, (6), where 6 E Br (K) is asso- ciated with D.

We can express the definition of i(D) in a slightly different way. The map x H y”xy-” is equal to F” on L and so is the identity. It follows that y” commutes with L and y” = c E L*. Now

UD(Y) = ; u,(f) = f UD(C) = 1 - c(c)* n

Hence we have u,(y) = (l/n&(c) = i/n where c = r&u.

Application Suppose that K’JK is an extension of degree n. By Theorem 3, Cor. 2,

an element 6 E Br (K) is killed by K’. Hence: any extension K’/K of degree n can be embedded in D as a maximal commutative sub$eld. This may be stated more spectacularly as: any irreducible equation of degree n over K can be solved in D.

EXERCISE

Consider the 2-adic field Qz and let H be the quaternion skew field over Q2. Prove that the ring of integers in H consists of the elements a+ bi+cj+dk where a, b, c, d E Z, or a, b, c, d E 3 (mod Z,). Make a list of the seven (up to conjugacy) quadratic subfields of H.

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LOCAL CLASSFIELD THEORY 139

2. Abelian Extensions of Local Fields 2.1 Cohomological Properties

Let L/K be a finite galois extension of local fields with Galois group G = G(L/K) of order n. We have seen (9 1.1, Theorem 3, Cor. 2.) that the group H2(L/K) = H’(G, L*) is cyclic of order n and contains a generator u,,, such that inv, (u~,,J = l/n E Q/Z. On the other hand, we know that H’(G, L*) = 0.

Now let H be a subgroup of G of order m. Since H is the Galois group of L/K’ for some K’ 1 K, we also have H’(H, L*) = 0 and H’(H, L*) is cyclic of order m and generated by u,,,,.

To go further, we need to know more about u,,,,. Now we have the restric- tion map Res : Br(K) --, Br(K’) and this suggests that u,,,, = Res (u&. To see that this is the case, we simply check on invariants. We have

invr(Resu,,,) = [K’ : K] inv, (u& = [K’ : K] . X = i = inv,, (uLIK, .

We can now apply Tate’s theorem (Chapter IV, 5 10) to obtain:

THEOREM 1. For all q E Z, the map cc H u. uLIK given by the cup-product is an isomorphism of fiq(G, Z) onto l?Q+2(G, L*).

A similar statement holds if H is a subgroup of G corresponding to an extension L/K’. The mappings Res and Cor connect the two isomorphisms and we have a more explicit statement in terms of diagrams.

STATEMENT. The diagrams

flq(G, Z> , @+ 2(G, L*) “ux fiq(G, Z> __, “L/X #+2(G, L*)

Res 1

Rcs I

COI T

C0r T

AyEI, Z) “L’n’ , m+2(H, L*) lP(H, Z) “WC’ l Aq+2(H, L*)

are commutative.

Proof. As above, uLIK. = Res (u&. We must show that

Res,,,. (uLIR.@) = u,,,, .Res,,, (a).

The left-hand side is Res,,,. (u& . Res,,,. (a) (see Cartan-Eilenberg, “Homo- logical Algebra”, Chap. XII, p. 256) and so commutativity with Res is proved.

For the second diagram we have to show that Cor (uLIK,. p) = uLIK . Car(P). Now Cor (uLIK,. /I) = Cor (Res (u,&.P) = u,,, . Cor (p) (Cartan-Eilenberg, lot. cit.) and this proves the commutativity of the second diagram.

2.2 The Reciprocity Map

We shall be particularly concerned with the case q = -2 of the foregoing discussion. By definition fi-‘(G, Z) is H,(G, Z) and we know that

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140 J-P. SERRE

H,(G, Z) = G/G’ = Gab. On the other hand, fi’(L/K) = K*/NLIICL*, where N,,, denotes the norm. In this case, Theorem 1 reads as follows.

THEOREM 2. The cup-product by uL/K defines an isomorphism of Gab(L/K) onto K*/NLIKL*.

We give a name to the isomorphism just constructed, or rather to its inverse. Define 8 = t&IK to be the isomorphism of K*/NLIKL* on to Gab, which is inverse to the cup-product by ULIx. The map 0 is called the local reciprocity map or the norm residue symbol.

If a E K* corresponds to 5? E K*/NL,KL*, then we write eL,K(a) = (a, L/K). The norm residue symbol is so named since it tells whether or not a E K+ is a norm from L*. Namely, (a, L/K) = 0 (remember that 0 means 1 !) if and only if a is a norm from L*.

Observe that if L/K is abelian, then Gab = G and we have an isomorphism 8 : K*/NL,KL* --) G.

2.3 Characterization of (a, L/K) by Characters

Let L/K be a galois extension with group G. We start from an a E K* and we seek a characterization of (a, L/K) E Gab. For ease of writing we set % = (a, L/K). Let 1 E Horn (G, Q/Z) = H’(G, Z) be a character of degree 1 of G and let 6~ E H’(G, Z) be the image of x by the coboundary map 6 : H’(G, Q/Z) + H2(G, Z) (cf. 9 1.1) Let

E E K*/NLIK(L*) = &‘(G, L*)

be the image of a. The cup-product Cr. 6~ is an element of H2(G, L*) c Br (K).

PROPOSITION 1. With the foregoing notation, we have the formula

x(sA = invx (z .6x). Proof. By definition s;.u,,, = E E fi’(G, L*), s, being identified with an

element of H- 2(G, Z). Using the associativity of the cup-product, this gives ti.6X = u,,,.S,.6X = u,,,.(&.6X) = u,,,.‘+,.X) with S,.X E fl-‘(G, Q/Z). Now I?-‘(G, Q/Z) -!+ fi’(G, Z) = Z/nZ and we identify fi-‘(G, Q/Z) with Z/nZ. Moreover, the identification between Hm2(G, Z) and Gab has been so made in order to ensure that s,. x = x(s,) (see “Corps Locaux”, Chap. XI, Annexe pp. 184-186). Write s,.x = r/n, r E Z. Then &r/n) E fi’(G, Z) and 6(r/n) = r. Hence u,lK.(s.bx) = r.uLKK and the invariant of this cohomology class is just r/n = x(s,J. So Proposition 1 is proved.

As an application we consider the following situation. Consider a tower of galois extensions K c L’ c L with G = G(L/K) and H = G(L/L’). Then, if x’ is a character of (G/H)“b and x is the corresponding character of Gab, and if cc E K* induces s, E G’b and si E (G/Hyb under the natural map s, w s:, we have x(s,) = x’(si). This follows from Prop. 2 and the fact that the inflation map transforms x’ (resp. Sx’) into x (resp. Sx).

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LOCAL CLASSFIELD THEORY 141

This compatibility allows us to define s, for any abelian extension; in particular, taking L = K”b, the maximal abelian extension of K, we get a homomorphism OK : K* --f G(Kub/K) defined by c( H (E, Kab/K).

2.4 Variations with the Fields Involved

Having considered the effect on (a, L/K) of extensions of L we turn now to consider extensions of K. Let K’/K be a separable extension and let Kab, Kfab be the maximal abelian extensions of K, K’ respectively.

We look at the first of the diagrams in the Statement of $2.1 and the case q = -2. Taking the projective limit of the groups involved, we obtain a commutative diagram :

K* eK f G”K” incl VI

Here V denotes the transfer (Chapter IV, 5 6), G$ denotes G(K’Ob/K) and Gib denotes G(Kab/K) = Gab(Klab/K).

Similarly, using the second of the diagrams in the Statement, we obtain a commutative diagram :

K’* “IC + G”,s

NIL/K I

I 5

K* eK l G”K”

where i is induced by the inclusion of G,, into G,. [Note that if K’/K is an inseparable extension, then in the first of these

diagrams the transfer, V, should be replaced by qV where q is the inseparable factor of the degree of the extension K’/K. The second diagram holds even in the inseparable case.]

2.5 UnramiJied Extensions

In this case it is possible to compute the norm residue symbol explicitly in terms of the Frobenius element :

PROPOSITION 2. Let L/K be an unramified extension of degree n and let

F E GLIR be the Frobenius element. Let a E K* and let v(u) E Z be its nor- malized valuation. Then (a, L/K) = PVC’).

Proof. Let x be an element of Horn (GLJK, Q/Z). By Prop. 1, we have:

x((cc, L/K)) = inv, (Z. 8~).

The map inv,: Z!Z2(GLIK, L*) + Q/Z has been defined as a composition:

H2(G~,~, L*) -2 H’(GL,K, Z)“: H1(G,m Q/Z> 2 Q/Z. We have ~(5.6~) = v(a). 6x, hence :

inv, (E. 6x) = y&‘ov(E. 6x) = v(a) .7(x) = v(a)x(F) = x(F”(“)).

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142

This shows that

J-P. SERRE

X((% L/K)) = X(F”‘“‘) for any character x of GLIR; hence (a, L/K) = F”(‘).

COROLLARY. Let E/K be a finite abelian extension. The norm residue symbol K* + GEIR maps U, onto the inertia subgroup T of GEIK.

Proof. Let L be the sub-extension of E corresponding to T. By Prop. 2, the image of UK in GLIK is trivial; this means that the image of UK in GEIR is contained in T. Conversely, let t E T, and let f = [L : K]; there exists a E K* such that t = (a, E/K). Since t E T, Prop. 2 shows that f divides u,(a); hence, there exists b E E* such that UK(a) = v,(Nb). If we put u = a. Nb-‘, we have u E UK and (u, E/K) = (a, E/K) = t.

2.6 Norm Subgroups

DEFINITION. A subgroup M of K* is called a norm subgroup ly there exists a finite abelian extension L/K with M = NLIKL*.

Example: Let m > 1 be an integer, and let M,,, be the set of elements a E K* with OK(a) E 0 mod m ; it follows from Prop. 2 (or from a direct computation of norms) that M,,, is the norm group of the unramified exten- sion of K of degree m.

Norm subgroups are closely related to the reciprocity map

0, : K* -+ Gzb = G(lPb/K)

defined in 9 2.3. By construction, 0, is obtained by projective limit from the isomorphisms K*/NL* + GLIK, where L runs through all finite abelian extensions of K. If we put:

R = lim.K*/NL*, -

we see that 13, can be factored into

K’XG$

where i is the natural map, and 0 is an isomorphism. Note that R is just the completion of K* with respect to the topology defined by the norm subgroups.

This shows that norm subgroups of K* and open subgroups of GKb corre- spond to each other in a one-one way: if U is an open subgroup of Ggb, with fixed field L, we attach to U the norm subgroup 0; ‘( U) = NLIRL* ; if M is a norm subgroup of K*, we attach to it the adherence of O,(M); the corresponding field L, is then the set of elements in Kab which are invariant by the O,(a), for a E M. We thus get a “Galois correspondence” between norm subgroups and finite abelian extensions; we state it as a proposition:

PROPOSITION 3. (a) The map L H NL* is a bijection of the set of finite abelian extensions of K onto the set of norm subgroups of K*.

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LOCAL CLASSFIELD THEORY 143

(b) This bijection reverses the inclusion. (c) N(L.L’) = NL n NL’ and N(L n L’) = NL.NL’. (d) Any subgroup of K* which contains a norm subgroup is a norm sub-

group.

(For a direct proof, see “Corps Locaux”, Chap. XI, 5 4.)

Non-abelian extensions give the same norm subgroups as the abelian ones :

PROPOSITION 4. Let E/K be a Jinite extension, and let L/K be the largest abelian extension contained in E. Then we have:

Proof This follows easily from the properties of the norm residue symbol proved in 5 2.4; for more details, see Artin-Tate, “Class Field Theory”, pp. 228-229, or “Corps Locaux”, p. 180. (These two books give only the case where E/K is separable; the general case reduces to this one by observing that NL = K when L is a purely inseparable extension of K.)

COROLLARY (“Limitation theorem”). The index (K* : NE*) divides [E : K]. It is equal to [E : K] if and only if E/K is abelian.

Proof This follows from the fact that the index of NL* in K* is equal to [L: K].

2.7 Statement of the Existence Theorem

It gives a characterization of the norm subgroups of K*:

THEOREM 3. A subgroup M of K* is a norm subgroup if and only if it satisfies the following two conditions:

(1) Its index (K* : M) is$nite. (2) M is open in K*.

(Note that, if (1) is satisfied, (2) is equivalent to “ M is closed”.)

Proofof necessity. If M = NL*, where L is a finite abelian extension of K, we know that K*IM is isomorphic to G,,,; hence (K* : M) is finite. More- over, one checks immediately that N: L* + K* is continuous and proper (the inverse image of a compact set is compact); hence M = NL* is closed, cf. Bourbaki, “Top. G&r.“, Chap. I, $10. As remarked above, this shows that M is open. [This last property of the norm subgroups may also be expressed by saying that the reciprocity map

tJx:K*-+G;b

is continuous.]

Proof of suficiency. See § 3.8, where we shall deduce it from Lubin Tate’s theory. The usual proof, reproduced for instance in “Corps Locaux”, uses Kummer and Artin-Schreier equations.

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144 J-P. SERRE

We give now some equivalent formulations. Consider the reciprocity map 0x : K* -+ Gib. By Prop. 2, the composition

K* 2 Ggb + G(K,,/K) = 2

is just the valuation map u : K* + Z. Hence we have a commutative diagram :

04JK+K*+Z-+0 JO JO lid.

04 I, -+G~b-&-+O,

where I, = G(Kab/K,,) is the inertia subgroup of Ggb, and G(K,,/K) is iden- tified with 2.

The map 0 : UK + IK is continuous, and its image is dense (cf. Cor. to Prop. 2); since UK is compact, it follows that it is surjectiue.

We can now state two equivalent formulations of the existence theorem.

THEOREM 3a. The map 8 : U, + IX is an isomorphism.

THEOREM 3b. The topology induced on U, by the norm subgroups is the natural topology of U,.

The group IK is just lim. U,/(M n U,>, where A4 runs through all norm

subgroups of K*; the ezvalence of Theorem 3a and Theorem 3b follows from this and a compacity argument. The fact that Theorem 3 * Theorem 3b is clear; the converse is easy, using Prop. 2.

COROLLARY. The exact sequence 0 -+ U, + K* + Z -+ 0 gives by com- pletion the exact sequence:

04J,-&-+2+0.

Loosely speaking, this means that fz is obtained from K* by “replacing Z by 2”.

2.8 Some Characterizations of (a, L/K)

Let L be an abelian extension of K containing K,,, the maximal unramified extension. We want to give characterizations of the reciprocity map 9 : K* + GLIK.

Since K, c L, we have an exact sequence 0 + H + GLIR + 2 -+ 0 where H = G(L/K,,,) and 2 is identified with G(K,,,/K). Choose a local uniformizer rc in K and write cm = t?(n) = (n, L/K) E GLIK. We know that b, maps onto the Frobenius element FE GK,,,k. Moreover, we can write GLIK as a direct product of subgroups GLIK = H. I, where I, is generated by un. Corresponding to this we have L = K,, @ K,, where K, is the fixed field of CT, = 0(x). In terms of a diagram, the interrelationship between the fields is expressed by

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LOCAL CLASSFIELD THEORY 145

/L\ L< ‘K /”

\K/

where K,, and K, are linearly disjoint.

PROPOSITION 5. Let f: K* -+ G be a homomorphism and assume that:

(1) the composition K* : G --) G(K,,/K), where G + G(K,,,/K) is the natural map, is the valuation map v : K* + Z;

(2) for any untformizing element n E K, f(n) is the identity on the corre- sponding extension K,.

Then f is equal to the reciprocity map 8.

Proof. Note that condition (1) can be restated as: for CI E K*, f(m) induces on K,, the power of the Frobenius element, F”@‘.

We know that f(n) is F on K,,, and that 0(x) is F on K,,,. On the other hand, f(n) is 1 on K, and e(n) is 1 on K,. Hence f(n) = e(n) on L.

Now K* is generated by its uniformizing elements rru (write rc”u as (7m).7r1 ). Hence f = 8.

PROPOSITION 6. Let f: K* + G be a homomorphism and assume that (I) of Proposition 3 holds, whilst (2) is replaced by:

(2’) if M E K*, if K’IK is a finite sub-extension of L and if u is a norm from K’*, then f(a) is trivial on K’.

Then f is equal to the reciprocity map 8.

Proof. It suffices to prove that (2’) implies (2). That is, we have to prove that if rc is a uniformizing element, then f(n) is trivial on K,. Let K’/K be a finite sub-extension of K,. We want to prove that n E NK’*. But e(n) is trivial on K, and so on K’. This implies n E NK’*.

2.9 The Archimedean Case

For global class-field theory it is necessary to extend these results to the (trivial) cases in which K is either R or C. Let G = G(C/R). In the case K = C, the Brauer group is trivial, Br (C) = 0. On the other hand, Br (R) = H’(G, C*) = R*/RP and so Br (R) is of order 2.

The invariant inv, : Br (R) + Q/Z has image (0, l/2} in Q/Z and inv, : Br (C) + Q/Z has image (0). The group H’(G, C*) = H’(C/R) is cyclic of order 2 and is generated by u E Br (R) such that inv, (u) = l/2.

Under the reciprocity map (or rather its inverse) we have an isomorphism G = H-‘(G, Z) + H’(G, C*) = R*/R*,.

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146 J-P. SERRE

3. Formal Multiplication in Local Fields

The results given in this chapter are due to Lubin-Tate, Annals of Muthe- matics, 81 (1965), 380-387.

For our purposes, the main consequences will be: (1) the construction of a cofinal system of abelian extensions of a given local field K; (2) a formula giving (a, L/K) explicitly in such extensions; (3) the Existence Theorem of fj 2.7.

In order to illustrate the ideas involved, we begin with the case K = Q,. The results to be proved were already known in this case (but were not easily obtained) and they will be shown to be trivial consequences of Lubin-Tate theory.

3.1 The Case K = Q,

THEOREM 1. Let QT’ be the field generated over Q, by all roots of unity. Then Qiyc’ is the maximal abelian extension of Q,.

In order to determine (CI, L/K) it is convenient to split QFc’ into parts. Define Q., to be the field generated over Q, by roots of unity of order prime to p (so Q,, is the maximal unramified extension of Q,) and define QPOD to be the field generated over Q, by p”th roots of unity, u = 1, 2,. . . (so QPm is totally ramified). Then Q., and QPm are linearly disjoint and

Q;yc' = Q,. Qpm = Q,, 63 Qp-. We have a diagram:

\a,’

Now G(Q,,/Q,) = 2 an d ‘f 1 rr E G(Qpm/Qp) then c is known by its action on the roots of unity. Let E be the group ofp”th roots of unity, u = 1,2,. . . . As an abelian group, E is isomorphic to le Z/p”Z = Q,/Z,. We shall

view E as a Z,-module. There is a canonical map Z, + End (E), defined in an obvious way and this map is an isomorphism. The action of the Galois group on E defines a homomorphism G(Qpm/Qp) + Aut (E) = U,, and it is known that this is an isomorphism. (See Chapter III, and “Corps Locaux”, Chap. IX, § 4, and Chap. XIV, § 7.) If u E Up, we shalldenote by [u] the corresponding automorphism of Qpm/Q,.

THEOREM 2. If a = p” . u where u E Up, then (a, QF”/Q,) = a, is described by:

(1) on Qnr, a, induces the nth power of the Frobenius automorphism; (2) on Qp-, a, induces the automorphism [u-l].

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LOCAL CLASS FIELD THEORY 147

Of these (1) is trivial and has already been proved in § 2.5, Prop. 2. The assertion (2) can be proved by (a) global methods, or (b) hard local methods (Dwork), or (c) Lubin-Tate theory (see § 3.4, Theorem 3).

Remark. Assertion (2) of Theorem 2 is equivalent to the following: if w is a primitive p”th root of unity and if u E Up then

b”(W) = WY-l = 1 + XD,

where w = 1+x.

3.2 Formal Groups

The main game will be played with something which replaces the multi- plicative group law F(X, Y) = X+ Y+XY and something instead of the binomial expansion. The group law will be a formal power series in two variables and we begin by studying such group laws.

DEFINITION. Let A be a commutative ring with 1 and let FE A[[X, Y]]. We say that F is a commutative formal group law $-

(4 FGK F(K 2)) = JfSK Y), 2); (b) F(0, Y) = Y and F(X, 0) = X; (c) there is a unique G(X) such that F(X, G(X)) = 0;

(4 W, Y) = I;(K W; (e) F(X, Y) 3 X+ Y (mod deg 2). (In fact one can show that (c) and (e) are consequences of (a), (b) and (d). Here, two formal power series are said to be congruent (mod deg n) if

and only if they coincide in terms of degree strictly less than n. Take A = Ox. Let F(X, Y) be a commutative formal group law defined

over Ox and let mK be the maximal ideal of OK. If x, y E mx then F(x, y) converges and its sum x*y belongs to Ox. Under this composition law, m, is a group which we denote by F(Q).

The same argument applies to an extension L/K and the maximal ideal mL in 0,. We then obtain a group F(ntJ defined for any algebraic extension of K by passage to the inductive limit from the finite case.

If F(X, Y) = X+ Y+ XY then we recover the multiplicative group law of l+m,.

The elements of finite order of F(m,,) form a torsion group and G(KJK) operates on this group. The structure of this Galois module presents an interesting problem which up to now has been solved only in special cases.

3.3 Lubin-Tate FormaI Group Laws

Let K be a local field, q = Card (k) and choose a uniformizing element ITCOp Let & be the set of formal power series f with:

(1) f(X) s aX (mod. deg. 2); (2) f(X) E XQ (mod. n).

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148 J-P. SERRE

(Two power series are said to be congruent (mod. 7~) if and only if each coefficient of their difference is divisible by 71. So the second con- dition means that if we go to the residue field and denote by j(X) the corresponding element of k[[X]] thenf (X) = X4.)

Examples.

(a) f(x) = 72x+ xq;

(b) K = Qp, ‘II = P, (X) = px + 0 ; XZ+...+pXP-‘+XP.

The following four propositions will be proved in 5 5 as consequences of Prop. 5.

PROPOSITION 1. Letfcz 5,. Then there exists a unique formal group law F;- with coeficients in A for which f is an endomorphism.

(This means f(F/(X, Y)) = FJf(X), f( Y)), that is f O Ff = F, 0 (f xf).)

PROPOSITION 2. Let f E 5, and Ff the corresponding group law of Prop. 1. Then for any a E A = 0, there exists a unique [a]/ E A[[X]] such that:

(1) [aIf commutes with f, (2) [a]/ = aX (mod. deg. 2). Moreover, [a], is then an endomorphism of the group law Ff.

From Prop. 2 we obtain a mapping A + End (Ff) defined by a H [aIf. For example, consider the case

K = Qp, f=pX+ 0 ; xz+...+xp;

then F is the multiplicative law X+ Y+XY, and

[alf=(l+X)“-l=f ‘: X’. 0 i=l z

PROPOSITION 3. The map a H [a]/ is an injective homomorphism of the ring A into the ring End (F/>.

PROPOSITION 4. Let f and g be members of 3,. Then the corresponding group laws are isomorphic.

3.4 Statements

Let K be a local field and let z be a uniformizing element. Let f E 5, and let I;;- be the corresponding group law (of Prop. 1). We denote by Mf = Ff(mKS) the group of points in the separable closure equipped with the group law deduced from F. Let a E A, x E M/ and put ax = [al/x. By Prop. 3, this defines a structure of an A-module on M/. Let E, be the torsion sub-module of MI; that is the set of elements of M/ killed by a power of rc.

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LOCAL CLASSFIELD THEORY 149

THEOREM 3. The following statements hold.

(a) The torsion sub-module EJ is isomorphic (as an A-module) with K/A. (b) Let K, = K(E/) be the jieId generated by E/ over K. Then K, is an

abelian extension of K. (c) Let u be a unit in K*. Then the element o, = (u, K,/K) of G(K,/K)

acts on EY via [u-l]/. (d) The operation described in (c) defines an isomorphism U, -+ G(KJK). (e) The norm residue symbol (71, KJK) is 1. (f) The fields K,,, and K, are linearly disjoint and Kab = K,,, . K,.

We may express the results of Theorem 3 as follows. We have a diagram:

Pab \

Id .,\ ,Kz

\K/

Here G(K,,,/K) = $ and G(K,/K) = U,. Moreover every u E K* can be written in the form c( = rc”.u and on gives o (the Frobenius) on K,,/K whilst o, gives [u- ‘1 on KJK.

Example. Take K = QP, rc = p and f = pX + 0 : X2+...+XP. The

formal group law is the multiplicative group law; E, is the set of p”th roots of unity; K, is the field denoted by Qpm in 9 3.1-and we recover Theorems 1 and 2.

3.5 Construction of Ff, [a],

In this section we shall construct the formal group law Ff and the map a i--+ [al/.

PROPOSITION 5. Let f, g E &, let n be an integer and let 41(Xl,. . ., X,) be a linear form in Xl,. . ., X, with coeficients in A. Then there exists a unique 4 E A[[X,, . . . , X,]] such that:

(a) 4 = r$r (mod deg 2); (b)f,$=4.(gx...xg). Remarks. (1) The property (b) may be written

f(ddXl, * * * 9 X”>> = 4MXl),. ’ * 3 dX”N. (2) The completeness of A will not be used in the proof. Moreover, the

proof shows that 4 is the only power series with coefficients in an extension of A, which is torsion free as an A-module, satisfying (a) and (b).

Proof. We shall construct I$ by successive approximations. More pre- cisely, we construct a sequence ($‘p’) such that 4(P) E A[[X,, . . . , X,]], #p) satisfies (a) and (b) (mod deg p+ l), and c#@) is unique (mod deg pf 1).

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150 J-P. SERRE

We shall then define 4 = lim 4(P) and this will be the 4 whose existence is asserted.

We take #‘I = 4i. Suppose that the approximation 4r + . . . +4, = #p) has been con-

structed. That is, f 0 4(P) z 4(P) 0 (g x . . . x g) (mod deg p + 1). For con- venience of writing, we shall replace g x . . . x g by the single variable g. Now write $(P+‘) = #p)+ 4,+ 1. Then we may write

fo c##‘) E c$(p) o g+E p+l (mod deg p+2),

where E,,, (“the error”) satisfies Ep+ 1 E 0 (mod degp+ 1). Consider $Jp+l); we have

f. ~#&p+l) =fo (t#@)+~$,+~) =fo ~$(p)+n~~+~ (mod degp+2)

(the derivative off at the origin is z) and

4(P) 0 g++p+l 0 g = 4(P) 0 g+“p+14p+l (mod deg p +2).

Thus

f. @‘+l)-~(p+l) o g s Ep+l+(n-~~+l)~p+l (mod deg p+2).

These equations show that we must take

4 P+l = -E,+,/~(l-~3.

The unicity is now clear and it remains to show that 4,+i has coefficients in A. That is, E,,, = 0 (mod n). Now for 4 E F,[[X]J, we have r#&#?) = (4(X))q and together with&Q E X4 (mod rr) this gives

f 0 ~#(~)-4(~) .fe (cjcp)(X))4-#P)(X4) = 0 (mod it).

So, given 4(p) we can construct a unique #p+l) and the proof is completed by induction and passage to the limit.

Proof of Proposition 1. For each f E $J,, let F/(X, Y) be the unique solution of F/(X, Y) 3 X+ Y (mod deg 2) and f 0 F/ = F, 0 (f x f) whose existence is assured by Prop. 5. That F, is a formal group law now requires the veri- fication of the rules (a) to (e) above. But this is an exercise in the application of Prop. 5: in each case we check that the left-and the right-hand sides are solutions to a problem of the type discussed there and we use the unicity statement of Prop. 5. For example, to prove associativity note that both F/(F,(X, Y), 2) and F/(X, F,( Y, Z)) are solutions of

H(X,Y,Z)=X+Y+Z (moddeg2) and

fwcn f(Y), f(Z)) = f(H(X, r, Z)). Proof of Proposition 2. For each u E A and f, g E 5, let [u],,Q) be the

unique solution of

C&X0 = UT (mod deg 2)

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LOCAL CLASS FIELD THEORY 151

and SC+, ,ux = C4/, &mh

(that is fo [a]/,e = bl,, 0 id. WAe k$ = blf,f. Now we have

~f<M/, em Cd/, go--N = ld/,,wx9 Y)). For each side is congruent to aX+a Y (mod deg 2) and if we replace X by g(X) and Y by g(Y) in either side, then the result is the same as if we sub- stitute the sides in question in $ Thus [a],,@ is a formal homomorphism of Fg into Fp If we take g = f, this shows that the [+‘s are endomor- phisms of F,.

Proof of Proposition 3. In the same way as outlined above, one proves that

I3 + blf, IJ = Ff o m/.,x CKIf.3 and

Wlf,h = b1f.e o C%* It follows from this that the composition of two homomorphisms of the type just established is reflected in the product of corresponding elements of A. Taking f = g, we see that the map a H [a]/ is a ring homomorphism of A into End (E/>. It is injective because the term of degree 1 of [a]/ is aX.

Proof of Proposition 4. If a is a unit in A, then [a]/,, is invertible (cf. the proof of Prop. 2) and so FB r Fi by means of the isomorphism [a],,.

Note that [z]/ = f and [l]/ is the identity (proved as before).

This completes the proofs of the propositions 1, 2, 3, 4.

3.6 First Properties of the Extension K, of K

From now on, we confine our attention to subfields of a fixed separable closure K’ of K. Given f E &, let F/ be the corresponding formal group law and let EJ be the torsion submodule of the A-module F’(m,). Let E; be the kernel of [n”],; so that E, = v Ej’. Let K,” = K(E;) and K, = u K:. If Gn,n denotes the Galois group of K(E/3 over K, then G(K,/K) = lim G,,“.

PRO~OSI~O% 6. (a) The A-module E/ is isomorphic to K/A;

(b) the natural homomorphism G(KJK) + Aut (EI) is an isomorphism.

Proof. We are free to choose f as we please, since, by Prop. 4, different choices give isomorphic group laws. We take f = .zX+ X4. Then a E EJ if and only if f cm)(a) = 0, where f (“) denotes the composition f. . . . Of n times; that is f (“) = [z”]~

If a E tttK, then the equation rcX+ X4 = a is separable and so solvable in KS, its solution belonging indeed to Q,. This shows that iUJ is divisible.

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Hence EJ is divisible also. This already implies that E, is a direct sum of modules isomorphic to K/A.

Let us consider the submodule E; of Ef consisting (see above) of those a E M, such that [~]~a = 0. The submodule Ej is isomorphic with A/mKS, since it is an A-module with q elements. This is enough to show that Ef is isomorphic to K/A.

An automorphism u E G(K,/K) induces an automorphism of the A-module E,. But since Ef 2 K/A and End, (K/A) = A this gives a map G(K,JK) -+ Aut (E,) = U,. This map is injective by the definition of K, and it remains to be proved that it is surjective.

Take n 2 1 and define E; and Ki as above. We have an injection G(Kz/K) -+ UK/U{, where Ug = 1 +d’A. Let M E ET be a primitive element; that is an element of E; such that [n”lfa = 0, but [x”-‘]~cx # 0. Finally, we define C#I as follows:

4 =p/p-1) =f(f’“-“)/f’“-1’.

Nowf = X4+71X; sof/X = Xq-‘+n. Hence

fP ‘9 which is of degree qn~qf~“-” = (f’“-lYoq-l+n~

n-1 and which is irreducible, since it is an Eisenstein polynomial. All primitive elements CI are roots of 4. Thus the order of G(K,“/K) is at least (q - l)q“- ‘. On the other hand, this is actually the order of the group U,lUL. Hence G(K”/K) = U,llJ;. It follows that

G(K,/K) = lim G(K:/K) = lim U&J: = UK, - -

and this completes the proof of Prop. 6. The same proof also yields:

COROLLARY. The element TT is a norm from K(a) = K:.

Proof. The polynomial C/J constructed above is manic and ends with 71. Hence iV(-cc) = 71.

3.7 The Reciprocity Map

We shall study the compositum L = K,,,K% of K,,, and K,, and the symbol (a, L/K), CI E K”. We need to compare two uniformizing elements rr and co = nu,uE U,.

Let R,, be the completion of K,, (remember: K,,, is an increasing union of complete fields but is not itself complete) and denote by A^,, the ring of integers of I?,,. By definition i?,, is complete; it has an algebraically closed residue field and rc is a uniformizing parameter in I?,,. We take f E 5. and g E 3,.

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LOCAL CLASS FIELD THEORY 153

LEMMA 1. Let a E G(KJK) be the Frobenius automorphism and extend it to I?,,, by continuity. Then there exists a power series 4 E A,,[[X]] with

d(X) s EX (mod deg 2) and E a unit, such that

(4 “4 = 4 0 blf; (b) 4oF/ = Fg4W; (c) f#~ o [a], = [a], O 4 for all a E A.

Proof. Since cr- 1 is surjective on & and on firm, (cf. “Corps Locaux”, p. 209), there exists a 4 E &[[X]] such that d(X) E EX (mod deg 2) where E is a unit and “4 = I$ Q [u]~. This is proved by successive approximation and we refer the reader to Lubin-Tate for the details. This particular 4 does not necessarily give (b) and (c) but can be adjusted to do so; the com- putations are given in Lubin-Tate (where they appear as (17) and (18) in Lemma 2 on p. 385). Note that together the above conditions express the fact that $J is an A-module isomorphism of I;; into F#.

Computation of the norm reciprocity map in L/K.

Let L, = K,, . K,. Since K,,, and K, are linearly disjoint over K, the Galois group G(LJK) is the product of the Galois groups G(KJK) and G&,/K). For each uniformizing element rc E A we define a homomorphism r, : K* --f G(L,/K) such that:

(a) r,(x) is 1 on K, and is the Frobenius automorphism o on K,,; (b) for u E UK, m(u) is equal to [u-I]~ on K, and is 1 on K,,.

We want to prove that the field L, and the homomorphism r, are independent of rr. Let w = nu be a second uniformizing element.

First, L, = L,. For by Lemma 1, Ff and FB are isomorphic over R,,. Hence, the fields generated by their division points are the same. So

R,,. K, = R,,. K,. On taking completions we find that Kz = Kz. In order to deduce that K,, . K, = K,, . K, from this, we require the following:

LEMMA 2. Let E be any algebraic extension (finite or infinite) of a local field and let CI E l?. Th en, if cc is separable algebraic over E, a belongs to E.

Proof. Let ES be the separable closure of E and let E’ be the adherence of E in ES. We can view a as an element of E’. Hence it is enough to show that E’ = E.

Let s E G(E,/E). Since s is continuous and is the identity on E, it is also the identity on E’. Hence G(EJE) = G(E,/E’) and by Galois theory we have E’ = E.

It follows from Lemma 2 that L, = L, and so L, = L is independent of n.

We turn now to the homomorphism r,: K* + G(L/K). We shall show

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154 J-P. SEW I

thar r&o) = rJo). This will imply that m(o) is independent of n and so the r,‘s coincide on the local uniformizers. Since these generate K*, the result will follow.

We look first at rD(w). On K,,,, ~~(0) is the Frobenius automorphism Q. On K, it is 1. On the other hand, r%(w) is c on K,,; so we must look at T,(O) on K,.

Now K, = K(E,), where g E &,. Let 4 E J[[X]] be as in Lemma 1; 4 determines an isomorphism of Ef onto E,,. So if 3, E E,, then we can write Iz = 4(p) with p E Ep We look at r-,(0)3, and we want to show that this is 1. As already remarked, rJcl))(l) = r,(o)+@). Write s = m(w). We want to show that “A = rl; that is “40 = &). Now, r,(o) = m(x). r%(u) and the effects of m(n) and r&) are described in (a) and (b) above. Since 4 has coefficients in R,,, “4 = “4 = 4 0 [u]/ by (a) of Lemma 1. But

“(4Q> = w/4 = “dau-- ‘I&N. Hence

wPl= 4 0 Cd, 0 b--‘l/O = 4w So r, is the identity on K, and it follows that r, is independent of 71. Thus r : K* + G(L/K) is the reciprocity map 8 (5 2.8, Prop. 5).

All assertions of Theorem 3 have now been proved except the equality L = Kab, which we are now going to prove.

3.8 The Existence Theorem

Let Kab be the maximal abelian extension of K; it contains K,,. The Existence Theorem is equivalent to the following assertion (9 2.3, Theorem 3a). If Ix = G(K”b/K,,) is the inertia subgroup of G(Kub/K), then the reciprocity map 0 : U, + ZK is an isomorphism.

Let L be the compositum K,. K, and let ZL = G(LIK,,) be the inertia subgroup of G(L/K). Consider the maps

6 e tJ,+Z,-+I; j

where 0 is the reciprocity map and e is the canonical map ZK + ZL. Both 0 and e are surjections.

On the other hand, the composition e 0 0 : UK --) Ix has just been com- puted. If we identify Z& with Ux it is u H u-l. Hence the composed map e 0 8 is an isomorphism. It follows that both 8 and e are isomorphisms.

As we have already noted, the first isomorphism is equivalent to the Existence Theorem. The second means that L = Kab, since both L and K”” contain K,,.

[Alternative Proof. Let us prove directly that every open subgroup M of K*, which is of finite index, is a norm subgroup corresponding to a

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LOCALCLASSFIELDTHEORY 155

finite subextension of L. This will prove both the existence theorem (3 2.7, Theorem 3) and the fact that L = K“‘.

Since A4 is open, there exists n > 1 such that Uz c 44; since it4 is of finite index, there exists m > 1 such that rP E M; hence M contains the subgroup V,,, generated by U,” and n;“‘. Now let K,,, be the unramified extension of K of degree m, and consider the subfield L,,,, = Ki. K,,, of L. If u E UK, and a E Z, we know that (ux”, L&K) is equal to [u-l] on Ki and to the a-th power of the Frobenius element on K,,,; hence (an”, L,,,/K) is trivial if and only if u E U< and a 3 0 mod m, i.e. if and only if urc’ E V,,.,. This shows that V.,, = NL,,,, and, since M contains V,,,, M is the norm group of a subextension of L,,,, Q.E.D.]

4. Ramification Subgroups and Conductors

4.1 Ram$cation Groups Let L/K be a galois extension of local fields with Galois group G(L/K). We recall briefly the definition of the upper numbering of the ramification groups. (For details, the reader should consult Chapter I, 9 9, or “Corps Locaux”, Chap. IV.)

Let the function ic: G(L/K) -+ {Z u co} be defined as follows. For s E G(L/K), let x be a generator of OL as an OR algebra and put i&s) = v~(s(x)-x). Now define GU for all positive real numbers u by: s E G, if and only if i,Js) > u+ 1. The groups G, are called the ramification groups of G(L/K) (or of L/K). In order to deal with the quotient groups, it is necessary to introduce a second enumeration of the ramification groups called the “upper numbering”. This new numbering is given by G” = G,, where u = 4(u) and where the function I$ is characterized by the properties:

(a> 4(O) = 0; (b) 4 is continuous; (c) 4 is piecewise linear; (d) 4’(u) = l/(G, : GJ when u is not an integer.

The G”‘s so defined are compatible with passage to the quotient: (G/H) is the image of G” in G/H (“Herbrand’s theorem”). This allows one to define the G”‘s even for infinite extensions.

On the other hand, we have a filtration on UK defined by Uz = 1 +m$. We extend this filtration to real exponents by Ul = Ufl if n- 1 < u < n. (It should be noted that v in this context is a real number and is not to be confused with the valuation map!)

THEOREM 1. Let L/K be an abelian extension with Galois group G. Then the local reciprocity map 9 : K* + G maps U,V onto G” for all v 2 0.

Proof. (1) VeriJication for the extensions K,” of 5 3.6. Let u E Vi with i < n and u 4 Uk”. Let s = 0(u) E G(c/K). We have

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156 J-P. SERRE

i(s) = +&L-2), where I is a uniformizing element. We choose a primitive root a for 1; that is, an a satisfying [?]/a = 0 but [n”-‘lfa # 0. Observe that s,(a) = [u- ‘If a and u -r = l+n’v (see 8 3.3, Theorem 3), where v is a unit. These imply that

s,ct = [l +7civ]fa = F&t, [n’v]p).

Jf we write p = [nivlfcr, then /J is a primitive (n - i)th root (that is, [n*-‘lz/I = 0, [Y-l-i]f/I # 0), and we have

for some yij E OK. Accordingly,

S,(O()--c( = P + C Yija’Pj

and

Now a is a uniformizing element in K,” whilst /? is a uniformizing element

Figure 1.

in K”,-’ and K;/KEsi is totally ramified. Its degree is ql. So we have deter- mined the i function of 19(u); namely, if UE U’ but u 4 Uifl, then i@(u)) = qi. This says that if q’-’ - 1 < u < qi- 1 then the ramification group G, is e( U&).

We turn now to the upper numbering of the G,‘s. That is, we define a function 4 = &F/R, corresponding to the extension K;, which satisfies the conditions (a) to (d) above. Namely,

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LOCAL CLASS FIELD THEORY 157

Then G” = GU with v = 4(u). The graph of d(u) is shown in Figure 1. Ifq’-‘-1 < u < q’-l,then@(u) = l/(q’-q’-l)and(U,: Ui)=q’-q’-‘.

So if i- 1 < v < i, then G” = 0(&) for o < n.

The general case

(2) Verzjication in the general case. Having proved Theorem 1 for K; it follows for K, = u K,” by taking

projective limits. Hence also for K,. K,,, since both extensions have the same intertia subgroup. Since K, . K,, is the maximal abelian extension, the result is true in general.

This concludes the proof of Theorem 1.

COROLLARY. The jumps in the jiltration (G”} of G occur only for integral values of v.

Proof. This follows from Theorem 1, since it is trivial for filtrations of U, and Theorem 1 transforms one into the other.

[This result is in fact true for any field which is complete with respect to a discrete valuation and which has perfect residue field (theorem of Hasse-Arf), cf. “Corps Locaux”, Chap. IV, V.]

4.2 Abelian Conductors Let L/K be a finite extension and let 8 : K* --, G(L/K) be the corresponding

reciprocity map. There is a smallest number n such that &Vi) = 0. This number n is called the conductor of the extension LJK and is denoted by fWK)-

PROPOSITION 1. Let c be the largest integer such that the ramification group G, is not trivial. Then f(L/K) = $LIR(~)+ 1.

Proof. This is a trivial consequence of Theorem 1 and the fact that the upper numbering is obtained by applying 4.

Now let L/K be an arbitrary galois extension. Let x : G + C * be a one- dimensional character and let L, be the subfield of L corresponding to Ker (x). The field Lx is a cyclic extension of K and f(LJK) is called the conductor of x and is denoted by f(x).

PROPOSITION 2. Let {G,} be the ramzjication subgroups of G = G(L/K) and write gr = Card (Gi). Then

(XI> = izo E Cl- X(Gi)) where x(GJ = g; 1 c x(s) is the “mean value” of x on G,.

SGG,

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Proof. We have x(GJ = 1 if x is trivial on Gi (that is, equal to 1 every- where) and x(G,) = 0 if x is non-trivial on Gi. Hence (the reader is referred to “Corps Locaux”, Chaps. IV and VI for the details)

where cX is the largest number such that the restriction x]Gfz # 1. Now f(x) = f(L,/K) is equal to &,,x(c) + 1, where c is defined as m Prop. 1 for the extension LJK. Since &,x is transitive, it suffices to show that c = ~L,Lz(cX) and this is a consequence of Herbrand’s theorem (§ 4.1).

4.3 Artin’s Conductors

Let L/K be a finite galois extension with Galois group G = G(L/K). Let x be a character of G (that is, an integral combination of irreducible characters). Artin defined the conductor of x as the number

for> = igo E (X(l) - X(GJ)*

If x is irreducible of degree 1, this j’(x) coincides with the previous f(x). We define Artin’s character a, as follows. For s E G, set

u&s) = -f. iG(S) ifs#l

adl) = f,J GAS).

Here f is the residue degree [I: k] (not to be confused with the conductor!), and io is the function defined above.

PROPOSITION 3. Let g = Card (G). Then

f(x) = (% xl = ~s~ox(s)aGo.

Proof. The proof depends on summation on successive differences GrGi+l and is left as an exercise. (See “Corps Locaux”, Chap. VI, 9 2.)

PROPOSITION 4. (a) Let K c L’ c L be a tower of galois extensions, let x’ be a character of G(L’/K) and let x be the corresponding character of G(L/K). nenfol) = f&9.

(b) Let K c K’ c L and let JI be a character of G(L/K’) and let II/* be the corresponding induced character of G(L/K). Then

m*> = w . %@K~,K) +fx*,R: x4% where fKpIK is the residue degree of K’IK and bKpIK is the discriminant of K’/K.

Proof. The proof depends on properties of the io function and on the relation between the different and the discriminant, and can be found in “Corps Locaux”, Chap. VI.

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LOCAL CLASSFELDTHEORY 159

THEOREM 2 (Artin). Let x be the character of [I representation of G. Then f(x> is a positive integer.

Proof. Let x be the character of the rational representation M of G. It follows from representation theory that

x(1) = dimM and

x(GJ = dim MO’. Thus in

each term is positive (2 0) and so f 01) 2 0. It remains to be proved that f(& is an integer. According to a theorem

of Brauer, x can be written x = c m i e: where ml E 2 and $: is induced by a character $r of degree 1 of a subgroup H, of G.

Hence, since f($:> = ICI I(l)vK(bRpIK) +fKpIK .f($J, f($:> is an integer provided that f($J is. But since $r has degree 1, f($J may be interpreted as an abelian conductor and so is obviously an integer. This proves Theorem 2.

4.4 Global Conductors

Let L/K be a finite galois extension of number fields and let G = G(L/K) be the Galois group. If x is a character of G, then we define an ideal f(x) of K, the conductor of x, as follows. Let p be a prime ideal in K and choose a prime ideal !J3 in L which divides p. Let Gp = G(L.s,/KJ be the corre- sponding decomposition subgroup. Let f(x, p) be the Artin conductor of the restriction of x to G+, as defined above. We have f(x, p) = 0 when p is unramified. The ideal

f(x) = g pf’(X*P)

is called the (global) conductor of x. In this notation, Prop. 4 gives:

PROPOSITION 5. Let K’/K be a sub-extension of L/K. Let 9 be a character of H = G(L/K’) and let @” be the induced character of G(L/K). Then

fob*) = e:Ii. hr,dfW where bRnIIL is the discriminant of K’IK.

We apply Prop. 5 to the case $ = 1 and we denote the induced character #* by sc,, (it corresponds to the permutation representation of G/H). Since f($) = (1) we obtain:

COROLLARY. We have f(sc,a, L/K) = bK.,*

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In the case H = 1 we have s,,, = ro, the character of the regular repre- sentation of G, and the corollary reads

bL,K = n f w”‘, n

where x runs through the set of irreducible characters of G. This is the “Fiihrerdiskriminantenproduktformel” of Artin and Hasse, which was first proved by analytical methods (L functions). In the abelian case it reads :

b L/K =x JI,xx~*

In the quadratic case it reduces to the fact that the discriminant is equal to the conductor.

4.5 Artin’s Representation

We return to the local case. THEOREM 3. Let L/K be aJinite galois extension of localjelds with Galois

group G. Let a, be the Artin character of G dejined above (cJ 3 4.3). Then ao is the character of a complex linear representation of G called “the” Artin representation.

Proof. The character a, takes the same values on conjugate elements and so is a class function. It follows that ao is a combination c, m,x, with complex coefficients mX, of the irreducible characters x. Since

mx = caG9 x) =fo1),

we know (Prop. 3 and Theorem 2) that m, is a positive integer. Hence the result.

Now let V, be an irreducible representation corresponding to x. We can define Artin’s representation Ao by:

AC = ~fol)- v,,

where the summation is over all irreducible characters x. Remark. This construction of Ao is rather artificial. Weil has posed the

problem of finding a “natural” Ao.

THEOREM 4. Let I be a prime number not equal to the residue characteristic. Then the Artin representation can be realized over Q,.

Proof. See J-P. Serre, Annals of Mathematics, 12 (1960), 406-420, or “Introduction a la theorie de Brauer” Uminaire I.H.E.S., 1965166.

Examples exist where the Artin representation cannot be realized over Q, R or Qp, where p is the residue characteristic. This suggests that there is no trivial definition of the Artin representation.

Assume now that L/K is totally ramified. Let u, = r,- 1; we have

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LOCAL CLASS FIELD THEORY 161

Q(S) = -1 ifs # 1 and U&Y) = Card (G)-1 ifs = 1. Now a, = u,+b, where b, is a character of some representation.

Note: a, = uo if and only if L/K is tamely ramified. So b, is a measure of how wild the ramification is.

THEOREM 5. Let I be a prime number not equal to the residue characteristic. Then ‘there exists a finitely generated, projective Z,[G] module Bo,, with character b, and this module is unique up to isomorphism.

Proof. This follows from a theorem of Swan, “Topology”, 2 (1963), Theorem 5, combined with Theorem 4 above and the remark that b,(s) = 0 when the order of s is divisible by 1. [See also the I.H.E.S. seminar quoted above.]

For applications of Theorem 5 to the construction of invariants of finite G-modules, see M. Raynaud, “S&m. Bourbaki”, 1964/65, expose 286. These invariants play an important role in the functional equation of the zeta functions of curves.

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CHAPTER VII

Global Class Field Theory

J. T. TATE

1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12.

Action of the Galois Group on Primes and Completions . Frobenius Automorphisms . . . . . . . . . . Artin’s Reciprocity Law Chevalley’s Interpretation b; Ideles 1 : 1 : : : : Statement of the Main Theorems on Abelian Extensions . Relation between Global and Local Artin Maps . . . . Cohomology of Ideles . . . . . . . . . . . Cohomology of Idele Classes (I), The First Inequality . . Cohomology of Idele Classes (II), The Second Inequality . Proof of the Reciprocity Law . Cohomology of Idele Classes (III), ‘The Fundamental Class’ Proof of the Existence Theorem .’ . . . . . . .

. . 163

. . 164

. . 165

. . 169 . . 172 . . 174 . . 176 . . 177 . . 180 . . 187 . . 193 . . 201

List of symbols . . . . . . . . . . . . . . . . . 203

Throughout these lectures, K will be a global field, as has been defined in Chapter II, § 12. We treat the number field case completely, but in the function field case there is one big gap in our proofs, in that the second inequality and the accompanying key lemma for the existence theorem are proved only for extensions of degree prime to the characteristic. (The reader interested in filling the gap can consult the Artin-Tate notes,? pp. 29-38.)

As in Local Class Field theory,,there are several aspe : (1) The cohomo- logy theory of Galois extensi extensions of K. (3) Gserie oqd

To s of K. (2) The determinat’ n of the abelian analysis.

We will discuss the first two, leaving the third to Heilbronn (Chapter VIII), except for a few remarks.

Sections l-6 constitute a statement and discussion of the reciprocity law and the main theorems on abelian extensions, with no mention of cohomo- logy. We hope that this preliminary discussion will serve both as orientation and bait for the reader. In sections 7-12 we give the main proofs, based on the determination of the Galois cohomology of idele classes and the Brauer group of K.

This chapter is strictly limited to the central theorems. In the exercises at the end of the book the reader will find a few concrete examples and further

t Harvard, Dept. of Mathematics. 1961. 162

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GLOBAL CLASSFIELD THEORY 163

results. There are some references to recent literature scattered in the text but we have made no attempt to give a systematic bibliography. A list of symbols used in this chapter is given at the end.

1. Action of the Galois Group on Primes and Completions

Let L be a finite Galois extension field of Kwith Galois group G = G(L/K).

1.1. First of all we have a few lines on our notation and language. If a E L and a E G, then the action of a on a will be denoted by aa or a*, according to the situation. If r E G we use the convention a(zu) = (ar)a and so (a’)” = &Jr).

A prime is an equivalence class of valuations, or a normalized valuation, of K; we usually denote a prime by the letter u or w. A prime may be either archimedean or discrete; if v is discrete we write 0, for its valuation ring and !& for the maximal ideal of 0,. We reserve the symbol ‘$3 for prime ideals..

Let w be a prime of L, then with the definition ]&,, = ]a-‘& it follows that aw is another prime of L and a(rw) = (ar)w. If Dw is the valuation ring of w, then an,,, = Daw. A Cauchy sequence for w, acted on by a, gives a Cauchy sequence for aw and conversely a Cauchy sequence for aw, acted on by a-i, gives a Cauchy sequence for w; so 0 induces by continuity an isomorphism a,,, : L, ‘$I L,, of the completions of L with respect to the primes w and aw respectively. If w is over the prime o of K so is aw and this map is a K,-isomorphism. Clearly, a, o r,,, = (ar),.

The decomposition group G, of w is the subgroup

G,={aoGIaw=w} , of G. Note that

(1) G,, = {a E Glarw = rw> = rG,r-’ ,y” _- , “\ thus the decomposition group of w is determined p to conjugacy by the’\ prime v. By what we have said a is a K,-aut 9ff orphism of L,,, if a E G, and so we have an injection i of G, into G(L,/K,).

1.2. PROPOSITION. (i) LJK, is Galois and the injection i : G, + G(L,/K,) is an isomorphism.

(ii) If w and w’ are two primes of L over the prime v of K, there exists a a E G such that aw = w’.

Proof. Letting [X] denote the cardinality of a set X, we have

CC,.1 G W,/~,) G CL, : &I, and these inequalities are equalities if and only if (i) is true. Let r = (G : G,) and let (a,), 1 < i < r, be a system of representatives for the cosets alGw of G, in G. Put w1 = aiw for 1 < i < r.. These are distinct primes of L lying over v; let w1 for r+ 1 < i < s be the remaining such, if any. Then

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164 J. T. TATE

[G] = r[Gw] = i~l[G,,J <&CL,,., : KJ G i&[L-‘ : KU] = [L : K] = cc]1

so we have equality throughout. Hence r = s, which implies (ii) and [G,] = [L, : K,], which implies (i). [The fact that the sum of the local degrees is equal to the global degree follows from the bijectivity of the

map L 63 K, + fi L,, on taking dimensions over K,; see Chapter II 5 10.

The surjzctivity,‘w&ch is all we have used, is an easy consequence of the weak approximation theorem.]

Write !U& for the set of primes of K. Then, since %XL + ‘9JIK is surjective (every prime D of K can be extended to a prime w of L), Proposition 1.2 amounts to saying that 5?& N 9&/G, i.e. the primes of K are in 1-l corre- spondence with the orbits under G of the primes of L, and for each prime w of L, its stabilizer G, is isomorphic to the Galois group of the corresponding local field extension L,/K,.

2. Frobenius Automorphisms

2.1. Suppose that w is a discrete, unramified prime of L over the prime v of K. (This is true of “almost all” primes V, w, i.e. for all but finitely many.) Then

(1) G = G, N- WXG) = W(wYWN, where k(v) (resp. k(w)) denotes the residue class field of K (resp. L) with respect to 2) (resp. w). Since these residue class fields are finite the Galois group G(k(~)/&)) is cyclic with a canonical generator,

F: XHXNU, --- ~-~--- where NV = [k(v)] is the “absolute norm”. Henee we see from (1) that t@ere is a unique element trw E G, which is characterized by the.property

ew E G, and saw = !aN” (mod ‘p,)

for all a E D,. This automorphism ow is called the Frobenius automorphism associated with the prime w. An immediate consequence of this definition is

2.2. PROPOSITION

a,, = z-1o,T.

Thus the Frobenius automorphism is determined by o up to conjugacy and we define

F&U) = (conjugacy class of ew, w over v) = (the set of a,‘s for w over u).

If S is a finite set of primes of K containing the archimedean primes and the primes ramified in the extension L/K, then FLIK is a map of 1111, - S into the conjugacy classes of G(L/K).

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GLOBAL CLASS FIELD THEORY 165

2.3. PROPOSITION. Let (T E F&U) have order f, so that it generates the sub- group <o> = (1, o,. . ., of-l }. Then in L, v spZits into [G : <o> ] factors, each of degree f = [k(w) : k(v)]. In particular, v splits completely zy and only if FLJ&) = 1, the identity element of G.

2.4. Remarks. This proposition tells us that knowledge of FLIR gives the decomposition law for unramified primes, and more, since it chooses a definite generator for the decomposition group.

Since FL,= is a function to classes of G, to know FLIK it is enough to know ;((F&u)) for all characters x of G. Accordingly, Artin was led to define his non-abelian L-series in terms of x(F), by means of which one can prove the fundamental Tchebotarev (= tiebotarev) Density Theorem: Let %? be a conjugacy class in G; the primes v with F(u) = %’ have density [%‘]/[G]. In particular, for each conjugacy class %‘, there exists an infinite number of primes o of K such that FLIK(u) = W.

In the cyclotomic case Tchebotarev’s theorem is equivalent to the Dirichlet theorem on primes in arithmetic progressions (see 3.4 below).

From Tchebotarev’s theorem it follows almost trivially that a finite Galois extension L of K is uniquely determined (up to isomorphism) by the set Spl (L/K) of primes of K which split completely in L (cf. exercise 6). Unfortunately one knows no way to characterize directly, in terms of the arithmetic of K itself, those sets T of primes of K which are of the form Spl (L/K), except in case L is abelian. The decomposition law for abelian extensions, together with the complete classification of such extensions, is given by the main theorem below (9 5); but no such theorem is known for non-abelian extensions, i.e. “non-abelian class field theory” does not exist. From the abelian theory one can derive decomposition laws of sorts for some soluble extensions (cf. Exercise 2) but this is not what is sought. Recently Shimura (“A reciprocity law in non-solvable extensions”, Crelle’s Journal, 221 (1966), 209-220) has given an explicit decomposition law for certain non-soluble extensions obtained by adjoining to Q the points of order I on a certain elliptic curve. The idea is to relate the behaviour of primes in

,/ I

those extensions to the zeta-function of the curve, and to identify that ’ zeta-function with a modular function, the coefficient of whose q-expansion can be calculated explicitly. The degree of generality of such examples, and whether they will point the way to a general theory, is unclear; but at any rate they are there to test hypotheses against.

3. Artin’s Reciprocity Law

3.1. First of all we give some notation. S will usually denote a finite set of primes of K including all the archimedean primes. If we are considering a particular finite extension L/K, then S will also include the primes of K

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166 J. T. TATE

ramified in L. We will denote by Is the free abelian group on the elements of %Rx-S (a subgroup of the group of ideals, see Chapter II § 17).

Assume now that L/K is a finite abelian .extension. Then the comugacy classes of G = G(L/K) are single elements and so l$x is a map from !32x - S into G. By linearity we can extend this to a homomorphism (to be denoted by FLIR also) of 1’ into G, putting

where the n,, are integers and n, = 0 except for a finite number of the V. The first proposition of this section concerns the change in the map

FLlr when the fields are changed. Suppose that L'/K' and L/K are abelian field extensions with Galois group G’ and G respectively, such that L’ 3 L and K' 3 K, and let 8 be the natural map G’ + G (every automorphism of L'IKI induces one of L/K). Let S denote a finite set of primes of K in- cluding the archimedean ones and those primes ramified in L' and let S’ be the set of primes of K' above those in S. Then

3.2. PROPOSITION. The diagram

IS’ h*/r*

-- G’

commutes, where N denotes “norm”.

Proof. By linearity, it is clear that it is enough to check that

~&~,x:‘V~ = FL,K(&,R ~‘1 for an arbitrary prime u’ of K' such that u’ 4 S’. Let NK~,xuu) = fu, where u is the prime of K below u’; thus f = [k(u’) : k(u)]. Let c’ = FLp,Rr(u') and o = F&u). We must show &a’) = 0’. Now Q and c1 are determined by their effect on the residue fields. Let w’ be a prime of L’ above u’ and let w be the prime of L below w’. For x E k(w) c k(w’) we have

f’ = XNu’ = -JNv)f = a/ , as required.

If a E K* (i.e. a is a non-zero element of K), then we write

W = J& us

where n, = u(a) for all u 4 S; thus (a)” is an element of Is. We can now state the reciprocity law in its crudest form

3.3. RECIPROCITY LAW (Crude form). If LJK is a finite abelian extension, and S is the set of primes of K consisting of the archimedean ones and those

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GLOBAL CLASSFIELD THEORY 167

ramified in L, then there exists E > 0 such that if a E K* and [a- 11, < e for aN I) E S, then F((a)‘) = 1.

In words, if a E K* is sufficiently near to 1 at all primes in a large enough set S, then F((a)s) = gsF(u)“@) = 1.

In the number field case the subgroup (K:)” is open in K* for all n > 0, and we claim that the condition ]a- I], < E can be replaced by a E (K:) for v E S, where n = [L : K]. Indeed if the latter is satisfied then, by the weak approximation theorem, there exists b E K* such that lab-“- 11, < e for all v E S, and then

F((a)S) = F((b”ab-“)s) = F((b)S)“F((ab-“)S) = 1.

Thus, in the case of number fields, although the set S depends on L, the neighbourhoods of 1 at the primes of S depend only on the degree n of L over K. In particular, for archimedean primes there is no condition needed unless v is real and n is even, in which case the condition a > 0 in K, is sufficient.

Using the approximation theorem in L instead of K, one can replace the condition “a is a local [L : K]-th power in S” by “a is a local norm from L to Kin S”, but for that we shall use the technique of idtles (see 4.4 and 6.4 below). The shift in emphasis from n-th powers to norms was decisive, and is due to Hilbert.

3.4. Example. The reciprocity law for cyclotomic extensions. This reciprocity law may be verified directly in the cyclotdmic case k = Q, L = Q(n, where [ is a primitive m-th root of unity. This particular result will be used later in one of our proofs of the general result (see 5 10, below) so we give some details. In the rational case it is conventional to denote primes by p and the associated valuations by up. The set Swill consist of the archimedean prime and those primes p which divide m (see Chapter III). For p 4 S and w above p, the powers of C have distinct images in the residue class field k(w). so we have

PROPOSI~ON.

F(v,)C = cp for all p $ S. From this we deduce

COROLLARY. If a E Z, a > 0 and (a, m) = 1, then F((a)sil: = c.

Consequently, if a is a positive rational number with ]a- 11, < ]m], for all p E S, then a is a p-adic integer for all p dividing m, and we can write a = b/c with (b, m) = (c, m) = 1, and b z c (mod m); hence lb,= c’ and so F((b)‘)I; = [* = Cc = F((c)‘)C. By linearity this gives F((a)‘)C = f so that‘ F((a)‘) = 1.

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168 J. T. TATE

For another example of an explicit description of FLIR, and for the con- nection between Artin’s general reciprocity law and the classical quadratic reciprocity law, see exercise 1.

3.5. Remark. The cyclotomic case was easy because one can use the roots of unity to “keep book” on the effect of the F(u)‘s for variable u. A similar direct proof works for abelian extensions of complex quadratic fields using division points and modular invariants of elliptic curves with complex multiplication, instead of roots of unity. In the general case no such proof is known (cf. the 12th problem of Hilbert), although Shimura and Taniyama and Weil have made a great contribution, using abelian varieties instead of elliptic curves. (See Shimura-Taniyama, “Complex Multiplication of Abelian Varieties and its Applications to Number Theory”, Publ. Math. Sot. Japan, No. 6, 1961, and more recently Shimura, “On the Field of Definition for a Field of Automorphic Functions: II”, Annals of Math, 81 (1965), 124-165.) The proof of the reciprocity law in the general case is very indirect, and can fairly be described as showing that the law holds “because it could not be otherwise”.

3.6. Remark. In the function field case, as Lang has shown ((‘Sur les series L dune variete algebrique”, RUN. Sot. Math. Fr. 84 (1956), 385407), the reciprocity law relates to a geometric theorem about the field K = k(C) of a curve C. Serre has carried out in detail the program initiated by Lang. In his book, “Groupes algebriques et corps de classes”, Hermann, Paris (1959) an analogue of the reciprocity law is described as follows. Let f : C --) G be a rational map of a non-singular curve C into a commutative algebraic group G; let S be the finite set of points of C where f is not regular. Then f induces a homomorphism of the group of divisors Is into G and

THEOREM. If #J E K takes the value 1 to a high order at each point of S, then f((+)) = 1.

--~-

This theorem is due to Rosenlicht, and independently, but later, Serre. It was Serre and Lang who applied it to class field theory.

3.7. Definition. Let K be a global field, S be a finite set of primes of K including all the archimedean ones and G a commutative topological group. A homomorphism 4 : Is + G is said to be admissible if for each neighbour- hood N of the identity element 1 of G there exists E > 0 such that &(a)“) E N whenever a E K* and la- 1 Iv < E for all v E S.

If G is a discrete group, we simply take N to be (1). Thus

3.8. Reformulation of the Reciprocity law: FLIK is admissible.

In this context the finite group G = G(L/K) is discrete. If G is the circle group, then 4 is admissible if and only if it is a Grijssencharakter (if it maps into a finite subgroup, it is a Dirichlet character). Dirichlet and Hecke

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GLOBAL CLASS FIELD THEORY 169

formed their L-series with such characters; Artin was originally forced to produce his reciprocity law in order to show that in the abelian case his L-series defined in terms of characters of the Galois group were really Weber L-series, in other words that #(u)) was admissible for each linear character x of the abelian Galois group.

4. Chevalley’s Interpretation by Id&es

The set of elements of the idele group JR (see Chapter II 5 16) which have the value 1 at all the v-th components, v E S, is denoted by J,“. If x E JR it has a non-unit component at only a finite number of v-components; if the (additive) valuation of the v-th component X, of x is n, E Z we write

(x)S =“~sn”u E IS.

4.1. PROPOSITION. Let K and S be as before, G be a complete commutative topoIogica1 group and 4 an admissible homomorphism of Is into G.

Then there exists a unique homomorphism $ of JR + G such that

(i) $ is continuous; (ii) +(K*) = 1 ; (iii) I&) = 4((x)“) for all x E J&

Conversely, if $ is a continuous homomorphism of JR + G such that $(K*) = 1, then J/ comes from some admissible pair S, 4 as defined above, provided there exists a neighbourhood of 1 in G in which (1) is the only sub- group.

Remark. It is clear that if such a J/ exists then it induces a continuous homomorphism of the idele class group C, N J,/K* into G. This induced homomorphism will also be denoted by I,+. Furthermore, if such a 1,4 exists for a given 4 and S, then by the unicity statement, it is unchanged if S is enlarged to a bigger set S’ and 4 replaced by its restriction 4’ to Is’ c Is. Similarly, two 4’s on Is which coincide on I” for some finite S’ 3 S are actually equal on Is (cf. Exercise 7).

For applications G can be thought of as a discrete group or the circle group.

Proof. Suppose we have an admissible map 4 : Is + G. If such a $ were to exist, for any a E K* and x E JK we would have

ti(x> = tW4 = 4V@x>dJ/((~&),

where (ax)l is the idele with the same v-component as ux for all v E S and value 1 elsewhere and (a~)~ is the idele with the same u-component as ax for all v 4 S and whose v-component is 1 at all v E S (thus (ux)~ E JE). By the (weak) approximation theorem (see Chapter II, 96) we can find a sequence

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170 J. T. TATB

(4) of elements a, E K*, such that a, -+ x-l, as n -+ co, at all o E S. Then

tw = :; 4w(a, x)1). Ma, 47 = ii+: md>.

Hence given I$ we define a function t+Q by

(1) Hx) = Lt 44% 4”).

As n, m -+ co we have a,Ja,,, + 1 at all primes v E S, and consequently

4((%4S) S

f$((a,x)s) = : + l 4 >>

in G, because C$ is admissible. Thus the limit exists, since G is complete, and the limit is independent of the sequence {a,}, because it exists for all such sequences. Moreover $ is continuous. If the components of x are units at primes Y # S, then we have $(x) = lim &(u.)~); and if in addition the components of x are sufficiently close to 1 at primes u E S, then so will be those of a, for large n, and by admissibility, 4((u,)“) will be close to 1 in G. The last two conditions (ii) and (iii) are trivially verified by taking a, = x-l and 1 for all n respectively.

Now suppose we are given a continuous homomorphism 1,9 : JR + G such that J/(K*) = 1. We will find a set S so that (a) the restriction of tj to JR comes from a function on 1’ and (b) if we call this function 4, then C$ is admissible.

For any finite set S of primes of K let Us be the set of idbles in JK for which the u-component is 1 at all v E S and a unit of K, for u 4: S. By taking S arbitrarily large we can make Us an arbitrarily small neighbourhood of the identity of JR. If N is a neighbourhood of (1) we can choose S sufficiently large so that I,+( Us) -C N, since 1,4 is continuous. Then taking iV small enough, we see that $(U’) = (1) for some set S by the “no-small-subgroup” hypo- thesis. We choose such a set S. Now Ji/U’ is canonically isomorphic to 1’ and so II/, when restricted to J,“, induces a continuous homomorphism 4 of Is into G.

It remains to verify that 4 is admissible; in words, given a neighbourhood iV, then ~((a)‘) E N whenever a E K* is near enough to 1 at all o E S. But in this case (a>” is near to a in JR and so by continuity tj((a)‘) is near to

l +(a), which is 1 since a E K*.

4.2. COROLLARY. The reciprocity law holds for a finite ubelian extension L of K fund only if there exists a continuous homomorphism 1,4 of JR + G(L/K) such that

(i) 1+4 is continuous. (ii) J/(K*) = 1.

(iii> 444 = &~K(W) f or a N x E J,“, where S consists of the archimedean primes of K and those ramified in L.

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GLOBAL CLASS FIELD THEORY 171

Such a map $ = rtQLIK whose existence we have just postulated is called the Artin map associated with the extension L/K. It has been defined as a map JR + G(L/K); but since it acts trivially on K* it may be viewed as a map of the idble class group C, = J,/K* into G(L/K).

The reciprocity law for finite abelian extensions will be proved later (see 5 10). In the meantime certain propositions will be proved and remarks made which depend on its validity. Suppose that L’IK’ and L/K are abelian field extensions with Galois groups G’ and G respectively and that L’ 3 L, K’ I> K. Let 8 be the natural map G’ + G. Then in terms of idtles and Artin maps Proposition 3@becomes

4.3. PROPOSITION. If the rec$rocity law holds for L/K and L’IK’, then !h/K’

JK - G’

is a commutative diagram. Proof. Let S be a large finite set of primes of K, and S’ the set of primes

of K’ above S. We have then a diagram

The non-rectangular parallelograms are commutative by the compatibility of ideal and idele norms, and by Proposition 3.2. The triangles are commuta- tive by (4.2)(iii). Thus the rectangle is commutative, i.e. the restrictions of JI NIYJK L/K 0 and 8 0 JIL.,x. to JR ‘1 coincide. But those two homomorphisms take the value 1 on principal ideles by 4.2 (ii), so they coincide on (K’)*J$, which is a dense subset of JK’ by the weak approximation theorem (Chapter II, 5 6). Since the two homomorphisms are continuous, they coincide on all of J,* which is what we wished to prove.

For proving Proposition 6.2 below, which is in turn needed for the first of our two proofs of the reciprocity law in $10.4, we need the following

VARIANT. Suppose L/K satisfies the reciprocity law, and K c iU c L. Then rl’,,rW~,r Jd = W/M)-

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172 J. T. TATE

Consider diagram (2) with L’ = L, K’ = M, but with the upper hori- zontal arrow $L,,KK’ = @L,M removed. It shows that

~,KWM,RJ%) = G’ = W/W. Consequently the same is true with Js replaced by M*J& and since that set is dense in JM we are done.

4.4. COROLLARY. If the reciprocity law holdsfor L/K, then

~,RWL,RJL) = 1. It follows that I,~~,~(K*~V~,~J~) = 1; the next theorem states (among other

things) that K*NLIKJL is the kernel of $L,R.

5. Statement of the Main Theorems on Abelian Extensions

5.1. MAIN THEOREM ON ABELIAN EXTENSIONS (Takagi-Artin).

(A) Every abelian extension L/K satisfies the reciprocity law (i.e. there is an Artin map $&.

(B) The Artin map I+?~,~ is surjective with kernel K*NLIK(JL) and hence induces an isomorphism of CK/NLIK(CL) on to G(L/K).

(C) IfM 2 L r> K are abelian extensions, then the diagram

SMIK W%,K CM -- G(M/K)

j, ,e

commutes (where 0 is the usual map and j is the natural surjective map which exists because NMIKCM c NLIKCL).

(D) (Existence Theorem.) For every open subgroup N of finite index in C, there exists a unique abelian extension L/K (in a fixed algebraic closure of K) such that NLIKCL = N.

The subgroups N of(D) are called Norm groups, and the abelian extension L such that NLIRCL = N is called the class field belonging to N. In the number field case every open subgroup of C, is of finite index in C,.

5.2. A certain amount of this theorem may be deduced readily from the rest. First, given (A) and (B), then (C) is a special case of 4.3 (put K’ = K and L’ = A4).

5.3. Secondly, the uniqueness, though not the existence, of the correspon- dence given in (D) follows from the rest. Given the existence, let L and L’ be two finite abelian extensions of K in a fixed algebraic closure of K and let M be the compositum of L and L’ (which is again a finite abelian extension of K). Now consider the commutative diagram above, under (C). Since the horizon- tal arrows are isomorphisms (by (B)) we see that Ker 0 = G(M/L) is the iso-

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GLOBAL CLASS FIELD THEORY 173

morphic image, under I++~,~, of the group NLIKCL/NM,K CM. Thus L, as the fixed field of the group Ker 0, is uniquely determined as a subfield of M, by NL,,C,. Applying the same reasoning with L replaced by L’, we see that if NLfIRe CL, = NLIKCL, then L = L’.

For some special examples of class fields (Hilbert class fields) see exercise 3. For the functorial properties of the Artin map when the ground field K

is changed, see 11.5 below.

5.4. The commutative diagram of (C) allows us to pass to the inverse limit (see Chapter III, Ql), as L runs over all finite abelian extensions of K. We obtain a homomorphism

t,bK : CK + lim G&/K) II G(Kab/K), .L

where Kab is the maximal abelian extension of K; and then, by (D), G(Kab/K) N lim (&IN),

‘N where the limit is taken over all open subgroups N of finite index in C,. Thus we know the Galois groups of all abelian extensions of K from a knowledge of the idele class group of K. The nature of the homomorphism $R : C, + G(Kab/K) is somewhat different in the function field and number field cases. The facts, which are not hard to derive from the main theorem, but whose proofs we omit, are as follows:

5.5. Function Field Case. Here the map eg is injective and its image is the dense subgroup of G(Kab/K) consisting of those automorphisms whose restriction to the algebraic closure & of the field of constants k is simply an integer power of the Frobenius automorphism F; (see Artin-Tate notes, p. 76).

5.6. Number Theory Case. Here $K is surjective and its kernel is the con- nected component D, of C,. So we have obtained a canonical isomorphism CK/DK N G(Ksb/K).

However, as Weil has stressed (“Sur la theorie du corps de classes”, J. Math. Sot. Japan, 3, 1951), we really want a Galois-theoretic interpretation of the whole of C,. The connected component DK can be very complicated (see Artin-Tate, p. 8.2).

5.7. Example. Cyclotomic Fields. Consider Q-/Q, the maximal cyclotomic extension of Q. Let $ = lim Z/nZ; by the Chinese remainder theorem this

- is isomorphic to fl Z,, whir-e Z, is the ring of p-adic integers. 2 acts on

any abelian torsioi group (for ZlnZ operates on any abelian group whose exponent divides n) and the invertible elements of 2 are those in n UP,

P

where UP is the set ofp-adic units in Z,.

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174 J. T. TATE

Now consider the torsion group ,u consisting of all roots of unity. If [ E ,Y we can define r” for all u E n UP; u induces an automorphism on ,u.

The idble group JQ is isomorphic ti the direct product Q* x R*, x fl UP. (In

fact, if x = (xoo, x2, x3,. . . } E JQ we have x = a. {t, u2, us,. . . }, khere a = (sign x,) n pup(+) E Q*,

and where t > 0, and up E UP forp = G, 3,. . . ; moreover, this decomposition is unique, because 1 is the only positive rational number which is a p-adir unit for all primes p.) Hence Co is canonically isomorphic to RT x n UP,

so there is a map of Co onto n UP, which is the Galois group of the ma$mal P

cyclotomic extension. What in fact happens is the following.

then c@(X) = p-’ (this re If x E Co and x H u by this map,

su It is an easy exercise, starting from 3.4, and is independent of parts (B) and (D) of the main theorem). Thus the kernel of e is RT, which is the connected component D, of Co. We have now used up the whole of C,/D,; so if we grant part (B) of the main theorem, we see that every abelian extension of Q must already have appeared as a subfield of Q”“, and that part (D) holds for abelian extensions of Q.

The connected component R*, of Co is uninteresting; similarly, C, has an uninteresting connected component when K is complex quadratic, essentially because there is only one archimedean prime. It may well be that it is the connected component that prevents a simple proof of the reciprocity law in the general case.

6. Relation Between Global and Local Artin Maps We continue to deduce results on the assumption that the reciprocity law

(but not necessarily the whole main theorem of 3 5) is true for an abelian extension L/K.

6.1. For each prime u of K, we let K, denote the completion of K at 1). If L/K is a finite Galois extension, then the various completions L, with w over v are isomorphic. It is convenient to write L” for “any one of the com- pletions L,,, for w over u”, and we write G” = G(L”/K,) for the local Galois group, which we can identify with a decomposition subgroup of G (see 1.2). In the abelian case this subgroup is unique, i.e. independent of the choice of w.

Assume that L/K is abelian and that there is an Artin map J/L,K : Jg + G(L/K) = G.

For each prime v of K we have

Kc&J, -k=--a G, JU

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GLOBALCLASSFIELDTHEORY 175

where i,, is the mapping of an x E Kc onto the element of Jg whose v com- ponent is x and whose other components are 1, andj, is the projection onto the v-th component. Call $, = $=,k 0 i,,; so 9, : Kt + G. In fact

6.2. PROPOSITION. If K, c &! c L”, then J/u(N4,&*) c G(L”/d). In particular, JI,(Kz) c e, and JIu(NLv/K,(L”)*) = 1.

Proof. Let A4 = L n JX be the tied field of G(L”/&) in L, so that G(L/iU) is identified with G(L’/d) under our identification of the decomposition group with the local Galois group. Then 4 = M,,,, where w is a prime above v, and the diagram

L

“:&IT 7 JfM,K

K” -- JK

is commutative. By the ‘variant” of 4.3 we conclude that

VL(N,,K~~*) = $L,K%,K = W/M) = G(LDl-4,

6.3. We shall call I/I”: KC: -+ G” the local Artin homomorphism, or by its classical name: norm residue homomorphism. If x = (x,) E Jg, then we have

and consequently, by continuity, we have

(this product is actually finite since if x, is a v-unit and v is not ramified, then it is a norm of L”/K,). Thus knowledge of all the local Artin maps $, is equivalent to knowledge of the global Artin map eLIR. Classically, the local maps Ic/, were studied via the global theory and, in particular, were shown to depend only on the local extension L”/K,, and not on the global extension L/K from which they were derived. Nowadays one reverses the procedure, giving fist a purely local construction (cf. Chapter VI) of maps 8, : K: + G, = G(L”/K,). We will take these maps 8, from Serre and show that n 8, satisfies the characterizing properties for II/, in particular,

that n 6,(u) = “1 for all u E K* (see § 10).

The” local theory tells us that the Main Theorem of 5.1 is true locally if we replace C, by K:, r/j by $, and G(L/K) by G(L”/K,). In particular

K:/NL”* 2: G(E’/K,)

and-in this isomorphism the ramification groups correspond to the standard filtration of K:/NL”*. Going back to the global theory we get a complete

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176 J. T. TATE

description of prime decomposition in terms of idele classes, even in the ramified case.

For the question of abelian and cyclic extensions with given local behaviour and the Grunwald-Wang theorem, see Artin-Tate notes Chapter 10, and Wang, “On Grunwald’s Theorem”, Annals, 51 (1950), pp. 471-484.

6.4. We can now give an apparently stronger statement of the reciprocity law formulated in 3.3.

RECIPROCITY LAW (Strong form). Let L/K be abelian, and let S consist of the archimedean primes of K and those ramified in L. If an element a E K+ is a norm from L” for all v E S, then FLIK((a)‘) = 1.

For ifj”(a) is a norm for v E S we can write j”(a) = iVLYIR,(b”) for some b” EL”. Then by Corollary 4.2

1 = $((a)“>. v~shM4) = FL,K((~)~). v~s$VWLY~dbv~~ = FLlK((4S>9

by 6.2. For the concrete description of the local Artin maps $, by means of the

norm residue symbols (a, b)” in case of Kummer extensions, and the appli- cation to the general n-th power reciprocity law, see exercise 2.

7. Cohomology of Id&s

7.1. L/K is a finite Galois extension (not necessarily abelian) with Galois group G. Write AL for the addle ring of L; then JL is the group of invertible elements in AL = L OKAK, and G acts on L BKAK by ~JHQ 0 1; so G acts on JL.

However, we want to look at the action of G on the Cartesian product structure of JL. Suppose x E JL, then x = (x,), where w runs through !&; 0 E G induces 6, : L, --+ L,, (see 1.1) and (ax),, = c,,,x,, that is, the diagrams

L;L, L*,, an-

L*

5” -Ltw iv+ 1 ~ i-W lwl ~ ye

JL -- JL JL - JL

commute. (Note that the image of L$ in JL is not a G-invariant subgroup; the smallest such subgroup containing Lz is wg LE.)

7.2. PROPOSITION. Let v E !?& and w” E 91L with w” over v. Then there are mutually inverse isomorphisms

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GLOBAL CLASSFTELD THEORY 177

and

H’(G 2 Uw) + W%w Uwo>,

where U, denotes the group of units in L,. The assertions remain valid when W is replaced by A’. The proof is immediate from Shapiro’s lemma (see Chapter IV, Q 4) in

view of Proposition 1.2 of $ 1. Thus the cohomology groups H’(G,, Lz) are canonically isomorphic for

all w over v, so it is permissible to use the notation H’(G”, (L”)*) for any one of these.

7.3. PROPOSITION. (a) JR N Jf, the group of id2les of L left fixed by all elements of G.

@I B’(G JL) =” &~W’, WI*),

where u denotes the direct sum. Proof. (a) is clear from Chapter II, 0 19. To prove (b) we observe that

and S is a finite set of primes of K containing all the ramified primes in L/K and the archimedean primes. The limit is taken over an increasing sequence of S with lim S = !l.& The cohomology of finite groups commutes with direct limits, and any cohomology theory commutes with products, so it is enough to look at the cohomology of the various parts. By 7.2 and Chapter VI, 5 1.4, n n U

v#S Lb 9 has trivial cohomology if S contains all the

ramified primes. Hence

B’(G, JL, s> = ~Jj-,Xf”G”, (Lu)*),

by 7.2. Let S -+ %Xx; we find fi’(G, Jd E u &(G”, (L”)*).

7.4. COROLLARY.

(a) H’(G, Jd = 0.

(b) H’(G, JJ II v (iZ/Z), where n, = [Lo : K,].

Here, the determination of H’ is just Hilbert’s “Theorem 90” for the local fields (see Chapter V, 0 2.6 and Chapter VI, 0 1.4). The second part follows from the determination of the Brauer group of K, in Chapter VI, 0 1.6.

8. Cohomology of Idkle Classes (I), The First Inequality We recollect the exact sequence 0 + L* + JL + C, + 0. The action of

G on CL is that induced by its action on J,.

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178 J. T. TATE

8.1. PROPOSITION. C, N C”,.

Proof. The above exact sequence gives rise to the homology sequence

O-B H’(G, L*) -, H’(G, JL) -+ H’(G, CL) + H’(G, L*), that is

O+K*-+J,-+C,G-+O.

8.2. Remark. Our object in the abelian case is to define

+ L/K: CKINL,KCL-+WIK) = G.

By the Proposition above CK/NLIKCL = fi’(G, CL), and on the other hand G = Ae2(G, Z). Comparison with Chapter VI, 9 2.1, suggests that the global theorem we want to prove about the cohomology of CL is essentially the same as the local theorem Serre proves about the cohomology of L*. This is in fact the case. Abstracting the common features, one gets the general notion of a “class formation”. [cf. the Artin-Tate notes.]

We recollect that if G is cyclic and A a G-module the Herbrand quotient is defined by h(G, A) = [H2(G, A)]/[H’(G, A)] if both these cardinalities [H’(G, 41 and W1(G 41 are finite (see Chapter IV, f 8).

8.3. THEOREM. Let LJK be a cyclic extension of degree n. Then h(G, CL) = n.

Proof. We take a finite set S of primes of K so large that we can write JL = L* . JLss, where

More precisely, S is to include the archimedean primes of K, the primes of K ramified in L and all primes of K which “lie below” some primes whose classes generate the ideal class group of L. Denote by T the set of primes of L which are above primes in S. Hence

CL N JL/L* N JL,s/(L* n JL s) = JL s/L,

where LT = L* n JL,s is the set of T-units of L, i.e. those elements of L which are units of L, for w 4 T. It follows that

MC,) = NJ,, ~)/WT), if the right-hand side is defined (we note that it is impossible to use the above equation with the S missing, since then the right-hand side is not defined).

First of all we determine h(J,,,). Since S contains all ramified primes, the group fl fl U, has trivial cohomology, as remarked in 7.3. Hence

( ) UC.9 vlw ~(JL,,) = h fl fl L*

(vESL,” d> =L!:Kl”“); so by 7.2 we have h(JL,s) = n n,, where the n, are the local degrees (see

USS

Chapter VI, 0 1.4). This was the “local part” of the proof. The “global part” consists in determining h(L,); in order to prove that

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GLOBAL CLASSFIELD THEORY 179

h(C,) = n we have to show that nh(L,) = fl n,. We do this by constructing VSS

a real vector space, on which G operates, with two lattices such that one has Herbrand quotient nh(L,) and the other has quotient n n,.

v-E.9 Let V be the real vector space of maps f: T --* R, so V N R’, where

t = [T], the cardinality of T. We make G operate on V by defining (OS>(W) = f(a-‘w) (so that (af>(crw) = f(w)), for all f E V, e E G and w E T.

Put N = {f E VJ f(w) E Z for all w E T}. Clearly, N spans V and is G-invariant. We have N N

i!s($! zw)9 where Z, N Z for all w, and the

action of G on N is to permute the Z, for all w over a given u E S. Hence.

by Shapiro’s lemma again. Therefore

h(N) =v~Cfio(G”, Z>l/CfJ’(G”, Z>l = n n,. VOS

Now define another lattice. Let 1 be a map: L, + V given by A(a) = f,, where f,(w) = log ]& for all w E T. The unit theorem (or at any rate its proof!) tells us that the kernel of A is finite and its image is a lattice M” of V spanning the subspace V” = {f E VIE f(w) = 01.

Since the kernel of A is finite, h(L,) = h(M”) (see Chapter IV, 5 8). Now V = V” +Rg, where g is defined by g(w) = 1 for all w E S,. We define the second lattice M as M’+Zg. Then M spans V and both M” and Zg are invariant under G. Hence h(M) = h(M’).h(Z) = nh(M”) = nh(L,).

Now A4, N are lattices spanning the same vector space, so h(N) = h(M) by Chapter IV, 5 8. Hence n n, = h(N) = h(M) = nh(L,), as required.

” 8.4. CONSEQUENCE. If LJK is cyclic of degree n, then

CJKIK*N~IXJLI b n. This inequality, called in the old days the second inequality, was always

proved by non-analytic methods having their origins in Gauss’ theory of the genera of quadratic forms, of which our present ones are an outgrowth. For us it is theJirst inequality, since the other inequality is deduced with the aid of this one.

8.5. CONSEQUENCE. If L/K is a finite abelian extension and D is a subgroup of JK such that

(a> D = h&JL9 (b) K * D is dense in JK,

then L = K.

Proof. We may suppose that L/K is cyclic, since if L 2 L’ 2 K and L’/K is cyclic, then D c N=,,J, c NL,,KJLS. Serre has proved that locally the

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180 J. T. TATE!

norms &,I~~ Lz are open subsets of K: which contain U, for almost all a; so NLIRJL (which is simply fl NLwlxK. Lz,) and K*NLIKJL are open, hence

closed in JR, and the latter is wdense since its subset K*D is dense. So it is the whole of JR, that is, [JK/K*NLIRJL] = 1; so n = 1 by the previous consequence.

8.6. Remark. We emphasize that in the Galois case an element x = (x.) E JX is in N ,.,KJL if and only if it is a local norm everywhere, i.e. x,, E N,,,,V(L”)* for all u E !l&.

8.7. CONSEQUENCE. If S is a finite subset of lozK and L/K is a finite abelian extension, then G(L/K) is generated by the elements FLIR(v) for v 4 S (i.e. the map FLIR : Is -+ G(L/K) is surjective; I$ 3.3).

Proof. Take G’ as the subgroup of G(L/K) generated by the F,,,(v) with u # S; let M be the fixed field of G’. For v 4 S, the FLIR(v) viewed in GWIK) N G/G’ are all trivial, so for all v # S, M,,, = K, if w E DIM is over v. Trivially, every element of K,* is a norm of this extension.

Take D = Ji (ideles with X, = 1 for v E S) ; every element of D is a local norm, i.e. D c NIMIKJM. By the weak approximation theorem (see Chapter IT, 5 6) K*J,” is dense in JK. So by 8.5 we have M = K and G’ is the whole of G.

8.8. COROLLARY. If L is a non-trivial abelian extension of K, there are infinitely many primes v of K that do not split completely (i.e. for which ~L,rm z 1).

For we have just seen that such primes exist outside of any finite set S.

9. Cohomology of Idkle Classes (II), The Second Inequality

Here we deduce what in the non-analytic treatment is the second inequality. This inequality can be proved very quickly and easily by analysis (see Chapter VIII, Theorem 5), and classically was called the first inequality. We give Chevalley’s proof (Annals, 1940).

9.1. THEOREM. Let L/K be a Galois extension of degree n, with Galois group G.

Then

(1) [Z?‘(G, CJ] and [I?‘(G, C,)] divide n, (2) fi’(G, CL) = (0). Proof. The proof will be in several steps.

Step 1. Suppose that the theorem has been proved when G is cyclic and n is prime. By the Ugly Lemma (see Chapter VI, $ 1.5) it follows that [Z?‘(G, CL)] d’ ‘d 1v1 es n and A’(G, C,) = (0). Using this triviality of fi’, it follows, again by the ugly lemma, that [fi’(G, C,)] divides n.

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GLOBAL CLASS FIELD THEORY 181

Step 2. Now we assume that G is cyclic of prime order n; in this case we know that A0 N A2 and by the first inequality 8.3 that [A’] = n[fi’]; so it will be enough to show that [fi’(G, CL)] = [C, : NLIKCL] divides n.

We will make the one assumption that in the function field case n is not equal to the characteristic of K. (The other case is treated in the Artin-Tate notes, Chapter 6.)

Step 3. We now show that we may further assume that K contains the n-th roots of unity.

In fact, if we adjoin a primitive n-th root of unity c to K, we get an extension K’ = K(c) whose degree m divides (n-l), and so is prime to the prime n. So

L n K

The degree of LK’ over K’ is n, and L and K’ are linearly disjoint over K. So there is a commutative diagram with exact rows (we drop the subscripts since they are obvious):

CL -4 CK -+ &/NC, ----+O

COII

1 COII

1 COll

1 CL -------a CK’ --------, CKc/NCL. --------0

I P N

I

I

“1. CL __- - G -+ C,INC, -.--+O

Here Con is the Conorm map; and the composite map N.Con is simply raising to the mth power (see Chapter II, 4 19, for this and the definition of the Conorm map). The group &/NC, is a torsion group in which each element has order n, for if a E C,, then a” is a norm, i.e. a” E NC,. Thus the map NKPIR Con K.IR : &/NC, -P C,/NC, is surjective since (m, n) = 1. Hence the map NK,IR : C,.JNC,. + C,/NC, is surjective; so if [C,, : NC,,] divides n so does [C, : NC,].

Step 4. We are thus reduced to the case where n is a prime and K contains the n-th roots of unity. In fact we shall prove directly in this case the more general result :

Let K contain the n-th roots of unity and L/K be an abelian extension of prime exponent n, with say G(L/K) = G N (Z/nZ>‘. Then

(1) [C, : N,,,C,] divides [L : K] = nr.

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182 J. T. TATE

For although, as we have just seen, the case of arbitrary r does follow from the case r = 1, yet the method to be used does not simplify at all if one puts r = 1, and some of the constructions in the proof are useful for large r (see 9.2 and 9.5).

By Kummer Theory (see Chapter III), we know that L = K(qa,, . . . , y/a,) for some ai, a,,. . ., a, E K. Take S to be a finite set of (bad) primes, such that

(2) (i) S contains all archimedean primes, (ii) S contains all divisors of n,

(iii) Jx = K*JK,s (by making S contain representatives of a system of generators for the ideal class group),

(iv) S contains all factors of the numerator and denominator of any a,.

Condition (iv) just means that all the ai are S-units, that is, they belong to KS = Kn JK,s: they are units for all t, 4 S.

Write M = K(qK,) for the field obtained from K by adjoining n-th roots of all S-units. By the unit theorem the group KS has a finite basis, so this extension is finite, and M is unramified outside S by Kummer Theory and condition (ii). Now M 3 L 3 K and

K,=M*“nK,~L*“nK,~K*“nK,=K~.

By Kummer theory with [M: L] = n’, [L : K] = nr (given) and [M: K] = ns we have

(3) [KS : L*” n KS] = n’, [L*” n K, : Ki] = nr and [KS : Ki] = n’

respectively. We claim that s = [S], the cardinality of S. By the unit theorem, there are [S]- 1 fundamental units, and the roots of unity include the n-th roots; so KS N Zcsl-’ x (cyclic group of order divisible by n) and

(4) [KS : Kg] = nLsl = n’, where s = t + r.

We recall we want to show that [C, : NLIIcCL] divides d, i.e. divides [L*” n KS : Ki]. So we need to show that NLIKCL is fairly large-we have to provide a lot of norms.

If w is a prime of L above a v # S, then, since M/K is unramified outside S, the Frobenius map -FIcIIL(w) is well-defined. By consequence 8.7, the F&W) generate G(M/L). Choose wl,. . ., w, so that P&w,) (i = 1,. . ., t) are a basis for G(M/L), and let vi,. . ., v, be primes of K below them. We assert that F,,,(w,) = F&vi) (i = 1,. . ., t). (In fact, each of the V’S is unramified, so F,,,(vi) is defined). The M/K decomposition group G,(M/K) is a cyclic subgroup of (Z/nZ>s, so is either of prime order n or trivial. The w’s were chosen so that the F.&w) were non-trivial, so the M/L decomposition group G,(M/L) is non-trivial; so the L/K decomposition group

G,GIK) = WWKYG,(MIL)

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GLOBAL CLASS FIELD THEORY 183

is trivial, i.e. v splits completely in L (see Proposition 2). Therefore, G,, = G,, and it is generated by the FM,L(ui) = am,,.) Notice also that we have L,,=K,,foralli= l,..., t.

Write T = {or,. . ., D,}. We claim that

(5) (L*)” n KS = (a E I(& E Ki for all D E T).

In fact, since L, = K, for all D E T and w above U, it follows trivially that L*” n KS is contained in the right-hand side. Conversely, if a E KS, then ;/u E M. If further a E K”, for all v E T, then ;/a E K, for all v E T and so is left fixed by all FM,,(u) = FMIL(w); these generate G(M/L) so ;/a E L. This proves (5).

Let

where U, is the set of v-units in K,; so E c JK s u =. Also E c NLIKJL (see Remark 8.6)-for every element of K:” is a norm, since K:/NLE N G, (see Chapter VI, 0 2.1), which is killed by n; we have K,* = Lz for all u E T, and so all the elements of these K: are norms, and the elements of U, are all norms for v unramified (see Chapter VI, $ 1.2, Prop. 1).

Now L-WC/K Gl = [J&*&,K JLI

divides [JR : K*E] because E c NLIxJL. The set S was chosen ((2)(iii)) so that

JR = K*Jx,s = K*JR,sv T therefore [CK/N,I,CJ divides [K*JK,S”T: K*E]. A general formula for indices of groups is

[CA:CB][CnA:CnB]=[A:B],

so to prove (1) it will be enough to show that

(7) [JK,SVT:EI/[KS,T:KnE]=n’ (where KS” T = K * nJK,@T).

First, we calculate [JK,SVT : E].

so [J~,~ v T : E] = n [Kz : (K,*>n], by (5). From Chapter VI, 0 1.7 (cf. also “ES

the Artin-Tate notes, p. xii), we see that the “trivial action” Herbrand quotient Ii = n/l& where ] 1, denotes the normed absolute value. But also h(K,*) = [K: : K,*“]/n because the n-th roots of unity are in K:. This means that [K* - K*”

(8) [JR. s"".T

” ] = n2/~& and

: E] = n*lg InI;’ = nZS, by the product formula

since In]” = 1 if 0 $ S.

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184 J. T. TATE

We will also need in a moment the formula

(9 [U” : u:-J = n/In]“.

which follows from the fact that h(U,) = l/l& (see Chapter VI, $ 1.7). By (8) we see that to prove (7), it will be enough to show that

(10) [KS,.: K* n E] = nzs- = nS+‘,

As in (4), replacing S by S u T, we have [I&, T : K,“, =] = nS+‘, so it will be enough to show that K* n E = K,“, T’

Trivially, K* n E 3 K,“, r, so it remains to prove

(11) K*nEcKi,.,

and this will result from the following lemma.

9.2. LEMMA. Let K contain the n-th roots of unity. Let S be a subset of %R, satisfying parts (i), (ii), (iii) of (2) in the above proof, and let T be a set of primes disjoint from S, and independent for KS in the sense that the map Ks -+ Vg U,/ Vi is surjective.

Suppose that b E K* is an n-th power in S, arbitrary in T, and a unit outside S v T. Then b E K*“.

Proof of Lemma. Consider the extension K’ = K(vb); it will be enough to deduce that K’ = K. Put

by arguments similar to ones used before (see after (6)), D c NRsIKJR,. Therefore, by the first inequality in the form of consequence 8.5, in order to prove that K’ = K it is sufficient to prove that K*D = JR. But by hypothesis, the map Ks + fl (U,/Un,> N J&D is surjective. Hence

Y6T

J K,S = KsD, and JR = K*JR,s = K*D as required. To deduce (11) from the lemma, we have to check that T is independent

for S in the sense of the lemma. Let H denote the kernel of the map

K, +&WJ3. T o P rove that map is surjective it suffices to show that

(Ks : H) = n (U, : Uz). This latter product is just n’ by (9), because VET

I& = 1 for v E T. On the other hand, by (5) we have H = Ks n (L*>“, and consequently (Ks : H) = n’ by (3).

The proof of the theorem is now complete.

9.3. Remark. Even the case of the Lemma 9.2 with T empty is interesting. “If S satisfies conditions (i), (ii), (iii) of (2), then an S-unit which is a local

n-th power at all primes in S is an n-th power”.

9.4. CONSEQUENCE. If L/K is abelian with Galois group G, and there is an Artin map I/I : I?‘(G, C,) = C,/NC, + G, then $ must be an isomorphism.

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GLOBAL CLASSFIELD THEORY 185

In fact, consequence 8.7 of the first inequality already tells us that JI has to be surjective; if now [fi’(G, C,)] < [G], then $ can only be an isomorphism!

9.5. CONSEQUENCE. (Extracted from the proof of Theorem 9.1.) Let n be a prime and let K be afield, not of characteristic n, containing the n-th roots of unity. Let S be a$nite set of primes of K satisfying the conditions (i), (ii), (iii)hzS), and let M = K(vKs). Th en, if the reciprocity law holds for M/K,

(12) K*NMIKJM = K*E, where E = n (K,*)” x fl U,. UES u+s

Consider the case L = A4 of the proof of 9.1 (so that T is empty, 1 = 0 and s = r). Then the E of that proof is as given in (12), and E c N,,,J,. By (7) with L = M, we have [JK: K*E] = ns = [M: K]. On the other hand, if the reciprocity law holds, we know that

[C, : N~,&J = [JK : K*NM,KJ&J = n”;

hence (12) must hold. This result we put into the refrigerator; we will pull it out for the proof

of the “existence theorem” in the final section $ 12.

9.6. CONSEQUENCE. Let L/K be a finite (not necessarily abelian) Galois exten- sion. Since H’(G, C,> = 0, the exact sequence 0 + L* + JL --f C, + 0 gives rise to a very short exact sequence 0 + H’(G, L*) -P H’(G, JL). Now H2(G, J,.) = u H’(G”,(L”)*), by Proposition 7.3, so there is an injection

VOrnK (13) 0 -+ H'(G, L*) -+" EgxH2(G", (I!')*).

We shall see later (from the fact that the arrow p1 in diagram (9) of 9 11 is an isomorphism, for example) that the image of this injection consists of those elements in the direct sum, the sum of whose local invariants is 0. We thus obtain a complete description of the structure of the group H2(G, L*).

In terms of central simple algebras, (13) gives the Brauer-Hasse-Noether Theorem, that a central simple algebra over K splits over K if and only if it splits locally everywhere. In particular, if G is cyclic, A2 N Go, and we have the Hasse Norm Theorem:

If a E K*, and LJK is cyclic, then a E NLIKL* if and only if a E NLyIgyLv* or all v E .!UI=.

Specializing further, take G of order 2, so L = K(Jb).

NL,&+yJb) = x2-by2,

so (if the characteristic is not 2) we deduce that a has the form x2- by’ if and only if it has this form locally everywhere. It follows that a quadratic form Q(x, y, z) in three variables over K has a non-trivial zero in K if and only ifit has a non-trivial zero in every completion of K. Extending to n variables,

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186 J. T. TATE

we may obtain the Minkowski-Hasse Theorem, that a quadratic form has a zero if and only if it has a zero locally everywhere, see exercise 4.

One may consider the general problem, “if a E K* and a E NLv* for all v, is a E NL* ?” Unfortunately, the answer is not always yes! (See 11.4.)

9.7. We return to the sequence (13). We write H’(L/K) for H2(G, L*) and H’(L”/K,) for H2(G”, L”*). Thus (13) becomes

(13’) 0 + H2(L/K) + 11 H’(L”/K,). ”

Serre (Chapter VI, 9 1.1, Theorem 3, Corollary 2) has determined H’(L”/K,); it is cyclic of order n, = [L”: K,], with a canonical generator. Thus

H'(G, JL) = u H2(E'/K,) N u (; Z/Z). " v "

and

0 + H’(L/K) + u (; Z/Z). v ”

If a E u H2(L”/K,), or u E H’(L/K), we can find its local invariants inv, (a) ”

(more precisely inv, (j,(a)), where j, is the projection on the v-component of or), which will determine it precisely.

We are interested in the functorial properties of the map inv,. Let L’ 2 L 1 K be finite Galois extensions with groups

G’ = G(L’/K), and

G = G(L/K) N G’/H,

where H = G(L’/L). If CI E H2(G, Ja, then infl (a) E H’(G’, JLS) and

(14) inv, (infl a) = inv, (cx).

Indeed, choosing a prime w’ of L’ above a prime w of L above v, one reduces this to the corresponding local statement for the tower L$ I> L, 3 K,; cf. Chapter VI, 0 1.1.

Thus nothing changes under inflation so we can pass in an invariant manner to the Brauer group of K, and get the local invariants for c1 E Br (K) = H’(K/K), where R is the algebraic closure of K (see Chapter VI, 5 l), and more generally for

a E H2(GIK, JR) = $ H2(GLIK, J3,

where JR = lim JL, by definition, the limits being taken over all finite Galois T? .

extensions L of K; cf. Chapter V.

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GLOBAL CLASSFIELD THEORY 187

If now a E H’(G’, JLP), then res ga E H2(H, JLP) and

(15) inv, (res ga) = n,/” inv” (a),

where w E ‘D, lies above u E ‘%RK and nwl” = [L, : K”] (again one reduces immediately to the local case, for which see Chapter VI, 9 1.1, or Serre’s “Corps Locaux”, Hermann, (1962), p. 175). Moreover, L/K need not be Galois here.

Finally we mention the result for corestriction, though we will not use it. Again, L/K need not be Galois. If a’ E H’(H, JLS), then car gal E H’(G’, JLP) and

(16) inv” (car ga’) = C inv, (a’), WI”

where the sum is over all primes w E ‘%lZL over v E 1)32, (see “Corps Locaux”, p. 175).

9.8. COROLLARY. Let CI E Br (K) or H2(G(R/K), JR), where K is the separable algebraic closure of K. Let L be an extension of K in x. Then res f(a) = 0 if and only if [L, : K,] inv” (a) = 0 for every w over v (this is only a finite condition, since almost all the inv” (E) are zero).

In the case when L/K is Galois, there is an exact sequence

0 -- H2(L/K) infl Br (K) ZL Br (L),

and a E H’(L/K) if and only if the denominator of inv” (cc) divides [L, : K,] for all w over 0.

10. Proof of the Reciprocity Law

10.1. Now let L/K be a finite abelian extension with Galois group G. We recall our discussion in $6 on local symbols in which we noted that if a global Artin map existed we were able to reduce it to the study of local symbols and remarked that, conversely, if the local Artin maps are defined we could obtain a global Artin map. We propose to carry out this latter program here using the local Artin (“norm residue”) maps defined in Chapter VI, Q 2.2.

Let the local Artin maps be denoted by 0” : K,* + G”; we define a map

e:J,+G

by e(x) =” ~~KUd, x E JK.

This is a proper definition, since (by Chapter VI, 0 2.3) 0,(x”) = FLy,K,(v)“(Xy) (v(x”) being the normalized valuation of x”) when v is unramified, and v(x”) = 0 if x” E U”; so 0,(x”) = 1 for all but finitely many v. (Indeed, even if L/K were an infinite extension, the product for 0(x) would be con- vergent.) It is clear that 0 is a continuous map.

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188 I. T. TATE

Take S,, c %JIK as the set of archimedean primes plus the primes ramified in L/K; then x E Jscso implies e(x) = ?‘((x)‘~). Thus, 8 satisfies two of the conditions for an Artin map ((i) and (iii) of Corollary 4.2); the other con- dition ((ii) of 4.2) is that

e(a) = v EQ,O,(a) = 1 for all a E K*.

So if we can prove this, we will have proved the reciprocity law.

10.2. To prove the reciprocity law, it will be convenient to state two related theorems, and to prove them both at once, gradually extending the cases for which they are true.

THEOREM A. Every finite abelian extension L/K satisfies the reciprocity law, and the Artin map 8 : JR + G(L/K) is given by 8 = n 8..

cl THEOREM B. If u E Br (K), then c inv, (a) = 0.

06 VRK Remarks. After what has been said above, Theorem A has been whittled

down to the assertion that

(1) n e,(a) = 1 for all a E K*. U~BRX

The sum of Theorem B is finite since inv, (a) = inv, (j,. CQ = 0 for all but finitely many a.

If a E Br (K), then ~~EH~(L/K) for some finite extension L/K, i.e. tl is split by a finite extension of K.

Logically, the proof is in four main steps. Step 1. Prove A for an arbitrary finite cyclotomic extension LJK. Step 2. Deduce B for tl split by a cyclic cyclotomic extension. Step 3. Deduce B for arbitrary u E Br (K). Step 4. Deduce A for all abelian extensions.

In practice, we first clarify the relation between Theorems A and B and deduce (step 2) that A implies B for cyclic extensions and (step 4) that B implies A for arbitrary abelian extensions. Then we prove Step 1 directly, and finally push through Step 3, by showing that every element of Br (K) has a cyclic cyclotomic splitting field. 10.3. Steps 2 and 4. The relation between A and B. A is about I? and B is about H2, so we need a lemma connecting them.

Let L/K be a finite abelian extension with Galois group G. Let x be a character of G, thus x E Horn (G, Q/Z) = H’(G, Q/Z), where Q/Z is a trivial G-module. If v E ‘&Xx, denote by xv the restriction of x to the decom- position group G”. Let 6 be the connecting homomorphism

6 : H’(G, Q/Z) --) H’(G, Z). If x = (x,) E JR, let X be its image in JK/NLIKJL N f?‘(G, JL). Then the cup product (see Chapter IV, § 7) z.6~ E H2(G, J&

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GLOBAL CLASS FIELD THEORY 189

LEMMA. For each u we have

inv, (z. 6x1 = x~(W,))~ and so

T inv, (a. Sx) = x(0(x)).

Proof. We refer to Chapter VI, 5 2.3. The projection j,: J, -+ (LO)* induces a map

j, . res: y : H’(G, JJ --, H’(G,, Jd -+ HZ&L CL”)*), and as restriction commutes with the cup product, so

inv, (X. 6~) = inv, (j,. resG,” (X. 6~))

= inv, (( 1,. Z) . Sx,)

= inv, (2,. 6~“)

= x”(e”b”)),

the final step coming from Chapter VI, 5 2.3. Tt follows immediately that

x(eW> = x(IJ WJ) = T xMxJ> = T inv,WW.

To check Step 4, apply the lemma with x = a E K* c J= Denote by a” the image of a in fi’(G, L*). Then a”. 6% E A2(G, L*) c Br (K), as we need. The image of a”. 6~ in H2(G, JL) is a.&, where d is the image of a in fl”(G,J,), and by the above lemma, c inv, (a.&) = x(6(u)); so if Theorem

B is true for all a E Br (K), it follois that x(&z)) = 0, and since this is true for all x, that e(u) = 0. This is Theorem A.

To check Step 2, take L/K cyclic. Choose x as a generating character, i.e. as an injection of G into Q/Z. Then cupping with 6~ gives an isomor- phism fro 3 H2, so every element of H’(L/K, L*) is of the form c.6~. If Theorem A is true, then by the above lemma

T inv, (Z .6x) = x(e(a)) = 0

for all a E K*, which is Theorem B.

10.4. Step 1. (Number Field Case.) We want to prove that if L/K is a cyclo- tomic extension, then n e,(a) = 1 for all a E K*.

Let L/K be a finite “cyclotomic extension. Then we have L c K(c) for some root of unity 5, and it will suffice to treat the case L = K(c) because of the compatibility of the local symbols 8. relative to the extensions (K([))u/K, and L”/K,; cf. Chapter VI, 5 2.4.

Next we reduce to the case K = Q. Suppose that M = K(n, with Galois

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190 J. T. TATE

group G’ ; define L = Q(l), with Galois group G. Then h4 = LK, and there is a natural injection i : G’ + G and norm map N: JK + JQ. The diagram

B’ JK - G’

(where 0’ = n &,, and 0 = n 6,) is commutative, since Y

('K,Q *)P = 3 NK./Q~ *,

(see Chapter II, $ 11, last display formula) and since the diagrams

KJ -, G:

N 1 i

-1 i QP - GP

are commutative whenever v is above p (see Chaper VI, 3 2.1, cf. Proposition 3.1 above). Thus i O O’(X) = B(Nx,o(x)) for all x E JR and so, in particular, i O e’(a) = e(N,o(U)) for all a E K. If Theorem A is true for L/Q, then O(b) = 1 for all b E Q, SO O’(o) = 1 for all a E K, because i is injective.

First Proof for L/Q cyclotomic. We know that the reciprocity law, in the sense of 3.8, holds for L/Q by computation (see 3.4), i.e. we have an admissible map F: Is + G(L/Q) = G, for some S. Using the Chevalley interpretation (Proposition 4.1) we get an Artin map $ : JQ + G, and we obtain induced local maps JI, : Q: + G, (see 3 6). Using Proposition 4.3 we can pass to the limit and take L as the maximal cyclotomic extension Q”” of Q. This gives us local maps 9, : Qy + G(Qy/Q$, for all primes p. We want to show that these $p’s are the same as the Op’s of Chapter VI, 3 2.2; we do this by using the characterization given by Chapter VI, 5 2.8, Proposition 3.

We have to check three things. Firstly, that Qy contains the maximal unramified extension Qy of Q,; this follows from Chapter I, 9 7, Application. Secondly, if a E Qp, then tip(a) 1 Q, = F’p(‘), where r+,(a) is the normalized valuation of a, and F the Frobenius element of G(Qy/Q,); this is clear. Thirdly, if A/Q, is a finite subextension in Qy, and cc E N-lllo,,~Z*, then I/J,@) leaves .& pointwise-fixed; this follows from Proposition 6.2. Hence

+P = 0, for all finite primes p. We must not forget to check that $, is the same as 8, (see Chapter VI, $2.9). By Proposition 6.2, $m is a con-

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GLOBAL CLASS FIELD THEORY 191

tinuous homomorphism of R* into G, = G(C/R) E { f I}, and $,(ZVo,&*) = 1. Hence @m and 8, induce maps of R*/R$ into G(C/R) and 8, is onto, so we just have to check that $, is onto-in other words, we have to check that $, is not the null map.

C = R(i), so consider the effect of $ on Q(i) 2 Q; the only ramification is at 2 and co. Therefore

1 = $(-7) = 1c/2(-7)1c1,(-7)~,(-7) = $,(7)*ti,(--),

since -7 is a 2-adic norm and so q?2(- 7) = 1. Now 1,4,(7) is the map i+i7= - i, so $,( - 1) is also the map i + -i, i.e. is non-trivial.

Second Proof for L/Q cyclotomic. We may proceed entirely locally, without using our results of the early sections, but using the explicit local computation of the norm residue symbol in cyclotomic extensions, due ori- ginally to Dwork.

Let c be a root of unity; by Chapter VI, $2.9

(1) c em(x) = [Sign (Xl, for x E R*,

and by Chapter VI, 5 3.1, if x E Qt, x = p’u, with u a unit in Q, and v an integer

{

py (2) c

e,hw = g, when f has order prime to p. when c has p-power order.

We need to check that flt9,(a) = 1 for all a E Q* and to do this it is

sufficient to show that fi 0,(q) = 1 for all primes q > 0, and that

n 0,( - 1) = 1. Furthermire, it is enough to consider the effect on c, an

Zrth power root of unity (Z a prime). One checks explicitly that the effect is trivial, using the tables

c-1, p = co “W-- 1) = c c-1, p = 1

t-9 P z 1, 00

c, p=4=

pw = [, p # 1, p # q (including the case p = 03)

tIq-l, P = 1, P # 4 cq, P f I, P = 4

(Since the Galois group is abelian, it does not matter in what order one applies the automorphisms 9,(- l), resp. e,(q).)

10.5. Step 3. (Number FieId Case.) It is enough to show that every element of Br (K) has a cyclic, cyclotomic splitting field. In other words, for every a E Br (K), there is a cyclic, cyclotomic extension L/K such that for every

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192 J. T. TATE

o E ‘9Xx, the local degree [L” : K,] is a multiple of the denominator of inv, (tl) (see Corollary 9.8). Now inv, (a) = 0 for all but a finite number of primes and so we need only prove the

LEMMA. Given a number field K of finite degree over Q, a finite set of primes S of K, and a positive integer m, there exists a cyclic, cyclotomic exten- sion L/K whose local degrees are divisibIe by m at the non-archimedean primes v of S and divisible by 2 at real archimedean primes v of S (in other words, L is complex).

Proof. It is sufficient to construct L in the case K = Q (multiply m by the degree [K: Q]). Take r very large and q an odd prime. The extension L(q) = Qt’;/l> h as a Galois group isomorphic to the direct sum of a cyclic group of order (q- 1) and cyclic group of order qrml, so has a subfield L’(q) which is a cyclic cyclotomic extension of Q of degree q’-‘. Now

CL(q) : wdl = 4 - 19 and so on localizing at a fixed prime p # CO of Q we have

[L(q)@’ : L’(q)(P)] < (q - 1);

since [L(q)(p) : Q,] -+ co as r -+ co (this follows for example from the fact that each finite extension of Q, contains only a finite number of roots of unity), it follows that [L’(q)(p) : Q,] -+ co as r + 0~). Therefore, since [L’(q)‘p’ : Q,] is always a power of q, it is divisible by a sufficiently large power of q if we take r large enough.

Now let q = 2, and put L(2) = Q(2q1) for r large. L(2) has a Galois group isomorphic to the direct sum of a cyclic group of order 2 and a cyclic group of order 2’-2. Let c be a primitive 2’-th root of unity and set < = [-c-l and L’(2) = Q(c). The automorphisms of Q(c) over Q are of the form a,, : 5 H cP for p odd, and a,(r) = lP- c-“. Since Yzr-l = - 1, one sees that cr- p+2P-1(r) = o,,(t); since either p or -~+2*-l is E 1 (mod4), this implies that the automorphisms of Q(t)/Q are induced by those c,, where p EE 1 (mod 4) and that they form a cyclic group of order 2’-2. Also, since 6- 1 { = - <, Q(c) is not real, and so its local degree at an infiinte real prime is 2.

Now [L(2) : L’(2)] = 2, and the same argument as above shows that for p # co we can make [L’(2)‘p) : Q,] d ivisible by as large a power of 2 as we like by taking r large enough.

If now the prime factors of m are ql,. . . , qn and possibly 2, then for large enough r the compositum of L’(q,), . . . , L’(q,) and possibly L’(2) is a complex cyclic cyclotomic extension of Q whose local degree over Q, is divisible by m for all p in a finite set S.

Cyclic cyclotomic extensions seem to be at the heart of all proofs of the general reciprocity law. We have been able to get away with a very trivial

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GLOBAL CLASS FIELD THEORY 193

existence lemma for them, because we have at our disposal both cohomology and the local theory. In his original proof Artin used a more subtle lemma; see for example Lang, “Algebraic Numbers”, Addison Wesley, 1964, p. 60. (But notice that the necessary hypothesis that p be unramified is omitted from the statement there.)

We may prove the reciprocity law for function fields on the same lines, but the special role of “cyclic cyclotomic extensions” in the proof is taken over by “constant field extensions”.

Step 3 goes through, if we replace “cyclic cyclotomic extension” by “constant field extension”; we have only to take for the L in the lemma the constant field extension whose degree is m times the least common multiple of the degrees of the primes in S.

For step 1, we check the reciprocity law directly for constant field exten- sions; in fact, if we denote by Q the Frobenius automorphism of k/k, where k is the constant field of K, then for each prime D of K the effect of F(v) on Z is just odes “, where deg t, = [k(u) : k] is the degree of u. Hence the effect on li of e(u) is fl c”(‘) de* ” = &“(“) deg ” = cdeg * = 1, since deg a = 0

for all a E K* (the number of zeros of an algebraic function a is equal to the number of poles).

11. Cohomology of Id&le Classes (III), The Fundamental Class

11.1 Let E/L/K be finite Galois extensions of K; then we have an exact commutative diagram

0 0 0

I I I 0 p---d lP(L/K, L*) ------, fP(L/K, JL) -- H2(L/K, C,)

i I i (1) 0 -+ H 2(E/K, E*) -- H’(E/K, J& -- H2(E/K, C,)

I I I 0 -- H2(E/L, E*) ----a H’(E/L, JJ -+ H’(E/L, C,),

where we have written H’(L/K, L*) for H’(G(L/K), L*), etc. In this diagram, the vertical lines are inflation-restriction sequences; these are exact since H’(E/L, E*) = (0) (Hilbert Theorem 90, Chapter V, 5 2.6), H’(E/L, JE) = (0) (Corollary 7.4) and H’(E/L, C,) = (0) (Theorem 9.1) [see Chapter IV, 5 5, Proposition 51. The horizontal sequences are exact, and come from the sequence 0 + L* + JL + C, -+ 0, since again H’(L/K, C,) = (0), etc.

We pass to the limit and let E -+ K, where R is the algebraic closure of K, to obtain the new commutative diagram

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194 J. T. TATE

0 0 0

I 1 I 0-h H2(L/K, Id*) yI+ H2(L/K, JJ --El- H2(L/K, CL)

I 1 i (2) 0 ---+ H2(K,R*) A*+ H2(K,JpJ L-+ H2(K, C,)

i I I o-- H2(L,K*) 23% H2(L,JK) 2+ H2(L, C,),

where we have written H2(K, R*) for H’(G@/K),K*), etc. Certain of the maps with which we shall be concerned below have been labelled in the diagram. 11.2. We are going to enlarge the above commutative diagram.

For the Galois extension L/K we have the map inv, = c inv, : H’(L/K, JJ + Q/Z,

” and Theorem B of 10.2 tells us that the sequence

(3) o- H’(L/K, L*) y1-, H”(L/K, JJ inv&. Q/Z

is a comp1ex.t Since inv, (infla) = inv, (a) for all a E H’(L/K,J,) (see 9.7, (14)), we

have a map inv, : H’(K, .7x) + Q/Z such that the diagram H’(L/K, JJ -+ invt Q/Z

(4) infl

1 l

H’W, JK) inn

----, Q/Z is commutative, where i is the identity map. Furthermore, the sequence

(5) 0 --...-a H2(K, R*) A-.+ H’(K, .J=)‘“X Q/Z

is a complex. In a similar manner we have a complex

(6) 0 -------a H’(L, R*) --=+ H2(L, JK) 2% Q/Z.

But now, inv, (res a) = nwlv inv, (a), where a E H’(K, JK) and w is a prime of L over v of K, and nwlv = [L, : K,] (see 5 9.7, (15)). Thus we have the commutative diagram

H2K JR) inv2 - Q/Z

(7) rcs i

I f

HZ@, JK) in*3 - QIZ

as the sum of the local degrees c nwlv = n = [L : K]. wlu

t i.e. the image of each map is in the kernel of the next.

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GLOBAL CLASS FIELD THEORY 195

Now let Image of s2 = Im .s2 in H’(K, C,) be denoted by H’(K, C& and Im sg by H2(L, C&. It follows that we have a map p2 (resp. &) induced by inv, (resp. inv,) of H2(K, C& into Q/Z (resp. H’(L, C,& into Q/Z). Thus for a E H2(K, Cg&, we have j12(u) = k,(b), where s2(b) = a (this is independent of the choice of b). We have now explained the two lower layers in diagram (9) below.

We define

(8) H’(L/K, CJrep = {a E H’(L/K, CJjinfl a E H’(K, Cr&}.

Then nP2 infl a = 0, and so B2 induces a homomorphism

/II : H’(L/K, CJreo + - Z/Z n

such that B&J) = D2 W 4.

If a = sL b with b E H’(L/K, JL) then

Pi(a) = j12 (infl b) = inv, (infl b) = inv, (b).

(Note the difference in construction of pt and B2; the point is that H’(L/K, C& =I Im s1 but they will not in general be equal.)

We put all the information from (3)-(8) into (2) to obtain a new commu- tative (three-dimensional) diagram

O-

O-

(9)

in which i is the inclusion map, n is multiplication by n and the “bent” sequences are complexes, and the horizontal and vertical sequences are exact.

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196 J. T. TATE

11.2. (his) We propose to show that

(10) H2K Wren = H2W, Cd 1: Q/Z.

Now Im (invl) in i Z/Z is the subgroup i Z/Z, where no is the lowest

common multiple of all the local degrees of L/K, by Corollary 7.4, and so since Im /I1 3 (invi) we have the inequalities

n z [H’(L/K, Cd] 2 [H’(L/K, C&] 2 [ImA] 2 [Im WI)] = no, by the second inequality, Theorem 9.1. It follows that if n = no for this particular finite extension LJK, then we have equality throughout, so that /I1 is bijective and the sequence

(11) 0 --.-+ H’(L/K,L*) 2. H2(L/K,&.) 2% Q/Z

is exact (for if 0 = inv, (b) = /I1 s1 b, then s1 b = 0, and b E Im ri). Now if L/K is a finite cyclic extension, then n = no because the Frobenius

elements I;L,x(u), whose orders are equal to the local degrees n,, generate the cyclic group G(L/K) by Consequence 8.7. So if, in particular, the exten- sion L/K is cyclic cyclotomic, then (11) is an exact sequence. But the Lemma of $10.5 says that the groups H’(K, R*) and H’(K, JK) are the unions (of the isomorphic images under inflation) of the groups H’(L/K, L*) and H’(L/K, JL) respectively, where L runs over all cyclic cyclotomic extensions of K. Consequently, in our commutative diagram (9) the complexes

0 ----, H’(K, g*) --=-. H2(K, JR) ZL Q/Z

and 0-h H2(L, R*) y3-+ H’(L, Jx) .-=.‘. Q/Z

are exact. Therefore ker (inv2) = ker (s2), so p2 (and similarly /Is) must be injective maps into Q/Z. They are surjective, since there exist finite exten- sions with arbitrarily high local degrees and consequently even inv2 and inv, are surjective. Hence both p2 and p3 are bijective maps. Now, letting L be an arbitrary finite Galois extension, we conclude that pi is a bijection:

H2(LIK Cdreg -;z/z; but H2(L/K, CL)rel is a subgroup of H’(L/K, C,) which has order dividing n. So it is the whole of H2(L/K, C,>. Letting L --) K we see that

H2(LIK Wreg = H2(L, Cd. Thus we can remove the subscripts “reg” from our diagram (9).

Also, we have proved the following

RESULT. H’(L/K, C,> is cyclic of order n, and it has a canonical generator

uLlr with invariant t, i.e. inv, (u,,,) = X.

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GLOBAL CLASS FIELD THEORY 197

This element uL/x is called the fundamental class of the extension L/K. It was first exhibited by Weil (see the discussion in 11.6 below). The com- plete determination of the structure of H’(L/K, Cd is due to Nakayama. He and Hochschild were the first to give a systematic cohomological treat- ment of class field theory; see G. Hochschild and T. Nakayama “Cohomo- logy in Class Field Theory”, Annals, 1952, and the references contained therein.

The two lower layers of diagram (9) and the vertical arrows between them make sense for an arbitrary finite separable extension L/K of finite degree n, and in this more general case, that much of the diagram is still commutative, because the argument showing the commutativity of (7) did not require L/K to be Galois. Using this, and replacing L by K’, we see that if L 3 K’ 3 K with L/K Galois, then restricting uLix from L/K to L/K’ gives the fundamental class uL,x,.

11.3. Applications. The results we have obtained show that the idble classes constitute a class formation. In particular (cf. Chapter IV, 0 10) the cup product with the fundamental class #L/x gives isomorphisms

B’(G(LIK), Z) =s B’+‘(G(LIK), CJ, for - cc < r < co, such that for L 1 K’ 3 K with L/K Galois the diagrams

B’(G, Z) r fP+‘(G, C,) flr(G, Z) 7: l?+2(G, CJ (12) resJ resl and COIN car

&(G’, Z) r &+‘(G’, CL) @(G’, Z) 3 i?,,:,‘, C,) are commutative, where G = G(L/K) and G’ = G(L/K’).

Case r = -2. There is a canonical isomorphism (see Chapter IV, 6 3)

WIOb-t CK/NL,KCLS

which is inverse to the Artin map. Using this as a definition in the local case, Serre deduced the formula inv (a. 6~) = x(@(a)) in Chapter VI, $2.3 ; we have proved the formula in the global case, so one can reverse the argument. (The isomorphism Gab II Hm2(G, Z) is to be chosen in such a manner that for x E Horn (G, Q/Z) N H1(G, Q/Z) and u E G, we have x. d = x(a)

upon identifying i Z/Z with H-‘(G, Q/Z) as usual.)

Reversing the horizontal arrows in (12) with r = - 2, and letting L +K, we obtain the commutative diagrams

C K p G(Kab/K) rL ti

(13) C0”l I iy

?--

G(Kab/K)

i and N T

C Kp - G((K’>“b/K’) e C K’ - G((K’)“blK’), JI where the Il/‘s are the Artin maps and V is the “Verlagerung~“. The right-hand

t Called the “ transfer ” in Chapter IV, f 6, Note after Prop. 7.

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198 J. T. TATE

diagram expresses the so-called translation theorem, and in fact results also directly from 4.3, which in turn came from an almost obvious property 3.2 of the Frobenius automorphisms I;(v). The commutativity of the left-hand diagram (13) can also be proved by a straightforward but somewhat more complicated computation with the Frobenius automorphisms which was first made by Artin in connection with the “principal ideal theorem” (see exercise 3, and Serre, “Corps Locaux”, p. 130).

Case r = -3. This leads to an isomorphism used by Roquette in Chapter IX, $2.

11.4. Application to the Cohomology of L *. The general idea is to determine the cohomology of L* from a knowledge of the cohomology of the ideles and the idble classes.

Let L/K be a finite extension, with Galois group G. Then the exact sequence 0 + L* + JL + C, + 0 gives an exact sequence

+ 8’-‘(G, JJ : tl’-‘(G, CJ + &(G, L*) : B’(G, JJ + . . . in which the kernel off is isomorphic to the cokernel of g. We know

f;t’-l(~, ~3 =v~xP-l(~o,c*) =.&=P-~(G”, z).

(see Proposition 7.3), and 8’-‘(G, CJ = flr-3(G, Z);

so the kernel of f : l?(G, L*) + L1[ s’(G”, E*)

is isomorphic to the cokernel of g1 : u it’-3(Go, Z) + l?‘-3(G, Z).

It is easy to see that the map g1 is given by

gl(l+) =~c0r2zo.

Using the fundamental duality theorem in the cohomology of finite groups, which states that the cup product pairing

&(G, Z) x &‘(G, Z) + l?‘(G, Z) z Z/nZ is a perfect duality of finite groups, one sees that the cokernel of g1 is the dual of the kernel of the map

h : I?‘-‘(G, Z) -+ n fi3 -‘(GO, Z)

which is defined by (h(z)), = res g.(z) four all v E 9Xx. Case r = 0.

Kerf = ala E K*, a is a local norm everywhere

ala E K*, a is a global norm > ’ and coker g is dual to ker (H3(G, Z) E, n H’(G”, Z)). For example, if

I)

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GLOBAL CLASS FIELD THEORY 199

G = G” for some a, then this is an injection, so local norms are global norms. If G is cyclic, then H3(G, Z) z H’(G, Z) = 0, so local norms are global norms and we recover Hasse’s theorem, 9.6. On the other hand, if for instance G is the Vierergruppe, it is possible that G’ is always one of the subgroups of order 2, so fi3(G”, Z) = 0 but @(G, Z) = Z/22. Explicitly,

we can consider Q(,/13, ,/17)/Q; here ($(A!)= 1, and the extension

is unramified at 2 because 13 = 17 = 1 (mod 4), so all the decomposition groups are cyclic. Thus the set of elements of K* which are local norms everywhere is not the same as the set of elements of K* which are global norms (see exercise 5).

Case r = 3. H3(G, L*) is cyclic of order n/n,,, the global degree divided by the lowest common multiple of the local degrees, generated by 6u,,, (6 : H’(C,) + H3(L*)), the “Teichmtiller 3-class”. This can be killed by inflation (replace L by a bigger L’ so that the no for L’ is divisible by n); so H3(x/K, K*) = 0.

For a more precise description of the situation, announced at the Amsterdam Congress (Proc. 11,66-67), see Tate: “The cohomology groups of tori in finite galois extensions of number fields”, Nagoya Math. J. 27 (1966), 709-719.

Group Extensions. Consider extensions M/L/K, where L/K is Galois with group G, and M/K is Galois with group E and M is a class field over L with abelian Galois group A. So 1 + A + E + G + 1 is exact. By the Artin isomorphism A N CL/NM,LCM (see Theorem A of 10.2 and Con- sequence 9.4). We want to know about E.

11.5. THEOREM. (i) Let a E E have image Z E G. Let x E C,; then y(iix) = ay(x)a-l, where y: C,+A is the Artin map.

(ii) Let v E H*(G, A) be the class of the group extension E; then v = $*(uLIK), where Ic/* is the map: H*(G, Cd + H*(G, A) induced by @ : C, + A, and where u,,, is the fundamental class for L/K.

It is straightforward to see (i). As usual in such cases the situation becomes clearer if we consider an arbitrary field isomorphism (T: M -+ M’ rather than an automorphism. Denoting aL by L’ and the restriction of d to L by t?, we have the picture

M&-M’

and by transporting the structure of M/L to M’/L’ we see that, if x E CL and Y E M then (Y’GW>(~Y) = 4.44(~)>.

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200 J. T. TATE

(ii) is non-trivial; SafareviE did the local case (“On Galois Groups of P-adic Fields”, Dok&‘y, 1946), but it is really a general theorem about class formations (see Artin-Tate notes, p. 246). We do not prove it here, and will not make any use of the result in these notes.

11.6. Before the structure of H’(G, CL) was known, Weil (“Sur la 2’ ‘. du Corps de Classes”, J. Math. Sot. Japan, 3 (19X), l-35) looked Z,S’l*;*c. situation from the opposite point of view. Taking M to be Lab, the rnz$,, : ., , abelian extension of K (so that now A = G(Lab/L) is a profinite ab&‘ar” group), Weil asked himself whether there was a group extension G = G(L/K) by CL which fits into a commutative diagram of the sort

1 -+ c, ----,d ---,G ----.l

(14) I *L I 1

id

1 1 1 -+A --E-G ----, 1,

where E = G(Lab/K), and if so, to what extent was it unique? In the function field case, lc/L is for all practical purposes an isomorphism, and the existence of such a diagram is obvious. Moreover, in that case, the group- theoretical transfer map V (Verlagerung) from d to CL (which has its image t in C,) gives a commutative diagram

as follows from the commutativity of the left-hand diagram (13). Inspired by the case of function fields, Weil proved that also in the number field case a diagram. (14) did indeed exist, and was essentially uniquely determined by the condition that (15) (together with its analogues when K is replaced by an arbitrary intermediate field K’ between K and L) should be commu- tative. In particular, the class u E H’(G, CL) of such an extension 8 was unique, and that is the way the fundamental class was discovered.

Nowadays one can proceed more directly, simply constructing B as a group extension of G by CL corresponding to the fundamental class uL,x, and interpreting the unicity as reflecting the fact that H’(G, CL) = 0 (cf. Artin-Tate, Ch. 14).

The kernel of the map IV: 8 + E is the connected component DL of CL. As Weil remarks, the search for a Galois-like interpretation of d (or even a “natural” construction, without recourse to factor systems, of a group 8 furnished with a “natural” map IV: 8 + E) seems to be one of the funda- mental problems of number theory.

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GLOBAL CLASS FIELD THEORY 201

In support of the idea that 8 behaves like a Galois group, Weil also describes how to attach L.-series to characters x of unitary representations of 8. These L-series of Weil generalize simultaneously Hecke’s L-series “mit Griissencharakteren” (which are obtained from representations which

+?rough C, via the arrow V in (15)) and Artin’s “non-abelian” , (which are obtained from representations which factor through E,

2: %ticular through G ft

N E/A, via the arrow Win (15)). (The intersection ,I; ,$ke’s and Artin’s Lseries are those of Weber obtained from repre- -“I/$ons of Eab N C,lD,, i.e. from ordinary congruence characters.) “‘!:g Brauer’s theorem on group characters, Weil shows that his L-series

can be expressed as products of (positive or negative) integral powers of Hecke L-series, and are therefore meromorphic.

12. Proof of the Existence Theorem

We still have to prove the Existence Theorem (D) of 5.1. Our proof, more traditional than that used by Serre in Chapter VI, works just as well in the local case.

If His an open subgroup of C, of finite index [C, : H], we say temporarily ;hat H is normic if and only if there is an abelian extension L/K such that H = NLIKCL. The existence theorem asserts that every open subgroup H of finite index in C, is normic. (We have already shown that if L/K is abelian, then NLIKCL is an open subgroup of C, of finite index; in fact the normic subgroups are just the inverse images of the open subgroups of G(Kab/K) under the Artin map $k : C, + G(Kab/K).)

First, two obvious remarks: If Hi 3 H, and H is normic, then HI is normic (the field L corresponding to H has a subfield L1 corresponding to H,). If HI, H, are normic, so is HI n H, (take as the field the compositum Ll w

Next, we go to 9.5 in order to prove KEY LEMMA. Let n be a prime, and K ajield not of characteristic n containing

the n-th roots of unity. Then every open subgroup H of index n in C, is normic.

Proof. In fact, suppose H is open in C, with [Cx: H] = n. Let H’ be the inverse image of H in Jx. Then H’ is open in JK, so there is a finite set S c ‘3Xx such that H’ 3 n (1) x n U, = Us. Furthermore, H is of

VES VCS index n in C,, so H’ =I Ji. Therefore H’ 3 n K*” x n U,, = E, say.

VS.7 VLS Thus H = HI/K* I EK*JK*, and from consequence 9.5, it follows that H is normic.

If L is an extension of K, there is a norm map N: CL -+ C,; conversely, if we start with H c C, we get a subgroup N-‘(H) c C,.

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202 J. T. TATE

LEMMA. If L/K is cyclic and H c C.,, and if N$(H) c C, is normic for L, then H is nor&c for K.

Proof. Write H’ for N,ii(H) and let M/L be the class field of H’. We claim that M is abelian over K, and NIKIRCM c H, so H is normic. That N,,,C, c H is clear, since N is transitive; the difficulty is to show that M/K is abelian.

(In point of fact, if something is the norm group from a non-abelian extension, then it is already the norm group from the maximal abelian sub- extension (see exercise 8). But this has not been proved here-if it had, we would not need L/K to be cyclic, and we would be already finished with the proof of the lemma.)

M/K is a Galois extension since H’ is invariant under G(L/K). The Galois group E of M/K is a group extension, 0 -+ A -+ E + G + 1; since E/A N G is cyclic, it is enough to show that A = G(M/L) is in the centre of E.

We can use the first part of Theorem 11.5. Let I,+ be the Artin map C, -+ A. To show that A is in the centre, it is enough to check that

l)(x) = aljqx>a- l = $(0x)

for all x E C, and ts E E. Now @ : C, + A has kernel H’, so we want to check that OX/~ E H’, which is clear since N(cx/x) = 1.

Proof of the Theorem. (In the function field case we can only treat the case in which the index is prime to the characteristic; for the general case, see Artin-Tate, p. 78.)

We use induction on the index of H. If the index is 1 everything is clear. Now let n be a prime dividing the index. Adjoin the n-th roots of unity to

K to get K’, and replace H by H’ = N&(H). By the last lemma, it suffices to consider H’. The index of H’ divides the index of H; we can assume (C,. : H’) = (C, : H), otherwise H’ is normic by induction hypothesis.

So n divides (Cx, : H’). Take Hi so that Hi XI H and (Cx, : H’) = n. By the above Key Lemma, Hi is normic. Let L be its class field, i.e. Hi = NLIK, C,. Put H” = N& (H’). Then

[C, : H” ] < cc,* : H’] - [C, : H].

(For CJH” N4& C,#JH’ is an injection, whose image is Hi/H’, properly contained in C&/H’.)

Hence H” is normic by induction hypothesis; L/K’ is cyclic, so we can apply the above lemma again; so H’ is normic.

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GLOBAL CLASS FIELD THEORY 203

LIST OF SYMBOLS

The numbering refers to the section of this Chapter where the symbol was first used.

Note (i) [A] is the cardinality of the set A. (ii) “ 3 ” denotes inclusion with the possibility of equality.

1.1. K, K* (non-zero 4. JR, J; 7.3. u elements of K) w J K,Ss s = mK

G(L/K), G JLS, S = 9-G

XL ; w, WI,. . . 4.1. * No

VP, E-T,, 8.1. h(G, A), h(A) Ku, L us S-units = KS

GIV 4.2. h/K 9.1. Conorm = Con

1.2. m2, Im$

2.1.

2.2.

k(v) NV

&K(U)

S

5.1. Norm group 5.5. 2 5.7. Q”“, R*,

6.1. I+?,

9.7. inv, inv, (infl) inv, (res) inv, (car) Br (9

3.1. IS 7.1. w/v, n 10.1. 8,, 0 Artin maps. 3.2. NK*/K 40

wS 7.2. G” 10.3. ., cup-product L” G ab

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CHAPTER VIII

Zeta-Functions and L-Functions

H. HEILBRONN

1. Characters . . . . . . . . . . 204 2. Dirichlet L-series and Density Theorems . . . . . 209 3. L-functions for Non-abelian Extensions . . . . . 218 References . . . . . . . . . . . 230

The object of this Chapter is to study the distribution of prime ideals in various algebraic number fields, the principal results being embodied in several so-called “density theorems” such as the Prime Ideal Theorem (Theorem 3) and Cebotarev’s Theorem (Section 3). As in the study of the distribution of rational primes, L-series formed with certain group characters play an important part in such investigations.

Much of what we have to say goes back to the early decades of this century and is described in Hasse’s “Bericht tiber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlenkiirper” (Hasse 1926, 1927 and 1930), henceforward to be referred to as Hasse’s Bericht.

1. Characters

Throughout, let k be a finite extension of Q, of degree v. Let S denote a set consisting of the infinite primes (archimedean valuations) of k together with a finite number of other primes (non-archimedean valuations) of k. S is sometimes referred to as the exceptional set.

In slight modification of earlier notation, an id&le x of k will be written as

x =&p. * “xpo;xp.+l,. * ‘,xpb;xpb+i,. * .I, where pl,. . . , p,, are the infinite primes, pa+r,. . ., pb are the finite primes of S, and the remaining pi are the primes not in S. Let J denote the idele group.

By a character $ of the idele class group we understand a homomorphism from J into the unit circle of the complex plane satisfying

(1) $(x) = 1 if x E k* (i.e. xp = x for all p); (2) $(x) is continuous on J (in the idele topology); (3) $(x) = 1 if xp = 1 for p E S and Ix,&, = 1 for p 4 S.

As was pointed out in Chapter VII, “Global Class Field Theory”, $4, I/ generates a character of the ideal group Is (the free abelian group on the set of all p # S) in the following way:

204

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ZETA-FUNCTIONS AND L-FUNCTIONS 205

Let

a = JJ p:’ i>b

(ai E Z)

be a general element of Zs (so that almost all the exponents ai are 0). For each i > b, let n, be an element of k with IXilp, < 1 and maximal. With a we associate the idele x” having

(xyv, = {ii,, j 1;;‘; - .’ bs ,**-,

then we define x(4 = It/(x?.

We observe that although x” is not unique, x(a) is, nevertheless, well-defined in view of property (3) above. Clearly x is a multiplicative function in the sense that for all ideals a, b coprime with S,

xW = xWx@). Any ideal character x defined in’ this way is called a Grossencharacter,

and such characters were first studied by Hecke (1920) and later, in the idMe setting, by Chevalley (1940).

Let S, S’ be two exceptional sets and x, x’ characters of Is, Is’ respectively. We say that x and 1’ are co-trained if x(a) = x’(a) whenever both x(a), x’(a) are defined. This definition determines an equivalence relation? among characters. In the equivalence class of x there is a unique x’ corresponding to the least possible exceptional set s’ (s’ being the intersection of all exceptional sets corresponding to the characters co-trained with x); we refer to this character 2’ as the primitive character co-trained with x.

The principal character x0 of Is is defined by x0(a) = 1 for all a E Z’. The principal characters form a co-trained equivalence class, and the primitive member of this class is 1 for all non-zero finite ideals.

We shall now look more closely into the structure of a character of the ideal group Z”, with a view to describing the important class of Hilbert and Dirichlet characters.

We can write each idtle x in the form

x=nx<P> V

where each factor x(p) is the idele defined by

xw,, = { 71’ P = Pi , P f Pi*

t To prove the transitivity of the relation it suffices to show that if v(y) = 1 for all idtles y for which yqi = 1, i = 1,2,. . . , m, then I&) = 1 for all idkles x. However, for any E > 0 there is an a E k* such that

la - qh < 4 i= I,2 ,..., m, by the Chinese Remainder theorem and thus V(X) = &a-lx) tends to 1 as I + 0.

.

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206

Then

H. HEILBRONN

QKd = F $pW,

where $, is a character of J defined by

II/,(4 = W(Pji

and is referred to as a local component of $ (Chevalley, 1940). We note that I,++, is continuous and satisfies (3). We first follow up some consequences of the continuity of JI.

Let JV be a neighbourhood of 1 containing no subgroup of J/(J) except 1. By continuity of +, there exists a neighbourhood N’ of 1 in J, say

Ixp-llp <sp, PEE, (1.1)

lxplp = 1, P 4 E, U-2)

where E is some finite set of valuations, such that

l&v’) c N.

We choose N’ first to make the set E minimal, and then to make the E~‘S maximal; so that for the finite p in E we have sP < 1. For finite p, (1.1) is now equivalent to saying that xP E 1 +p”+’ where p,, is a positive integer. But since 1 +pPp is a group, so is $,(l +p”p), whence ti+,(l +p@) = 1, and this holds for no smaller integer than pP. We write

f.Y = p LLP”’ PGE and we refer to f, as the conductor of the character x derived from JI. If p is finite and belongs to E, (1.1) implies, by virtue of (3), that p E S. More- over, if p is finite and not in E, (1.2) shows that it is not necessary, in view of (3), to include p in S. Thus the finite p in E are just those non-archimedean primes in the exceptional set of the primitive character co-trained with x.

Let m be a given integral ideal of k and let x be any character such that f&n. Then if a = (a) where CI E 1 (mod m),

x(a)=ti(l,..., i;l,..., l;cr,a ,...)

= *(a-l,. . .,a-‘;a-‘,. . .,a-1; l,l,. . .

by (11, =+(a-‘,..., a-‘;1 ,..., l;l,l,...

since a-l E 1 +p’P for finite p E S. Hence

x(4 = ,JT,JlpW ‘1

where Se is the set of archimedean primes.

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ZETA-FUNCTIONS AND L-FUNCTIONS 207

We now restrict attention to those characters x for which $,(J) is a discrete sub-set of the unit circle for all p E S,,. Consider a particular p E S,. There exists an integer m such that

9,(x”) = 1 for all x E J,

and, in particular, for all x E k*. The completion k, is R or C according as p is real or complex, with each x of k* mapped onto the p-conjugate x, of x. However, a homomorphic mapping of C* into the unit circle and having finite order must map C* onto 1 and similarly, a homomorphic map of R*, of finite order, into the unit circle either maps R* onto 1 or maps R* onto & 1 by x + sgn X. Thus, if p is complex, I& is identically 1, and if p is real I,$, is either identically 1 or is f 1; in the latter case, for x E k* we have

If X(~) > 0 for all real p E Se, x is said to be totally positive and we write x $0. Thus, if $ has discrete infinite components, x is a character deter- mined by the subgroup of totally positive principal ideals = 1 (mod m). Such a character is called a Dirichlet character moduIo m; if m = 1, x is called a Hilbert character.

Conversely, we shall now show that any character x of the ideal group Is which is 1 on the subgroup HCm) of totally positive principal ideals congruent 1 modulo m (where S consists of the archimedean primes together with the primes dividing m) arises in this way from an idele class character JI with exceptional set S. Let x be such a character, and let x*, xi be the restrictions of x to the group AS of principal ideals coprime with m, and to the group B(‘“) of principal ideals 3 l(mod m) respectively. Then xi(~) is determined by the signs of the real conjugates of a. We extend this definition to all elements tc of k* in the obvious way. Then we write xz = x*x;’ ; clearly xz is a character of AS equal to 1 on B(“‘).

We define $ corresponding to x by stages. For any idele x, and each p not in S, define

Il/,w = X(P”P) where? @‘P]]x~. (1.3) This part of the definition ensures that $ satisfies (3).

We next consider together all the finite primes of S, that is, all p E S-S,,. We choose y E k* such that y = xp (mod p’p) for each p E S-S,, where p’pllm (this is possible by the Chinese Remainder Theorem). We then define

p J-Iso~pb~ = X2(Y) (1.4)

for those ideles x having 1x,1, = 1 for all p E S-S,,.

t pvjlxp means that xp E p” - @“+I.

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208 H. HEILBRONN

To complete the definition of II/ on this sub-set of J, we need to define tip(x) for P E So. For real p E S,, let x”‘) be the character on k* defined by fp) (a) = sgn a Then x1 is the product of some sub-set of these characters xcp), say

p.

Xl =pil$p) CT = So) (1.5)

and we define tip(x) for p E SO by

(1.6)

provided xp E k*, and we extend this definition to k: by continuity. We postpone defining $ for those ideles x for which [x,1, # 1 for some

PCS-SO. It is clear from our construction that $ is multiplicative. We proceed to

verify that $ satisfies (1) and (2), (3) having been satisfied by construction. Consider (1) first. Then

,g,Jlpw = x; %>

by (1.5) and (1.6),

p JIsotipw = x34

by (1.4), and

pIjJshc4 = x(4 = x*(4

by (1.3). Since x* = x1 xz, $ is seen to be 1 on k*. It remains to consider (2). But I&) = 1 if

Ixp-llp < 1, P E so, xpE1+p”p, P ES-S,,

lslp = 1, P 4 s, and this is an open set in the idele topology.

Finally, we need to extend the definition of $ to ideles x for which, for some p E S-S,, lx,j, # 1. Let x be such an idele and suppose that p E S-S,,. If p”pllx, (where up is, of course, not necessarily positive), choose a E k* such that

V”pIIa for each p E S- SO.

We now define

W = #(ax), noting that the right-hand side has already been defined. Hence Ic/ is well defined; for, if jI is another element of k* such that p-“~1 ID for each p E S-S,,,

ILO = WW~@x) = 4X4,

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ZETA-FUNCTIONS AND L-FUNCTIONS 209

since j?/u E k* and $ is multiplicative. This completes the definition of JI as an idHe class character.

We recall that Is is the group of all ideals of k relatively prime to S, AS denotes the subgroup of principal ideals in I’, B(“) the subgroup of AS consisting of those principal ideals (a) where ct E 1 (mod m), and HCm) the subgroup of B(“‘) consisting of those principal ideals (a) which satisfy

CL = 1 (mod m), a g 0.

We note that H(“) is the intersection of the kernels of all the distinct Dirichlet characters modulo m, and the number h, of such characters is thus the index of H+) in P.

The index of AS in Is is equal to the so-called absolute class number h. The index of BCm) in AS is the number of units in the residue class ring mod m, denoted by d(m); and the index of H’“’ in B”“’ is the number of totally positive units G 1 (mod m), equal to H/(h4,(m)) where 41(m) is the number of residue classes mod tn containing totally positive units and H is the number of ideal classes in the “narrow” sense, i.e. relative to the subgroup of 1’ consisting of all totally positive principal ideals of k. Thus the number h, of distinct Dirichlet characters mod m is given by

2. Dirichlet L-series and Density Theorems In this section we shall take “character” to mean “Dirichlet character”. If x is a character, we define x on S by x(p) = 0 if p E S.

Let a denote a general integral ideal, with the prime decomposition of a in k given by

a = JJ P”P, vP 2 0, vP = 0 for almost all p.

By N(a) we shall understand the absolute norm of a, i.e. N&a). We define the Dedekind zeta-function &(s) by

&) = & N&s =!(I-j&q.)-’ (s=d+it),

and with each character x we associate a so-called L-series

Since each rational prime p is the product of at most [k : Q] primes p, the convergence of both sum and product in each case, and the equality between them, for CJ > 1 all follow from the absolute convergence of sum and product fora> 1.

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210 H. HEILBRONN

We observe that L(s, x0) and c,(s) differ only by the factors corresponding to the ramified primes, and that &Js) is the L-series corresponding to the primitive character co-trained with x0.

As in classical rational number theory, L-functions are introduced as a means of proving various density theorems (such as Theorem 3 below). An important property of these functions is that their range of definition can be extended analytically to the left of the line G = 1. The continuation into the

region d > 1 - A can be effected in an elementary way (using only Abel

summation and ihe fact that c(s) is regular apart from a simple pole at s = 1) by virtue of the following

THEOREM 1

where K depends on the degree, class number, discriminant and units of k. We indicate briefly the proof of this result. Let C denote a typical coset of

I?“‘) in I’. Then the sum considered in the theorem is equal to

and since

it suffices to estimate the inner sum. Let 6 be a fixed ideal in C-‘. Then a6 = (a) where a is a totally positive element of k such that a E 1 (mod m). As a varies, a is a variable integer in the ideal 6. Also,

IW>I = [N(a))1 = WOJ@),

so that the inner sum may be written in the form

,N(&TXN(b)l aeb

where the asterisk indicates that a 3 1 (mod m), a 9 0, and that in each set of associates only one element is to be counted. The integers a of the ideal 6 may be represented by the points of a certain n-dimensional lattice, and the problem is essentially the classical one of estimating the number of these points in a certain simplex (Dedekind, 1871-94; Weber, 1896 and Hecke, 1954). The estimate takes the form?

t A better error term can be derived as in Landau (1927), Satz 210.

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ZETA-FUNCTIONS AND L-FUNCTIONS 211

A result of this type may also be derived for Grossencharacters, but the proof is complicated (Hecke, 1920).

As in classical theory, deeper methods lead to analytic continuation of L-functions into the whole complex plane, and thence to a functional equation for such functions. In the case of the ordinary C-function, one obtains?

co

72 -sqY 0 ; i(s) Ll 1

-;+ (x s

-*“-~+x~s-l)~(e(x)- 1)dx = Q(s), 1

where

O(x) = g e-nm2x, WI=-03

and one derives at once the functional equation

Q(s) = ar(l -s).

Hecke (1920) extended this classical result in a far-reaching way to L-series formed with Grossencharacters; and, more recently, Tate (Chapter XV) generalized Hecke’s result to general spaces. For Dirichlet characters, Hecke’s result is summarized below (Hasse’s Bericht, 1926, 1927 and 1930).

Let

then Q(s,x) is meromorphic and satisfies the functional equation

w, x> = w’(xw - s, 2) where W is a constant of absolute value 1, and

d denotes the discriminant; riFrz are the numbers of real and complex valuations respectively; a4 = 0 or 1 according as the value of x in the domain of all principal ideals

(N) with a E 1 (mod fJ does or does not depend on the sign of the qth real conjugate of a.

An explicit expression for W is given, e.g. in Hasse’s Bericht (Section 9, Satz 15); its presence in the functional equation leads to interesting informa- tion of an algebraic nature, for instance, about generalized Gaussian sums (Bericht, Section 9.2 et seq). It follows from the proof of the functional equation that L&x) is an integral function, except for a simple pole at s = 1 when x = x0.

We shall see that information about the zeros of L-functions plays an important part in applications to density results. The distribution of zeros in the “critical” strip 0 < u < 1 is particularly important. According to the

t Titchmarsh (1951).

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212 H. HEILBRONN

generalized Riemann hypothesis, L(s, x) # 0 if u > 3; but this conjecture is far from being settled.

Nevertheless, significant information can be extracted from the much weaker

THEOREM 2 us, xl # 0 if aal.

Proof. In view of the product representation of L for u > 1 we may restrict ourselves to the case cr = 1. The proof falls into two parts, the first due to Hadamard, the second to Landau. We suppose on the contrary that 1 + it is a zero of L(s, x).

(i) (Hadamard) Assume that x2 #x0 if t = 0. If CJ > 1, we have, from the product representation, that

L(s, x) = exp 1

Fmzl ~N(p)-mue-i’miog No >

.

When p is prime to fx, we write x(p) = e’ “p, & = t log N(p) + cp and consider the function

IL3(a, xo)L4(a+ it, x)L(a+2it, x2>/

= exp 1

1 f -!- N(p)-““(3 +4 cos m/3, + cos 2mfl,) pprimetofxm=l m 1

since 3 +4 cos o +cos 20 > 0 for all real w. Keeping t fixed, we now let CJ -+ 1 +O. Of the terms on the left, the first is O((a- 1)-3), the second is @(a-- 1)4) by hypothesis and the third is O(1) because if t = 0, x2 # x0. Hence the expression on the left tends to 0 as 0 + 1 +0, and we arrive at a contradiction.

(ii) (Landau) It remains to consider the case when x2 = x0 (so that x is real) and, if possible, L(1, x) = 0. We consider the product

&(s)L(s, x) = T N(a)-“l(a) = 5 a,. u-’ u=l

where a, = C 44

N(a)=” and

%a) = C x(b) = n (1 +X(P) + . - - x(P”)) 610 PTb

where p”lla signifies that m = mp is the highest exponent to which p divides a. Moreover, if mp is even for all p dividing a,

n(a) > 1.

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ZETA-FUNCTIONS AND L-FUNCTIONS 213

Hence, if CJ is real and both series converge, c N(a)-“ql) 2 T N(a)-% a

(2.1)

We now utilize the following simple result from the theory of Dirichlet series :

A Dirichlet series of type

f(s) =“~l%.c a,20 (24=1,2,...),

has a halJlplane Re s > Q,, as its domain of convergence, and ij c,-, is finite, then f(s) is non-regular at s = ran. (Titchmarsh, 1939, see the theorem of Section 9.2.)

By hypothesis, f(s) = Ck(s)L(s, x) is everywhere regular (the hypothetical zero of L(s, x) at s = 1 counteracting the pole of &(s)); hence the series on the left of (2.1) converges for all real rr. However, the series on the right of (2.1) is equal to &J20) which tends to + co as c + 4 + 0. Hence we arrive at a contradiction.

The Landau proof is purely existential, whereas Hadamard’s argument can be used to yield quantitative results about zero-free regions and orders of magnitude of L-functi0ns.t

We are now in a position to prove

THEOREM 3 (Prime Ideal Theorem)

x = x09

NJ< F) = x f x0*

Proof. For Re s > 1, form the logarithmic derivative of L(s, x), namely

- ;’ = ;m~lNP)-mslog N(p). x(P”) ,

where g is a function regular for Re s > 3. Our first, and major, step will be to estimate the coefficient sum

& xx(~> log N(p).

One way of doing this is to express the sum as a contour integral of

2 L’(s, xl -~ s J&x)

t See Estennann (1952) and Landau (1907).

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214 H.HEILBRONN

along the line (c-ice, c+ ico) with some c > 1, and then to shift the line of integration to the parallel line Re s = 1, with an indent round s = 1. This we now know to be the only point at which a pole can occur (and does occur precisely when x = x0, the residue in this case then leading to the dominant term). For an account of this treatment, see Landau (1927). It is clear from even this sketch that the non-vanishing of L(s, x) on Re s = 1 is vital.

An alternative approach is via the Wiener-Ikehara tauberian theorem : Suppose that

are DirichIet series, convergent for Re s > 1, regular on Re s = 1 with simple poles at s = 1, of residue 1 in the case off, and q in the case of g, where 1 may be 0. Assume that there exists a constant c such that lb,,,] =S ca,. Then

m&h - v-6 as x + 00.

We apply this result with

f(s) = - c’(s> C(s) ’

us, xl g(s) = - - L(s, x>’

c=Fk:

Clearly q = 1 when x = x0, and otherwise q = 0.

Finally, it is an easy matter to show that

e.g. by partial summation; and this proves the theorem. (The tauberian approach is described in Lang (1964).)

It is worth remarking that the analytic method can be made to give sharper estimates of the error terms (Hecke, 1920). Both methods extend to Grtissencharacters.

In close analogy to Dirichlet’s famous proof of the infinitude of rational primes in arithmetic progressions, one can show that every Dirichlet ideal class C contains infinitely many primes p of k. Indeed, by means of the prime ideal theorem we can show that the primes p are equally distributed among the classes C; we have, in particular, that

THEOREM 4

N(g - h’- lg? x+00.

m VGC

Proof: We have only to remark that

; j(C)&) =

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ZETA-FUNCTIONS AND L-FUNCTIONS

By virtue of this orthogonality formula, we have

215

on the other hand, the sum on the right is asymptotic to x/log x as x + co by the prime ideal theorem.

If B is any set of ideals in k, and

lim 10gx - card (p E BIN(p) < x> = x+co x

exists, then I is called the density of primes in B. Thus the density of the set of all primes of k is 1.

To investigate the density of primes in a given set, it suffices to consider only primes p of absolute first degree, that is, those p for which N&p) is a rational prime; for all primes p whose norms N(p) are powers of rational primes greater than the first, and satisfy N(p) < x, are 0(x*) in number. For the same reason we may clearly disregard the finite number of ramified primes.

We shall use this remark in proving the next result, known in classical parlance as the first fundamental inequality of class Jield theory (see Chapter VII, “Global Class Field Theory”, $9, and Chapter XI, “The History of Class Field Theory”, $1).

THEOREM 5. Let K be a finite normal extension of k, and denote [K : k] by n. Let S be the exceptional set in k. K determines the subgroup Hs of I’, offinite index hs in I’, composed of those cosets of H” which contain norms (relatiue to k) of ideals of K coprime with S. Then

h, < n.

Proof. We identify l/n and l/h, as the densities of two sets of primes in k. First of all, if C is the set of all primes p (of k) in Hs, then, by the pre-

ceeding theorem, the density of C is l/h,. Next, since K/k is normal, the prime decomposition of any p in K is of the

form P=(r)1 .-.rpr)

where all the primes ‘$3, have the same degree f relative to y, and

efr = n.

Since the number of ramified p is finite, we need consider only primes p for which e = 1. Now consider the set B of all primes p whose prime decom- position in K is characterized by e =f = 1, that is, by r = n. On the one hand, by the prime ideal theorem for K (noting that the primes I are all the primes of K of absolute first degree) the density of B is clearly l/n; on the other

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216 H. HEILBRONN

hand, if p c B and ‘Qplp, then N,&J) = p, so that B c H,. The result follows at once.

Note. It is worth remarking that Theorem 5 can be proved without appeal to the relatively deep Theorems 3 and 4 and, in particular, to the non- vanishing of L(s, x) on Re s = 1. A more elementary argument, using only real variable theory, runs as follows:

Let s be real and > 1. By Theorem 1 and partial summation we have

(a>

and

lim (s - l)L(s, x0) = 1c s-1

@I lim L(s, x) exists and is finite if x # x0. s-1

From the product representation of L(s, x) it follows that

log us, x> = c g$ + d% x> where g(s, x) is a Dirichlet series absolutely convergent for s > 3. Using the orthogonality properties of the characters x, we conclude that

where

M-(s) = x;xoIlog Vs, xl - ds, xl> + log (s - WXs, xo) -ids, x01.

By (a) and (b), limsupf(s) is not + co and, indeed, is finite unless one of the S-r1

L(s, x) (x # x0) vanishes at s = 1. Now let K be as described in the statement of Theorem 5. We have,

analogously to (a), that liiy Cx(s).(s- 1) exists and is finite. Taking

logarithms of the product representation of &(s), gives

where lim G(s) exists and is finite. Subtracting (d) from (c), and using s-1

B c H,, we obtain

+f(s)-G(s)>0

for all s > 1. Letting s + 1 +0, the result follows. For the next theorem we require some results from class-field theory.

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ZETA-FUNCTIONS AND L-FUNCTIONS 217

THEOREM 6. Zf K is a finite abelian extension of k, then

MS) = lJ m, x, k) (2.2) x

where, in the notation of the preceding theorem, the product on the right extends over the primitive characters co-trained with the characters of the class group Is/H,.

Proof. The proof is carried out in terms of local factors, and we consider separately non-ramified and ramified primes p of k.

(i) Let p be a non-ramified prime of k, so that

P=%***rPr where Ipi,. . ., Yj3, are distinct primes of K. From class field theory,

NK/Q(rZi) = Nk/Q(P>f

where If= [K : k] = n. Thus the corresponding local factor on the left is

(1 -N(p)--fS)-“‘f,

whilst the corresponding local factor on the right is

IJ (1 -xtPvw9-“)-‘.

Since f is the least positive integer such that x(p/) = 1 for all x, we have the easily verifiable identity (take logs of both sides and use hs = n)

(l-J+“” = n (l-x(p).y)-‘; x

from this, with y = N(p)-“, equality of the local factors follows at once. (ii) The proof for ramified primes is more difficult, and depends on the

functional equations satisfied by the various L-functions. We begin by writing

M) = g(s) n us, x9 42 x

and prove that g(s) is identically 1. From above, g(s) is equal to the finite product over ramified p of the expressions

n: (1 -XtPYw-“)

“I-J (1 -NW”1 * If this product is not constant, then it has a pole or zero at a pure

imaginary point it,, t,, # 0. In view of the functional equations, g( 1 - s)/g(s) is a quotient of gamma functions and so can have only real poles and zeros. Thus 1 -it,, is also a pole or zero of g. But we know that 1 -it,, is not a singularity or zero of any of the L-series or of &(s). Hence g is constant, and so equal to 1.

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218 H. HEILBRONN

Example 1.7 Let k = Q, K = Q(o) where o is a primitive mth root of unity. Then

IAS) = C(s) n us, xl x

where the product on the right extends over the primitive characters co-trained with all non-principal rational characters mod m.

Proof. See Chapter III, $1.

Example 2. Let k = Q, K = Q(Jd) w h ere d is the discriminant of K. Then

where 0

g m

denotes the Kronecker symbol (Hecke, 1954, Chapter VII).

We touch upon another consequence of this theorem. We recall that the functional equation of an L-function contains a factor

{pp(f,v’2. Applying the functional equation to (x(s) on the left of (2.2) and to each of the L-series on the right, one can derive the relation

where D is the discriminant of K (Hasse’s Bericht, Section 9.3). If we assume for the moment that k = Q, we obtain

n fx = Disc (K/k). (2.3)

This inference is not as ea,“, to justify in the general case when k # Q, because two distinct ideals in k can have equal norms. However, it can be proved (Hasse’s Bericht, Section 9.3, formula (12) and Note 44) that (2.3) is valid in general.

3. L-functions for Non-abelian Extensions

Suppose now that K is a finite, normal but not necessarily abelian extension of k of degree n. The problem is to develop in this case an analogue of the above theory of L-series formed with abelian group characters.

As usual, let G be the Galois group of K over k. Let {Mh)},, o be a repre- sentation of G into matrices over the complex field. Thus to each element p of G corresponds a matrix Mb). The character ~(,a) of p is defined to be

t By (2.2), since &Jr) and L(s, x0, k) = C*(S) have simple poles at s = 1, and the remaining L(s, x, k) on the right are regular there, it follows that these L(s, x, k) do not vanish at s = 1. In particular, in the situation of Example 1, it follows that ail the L-func- tions formed with non-principal Dirichlet characters mod m do not vanish at s = 1. This is an alternative proof of the key step in the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions mod m.

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ZETA-FUNCTIONS AND L-FUNCTIONS 219

the trace of il4@) and depends only on the conjugacy class (p) in which /J lies.

Two representationst {M&)},,c, (Nb)},, o are said to be equivaht if there exists a non-singular matrix P such that

PMQP-1 = N(p) for all p E G.

A representation {Mb)} is said to be reducible if and only if it is equivalent to a representation {AQ)} such that

for all p E G,

where (N(‘)Q}, {N(2~Q) are themselves representations of G. The character of an irreducible representation is said to be simple. From

the general theory of group representations (for an account of this theory see e.g. Hall, 1959) the number g of simple characters of G is equal to the number of conjugacy classes of G, and the following orthogonality relations are known to hold:

(3.1)

in particular

&xQ = {;* ,

;;;;z;, ,

where the principal character is the character of the representation M(P) = 1 for all p E G, to be denoted henceforward by x0. Also, if el, $2,. . . , $,, are the simple characters of G, then

(3.2)

where I,, is the number of elements in the conjugacy class <p) of p. In particular, taking p’ = p = 1,

(3.3)

where n, is the degree of JI, (that is, $i is the character of a representation by nl x ni matrices).

If G is abelian, then each I,, = 1, so that g = n and each n1 = 1. Thus each character er is a homomorphism of G into the unit circle, and so an abelian character.

Let ‘$3 be a non-ramified prime of K, and let p be the prime of k lying under ‘p. We shall denote the Frobenius automorphism of K/k relative to ‘$ (for

t It should be clear from the context that N is not used here to denote a norm1

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220 H. HEILBRONN

definition see Chapter VII $2) by

is the characteristic polynomial of M(p)-so that I denotes the unit matrix- we take as local factor of our L-function the expression

We note that this definition cannot be extended to ramified ‘$3 since there is no corresponding Frobenius automorphism.

It is important to note that this local factor depends only on the character x of the representation, and not on the explicit representation matrix. For any similarity transformation of Mb) leaves its characteristic polynomial invariant and, since M@) has finite order, the Jordan canonical form of M(p) must therefore be a diagonal matrix all of whose diagonal elements are roots of unity. Thus, without loss of generality,

may be taken to be

The local factor is then equal to

b

since c E;” is equal to the trace of M - i=l

{ r[] )“which, by definition, is equal

K/k m tax -

([ I> v .

We collect up the local factors corresponding to non-ramified ‘$3 and define the so-called Artin Gfunctiont essentially by

L(s, x) = L(s, x, K/k) =

P we shall introduce later a factor derived from the ramified primes.

t See also Artin (1930).

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ZETA-FUNCTIONS AND L-FUNCTIONS 221

We now list a number of observations about L(s, x): (I) L(s, x) is regular for u > 1, since the product is absolutely and uniformly

convergent in every closed sub-set of the half-plane Re s > 1. (II) If K/k is abelian, and if x is simple then, apart from the factor corre-

sponding to the ramified primes, this definition coincides with that given in Section 2 above.

(III) Suppose that n is a field intermediate between K and k, normal over k. Let H = Gal(K/Q), so that H is a normal subgroup of G and

G/H = Gal (O/k).

Then if x is a character of G/H, it can be regarded in an obvious way as a character of G, and

L(s, x, K/k) = L(s, x, n/k).

Proof. Take the character over G defined by the representation

iw4= MPm We have

~I% E xN(P) (mod p) (3.5)

for all integers X E K, and in particular for all integers X E 0. Since Q is a normal extension of k,

XEa*x~ER for all p E G.

Hence (3.5) is a congruence in a, and since ‘$ is unramified, if ‘$ lies over q in R we have

Hence also

,c~l” E XNcp) (mod q),

i.e. Klk [ 1 - H is the Frobenius automorphism in R. cp

Thus

jl--M’( F])N(p)-“1 = II-M( F] H)N(&-‘1

= pvf (F]) N($-yl.

(IV) Suppose that x is a non-simple character of G, say x = xi +x2. Then

us, xl = us9 x&o, x2),

since log L is linear in x by (3.4).

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222 H. HEILBRONN

(V) Suppose again that R is a field intermediate between K and k, this time not necessarily normal over k. Let H = Gal (K/Q, and suppose that

is the partition of G into right cosets of H. To each character 2 of H there corresponds an induced character x* of G, given by

x*w = C XC% PC ‘13 PEG; (3.6) i

lt,pa,- 1 E a then

J% x*, K/k) = Us, x, W3. Proof.? Let !JJ be a non-ramified prime of K, and p the prime of k under ‘$3.

Suppose that, in R, p has the prime decomposition

P =ifJqi9 ft.

N*/Q(Si) = CNk/QP) 9 (3.7)

and suppose that r, is an element of G such that q1 lies under oils. Then (Hasse’s Bericht, Section 23, I) G can be decomposed in the form

where Klk po= -. [ 1 rp From the definition of induced character given above, we have

since (z~~~z;‘)~ E H if and only iff,lm. The logarithm of the p-component (p non-ramified) of L(s, x*, K/k) is

t For an alternative proof see Hasse’s Bericht, Section 27, VII.

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ZETA-FUNCTIONS AND L-FUNCTIONS 223

by (3.7), on writing Ott =f;t in the inner summation; and this is nothing but the sum of the logarithms of those q-components of L(s, x, K/Q) which, by (3.7), correspond to p.

We proceed to consider some special cases of(V). V(i) n = K. Then H = Gal (K/Cl) = (1) and we have only one character,

the principal character x0. In this case, then, L(s,x,,,K/Q reduces to [k(s). By (3.6) the corresponding induced character, x0*, of G is given by

xx4 = n if p is the unit element of G, 0 otherwise. (3.8)

Let $1, 11/2,. . . ,1,4# be all the simple characters of G. By (3.2), with p’ = 1, we have I1 = 1 and arrive, by (3.8), at

j$l$iQh(l) = XZPh (3.9)

We observe that, of course, $i(l) must be a positive integer. By repeated application of (IV), it follows that

CKts) = if! Us, $is KlkY”“s (3.10)

V(ii) R = k. By (III) (with R = k and therefore G = H),

Us, xo, K/k) = Us, xo, k/k)

= t-k(S).

We shall now deduce from the preceding theorems the remarkable result that a general Artin L-function L&x, K/k) can be expressed as a product of rational powers of abelian L-functions L(s, $, K/S2), where the R’s are fields intermediate between k and K with K/Q abelian.

With the notation used in (i), each character x of G can be written in the form

(3.11)

where the r,‘s are non-negative rational integers. Hence, by repeated application of (IV), it suffices to consider the simple Artin Gfunctions

Us, Icli, K/k).

Let H be any subgroup of G, and let l, run through the simple characters of H. Each r, induces a character c$’ of G, given, let us say, by

tjYPl= T rji$ioL> for all p E G, (3.12)

in accordance with (3.11). The restriction of $, to H is, itself, a character of H and therefore has an

expression of type (3.11) in terms of-the rj’S. Moreover, by the theory of

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224 H. HEILBRONN

induced characters, this representation actually takes the form

Iclitz> = T ‘;* tjCz)9 all r E H, (3.13)

where the coefficients rjl are the same as in (3.12). Now take H to be a cyclic subgroup of G. Then if R is the subfield of K

consisting of elements invariant under H, K/Cl is abelian and so, by (V),

Us, 579 Klk) = Us, tj, K/a),

which is an abelian L-function. To prove our result, it therefore suffices to express each ei as a linear combination of characters of type $.

Each element y of G generates a cyclic subgroup H,, of G and so, by (3.12), denoting the simple characters of H,, by &,, we have

Cr*;jQ =i$l~y;ji~ioc), all CC E G- (3.14)

The system of equations (3.14) is described by a matrix with each fixed pair (r,j) determining a row and each i a column and we shall prove that its rank is equal to g.

Suppose on the contrary that the rank is less than g. Then the columns are linearly dependent, i.e. there exist integers cl, c2,. . . , c8 not all zero, such that

$Try;ji = O for all y E G and all j.

Thus, by (3.13), interpreted in relation to a particular HY,

7 ci$i(r) = T ct T ru;ji &j(r), all 7 E H,,

= 0, all r E H,.

In particular, taking r = y, we arrive at

T C,$i(y) = 0 for all y E G, (3.15)

contradicting the linear independence of the simple characters $r, Jlz, . . . , $#. Following on from (3.1!9, multiply this relation by ij,Jr) and sum over all the elements y of G. From (3.1) it follows that

nc, = 0 (k = 1,2,. . .,g) (3.16)

whence c1 = c2 = . . . = ce = 0 (so that, incidentally, we have here another way of arriving at a contradiction). Either way, we have now that the system of equations (3.14) can be solved for pi, giving

@i =,TGT u7;ji$j

where the coefficients uy; ji E Q.

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ZETA-FUNCTIONS AND L-FUNCTIONS 225

The argument centering on (3.16) can be carried out modulo any prime p which does not divide n, whence it follows that the coefficients u,:,~ have denominators composed of prime factors of n.

We have thus proved the following

THEOREM 7. For any character x of G, the Artin L-function L(s, x, K/k) is given by

L(S9 X9 Klk) = JJ JJ L(s, 51j9 K/QIY’

where each Gal (K/a,) is cyclic, the &,‘s are (abelian) characters of Gal (K/R,) and each qj is rational with denominator composed of prime factors of n.

It has been proved by Brauer (1947) that the exponents in the above theorem can be taken to be rational integers, and it follows, in particular, that the Artin L-functions are meromorphic. Furthermore, if x is non- principal, then the r’s in the above product representation of L(s, x, K/k) can also be chosen non-principal. Hence Artin L-functions formed with non- principal characters are, in addition, regular and non-zero for 0 > 1.

If, on the other hand, x = x0 is the principal character, then L(s, x0, K/k) has a simple pole at s = 1. Artin has conjectured that, apart from this simple pole, in the case x = x0, the L(s, x, K/k) are integral functions.

This conjecture would imply that &(s)/&.(s) is an integral function when- ever k c K. If K/k is normal, this is true by (3.10), (V(ii)) and a fundamental theorem on group characters due to Brauer (see Lang, 1964 p. 139).

Artin’s conjecture has been verified when G is one of the following special groups: S3 and, more generally, any group of squarefree order; any group of prime power order; any group whose commutator subgroup is abelian (Speiser, 1927); S, (Artin, 1924). The conjecture has not been confirmed for G=A5.

By Theorem 7, each Artin’s L-function L(s, x) is seen to satisfy a functional equation by virtue of the fact that each abelian L-function occuring on the right satisfies its own functional equation. This equation will be of the form

w, xl = wx>w - s, Z) where W is a constant of absolute value 1, and Q, is of the form

(9(s,x) = A(X)SI- (;)‘or(f+)-L(s,x)

with a, b in Q and A a positive constant. At this point we recall that, in the definition of Artin’s L-function, we

omitted to include a factor corresponding to the ramified primes. Now we introduce on the right-hand side of (3.17) the ramified local factors corre- sponding to the abelian L-functions (see beginning of Section 2), and take the new product as the definition of L. Then the above functional equation is satisfied automatically by the redefined L-function.

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226 H. HEILBRONN

What is lacking in the theory of Artin Gfunctions is a non-local approach (provided, in the abelian case, by a series definition of L-functions). This lack causes the difficulties in extending these Gfunctions to the complex plane.

Example: The case G = S,. The elements of S, fall into three conjugacy classes

Cl : (1); c, : U,2,3), (3,2,1>; c, : &2), (2,3), (391). Hence there are three simple characters. Let 11/l be the principal character and 1,4~ the other character determined by the subgroup C, v C,. These are both of first degree and so, by (3.3), $3 is of degree 2. Thus, by (3.2) with p’ = 1, we have that

Hence the table of simple characters of S, can be set out as follows:

+1 $2 $3 Cl 1 1 2 c2 1 1 -1 c3 1 -1 0.

Working from (3.6), we obtain the induced characters x* corresponding to the characters x of

(i) H= A,:

Cl 2 c2 2 c3 0

(ii) H= {1,(1,2)}: xx

Cl 3 c2 0

C3 1

(iii) H= (1): x2

Cl 6 c2 0 c3 0.

From these tables we see that

x: =*I+$29

x:=IcI1+J13*

xt 2

-1 0

xs 3 0

-1;

x: 2

-1 0;

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ZETA-FUNCTIONS AND L-FUNCTIONS 227

Ss is the Galois group of any non-abelian normal extension K of k of degree 6. Let Q2,, sZz be the two intermediate fields tied under the subgroups A,,{(1),(1,2)} respectively. Thus R, is a quadratic extension and Qz a cubic extension of k, and K is therefore an abelian extension of each of them. Also, Sz, is an abelian extension of k. Thus, by (V(ii)), and (IV),

MS) 1 ~(s’~xo$/I<) = Us, +I+ tiz + % K/k)

llrl $1 JI3’

C,,(s) = Us, xo, K/W = Us, $I+ $2, K/k)

= 41 L*,,

L,<s> z ;G,,, WM = Us, @I +@a K/k)

91 $3’ and

C,(s) = Us, xo, K/k) = L,,. We remark that

L,, = Us, $2, K/k) = Us, x5, Wk)

by (III) and that L,, = Us, Jls, K/W = Us, xz, W-4)

by (V). Hence L,, and Le3 are integral functions. Incidentally, this example verifies At-tin’s conjecture for Gal (K/k) = S,. We conclude this brief survey of the non-abelian case by quoting cebo-

tarev’s density theorem: Let K/k be normal and let G = Gal (K/k). Let C denote a given conjugacy

class in G. Then the class of all non-ramified primes p, each having the

[ 1 Klk property that - rp

E C for some? prime ‘$3 of K lying over p, has density

equal to card C card

Using arguments of the kind described above (see Section 2), this result follows from the fact than an Artin L-function has no zeros on the line Res= 1.

Example. As an illustration of the theorem, let us consider the case of a non-normal cubic extension KS/k. A (non-ramified) prime p in k factorizes in K3 in one of the following three ways:

(1) P = rp,cp,% (2) p = !Q, !& where the deg !JJJp = 1, deg ‘pz/p = 2 (3) P = %

t In fact, C consists of the Frobenius automorphisms 5 corresponding to those c 1 Klk

$3, lying over 4.

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228 H.H&LBRO~

We wish to determine the density of primes p falling into each of these categories.

Let K6 denote the minimal extension of K3 that is normal over k. We study the factorization of p in K6, bearing in mind that in any decomposition of p into primes of KS, these all have equal order. Thus in (1) p decomposes either in six primes (in KS) of degree 1, or !&, ‘pz, !& are themselves primes of KS, each of degree 2. The latter possibility is ruled out because KS/k is not normal.

[Suppose, on the contrary that, each ‘$, remains a prime in Ke, and hence is of degree 2 in K8. Write

[ I Klk oi= -.

Pi

Then each crl is of order 2, and hence is a 2-cycle (note that Gal (K,Jk) = S,). Further, the u,‘s span a conjugacy class, and hence are distinct transpositions.

Now K3 is the set of elements of KS invariant under some 2-cycle, say ul. Thus

d’;Pi n KS) = ‘% n KS.

But since there are three distinct prime ideals in K3, and since each has a unique extension to Ke (for each remains a prime in K8),

a1’Pi = rp, (i = 1, 2, 3).

Let the subfields invariant under uz and a3 be Kj and Ki. Then, in a similar way we have (since Kj and Kg are the conjugates of K3)

OjVP = Pi (i,j, = 1, 2, 3).

But the transpositions generate S,. Hence

s,%Ji = IPr- Thus there is no automorphism z such that

4% = %

and this is impossible.]

In (2), we must have p = q1 q, ‘$J1 as the prime factorization over KS, each factor being of degree 2, with q, qz = !&. In (3), either p remains prime, or p decomposes into two primes of degree 3. The former would imply that

F-1 G/k generates S3, which is impossible. Hence the latter decomposition

P must hold.

We now apply the fact (Chapter VII, 92) that the order of - [ 1 K,lk q equals the degree of q. From this it follows for each prime q of K6 lying

[ I &lk over p that - is an i-cycle in case (i) (i = 1,2,3). Hence, by Cebotarev’s

9

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ZETA-FUNCTIONS AND L-FUNCTIONS 229

theorem, the required densities in cases (I), (2) and (3) are 6, 3 and 3 respectively.

The following result is a direct consequence of Kummer’s theorem (see Chapter III, Appendix, or Weiss (1963), 4.9 and in particular 4.9.2).

“Supposefg k[x], is irreducible, and 8 is a zero off. Let p denote a prime of.k. Then, for almost all primes p of k, p splits in k(8) asf(x) splits over the residue class field of k module p.”

Now let n(p,j’) denote the number of solutions of the congruence

f(x) = 0 (mod p);

then, for almost all p, n(p,f) may be regarded, alternatively, as the number of prime factors from k(B) of p having degree 1 relative to k. Applying the prime ideal theorem (Theorem 3) to primes of k(B), we arrive therefore at the following

THEOREM 8. Using the notation defined above,

&/(PYf) - -?- log x asx-,co.

Corollary I. More generally, if f is the product of I distinct irreducible factors, then

Corollary 2. Ift for almost all p, f splits completely mod p, then f factorizes completely over k[x].

Proof. If n’ denotes the degree off, then we are given that n(p,f) = n’ for almost all p. Hence, by the prime ideal theorem for k, I = n’.

It would seem plausible that if f has at least one linear factor mod p for almost all p, then f has at least one linear factor in k[x]. However, the following counter-examples show this to be false.

(0 f(x) = (x2-a)(x’-b)(x’-c)

where abc is a perfect square in Z, with none of a, b, c a perfect square. For, if a, b are quadratic non-residues mod p, then c is a quadratic residue

modp and hence f has two linear factors modp.

(ii) f(x) = (x2+3)(x3+2).

For if p s 1 (mod 3), x2 +3 factorizes mod p, and if p = 2 (mod 3), x3 + 2 factorizes mod p.

However, we do have the following,

THEOREM 9. If f is a non-linear polynomial over k which has at least one linear factor modulo p for almost all p, then f is reducible in k[x].

Proof. Assume that f is irreducible. Then it follows from Theorem 8 and

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230 H. HEILBRONN

the prime ideal theorem for k (Theorem 3) that

N(&xMP9f’- 1) = +&).

Hence, since n(p,f) > 1 almost everywhere, n(p,f) = 1 almost everywhere. Let K be the root field off over k, and q a prime ideal in k which splits

totally in K. Then q splits totally in k(8) and n(q,f) is equal to the degree of f, with a finite number of exceptions. These q have positive density by the prime ideal theorem (Theorem 3) in K. Hence f is linear.

REFERENCES

Artin, E. (1923). uber eine neue Art von L-Reihen. Abh. math. Semin. Univ. Hamburg, 3, 89-108. (Collected Papers (1965), 105-124. Addison-Wesley.)

Artin, E. (1930). Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. Abh. math. Semin. Univ. Hamburg, 8,292-306. (Collected Papers (1965), 165-179. Addison-Wesley.)

Brauer, R. (1947). On Artin’s L-series with general group characters. Ann. Math. (2) 48, 502-514.

Chevalley, C. (1940). La thCorie du corps de classes. Ann. Math. (2) 41, 394-418; and other papers referred to in the above.

Dedekind, R. Vorlesungen i.lber Zahlentheorie von P. G. Lejeune Dirichlet. Braunschweig, 1871-1894. Supplement 11.

Estermann, T. (1952). Introduction to modem prime number theory. Camb. Tracts in Math. 41.

Hall, M., Jr. (1959). “The Theory of Groups”. Macmillan, New York. Hasse, H. Bericht tiber neuere Untersuchungen und Probleme aus der Theorie der

algebraischen Zahlkdrper. Jber. dt. Mat Verein., 35 (1926), 36 (1927) and 39 (1930).

Hecke, E. (1920). Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen (Zweite Mitteilung). Math. Z. 6,11-51 (Mathematische Werke (1959), 249-289. Vandenhoeck und Ruprecht).

Hecke, E. (1954). Vorlesungen tiber die Theorie der algebraischen Zahlen, 2. Unverlnderte Aufl., Leipzig.

Landau, E. (1907). Uber die Verteilung der Primideale in den Idealklassen eines algebraischen Zahlkijrpers. Math. Annln, 63, 145-204.

Landau, E. (1927). Einftihrung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, 2. Aufl. Leipzig.

Lang, S. (1964). “Algebraic Numbers”. Addison-Wesley. Speiser, A. (1927). “Die Theorie der Gruppen von endlicher Ordnung”, 2. Aufl.

Berlin. Tit&marsh, E. C. (1939). “The Theory of Functions”, 2nd edition. Oxford

University Press, London. Titchmarsh, E. C. (1951). “The Theory of the Riemann Zeta-function”. Oxford

University Press, London. Weber, H. (1896). Uber einen in der Zahlentheorie angewandten Satz der

Integralrechnung. Giittinger Nachrichten, 275-281. Weiss, E. (1963). “Algebraic Number Theory”. McGraw-Hill, New York.

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CHAPTER IX

On Class Field Towers

PETER R~QUETTE

1. Introduction . . . . . . . . . . 231 2. ProofofTheorem3 . . . . . . . . : Z?? 3. Proof of Theorem 5 for Galois Extensions . . . . References . . . . . . . . . . . 248

1. Introduction Let k be an algebraic number field of finite degree and hk its class number, i.e. the order of the finite group Cl, of divisor classes of k. One of the most striking features of algebraic number theory-as compared to rational number theory-is the existence of fields k with class number ht > 1, i.e. whose ring of integers is not a principal ideal ring.

There arises the question as to whether every such k can be imbedded into an algebraic number field K of finite degree with hK = 1. We refer to this problem as the imbedding problem for k, and to K as a solution of the imbedding problem.

Let k, be the Hilbert class field of k. It can be defined as the maximal unramified abelian field extension of k. The Galois group of kJk is iso- morphic to Cl, via the reciprocity isomorphism of class field theory. In particular,

(k, : k) = hk. The principal divisor theorem states that every divisor of k becomes a prin- cipal divisor in k,. But there may be divisors of kl which are not principal; so let k2 be the Hilbert class field of kl. Continuing, we obtain a tower of fields

k c kl c k2 c k, c . . . in which each field is the Hilbert class field of its predecessor. This is called the Hilbert classfield tower of k. We denote by k, the union of the fields ki. This is an algebraic number field whose degree may be finite or infinite.

PROPOSITION 1. If the imbedding problem k c K with hK = 1 has a solution K, then k, c K. In particular, the degree of k, is then finite.

Conversely, if the degree of k, is Jinite, then hk, = 1 and hence k, is then the smallest solution of the imbedding problem for k.

Proof. (i) First we show k, c K for all i. Using induction we may assume i = 1. The field extension k,/k is unramified and has abelian Galois group.

231

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232 PETER ROQUJZTTE

Both these properties carry over to the composite field extension k, K/K. Hence k, K is contained in the Hilbert class field Kl of K. By assumption, (K,:K)=h,= 1. It follows ki c K.

(ii) Now we assume k, of finite degree. Then k, = k, if i is large. Hence h k, = hk, = (ki+i : ki) = 1. QED.

By Proposition 1 we see that the imbedding problem for k is equivalent to the problem of whether the Hilbert class field tower of k is finite. This problem has been posed by Furtwlingler and it is mentioned in Hasse’s Klassenkbrperbericht 1926, in connection with the (then yet unsolved) principal divisor theorem for k. Although the principal divisor theorem was proved already in 1930 (Furtwangler), it was not until 1964 that the finiteness problem of the Hilbert class field tower could be solved. The solution has been given by SafareviE (together with Golod) who showed that the answer in general is negative, i.e. the Hilbert class field tower can be in&rite.

The purpose of this course is to give a report about SafareviE’s results, together with the related work of Brumer.

Let p be a prime number. A field extension K/k is called a p-extension if it is Galois and if its Galois group is a p-group. (Warning: if K/k is not Galois it is not called a p-extension, even if its degree is a p-power.)

Let ky) be the maximal p-extension of k contained in k, ; this is called the Hilbert p-class field of k. Let kip) be the Hilbert p-class field of kp’. We obtain a field tower

k c k’p’ c kp’ c k$“’ c . . .

in which each field is the Hilbert p-class field of its predecessor. This is called the Hilbert p-class field tower of k. The union of the kp) is called k’,P).

It is easy to see that klp) c k, and in fact kjp) is the maximalp-extension of k which is contained in k,. It follows k’,P’ c k,. In particular, if k’,P’ is of infinite degree we see that k, is of infinite degree too.

Hence we ask: Under what conditions is k’,P’ of finite degree, ifp is a fixed prime? The fact that we shall deal with k’,P’ rather than with k, is due only to the fact that p-groups are easier to handle than arbitrary solvable groups.

The following proposition is the analogue to Proposition 1 for the p-class field tower and is proved similarly, using the fact that (ky) : k) = hip) is the p-power part of the class number h,. The proof is left to the reader.

PROPOSITION 2. Let p be a prime number. If the imbedding problem k c K with p ,/‘h, has a solution K, then k’,P) c K. In particular, the degree of k’,P’ is then finite.

Conversely, if the degree of k’,P’ is finite, then its class number is not divisible by p and hence k’,P) is then the smallest solution of the imbedding problem for k with respect to p.

Now we state our main result. First a definition: If G is any group, we

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ON CLASS FIELD TOWERS 233

denote by G/p the maximal abelian factor group of G of exponent p, regarded as a vector space over the field of p elements. We define the p-rank d@‘G to be the dimension of G/p:

dcP)G = dim G/p.

If G is finite abelian, then d@)G is the number of factors of p-power order which occur in a direct decomposition of G into cyclic primary components.

THEOREM 3 (Golod-SafareviE). There exists u function r(n) such that

dcp) Cl, < y(n)

for any algebraic number field k of degree n whose p-class field tower is finite.

Remark 4. As we shall see in the proof of Theorem 3, we have

(1) dcp) Cl, < 2 + 2,/(rk + SF))

where rk denotes the number of infinite primes of k and Sp) = 1 or 0 according to whether the pth roots of unity are contained in k or not. Since rk 6 n and @‘I 6 1 we see that we can take

r(n) = 2+2J(n + 1).

In order to formulate our next theorem, we introduce the following nota- tion. Let q be a finite prime in the rational number field Q, and let D be any extension of q to k with the corresponding ramification degree e(Q). Let us put

ek(d = tg$ e(Q)

where D ranges over the extensions of q to k (gcd means “greatest common divisor”). We shall call q completely ramified in k if e,(q) > 1. Let tP be the number of completely ramified q such that p divides e,(q).

THEOREM 5 (Brumer). There exists a function c(n) such that

dcP) Clk 2 tip’ - c(n)

for every algebraic number field k of degree n.

Remark 6. One can show that dcp) Cl, 2 tip’ - r,n

where tk has the same meaning as in Remark 4. Since rk 5 n we see that we can take

c(n) = n2.

Actually, following Brumer, we are going to prove here Theorem 5 only in a somewhat weaker version, insofar as we shall consider only Galois extensions k of Q. We shall exhibit a function c’(n) such that for every Galois extension k/Q of degree n the following inequality holds:

dfP) 51, 2 tp’ - C’(n).

Our proof of Theorem 5 in the general case requires the use of the Amitsur

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234 PJTI'ERROQUEXTE

cohomology in number fields which we do not want to assume known in this Chapter. For Galois extensions, we shall show that

d(P) Cl, zi: tP’ - L

F; + w,(n).bp >

where wp(n) denotes the exponent of p occurring in n. Since wp(n> s n- 1 we see that we can take

c’(n) = (n-1)+(n-1) = 2(n-1).

Combining Theorems 3 and 5 we obtain the

COROLLARY 7. If k is an algebraic number field of degree n and if

tip’ 2 y(n) + c(n)

then the p-classfield tower of k is infinite. In particular, for any given degree n > 1 and any prime p dividing n, there

exist injbtitely many algebraic number fields k of degree n with an infinite p-class jield tower, for instance the Jields

k=Q(;/(q,... a~)) with N 5 r(n) + c(n)

where the q, > 0 are different prime numbers in Q. Numerical examples: We consider the case n =p = 2 and use the

estimates given by formulae (1) and (2). We have Sp)= 1, w,(2) = 1. Furthermore, ti2) is just the number of finite primes of Q which ramify in k. It follows: A quadraticfield k has an infinite 2-classfield tower if the number of finite primes of Q which are ramified in k is 2 2 + 2J(rk + 1) + rk.

If k iS iIIXigiXUUy, we have r& = 1. Since 2+2&+ 1 < 6 it follows that any imaginary quadratic field with at least six ramified finite primes has an infinite 2-class field tower. A small numerical example is

k = Q(,/(-2.3.5.7.11.13)) = Q(,/(-30030)).

If k is real, we have rk = 2. Since 2 +2&+2 < 8 we see that in the real case we must have at least eight finite ramified primes in order to deduce the infinity of the 2-class field tower. A small numerical example is

k=Q(,/(2.3.5.7.11.13.17.19))

= Q&/(9699690)).

2. Proof of Theorem 3

We consider the following situation:

k an algebraic number field of finite degree; Cl& its divisor class group; C& its idkle class group; u& the group Of idele U!litS ;

Ek = u& n k the group of units in k;

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ON CLASS FIELD TOWERS 235

W, the group of roots of unity in k;

P a fixed prime number K = kcp) as explained in Section 1. G = GTK/k) the corresponding Gaiois group.

Under the assumption that (K: k) is finite we are going to prove the inequality (1) of Section 1.

The proof will be divided into two parts. In the first part, we shall reduce Theorem 3 to a purely group theoretical statement about finite p-groups. In the second part, we shall prove this group theoretical statement.

The unit theorem says that Ek is a direct product of the finite cyclic group W, and a free abelian group with r,- 1 free generators. (See Chapter II, $18.) This gives

dtp)Ek = (rk- 1) + dtp) W, = (r, - 1) + 69).

Therefore the inequality (1) to be proved can also be written as

d(P) Cl, < 2 + 2 J(d’P’ Ek + 1)

or equivalently, in rational form:

(3) *(d(P) Cld2 - dcp) Cl, < d’p’Ek.

First we observe that

dtP) Cl, = dtp) G.

Namely, we have d(P) G = d(P)(@)

where Gab is the maximal abelian factor group of G. This follows directly from the definition of the p-rank of G as given in Section 1. Now, Gab is the Galois group of the maximal subfield of K which is abelian over k; this subfield is the Hilbert p-class field kl (p). By the reciprocity isomorphism of class field theory, we have therefore

G cab = c~$P’

where Cl,$‘) is the p-Sylow group of Clk. By definition of the p-rank, d(P) C$‘) = d(P) Cl,.

This proves our contention. Therefore the inequality (3) to be proved may be written as

(4) $(dcP) G)’ - dcp) G < dcp’E,.

Any factor group of Ek has p-rank 5 d(P)&. In particular, this is true for the norm factor group:

W&&K) = fi”G E3. Hence it suffices to show that

(5) +(dfP’ G)2 - dcp) G < dcp’fio(G, I&).

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236 PETER ROQUETTE

To determine the right-hand side, we use the exact sequence

l-+EK-+UK-‘UK/ER+l.

Since K/k is unramified, UK is cohomologically trivial as a G-module. (For the proof, decompose UK into its local components and prove the similar fact for a local Galois extension and its unit group, cf. Chapter VI, 0 2.5,)t

It follows that .tl’(G, EK) = H- ‘(G, UK/&).

Now we use the exact sequence

1-+U~/E~-+C~+Cl~+1.

Here, the group Cl, is of order prime to p. This follows from the fact that K = k’,P) is the maximal field in the Hilbert p-class tower; see Proposition 2 in Section 1. Since G is a p-group, we conclude that Cl, is cohomologically trivial as a G-module. Therefore

H-‘(G, UK/EK) = H-‘(G, C,) = H-3(G,Z).

using Tate’s fundamental theorem of cohomology in class field theory (Chapter VII, 0 11.3).

It follows that the inequality (5) to be proved may be written as

(6) *(d(p) G)2 - dcp) G < d’P’H2(G, Z).

using the fact that H,(G, Z) = H-3(G, Z)

by definition of the cohomology groups with negative dimensions. The inequality (6) is now a purely group theoretical statement; we shaN see

that (6) is true for anyjinite p-group G. According to our notation introduced in Section 1, let Z/p denote the

cyclic group with p elements. The homology groups Hi(G, Z/p) are annihi- lated by p and may therefore be regarded as vector spaces over the field with p elements. Let us put

d@) G = dim H.(G, Z/p).

LEMMA 8. For any group k, there is a na;ural isomorphism H,(G, Z/p) = G/p. In particular, dip’ G = dcp) G.

LEMMA 9. For anyjinite group G, we have

d(P) H,(G, Z) = d$” G - d’P’ G.

Proofs. Consider the exact sequence

o-,z+z+z/p-+o P

where 7 denotes the map “multiplication byp”. Consider the corresponding

t Recall that the absence of ramification for an hfinite prime means the corresponding local extension is trivial, i.e. of degree 1.

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ON CLASS FIELD TOWERS 237

exact homology sequence H,(Z) --) Hi(Z) --) Hi(Z/p) j Hi- l(Z) --, Hi-,(Z).

(For brevity, we havz written H,(Z) instead of H,~~, Z).) In general, if A is any abelian group, let A, be the kernel of the map

A 7 A; its cokernel is A/p. With this notation, we obtain the exact sequence

(7) 0 --) Hi(Z)/p + Hi(Z/p) * Hi- l(Z), + 0.

First, take i = 1. We have H,(Z) = Z which has no p-torsion; it follows Km, = 0 and therefore

(8) HIWP = H,(Z/P).

Here, H,(Z) = H,(G, Z) = Gab is the maximal abelian factor group of G. (See Chapter IV.) By definition of G/p, we have G/p = Gab/p. This proves Lemma 8.

Secondly, take i = 2 in formula (7). Since G is finite, all groups occurring there are finite; hence they are finite dimensional vector spaces over the field with p elements. Comparing dimensions :

dy) G = dP)Hz(Z) + dim H,(Z) P

In general, if A is any finite abelian group, we have dim A, = dim A/p

which is proved by decomposing A into cyclic direct factors. Applying this to A = HI(Z) and using formula (8) we obtain

dimHi P = dimH,(Z/p) = dip’ G.

This proves Lemma 9. Using Lemmas 8 and 9, we see that the inequality (6) to be proved is

equivalent to the following.

THEOREM 10. Let G be a finite p-group, p a prime number. Then

d%’ G > *(dip’ G)’ ,

where dp’G = dim H.(G Z/p). 1 3 We use the following notations:

A = Z(G) is the integral group ring of G; I is the augmentation ideal of A, defined to be the kernel of the

augmentation map A + Z; A/p is the group ring of G over the field of p elements.

First we prove two preparatory lemmas.

LEMMA 11. Let G be a jinite p-group and A a G-module with pA = 0. Then the minimal number of generators of A as G-module equals

dim H,(G, A) = dim A/IA,

the dimension to be understood over theJield with p elements.

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238 PETER ROQUETTE

More precisely: Let a, E A. Then the a, generate A as a G-module if and only if their images a, in A/IA generate AJIA as a vector space.

Proof. Assume the a, generate A/IA. Let B be the G-submodule of A generated by the a,. Then the natural map B/IB + A/IA is epimorphic, which is the same as the map H,(B) + H,(A). The exact sequence

O-+B-+A-,A/B-+O

yields the exact homology sequence

H,(B) + H,(A) + H&W) --) 0

from which we infer that H,(A/B) = 0. Since A/B is annihilated by p and G a p-group, it follows A/B = 0, i.e. A = B. (See Chapter IV, Q 9.) QED.

LEMMA 12. Let G be ajinite p-group and A a G-module with pA = 0. Then there exists a resolution

. ..+Y.-+Y,+Y,-+A+O

with the following properties:

(i) Each Y, is a free module over Alp. (ii) The number of free module generators of Y, over h/p equals

dim H,,(G, A). (iii) The image im( Y,,+ 1) is contained in I. Y,.

Proof. Put d = dim H,(G, A). By Lemma 11 there is a free A-module X on d generators and an epimorphism X+ A. Since pA = 0 we obtain by factorization an epimorphism X/p + A. Put Y = X/p. Then Y is a free A/p-module on d generators. In particular,

Hi(Y) = Ofor i 2 1.

Let B be the kernel of Y+ A so that

O+B+Y+A+O

is exact. The exact homology sequence

. . . jH,+,(Y)jHl+,(A)jH,(B)-Hi(Y>j... yields

(9) Hi(B) = Hi+ I(A) for i 2 1. As to the case i = 0, we have the exact sequence

0 -B H,(A) + H,(B) --) H,(Y) + H,(A) + 0.

By construction, Y and A have the same minimal number of generators as G-modules; hence dim Ho(Y) = dim H,(A) by Lemma 11 which shows that H,,(Y) + H,,(A) is an isomorphism. We conclude that (9) holds also in the case i = 0.

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ONCLASSFIELDTOWERS

Since I&(Y) + H,,(A) is an isomorphism, the map

B/IB = H,(B) + l&,(Y) = Y,UY is zero; hence

(10) Bc1.Y.

239

Now put Y = Y,,. Then Y,-, + A --, 0 is the first step in the resolution to be constructed. The second step is obtained by applying the same procedure to B. We obtain YI + B+ 0 with a kernel C such that

Hi(C) = Hi+l(B) = Hi+2(A)

for i 2 0, and c c I. Yi.

YI is a free A/p-module on dim H,,(B) = dim H,(A) generators. If we define YI + Y0 to be the composite map Y, --f B + Y,, then (10) says that im( YJ c I. YO.

Continuing this process, we obtain Lemma 12 by induction. QED. Let us apply Lemma 12 to the case A = Z/p. Then H,(G,Z/p) = Z/p is

of dimension 1. Hence Y0 is free with one generator. By (iii), the kernel of Y,-, + Z/p is contained in I. YO. Since YJI. Y0 is of dimension 1, this kernel is precisely I. YO. Hence YI + I. Y0 --f 0 is exact. Changing notation, we obtain the

COROLLARY 13. Let G be afinite p-group and put d = d’,P) G, r = dp) G.

Then there is an exact sequence

R+D+I.E+Owithim(R)cI.D

where E, D, R denote the free Alp-modules with 1, d, r generators respectively. Proof of Theorem 10: We use the abbreviations of Corollary 13. For any finite G-module A with pA = 0 we introduce the PoincarC

polynomial

PA(t) = c C”(A). t” with c,(A) = dim Z”A/Z”+‘A. OSII

(Note that c,(A) = 0 if n is sufficiently large.) We put P(t) = PE(t).

Since c,(E) = dim E/IE = 1 we have P(t) - 1

Pm(t) = -y

Also, since D = Ed is the d-fold direct product of E, we have c,(D) = d. c,,(E) and therefore

PO(t) = d. P(t).

Similarly, PR(f) = r . P(t).

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240 PETER ROQUETTE

In general, if 0 c t c 1 is a real variable, then

PAM 1-t ! =o~~(A)t” with s,(A) =,~&c,(A) = dim A/I”+lA.

Also,

where we have to interpret s-,(A) = 0. From Corollary 13 we obtain an epimorphism I“+ ‘D + I”+‘E. Hence,

ifR,+l denotes the foreimage of I”+‘D in R then the sequence

0 + R/R,+I --, DIl”+‘D --) IE/I”+‘E -B 0

is exact. This gives

s,,(D) = s,,(IE) + dim R/R, + i.

By Corollary 13, image(R) c ID, hence image (PR) c Z”+‘D, I”R c R,, i. Therefore,

This gives

dim R/R,+l 4 dim RII”R = s,-,(R).

s,(D) 5 s,UE) + s, - I(R).

The numbers occuring in these inequalities are the coefficients of power series as given above. It follows that

or equivalently

P(t)- 1 d.PW I t - + r.t.P(t) if 0 <t < 1.

In other words: 1 s P(t).(rtG&+l) if 0 < t < 1.

Since P(t) has positive coefficients, we conclude that O<rt2--dt+l if o<t<1.

Substituting t + d we get 2r

r>)d’

d as contended. The substitution t + -

2r is permissible since we know from

Lemma 9 that d 5 r < 2r, hence 0 < f < 1. QED

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ON CLASS FIELD TOWERS 241

Remark 14. Actually, Golod and SafareviE have proved in their paper only dip)G > f(djP)G- 1)2. The inequality as given here has been obtained independently by Gaschiitz and Vinberg.

Remark 15. Theorem 10 is important not only for the class field tower problem but also for various other problems in the theory of p-groups. This is due to the fact that the homological invariants &“G and dp)G permit the following group theoretical interpretations if G is a finite p-group : dip’G is the minimal number of generators of G (Bumside’s basis theorem; see Chapter V, $ 2.5) and d(ZP) is the minimal number of relations among these generators which define G as a p-group (see Serre, “Cohomologie Galoisienne”).

3. Proof of Theorem 5 for Galois Extensions

Let k/Q be a Galois extension of finite degree n, and p a prime number. As said in Section 1, we are going to prove the inequality

d(P) Cl, 2 $0 - ( rL -1

P-l + w,(n). Sp)

> .

Again, the proof will be divided into two parts. First, we shall reduce the proof to a group theoretical statement about the cohomology of finite groups. Secondly, we shall prove this group theoretical statement.

LetK = kl be the Hilbert class field of k; G the Galois group of KJk; G* the Galois group of K/Q ; g the Galois group of kJQ.

Then g = G*/G and we have the inflation-restriction sequence of cohomo- logy groups

1 -R H’(g, &) + H’(G*,E,) + H’(G, I&)

where Ek as in Section 2 denotes the group of units in k. We conclude that

dP’H’(G*, EJ 6 dP’H’(G, EJ +dP’H1(g, EJ.

Hence the inequality (11) to be proved is an immediate consequence of the following three statements :

(12) H’(G, E3 = Cl,

03) dP’H’(G*, EK) = rp’

(14) r,-1

&‘H’(g, J&) 5 - P-1

+ w,(n). sp.

We are going to prove these three statements. For any algebraic number field K of finite degree we consider the com-

mutative and exact diagram:

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242 PETER ROQUETTE

i r t

l-+ EK -+UK-+U~IE~-+l

1 1 1 l+K” + I, -+ c, + 1

1 1 1 l-+P,-+D,-+ Cl, +1

1 1 1 1 1

where KX is the multiplicative group of K;

IK the idble group; DR the divisor group; PR the group of principal divisors ;

and the other groups have the same meaning as before in Section 2. The arrows denote the natural maps. (In Section 2 we have used already the first row and the last column of this diagram.)

If K is a Galois extension of a subfield k with Galois group G then all groups and maps of this diagram are G-permissible and we obtain a corre- sponding commutative and exact cohomology diagram:

z i i l+ EK --, Uk + (WEdG --) fJ’(G, Ed + H ‘(G, UK)

1 cl c JY d 1 Q l+ k” --* Ik -+ C, -+ 1

cl l+ j” + 0; : C$

i 1 = H'G &I + H'(G, U,> 1 (" 1 1 1

where we have twice used Hilbert’s Theorem 90:

H’(G,KX) = 1

and also the corresponding local statement :

H’(G,IJ = 1.

LEMMA 14. Let K be an algebraic number$eld offinite degree which is a Galois extension of a subfield k with Galois group G.

Assume 0: c P,, i.e. that every invariant divisor is principal. Then there is a natural exact sequence

l-+C1,+iY’(G,E&-+H1(G,UK)+l. Q

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ON CLASS FIELD TOWERS 243

ProoJ We use the cohomology diagram as explained above. Observe that the natural map

cp : H’ (G, EK) + H’ (G, U,)

occurs twice in this diagram. We have to show that, under the assumption of Lemma 14, cp is epimorphic and its kernel is Clk.

The assumption in Lemma 14 means Pg = Dg. Looking at the left lower corner of our cohomology diagram we see there-

fore that cp is epimorphic. On the other hand, Pg =I$ shows that a= 1. Hence tlOr=jIOq= 1.

Since q is epimorphic, /I = 1. Hence y is an isomorphism. Looking at the right upper corner of our diagram we see that

kernel (cp) = image (6) = (UK/,??K)G/image (E);

the isomorphism y shows that this equals

C,Jimage (y 0 s) = C,/image (rl 0 c) = IJk. lJ, = Cl,. QED.

Proof of (12): We use Lemma 14 in the case where (as in (12)) K is the Hilbert class field of k. Since K/k is unramified, we have H’(G, UK) = 1, as already observed in Section 2. In view of Lemma 14 it suffices therefore to show that the assumption of Lemma 14 is satisfied if K is the Hilbert class field of k.

Since H’(G, UK) = 1 we infer from our cohomology diagram above that c : I, + 0: is epimorphic. On the other hand, its image is I,JU, = Dk. Hence D, = DE. Now, the principal divisor theorem for the Hilbert class field says that Dk c PK. QED.

Proof of(13): We want to apply Lemma 14 to the Galois extension K/Q, where K (as in (13)) is the Hilbert class field of k. First we have to show that the assumption of Lemma 14 is satisfied for K/Q.

G* is the group of K/Q and G the group of K/k. We have therefore G c G* and Dr c 0:. The foregoing proof shows 0: c PK. Hence Dr c P,.

Since Cl, = 1 we infer therefore from Lemma 14 an isomorphism

H’(G*, EK) = H ‘(G*, U,).

For any finite prime q of Q let e,(q) denote the ramification index of some extension 61 of q to K. (Since K/Q is Galois, this does not depend on the choice of fllq.) Let Z/e,(q) denote the cyclic group of order e,(q). Then

11 ‘(G*, UK> = n Z/e,(q).

(For the proof, decompose U, into its loci1 components and prove the similar fact for a local Galois extension and its unit group.)

Now K/k is unramified and therefore

e&) = ekd

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244 PETER ROQUETTE

for every q. Hence we obtain

H’(G*, J%) = n Z/ek(d. 4

The p-rank of the direct product on the right-hand side is equal to the number of q with pie,(q), i.e. to tp’. QED.

Proof of (14): The torsion group of Ek is the group W, of roots of unity in k which is cyclic of p-rank Sip). Moreover, the unit theory says that EJ W, is a free abelian group on rk - 1 generators. Hence (14) is an imme- diate consequence of the following group theoretical statement :

LEMMA 15. Let G be a finite group of order n and p a prime number. Let A be a finitely generated G-module with torsion group tA, and let p(A) be the number of free generators of AJtA as an abelian group. Then

dP’H ‘(G ~(4

, A) 5 -- + w p-l p

(n). d(P)tA.

Proof. Consider the restriction map

H ‘(G, A) + H’(GfP’, A)

of G to its p-Sylow subgroup Gcp). This is a monomorphism of the p-primary components (Chapter IV, Prop. 8, Cor. 3). In particular,

dCP’H ‘(G, A) $ d”‘H ‘(GCp), A).

Hence we may assume in the following that G = Gtp) is a p-group. (Observe that w (n) = w (ntp)) where n(P) is the order of Gtp’.)

No; considb the exact sequence

O-+tA+A+A/tA+O

and the corresponding exact cohomology sequence

H ‘(G, tA) + H ‘(G, A) + H ‘(G, A/tA).

It follows that

dtP’H ‘(G, A) 5 dCP’H ‘(G, tA) + dCP’H ‘(G, A/tA).

We shall show that

dtP’H’(G tA) I w (n) dCP)tA , --p’

and

dCP’H ‘(G, A/tA) 4 p0 p-l’

In other words: We shall treat the following two cases separately: (I) A is a torsion module; (II) A is torsion free.

(I) The torsion case: We have

dCP)G = dim G/p 5 w,(n)

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ON CLASS FIELD TOWERS 245

and therefore it suffices to show:

(1% dtP’H ‘(A) S dtP’G. dtP’A.

(For brevity, we omit the symbol Gin the notation of the cohomology groups.) Let Z’(A) be the module of crossed homomorphisms/z G --) A. Let g,, . . . , gd be a minimal system of generators of G; Bumside’s basis theorem forp-groups tells us that d = d@)G. Every crossed homomorphismfis uniquely determined by its values J(gJ, 1 s i 6 d. In other words : The map

f+ tf(sl), - - * s&d))

is an injection of Z’(A) into the d-fold direct product Ad of A. It follows d’p’Z’(A) % d”‘(Ad) = d. (dcp’A) = dtP’G. dtP’A.

Since H’(A) is a factor module of Z’(A), we obtain (15). Remark 16. In the proof just given, we have not made use of the fact that

A is a torsion module. Hence the inequality (15) holds for every &itely generated G-module A. This means that we have proved instead of (14) the inequality

d(p)H ‘(9, EC) 6 wp(n)(rk- 1 + 6!p))

and therefore instead of (11) the inequality

dfp’CI, ;r tp) - wp(n)(rk - 1 + Sip’).

This would suffice to obtain a function c’(n) with d’P’C1, ;r: tp’ - c’(h) ,

as explained in Section 1. Namely, since w#) 5 n - 1, we could take

c’(n) = (n - 1). n.

The following proof, which permits to obtain a better estimate in the torsion free case, is needed only in order to obtain the better estimate c’(n) = 2(n- 1) as stated in Remark 6 of Section 1.

(II) The torsion-free case: We begin with LEMMA 17 (Chevalley). Let G be a group of order p and A a finitely

generated G-module. Then

~(4 - P - p(AG) d’P’H’(G,A)-d’P’H2(G,A) = p-l -.

Proof. (i) Let us put dl -,(A) = dcp)H ‘(A) - d(p)H ‘(A).

Note that p . H’(A) = 0 since G is of order p. Hence dfp)H’(A) is the dimension of H’(A) over the field of p elements; the order h’(A) of H’(A) is therefore p to the power d(P)H’(A). If we define the Herbrand quotient

h,z(A) = h’Wh2t4

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246 PETER ROQUETTE

then it follows that h,,,(A) = pd+2(A) .

We may therefore regard d,-,(A) as an additive analogue to the Herbrand quotient. The properties of the Herbrand quotient, as stated in Chapter IV, 8 8, can therefore be applied mutatis mutandis to dI -2. In particular, we have :

If 0 --* A + B + C + 0 is an exact sequence of G-modules, then

dl-AB) = 4-A-4+4--,(0

We say therefore that dI _ 2 is an additive function of G-modules. Let us say that two finitely generated G-modules A, B are rationally

equivalent if A @ Q and B @ Q are isomorphic as Q(G)-modules. (The tensor product is to be understood over Z and Q(G) = Z(G) QD Q is the group ring of G over Q.) We may regard A/tA as a submodule of A @ Q and in fact as a G-invariant lattice of the rational representation space A @3 Q of G. In view of Propositions 10 and 11, Section 8 of the cohomology course, we have:

If A and B are rationally equivalent, then d, --2(A) = d, _ JB). We say therefore that d,-,(A) is a function of rational equivalence classes.

(ii) For brevity, we define the function

T(A) = p(A)-P.P(A’) p-l *

We claim that z(A) too is an additive function of rational equivalence classes, and have to show that both p(A) and p(A’) are.

It follows from the definition of p that p(A) = dime A@Q and p(AG) = dimo AG@Q.

From the 8rst relation we infer that p(A) is a function of rational equivalence classes. From the second relation we infer the same for p(A’) since

AG@Q=(A@Q)O.

IfO-tA+B+C+Oisexactthen 0 + A@Q -P B@Q+ C@Q+ 0

is exact too. This shows the additivity of p(A). As to the additivity of p(AG), we have to use the fact that

O-,(A~Q)Gj(B~Q)G--t(C~Q)GjO is exact too, since H’(G, A @ Q) = 0 because A @ Q is uniquely divisible.

(iii) We have now seen that dI -2 and r are both additive functions of rational equivalence classes. Furthermore, an easy computation shows

d, emz(Z) = z(Z) = - 1

dl-&I) = t(A) = 0

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ON CLASS FIELD TOWERS 247

where we consider the integers Z and the group ring A = Z(G) as G-modules. Hence the contention of Lemma 17 follows from the following statement:

Every additive function f of rational equivalence classes of finitely generated G-modules A is uniquely determined by its values for the modules Z and A.

(iv) Proof of the statement in (iii): The exact sequence

O+I-FA+Z+O

shows f(l> =f(N -f(Z), so that f(I) is determined by f(Z) and f(A). Hence it remains to be shown that every finitely generated G-module A is rationally equivalent to a direct sum of G-modules which are isomorphic either to Z or to I. In other words:

A@Q=CAi@Q

where A , = Z or A, = I. Since A @ Q as a representation space of G over Q is a direct sum of irreducible representation spaces Vi, this amounts to showing that every irreducible representation space V # 0 of G over Q is either iso- morphic to Z@ Q = Q or to I@ Q.

Now every such V is isomorphic to a direct summand of Q(G) = A @ Q. Hence we have to determine the direct decomposition of Q(G). The exact augmentation sequence

0 + I@Q+Q(G) + Q + 0

splits; the corresponding map Q + Q(G) being given by

x+x.e (xEQ)

where e is the idempotent

e=l C g. Pu=G

We have therefore the direct decomposition

Q(G) = (IC3QNBQ.e

and it remains to show that 1@Q is irreducible, i.e. that I@ Q, as a Q-algebra, is a field.

Let K denote the field of pth roots of unity. Let x denote an isomorphism of G on to the group of pth roots of unity in K. By linearity, x extends uniquely to an algebra homomorphism of Q(G) on to K. Its kernel contains e since the sum of all the pth roots of unity in K is 0. Hence x defines an algebra homomorphism of Q(G)/e = I@ Q on to K. Since

dimoK=p-l=dim,i@Q

it follows that this is an isomorphism of 18 Q on to K. QED.

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248 PETER ROQUJZTTE

LEMMA 18. Let G be a finite p-group and A a finitely generated G-module which, as an abelian group, is torsion free. Then

d’P’H ‘(G, A) 5 p(A;;yG).

In particular, it follows that

d’P’H’(G, A) $ @ P-1

which we wanted to prove. Prooj First let G be of order p. Then Lemma 17 shows

P(A) - P . P(A3 dfP’H ’ (G A) S - 9 - P-l

+ dfP’H ‘(G, A).

Since G is cyclic, H’(G, A) = fi’(G, A) is a certain factor group of AC and has therefore p-rank 5 dCp)AG = p(A’), the latter equality holding since AC is torsion free.

It follows

d’P’H’(G,A) < p(A)-p’p(Ac_) + p(~G) -

P-l which proves our assertion if G has order p.

Now let G be of order 2 p2. Let U be a proper normal subgroup. The inflation-restriction sequence

1 + H ‘(G/U, AU) + H’(G, A) + H’(U, A)

shows

dtP’H ‘(G, A) S d’P’H’(G/U, A”) + dtP’H ‘(U, A).

Using induction, we may assume that

dCp’li ’ (G/U, A”) 5 PC& - dAG> P-l

dCP’H ‘(U, A) 5 ~(4 - PU”) . P-l

Adding, we obtain our contention. QED.

REFERENCES

Brumer, A. (1965). Ramification and class towers of number fields. A&h. moth. J. 12,129-131.

Brumer, A. and Rosen, M. (1963). Class number and ramification in number fields. Nagova math. J. 23,97-101.

Friihlich, A. (1954). On fields of class two. Proc. Land. math. Sot. (3) 4, 235-256. Frohlich, A. (1954). On the absolute class group of abelian fields. J. Lond. math. Sot.

29,211-217.

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ON CLASS FIELD TOWERS 249

Frohlich, A. (1954). A note on the class field tower. Q. JI. Math. (2) 5, 141-144. Friihlich, A. (1962). On non-ramified extensions with prescribed Galois group.

Mathematika, 9,133-l 34. Golod, E. S. and SafareviE, I. R. (1964). On class field towers (Russian). Zzv. Akud.

Nauk. SSSR, 28, 261-272. English translation in Am. math. Sot. Transl. (2) 48, 91-102.

Hasse, H. (1926). Rericht tiber neuere Untersuchungen und Probleme der Theorie der algebraischen Zahlkorper, Teil 1. Jber. dt. Matverein. 35, in particular p. 46.

Iwasawa, K. (1956). A note on the group of units of an algebraic number field. J. Math. pures appl. 35, 189-192.

Koch, H. (1964). Uber den 2-Klassenkorperturm eines quadratischen Zahl- korpers. J. reine angew. Math. 2141215, 201-206.

SafarevE, I. R. (1962). Algebraic number fields (Russian). Proc. Znt. Congr. Mufh. Stockholm, 163-176. English translation by H. Alderson in Am. math. Sot. Transl. (2) 31, 25-39.

SafareviE, I. R. (1963). Extensions with prescribed ramification points (Russian with French summary). Inst. Hautes Etudes Sci. Publ. Math. 18, 71-95.

Serre, J.-P. (1963). “Cohomologie galoisienne”. (Lecture notes, Paris 1963, reprinted by Springer, Berlin.)

Scholz, A. (1929). Zwei Remerkungen zum Klassenkorperturmproblem. J. reine angew. Math. 161, 201-207.

Scholz, A. and Taussky, 0. (1934). Die Hauptideale der kubischen Klassenkorper imaginiirquadratischer Zahlkorper usw. J. reine angew. Math. 171, 19-41.

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CHAPTER X

Semi-Simple Algebraic Groups

M. KNESER

Introduction . . . . . . . . . 1. Algebraic Theory

1. Algebraic groups ‘over an algebraically ciosed field : 2. Semi-simple groups over an algebraically closed field 3. Semi-simple groups over perfect fields . . .

2. Galois Cohomology . . . . . . . 1. Non-commutative cohomology . . . . 2. K-forms . . . . . . . . 3. Fields of dimension < 1 . . . . . 4. p-adic fields . . . . . . . 5. Number fields . . . . . . .

3. Tamagawa Numbers. . . . . . . Introduction . . . . . . .

1. The Tamagawa measure . . . . . 2. The Tamagawa number . . . . . 3. The theorem of Minkowski and Siegel . . .

References . . . . . . . . .

. 250

. 250

. 250

. 252

. 253

. 253

. 253

. 254

. 255

. 255

. 257 . 258 . 258 . 258 . 262 . 262 . 264

Introduction Section 1 contains the basic definitions and statements of some fundamental results of an algebraic nature. Sections 2 and 3 are devoted to certain arithmetical questions relating to algebraic groups defined over local or global fields. Throughout, very few proofs are given, but references are given in the bibliography.

For more information on recent progress in the theory of algebraic groups, the reader is referred to the proceedings of the following conferences:

Colloque sur la ThCorie des Groupes Algebriques, Bruxelles, 1962. International Congress of Mathematicians, Stockholm, 1962. Summer Institute on Algebraic Groups and Discontinuous Subgroups,

Boulder, 1965.

References to the bibliography are at the end of this Chapter.

1. Algebraic Theory Basic reference (Chevalley, 1956-58).

1. Algebraic groups over an algebraically closed jieId (Chevalley, 1956-58 and Rosenlicht, 1956)

Let k be an algebraically closed field. An algebraic group defined over k is 250

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SEMI-SIMPLE ALGEBRAIC GROUPS 251

an algebraic variety G defined over k, together with mappings (X,JJ)HXJJ of G x G into G and x H X-I of G into G which are morphisms of algebraic varieties and satisfy the usual group axioms. From now on, the words group, subgroup, etc., shall mean algebraic group, algebraic subgroup (i.e. closed subgroup with respect to the Zariski topology), etc.

An algebraic group G is linear if it is alline as an algebraic variety. For example, GL,(k) is a linear algebraic group, since it can be represented as the set of points (xr,,u) in k”‘+l which satisfy the equation y . det (xi,) = 1. Hence any (closed) subgroup of GL,(k) is a linear algebraic group, and it is not difficult to show that conversely every linear algebraic group is of this form. The additive and multiplicative groups of k, considered as one-dimensional algebraic groups, are denoted by GO and G,,, respectively; they are both linear (G,,, = GL,). A torus is a product of copies of G,,,. A linear algebraic group G is unipotent if every algebraic representation of G consists of unipotent matrices (i.e. matrices all of whose latent roots are 1). A connected algebraic group which is projective as an algebraic variety is an abelian variety. Every abelian variety is commutative. For example, any non-singular projective elliptic curve carries a structure of an abelian variety.

If G is any algebraic group, the connected component GO of the identity element of G is a normal subgroup of finite index. GO has a unique maximal connected linear subgroup G1, which is normal, and Go/G1 is an abelian variety (Chevalley’s theorem: proof in Rosenlicht, 1956). G1 has a unique maximal connected linear solvable normal subgroup Gz, called the radical of G1, and G1/G2 is semi-simple, i.e. it is connected and linear and its radical is (1). G, has a unique maximal connected unipotent subgroup G3, which is normal in G,, and Gz/G3 is a torus. G3 is called the unipotent radical of Gz.

Thus we have the following chain of subgroups of G:

Y finite connected G,

I abelian variety linear G,

semi-simple solvable G,

torus G3

unipotent (1)

Example. G = GL,,. G is linear and connected, hence G = Gr. The radical Gz is the centre of GL,, consisting of the diagonal matrices (z G,), and G/G2 is simple.

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252 M. KNESER

We shall concentrate our attention mainly on sem -simple groups.

2. Semi-simple groups over an algebraically closed field (Chevalley, 1956-58)

Let k be an algebraically closed field, and G a semi-simple group defined over k. A Cartan subgroup of G is a maximal torus, and a Bore1 subgroup of G is a maximal connected solvable subgroup. For example, if G is the special linear group SL,, then the group of diagonal matrices contained in G is a Cartan subgroup (it is clearly a torus, and it is maximal because it is equal to its centralizer), and the group of upper triangular matrices in G is a Bore1 subgroup.

A homomorphism cp: G --) His an isogeny if G, H are connected groups of the same dimension and the kernel of cp has dimension zero (i.e: is finite). It follows that cp is surjective, since cp(G) is connected and of the same dimen- sion as H. (For example, SL, + PGL, is an isogeny : its kernel is the group of nth roots of unity.) If the characteristic of k is zero, the kernel of rp is contained in the centre of G. In characteristic p > 0 there are unpleasant phenomena connected with the Frobenius automorphism. For example, let G = H = SL,, and let cp be the mapping x = (xii) I-+ (x;), which is an isogeny. If the xu are in the field k and q(x) = 1, then x = 1; cp is bijective but is not an isomorphism, since cp- ’ is not algebraic. If we allow the xi, to be, say, dual numbers (that is, elements of the k-algebra k[e], where s2 = 0), then x = 1 +sy is in the kernel of cp for every y E G, but is not in the centre. We want to exclude this type of phenomenon, so we define a central isogeny to be an isogeny whose kernel is contained in the centre, for points with coor- dinates in any k-algebra. If cp : G + H is a central isogeny, G is said to be a central covering of H.

Among the groups which centrally cover G there is a maximal one, G, which admits no proper central covering, and among the groups centrally covered by G there is a minimal one, G, which centrally covers no other group. 6 is called simply connected, and G is the aa’joint group of G. Example : G = SL,, G = PGL,,.

In order to classify semi-simple groups, it is enough to classify the simply- connected groups, and then to look for the possible central isogenies.

Every simply-connected group is uniquely expressible as a product of almost simple groups (i.e. groups G with finite centre C such that G/C is simple). SL, is an almost simple group.

The classification of almost simple groups is the same as for simple Lie groups :

4,: %,+I G: SP2.

B,, D,: isogenous to orthogonal groups E6, E ,,E8, F4, G, : exceptional groups.

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SEMI-SIMPLE ALGEBRAIC GROUPS 253

3. Semi-simple groups over perfect fields (For groups over non-perfect fields, see Demazure and Grothendieck, 1963-64)

Let k be a perfect field, with everything defined over k: this means that the coefficients of the defining equations lie in k. A k-torus is an algebraic group defined over k which over the algebraic closure k of k becomes a torus, i.e. is isomorphic to a product of groups G,,, = GL, ; it is split over k if it is iso- morphic to such a product over k. A semi-simple group is said to be split over k if it has a maximal k-torus which is split over k; it is quasi-split over k if it has a Bore1 subgroup defined over k. As this terminology indicates, a split group is quasi-split. For example, SL, is split, and the group of elements of norm 1 in a quaternion division algebra over k is neither split nor quasi- split.

In the decomposition of a simply-connected semi-simple group G as a product x G, of almost simple groups, the factors Gi and the isomorphism

~~~& are defined over li, not necessarily over k. The Galois group r = Gal (E/k) permutes the G,, and the product of the Gi belonging to a given transitivity class is a group defined over k. Thus we may assume that r permutes the G1 transitively. The isotropy group of G1 is a subgroup of finite index in l?, whose fixed field is a finite extension I of k, and Gr is defined over 1. G is said to be obtained from Gr by restriction of thejieid of definition from I to k : G = Rllk(G1) @Veil, 1961). This procedure allows one to reduce many questions to the case of almost simple groups, so we shall assume from now on that G is almost simple.

In view of the classification of almost simple groups over E mentioned in Section 2, it is natural to ask whether, for each of the types A,, . . ., G2, there exists a group defined over k. In fact it is true that for each type there is a group G defined over k and split over k; and G is unique up to k-iso- morphism if we require in addition that it should be simply-connected or adjoint (Chevalley, 1955; Demazure and Grothendieck, 1963-64). The classification of the various k-forms of G, i.e. of groups G’ isomorphic to G over k, is a problem of Galois cohomology.

2. Galois Cohomology

Basic reference (Serre, 1964).

1. Non-commutative cohomology

Let G be a group and let A be a G-group, i.e. a (not necessarily commutative) group on which G acts. (If G is profinite we require that the action of G should be continuous with respect to the discrete topology on A.) The action of s E G on a E A is denoted by “a.

We define H’(G, A) to be the group AG of G-invariant elements of A. Next

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254 M. KNESER

we define a cocycle to be a function SH a, on G with values in A which satisfies

a, = as.‘a, (s, f E G).

Two cocycles a,, b, are equivalent if there exists c E A such that

b, = c-l.as.‘c (s E G).

The set of equivalence classes of cocycles is the cohomology set H’(G, A). This is a set with a distinguished element, namely the class of the identity cocycle.- Both H’(G, A) and H’(G, A) are functorial in A.

~Ef 14 A + B + C + 1 is an exact sequence of G-groups and G-homo- morphisms, then one can define a coboundary map H’(G, C) + H’(G, A), and the sequence (of sets with distinguished elements)

1 + H”(G, A) + H’(G, B) + H’(G, C) + H’(G, A) --, H’(G,B) + H’(G, C)

is exact. (Exactness is to be understood in the usual sense, the kernel of a map being defined as the inverse image of the distinguished element of the image set.) If A is contained in the centre of B (so that A is abelian and therefore H’(G, A) is defined) then one can define a coboundary map H’(G, C) + H’(G, A) which extends the exact sequence one stage further.

Let K be a perfect field and L a Galois extension of K, with Galois group G. Let A be an algebraic group defined over K, and A, the group of points of A which are rational over L. Then AL is a G-group, and we write H’(L/K, A) in place of H’(G, AL). If R is the algebraic closure of K, we write H’(K, A) in place of H’(K/K, A).

2. K-forms Let V, V’ be algebraic varieties with some additional algebraic structure (for example, algebraic groups, or vector spaces with a quadratic form, or algebras) all defined over a perfect field K. Suppose that f: V+ v’ is an isomorphism for the type of structure in question, defined over a Galois extension L of K. Ifs is any element of the Galois group G of L/K, then s transforms f into an isomorphism “f: V-, V’, and a, = f -’ o “f is a cocycle of G with values in Aut (V). It is easily seen that such a “K-form” v’ of V is K-isomorphic to another K-form V” of V if and only if the corresponding cocycles are equivalent, and therefore we have an injective mapping of the set of K-forms of V (modulo K-isomorphism) into the cohomology set H’(L/K, Aut (V)). In many important cases (algebraic groups, vector spaces with a finite number of tensors) this mapping is bijective. In such cases the problem of classifying K-forms is solved by (i) classification over the algebraic closure R of K, and (ii) computation of H’(K, Aut (V)).

Consider for example the classification of K-forms of simply-connected semi-simple linear algebraic groups. As we have seen in Section 1, it is enough to consider almost simple groups, and then the classification over

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SEMI-SIMPLE ALGEBRAIC GROUPS 255

R is known. Let G be the split group of a given type; then the group of inner automorphisms of G is isomorphic to the adjoint group G rG/C where C is the centre of G, and the quotient Aut (G)/G is isomorphic to tne group Sym (G) of symmetries of the Dynkin diagram of G. So we have an exact sequence \\~

l+G+Aut(G)+Sym(G)-+l

and the derived cohomology sequence:

H’(K, Sym(G)) + H’(K, G) + H’(K, Aut (G)) I H’(K, Sym (G)).

The Galois group g of K/K operates trivially on Sym(G), so that a cocycle of g with values in Sym (G) is a homomorphism g + Sym(G), and therefore H’(K, Sym(G)) is a quotient of the set Horn (g, Sym(G)). In fact cp is surjective, and for each u EH’(K, Sym(G)) the fibre cp-‘(cr) contains a quasi-split K-form G, of G, which is unique up to K-isomorphism (Steinberg, 1959); and the elements of cp-‘(or) correspond to those K-forms G’ of G such that there exists an isomorphism f: G, + G’, defined over it, for which f - ’ O “f is an inner automorphism of G. for all s E g. By a twisting argument (cf. Serre, 1964, I-5.5) qo-‘(or) is seen to be a quotient of the cohomology set W(K, G,&.

3. Fields of dimension < 1

A field K is of dimension < 1 if the Brauer group Br(L) is zero for every algebraic extension L of K. Examples of such fields are all finite fields, and the maximal unramified extension K,, of a field K which is complete with respect to a discrete valuation.

THEOREM 1 (Steinberg, 1965). If K is a perfectfield of dimension < 1, and if G is a connected linear algebraic group defined over K, then H’(K, G) = 1.

In the case where K is finite, this theorem was proved earlier by Lang.

From Steinberg’s theorem and the results stated in Section 2, it follows that cp-‘(a) contains exactly one element, so that every K-form of G is quasi-split, and the K-forms are classified by H’(K, Sym (G)).

4. p-adic fields

If K is a p-adic field (i.e. a completion of a global field with respect to a discrete absolute value) it is no longer true that H’(K, G) = 1 for every connected group G. However, there is the following result:

THEOREM 2 (Kneser, 1965). If K is a p-adicjield and G is a simply-connected semi-simple linear algebraic group defined over K, then H’(K, G) = 1.

The proof in Kneser (1965) proceeds by reduction to the case of almost simple groups and then deals with each type A,, . . . , Gz separately. For the classical groups the theorem is closely related to known results on the

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256 M. KNESER

classification of quadratic, hermitian, etc., forms over a p-adic field. More recently Bruhat and Tits (1965) have indicated a uniform proof, based on results of Iwahori and Matsumoto (1965).

If G is the si-mply-connected covering group of an orthogonal group, Theorem 2 is essentially equivalent to the classification of quadratic forms over a p-adic field. Let q be a non-degenerate quadratic form in n > 3 variables over a p-adic field K of characteristic +2, let 0 be the orthogonal group of q, and SO the special orthogonal group of q. All non-degenerate quadratic forms in n variables are equivalent over the algebraic closure R of K, so the classification over K of non-degenerate quadratic forms in n variables is .equivalent to the determination of H’(K, 0).

We have an exact sequence 1+so+o+~~-+1

where p2 = (k 1); now the map H’(K, 0) + ZZ’(K,&, i.e. Ox--t p2, is surjective, hence we have an exact sequence (of sets)

13 H’(K, SO) : If’@, 0) : H’(K, /.Q. (1) By a twisting technique it may be shown that CI is injective. To compute H’(K,,u,) we use the exact sequence

2

2 1-+/l,+K*+K*+1

where R* + fz* is the map XI-+X~. This exact sequence gives rise to a cohomology exact sequence (of groups)

K* : K* + H’(K, p2) + 1 + 1 -+ H2(K, p2) -P Br (K) t Br (K)

(since H’(K, R*) = 1 by Hilbert’s Satz 90). We deduce that

H’(K, p2) E K*/K*2, H2W, p2)= Br WI2

where Br(K), denotes the group of elements of order 2 in the Brauer group Br(K). (Since BrQ = Q/Z, we have Br(K), = p2.) The exact sequence (1) now becomes

1 -+H1(K,SO):H’(K,0):K*/K*2.

Here d is essentially the discriminant: more precisely, if r E H’(K, 0), d(t) is the discriminant (modulo squares) of the quadratic form defined by r divided by that of q (and therefore d is surjective). The quadratic forms defined over K which have the same discriminant (modulo squares) as q are classified by H’(K, SO).

The simply-connected covering of SO is the spin group: Spin. We have an exact sequence

1+~2+Spin+SO+1

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SEMI-SIMPLE ALGEBRAIC GROUPS 257

and hence a cohomology exact sequence (of sets)

Spin, + Son: K*/K*’ + H’(K, Spin) + H’(K, SO): Br(K)2.

Here 6 is the spinor norm and-d is closely related to the Witt invariant of a quadratic form. From Theorem 2 we have H’(K, Spin) = 1; hence

(i) the spinor norm 6 : SOx --, K*/K*’ is surjective; (ii) every quadratic form with the same discriminant d and the same Witt

invariant as q is K-isomorphic to q. Conversely, (i) and (ii) imply that H’(K, Spin) = 1.

5. Number Jields

Let K be an algebraic number field, G a linear algebraic group defined over K. If v is any prime of K and if K, denotes the completion of K at v, we have mappings

H’(K, G) + H’(K,, G)

obtained by restriction together with the embedding Gx + GKV, hence a mapping

8: H’(K, G) + n H’(K,, G). (1) ”

The theorem of Minkowski-Hasse states that two quadratic forms defined over K are isomorphic over K if and only if they are isomorphic over K, for all v (discrete and archimedean). In terms of Galois cohomology, this theorem is equivalent to the injectivity of the map

6:H’(K,0)+~H1(K,,0) ”

where 0 is the orthogonal group of a quadratic form. The map 1’3 defined by (1) is not always injective, even if G is connected and

semi-simple (Serre, 1964). However, if G is simply-connected, the following theorem seems to hold:

THEOREM 3. Let K be an algebraic number field, G a simply-connected semi- simple linear algebraic group defined over K. Then

0: H’(K, G) + n H1(KU, G) ”

is bijective.

In fact, H’(K,, G) is trivial for all discrete v (Theorem 2), so that Theorem 3 is equivalent to

THEOREM 3’. The mapping

~‘KG)+)-‘jHIUG,G) “EC0

is bijective, where 03 denotes the set of archimedean primes of K.

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258 M. KNEWR

The proof of Theorem 3 proceeds by reduction to the case of almost-simple groups, and then by examination of each type A,, . . . , Gz separately. For the classical groups it is related to known results on quadratic, hermitian, etc., forms. For the exceptional groups other than Es see Harder (1965). The case E, is still oben.

3. Tamagawa N’umbers Basic reference (Weil, 1961).

Introduction

Let K be an algebraic number field, A the addle ring of K. The letter u will denote a prime (discrete or archimedean) of K; K, denotes the completion of K at v, and o, the ring of integers of K, if v is discrete.

Let G be a connected linear algebraic group defined over K. G is isomorphic to a subgroup of a general linear group GL,, and we may therefore regard G as a closed subset of affine m-space (m = nz + 1, see Chapter 1, Section 1) when convenient. The addle group GA of G is the set of all points in A” which satisfy the equations of G. We give GA the topology induced by the product topology on A”, and GA is then a locally compact topo- logical group. GA may also be defined as the restricted direct product of the groups GKU with respect to their compact subgroups G,“, where GKY (resp. G,) is the set of points of G with coordinates in K, (resp. 0,). At first sight, GA appears to depend on the embedding of G in affine space. But it is not difficult to show that if we change the embedding we change G,” at only a finite set of primes v, and hence GA is unaltered.

Examples. If G = G,, then GA = A. If G = G,, then G, is the idele group of K.

Since K is a discrete subgroup of A, it follows that G, is a discrete subgroup of GA. Since G, is locally compact, it has a left-invariant Haar measure, unique up to a constant factor. The product formula shows that right multiplication by an element of GK does not change the measure on GA, and therefore we have an induced left-invariant measure on the homogeneous space GA/GK.

There is a canonical choice of the Haar measure on GA, called the Tamagawa measure z (Section 1). Once this is defined there is the problem of computing the Tamagawa number z(G) = r(GA/GK) (Section 2). This, applied to the case where G is an orthogonal group, is equivalent to the classical arithmetical results of Minkowski and Siegel on quadratic forms (Section 3).

1. The Tamagawa measure

Let V be an algebraic variety of dimension n, defined over K. Let x0 be a simple point of V and let x1,. . . , x,, be local coordinates on V at x0 (not

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SEMI-SIMPLE ALGEBRAIC GROUPS 259

necessarily zero at x0). A differential n-form on V is defined in a neighbour- hood of x0 in V by an expression

co = f(x) dxl . . . dx,

where f is a‘ rational function on V which is defined at x0. The form o is said to be defined over K if f and the coordinate functions xi are defined over K. The rule for transforming o under change of coordinates is the usual one. If cp : W-+ V is a morphism of algebraic varieties, a differential form o on V pulls back to a differential form q*(o) on W.

Suppose now that V is a connected linear algebraic group G. The left translation 1, : x H ax (a E G) is a morphism V + V and therefore transforms a differential form w on G into a differential form n:(w). Thus we can define left-invariant differential forms on G, and the basic fact is that there exists a non-zero left-invariant differential n-form o on G, defined over K, and o is unique up to a constant factor c E K*.

Examples.

G=G,, o=dx; G = G,,, o = dx/x;

G = GL,, o = (n dxij)/(det xij>“.

We shall use o to construct measures o, on the local groups GKY. For this purpose we need to fix Haar measures K on the additive groups K,‘. When K = Q we fix pP (p a fi@te prime) by pJZ,> = 1, and p, we take to be ordinary Lebesgue measure on R. When K is an arbitrary number field, there are various possible normalizations: all we require is that

(i) p(,(o”) = 1 for almost all discrete v, and (ii) if p = n pL, is the product measure on A, then ,u(A,/K) = 1.

(One such normalization is to take ~~(0”) = 1 for all discrete v, and pV to be c, times Lebesgue measure for archimedean u, where the c, are positive real numbers such that

l-I c, = 2’2/dKI-*, "EC0

where 00 denotes the set of infinite primes of K, r2 is the number of complex infinite primes, and dK is the discriminant of K.)

To define oU, we proceed as follows. The rational functionfcan be written as a formal power series in tj = xi-x: with coefficients in K. If the xp are in K,, thenfis a power series in the xi with coefficients in K,, which converges in some neighbourhood of the origin in K,“. Hence there is a neighbourhood U of x0 in Gk, such that cp :xH(tl,. . . , t,,) is a homeomorphism of U onto a neighbourhood U’ of the origin in K,“, and such that the above power series converges in 27’. In U’ we have the positive measure If(t)l,dtl . . .dr, (where dtl . . . dt, is the product measure pV x . . . x pL, on K,“); pull this back

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260 M. KNESER

to U by means of cp and we have a positive measure w” on U. Explicitly, if g is a continuous real-valued function on Gx, with compact support, then

I s 9% = u

Lp-‘(0)lf(0l” dt1 * * * et.

The measure o” is in fact independent of the choice of local coordinates xi. In the archimedean case this follows from the Jacobian formula for change of variables in a multiple integral, and in the discrete case from its p-adic analogue.

If the product

“IJ%W

converges absolutely, we define the Tamagawa measure by

7= Y

a”-

Explicitly, if S is a finite set of primes such that co E S and if, for each u E S, U” is an open set in GK” with compact closure, then 7 is the unique Haar measure on GA for which

7 “4u”x l-h, (

= “6s ) = “US

TV x vIjlsdGoJ

If the product

does not converge absolutely, we have to introduce factors to make it converge. A family (A”) of strictly positive real numbers, indexed by the primes v of K, is a set of convergence-factors if the product

n G ‘c&L) “Cm

is absolutely convergent. The Tamagawa measure 7 (relative to the 2”) is then

7=l--J43D,. Y

In either case, 7 does not depend on the choice of the form w. For if we replace o by cw (c E K*) then (cw)” = Jcl”o”, and n ICI” = 1 by the product formula. ”

Let k(v) denote the residue field at v (v discrete) and let G(“) denote the algebraic group defined over the finite field k(v) obtained by reducing the equations of G modulo the maximal ideal p” of 0”. Then it can be shown by a generalization of Hensel’s lemma that for almost all v we have

o”(G,“) = (NV)-” card (G(“) ) kc”) ’ (1) where NV is the number of elements in k(v), and G$& is the group of points of G(“) rational over k(v) .

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SEMI-SIMPLE ALGEBRAIC GROUPS 261

Examples. If G = G,,

o,(GJ = 1. If G = G,,,,

~v(G,,.) = 1 - &; if G = GL,,, then

%(G.S=(l-k)...(l-&); if G = SL,,, then

Since

rI(1 - &) = MS)-1 ” is convergent for Re (s) > 1 but not for s = 1, it follows that the product n o,(G,,$ converges when G = SL, but not when G = GL,. In the latter

&se we may take

as convergence-factors.

PROPOSITION. If G is semi-simple, the product n w,(GJ is absolutely con-

vergent (and so convergence-factors are not needed).

In outline, the proof is as follows (for details, see Ono (1965)). From (1) we have to prove that the product

n {WV)-” card (G&,)1 ”

(2)

is absolutely convergent. Using the theorem (due to Lang) that isogenous groups over a finite field have the same number of rational points we can reduce to the case where G is simple over the algebraic closure R of K i.e. has no non-trivial algebraic normal subgroups defined over K. For such a group the order of Gg!, can be explicitly determined and is of the form

(Nvy{l+O((Nv)-2)~.

The convergence of (2) now follows from the convergence of the zeta-function t&(s) for Re (s) > 1.

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262 M. KNESER

2. The Tamagawa number

The Tamagawa number z(G) is defined to be r(GA/GK).

THEOREM 1 (Bore1 and Harish-Chandra, 1962). If G is semi-simple, z(G) is jinite.

For another proof of this theorem see Godement (1962-63).

THEOREM 2 (Ono, 1965). Let G be a semi-simple group, e its universal (i.e. simply-connected) covering, C the (finite) kernel of the covering map G + G, and let c denote the character group Horn (C, G,,,). Then

z(G) ho(c) -=- a i’(e)’

where ho(e) = Card H’(K, t?) and i’(e) is the number of elements in the kernel of the map H’(K, e) + n H’(K,, c).

Hence it is enough to com;ute one Tamagawa number in each isogeny class, for example r(c). In this direction, there is Weil’s

CONJECTURE. T(G)= 1.

This conjecture is proved for many of the classical groups [(Weil, 1961, 1964, 1965) for all groups of types B, and C,,, and some of types A, and D,; and (Mars, 196?; Weil, 1961) for some exceptional groups]. Langlands gave a non-enumerative proof for all split semi-simple groups (Bruhat-Tits (1965)).

Example. G = SO,, (n > 3). Here C has order 2, and it can be shown that h’(c)/i’(e) = 2. S o in this case z(G) = 1 is equivalent to z(G) = 2. As we shall see (Section 3), z(G) = 2 is equivalent to the theorems of Minkowski- Siegel on quadratic forms.

3. The theorem of Minkowski and Siegel Let K be an algebraic number field, I/ an n-dimensional vector space over K (n > 3), q a non-degenerate quadratic form on V defined over K, G = SO(q) the group of all automorphisms of the vector space V which preserve q and have determinant 1. We regard G as acting on V on the right. We shall identify V with K” by choosing a fixed basis of V; then the elements of G are n x n matrices.

A lattice M in V is a finitely generated o-submodule of V of rank n, where o is the ring of integers of K. In particular L = o” is a lattice, called the standard lattice (with respect to the chosen basis of V). Two lattices M, M’ are isomorphic if M’ = Mx for some x E GK. For each finite prime v, let M, = o, @ M, which is a lattice in K,“. It can be shown that M is uniquely determined by the M, [namely M = n (M, n V)]; that M, = 0; for

” almost all v; and that, conversely, if M, is a lattice in K,” for each finite u,

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SEMI-SIMPLE ALGEBRAIC GROUPS 263

such that M, = 0: for almost all U, then there is a unique lattice A4 in K” whose local components are the M,.

The addle group G,, operates on the set of lattices in K” as follows: if M is a lattice and x E GA, then (MA), = i&x, (observe that, since M, = 0: for almost all u, and X, E G,,” for almost all t), we have (MC), = 0: for almost all v). The orbit of M under GA is the genus of iki, which therefore consists of the lattices in V which are locally isomorphic to M at all finite primes. The class of M is the orbit of M under Gx, i.e. the set of all lattices globally isomorphic to M.

The stabilizer of the standard lattice L in GA is the open subgroup

where co denotes the set of archimedean primes of K. Hence the lattices in the genus of L are in one-one correspondence with the cosets GA-x in GA, and the classes in the genus with the double cosets G,,-xGK in GA. For any semi-simple group, the number of double cosets G,,xGx in GA is finite (Bore1 and Harish-Chandra, 1962). In our special case G = SO(q), this is the well-known finiteness of the number of classes in a genus. Let h denote this number.

We have

r(G) = ~GA/&c) = i+L, xi G&L)

= i$l7K %., xi WW (1)

since 7 is left-invariant. The group xi -’ GA,xI is the stabilizer of LXi in GA. Hence we need consider only one term in this sum, say r(GA, Gx/Gx). We have GA- Gx/Gx z GA-/G,, (since GA- n Gx = GO). Hence

~(GA, G/W = ~GJGo) = 70

where F is a fundamental domain for GO in GA-, i.e. a Bore1 set in GA, which meets each coset xG, exactly once. The projection GA, = G, x G, + G,, restricted to G,, embeds G, injectively in G,, and we may take F to be a set of the form G,x F,, where Fa is a fundamental domain for G, in G,. Hence we have

7(G,m WW = 7(G, x F-1

= vIjTmdG,J x wdG,/Go)

where

“co =“lJp.

If we replace L by Lx,, G,,” is replaced by XI,: G,“xi, V and therefore the term

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264 M. KNESER

o,(G,J is unaltered (since CO is right-invariant as well as left-invariant, for any semi-simple group) ; and GO = Gx n GA, is replaced by

G,(x,) = GK n x; ’ GA, xi.

Hence

‘5(X; lG~,xi GKIGK) =vSI_wu(Go,,l X ~m(Gce/Go(Xi))

and therefore, from (1) and (2),

(2)

Suppose now that the form q is totally definite, i.e. that each archimedean v is real and that q is definite at each archimedean completion K,. Then

G, = n G, 06 00

is a product of real orthogonal groups and is therefore compact, and G&K,) is the group of units E(Lxi) of the lattice Lx*, i.e. the group of all integer matrices which transform LXi onto itself, and is finite. (If S, is the matrix of q with respect to a basis of Lxi, then E(LXi) is the group of all integer matrices X such that xl&X = S,; since S, is definite, E(LXi) is clearly finite.) Hence (3) now takes the form

E l I= I Card WW)

=-Ll-+L=L. %A%) u I m dG,J 7GJ

(4)

When K= Q, this is equivalent to Minkowski’s formula for the weight of a genus of definite quadratic forms.

Similarly, we can recover Siegel’s theorem (1935-37) on the number of representations of one quadratic form (in n variables, say) by another (in m variables), by considering an m-dimensional vector space V endowed with a non-degenerate quadratic form q, an n-dimensional subspace W and the Tamagawa number of the group G of all q-orthogonal transformations which induce the identity on IV. For details see Kneser (1961) and Weil (1962).

REFERENCES

Bore], A. and Harish-Chandra (1962). Arithmetic subgroups of algebraic groups. AM. Math. 75, 485-535.

Bruhat, F. and Tits, J. (1965). A.M.S. Summer Institute on Algebraic Groups and Discontinuous Subgroups, Boulder. (To appear.)

Chevalley, C. (1955). Sur certains groupes simples. Tohoku math. J. 7, 14-66.

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SEMI-SIMPLE ALGEBRAIC GROUPS 265

Chevalley, C. (1956-58). Classification des groupes de Lie algebriques. SPminuire E. N.S.

Demazure, M. and Grothendieck, A. (196364). Schemes en groupes. Sbminaire de Giometrie Algibrique I.H.E.S.

Godement, R. (1962-63). Domaines fondamentaux des groupes arithmetiques. S&ninaire Bourbaki, No. 257.

Harder, G. (1965), l%er die Galoiskohomologie halbeinfacher Matrizengruppen, I. Math. 2. 90, 404-428; and II, (1966), 92, 396-415.

Iwahori, N. and Matsumoto, H. (1965). On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Publs. math. ht. Etud. scient., No. 25.

Kneser, M. (1961). DarstellungsmaBe indefiniter quadratischer Formen. Math. Z. 77, 188-194.

Kneser, M. (1965). Galois-Kohomologie halbeinfacher algebraischer Gruppen tiber p-adischen Korpern. Math. 2. 88,40-47; and 89, 250-272.

Mars, J. M. G. (196?). Les nombres de Tamagawa de certains groupes excep- tionnels. (To appear.)

Ono, T. (1965). On the relative theory of Tamagawa numbers. Ann. Math. 82, 88-111.

Rosenlicht, M. (1956). Some basic theorems on algebraic groups. Am. J. Math. 78, 401443.

Serre, J.-P. (1964). “Cohomologie Galoisienne”, Lecture notes, College de France, 1962-63, 2nd edition, Springer-Verlag.

Siegel, C. L. Uber die analytische Theorie der quadratischen Formen. Ann. Math. (1935), 36, 527-606; (1936), 37,230-263; (1937), 38,212-291.

Steinberg, R. (1959). Variations on a theme of Chevalley. Pucif. J. Math. 9,875-891. Steinberg, R. (1965). Regular elements of semi-simple algebraic groups. Publs.

math. Inst. ht. Etud. scient., No. 25. Weil, A. (1961). “Adeles and Algebraic Groups”, Lecture notes, Princeton. Weil, A. (1962). Sur la theorie des formes quadratiques. Colloque sur la ThCorie

des Groupes Algebriques, Bruxelles, 9-22. Weil, A. (1964). Sur certains groupes d’operateurs unitaires. Acta Math. Stockh.

111, 143-211. Weil, A. (1965). Sur la formula de Siege1 dans la theorie des groupes classiques.

Actu Math. Stockh. 113, l-87.

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CHAPTER XI

History of Class Field Theory

HELMUT HASSE

1. The notion of cZassfieZd is generally attributed to Hilbert. In truth this notion was already present in the mind of Kronecker and the term was coined by Weber, before Hilbert’s fundamental papers appeared.

Kronecker, in his large treatise “Grundziige einer arithmetischen Theorie der algebra&hen Griissen” of 1882, discusses at length the “zu assoziierenden Gattungen” (species to be associated). What he means by this is, in modern terminology, an algebraic extension K of a given algebraic number field k such that all divisors? in k become principal divisors in K. By his investi- gations he had found that for imaginary quadratic fields k the “singular moduli” generate such extensions K. By this notion Kronecker anticipates the principal divisor theorem of class field theory, stated and in special cases proved by Hilbert.

Weber (1891, 1897a, b, 1898, 1908), however, did not define the notion of class field on this basis, a basis which, as we know today, is unsuitable for building the theory. What he postulated is part of the law of decomposition. Whereas Hilbert in his later definition considered only the case of absolute divisor classes, Weber gave his definition in full generality, viz.

Let k be an algebraic number field and A/H a congruence divisor class group in k. An algebraic extension K/k is called cIassJield to A/H, if exactly those prime divisors in k of first degree which belong to the principaI ‘class H split completely in K.

In order to define Weber’s notion of a congruence divisor class group AJH in k, consider integer divisor moduli m in k, i.e. formal finite products com- posed of finite prime spots (prime divisors) of k with positive integer expo- nents and real infinite prime spots of k with exponents 1. Call a number or a divisor in k prime to m if it is prime to all prime divisors contained in nt, and understand congruence mod m as congruence modulo all prime divisor powers contained in m and equality of sign for all real prime spots contained

t I use throughout the term “divisor” instead of the classical term “ideal”, because valuation theory, derived from the Kronecker-Hensel notion of divisor, has been nowadays adopted widely as a more suitable foundation of algebraic number theory than Dedekind’s ideal theory.

266

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HISTORY OF CLASS FIELD THEORY 267

in nt. Further consider quotient groups A,,/H,,, where A,,, is the group of all divisors in k prime to m for a given integer divisor modulus m and H,,, is a subgroup containing all numbers a = 1 mod nt in k (considered as principal divisors in k). Call two such quotient groups A,,,,/H,,,, and A,,/H,,,, “equal” to each other if for the least common multiple m = [m,, nt2] (and hence for every common multiple of ml and ntJ there holds the equality Hm, n Am = H,,,, n A,,,, which implies the isomorphy A,,,/H,,,, LZ A,,/H,,,,. Each set of quotient groups A,,/H,,, A,,/H,,,,,. . . that are all “equal” to each other defines a congruence divisor class group A/H in Weber’s sense. The single quotient groups A,,/H,,, A,,/H,,,,,. . . are called the interpre- tations (Erklirungen) mod m,, mod m2,. . . of A/H. The lowest possible interpretation modulus (Erklirungsmodul) f, which turns out to be the greatest common divisor of all possible m,, m,,. . . is called the conductor (Fiihrer) of A/H. Occasionally the single interpretations are also called congruence divisor class groups.

Besides the theorems on class fields proved by Weber which will be adduced presently, in the cases he treated he further observed also the fundamental isomorphy theorem :

The Gulois group @(K/k) is isomorphic to the CIUSS group A/H, and hence is sureIy ubeliun.

Already Kronecker (1853, 1877) knew that the cyclotomic fields are class fields in the above sense. He stated his famous completeness theorem:

Every ubeliun field over the rational number field is a cyclotomic field, and hence is a ClussJield.

This was first proved completely by Weber (1886, 1887) and later more simply by Hilbert (1896, 1897); further proofs were given by Weber (1909), Speiser (1919) and Delaunay (1923).

Further Kronecker (1883-1890), by his investigations on modular func- tions and elliptic functions with “singular moduli”, had ascertained that their transformation and division equations generate relatively abelian fields K over imaginary quadratic fields k. It was his “liebster Jugendtruum” (1880) (dearest dream of his youth) to prove also in this case the completeness theorem, that every relatively ubeliun jield K over an imaginary quadratic field k is obtained by such transformation and division equations. This was proved later, first partially by Weber (1908) and Fueter (1914), then com- pletely by Takagi (1920) and once more by Fueter (with Gut) (1927).

Weber (1897, 1898) conceived his notion of class field from those examples. By making use of Dirichlet’s analytic means (L-series) he deduced from his class field definition thefirst? fundamental inequality of class field theory:

[A:H]=h<n=[K:k],

t In modern terfninology: second.

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268 HELMUT HASSE

further the uniqueness theorem:

H = H’cs K = K’,

the ordering theorem:

Hz H’csKEK’,

and the following presupposition for the isomorphy theorem:

KJk is normal.

From the introduction to Weber (1897, 1898) it is safe to conclude that he was convinced of the validity of the general existence theorem:

To every congruence divisor cIass group A/H in k there exists a class field K/k.

He points out that from the existence of a class field K/k follows the existence of an itinity of prime divisors in the single classes of A/H, i.e. a far-reaching generalization of Dirichlet’s famous theorem on primes in prime residue classes. He did not know, however, other examples for the existence of class fields than those from the theory of cyclotomic, modular and elliptic functions.

2. The preceding references to Weber’s share in the origin of class field theory are not meant to detract from the great merit of Hilbert in the development of this theory, but only to put it in the right light. Hilbert himself had a high opinion of Weber; he has repeatedly quoted and duly acknowledged his ideas and results on class fields.

The publications of Hilbert (1898, 1899a, b, 19OOa, b, 1902) on class field theory are apparently concerned only with the special case of absolute divisor classes, i.e. where the principal class H contains all principal divisors (without or with sign conditions). Moreover, he had carried through the proofs only for the reZativeIy quadratic number fields-i.e. n = 2-with class number h = 2. Before his mind’s eye, however, he had throughout the most general case. For instance, in his lecture (1899a) before the DMV (German Mathematical Association) meeting at Braunschweig (Brunswick) in 1897 he declared (in free translation):

“In this lecture we have restricted ourselves to the investigation of relatively abelian fields of second degree. This restriction however is only a provisional one, and since all conclusions in the proofs of the theorems are capable of generalization, it is to be hoped that the difficulties of establishing a theory of relatively abelian fields will not be insurmountable.”

In this sense he had in mind in full generality the main theorems of class field theory already mentioned (existence theorem, uniqueness theorem, ordering theorem, isomorphy theorem and decomposition law).

For Hilbert class field theory was not only, as it was for Kronecker and

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HISTORY OF CLASS FIELD THEORY 269

Weber, a means towards proving the completeness theorem and towards generalizing Dirichlet’s prime number theorem. As clearly stated in utter- ances like that quoted above and in the headline to his note (1898, 1902) in the G&finger Nachrichten, he rather regarded it as a “Theory of relatively abelian fields”. A further aim of his was the problem of the higher reciprocity laws, handed down from Gauss, Jacobi, Eisenstein and Kummer, and appearing in his famous Paris lecture of 1900 as Problem 9. Indeed, one of Hilbert’s most profound achievements is the conception of the reciprocity law as a product formula for his norm residue symbol

a,b n( >

1. P -7 “=

Here k is assumed to contain the nth roots of uinty and p runs through all prime divisors of k, including what we call today the infinite prime spots, introduced by him as symbols 1, l’, l”, . . . . He pointed out the analogy of this formula (given by him only in special cases) to the residue theorem for algebraic functions-the prime spots p with norm residue symbol # 1 corresponding to the ramification points of the Riemann surface.? This analogy was later splendidly justified by the subsummation of the product formula in algebraic function fields with finite constant field under the residue theorem by Schmid (1936) and Witt (1937). Moreover, Hilbert stated (again only in special cases) the norm theorem:

= 1 for all p c> a is relative norm from k(Jb),

so important for the further development of the theory. This theorem was proved later in full generality by me (193Oc).

Due to the restriction to the absolute divisor class group A/H, Hilbert’s list of class field theorems contains, in addition to those already mentioned, the discriminant theorem:

K/k has relative discriminant b = 1. He further stated the principal divisor theorem, already touched upon earlier in this Chapter:

AN divisors from k become principal divisors in K. Moreover, he asserted :

The class number of K is not divisible by 2,

and this not only under his original restriction, that k has class number h = 2, but also for h = 4.

t From today’s standpoint this is not quite correct. In number theory the norm residue symbol may be different from 1 also without ramification, namely by inertia. In algebraic function fields over the complex number field this does not occur, so that for them only ramification comes into question.

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270 HELMUT HA&SE

Excepting only this latter assertion, which is true for h = 2 but not necessarily for h = 4, all Hilbert’s assertions on class fields have proved true in the general case. For the principal divisor theorem, however, things turned out to be much more complicated than Hilbert apparently thought. One cannot help feeling the greatest admiration for his keen-mindedness and penetration, which enabled him to conceive with such accuracy general laws from rather special cases.

3. As to the reciprocity law, in 1899 the “Kiinigliche Gesellschuft der Wissen- schaften zu Giittingen”, presumably on Hilbert’s suggestion, had set as prize- subject for 1901 the detailed treatment of this law for prime exponents Z # 2. This problem was solved by Furtwgngler. In the prize-treatise (1902, 1904) itself he gave the solution only for fields k with class number not divisible by Z, and without distinguishing between the Z-l different non- residue classes. In further papers (1909, 1912, 1913, 1928), however, he proved the law of reciprocity for prime exponents 1 (including 2) in full generality. Not only did he prove it in the classical shape:

a b 0 0 i7,= a, if a and b are primary (and in case I= 2, totally positive),

but also in the shape of Hilbert’s product formula for the norm residue symbol. According to a very beautiful idea of his, the law in the classical shape-by passing to the extension field li = k(y(ab-‘)) over which Ii = E(vb) is unramified and hence contained in the absolute class field K-comes back to the decomposition law in ZQli. In other words, that

(ii= 0, over E depends only on the absolute class in E to which p

belongs. This decomposition law was at his disposal since he had previously proved the existence theorem for the absolute class field (1907).

4. A decisive step forward was taken by Takagi (1920, 1922) in two highly important papers on class field theory and the law of reciprocity. Takagi had been studying Hilbert’s papers on the theory of numbers at the turn of the century and had investigated the relatively abelian fields over the Gaussian number field. This resulted in his solution (1903) of Kronecker’s famous completeness problem over this field by means of the division theory of elliptic functions in the lemniscatic case (i.e. where the period parallelogram is a square).

In his big class field paper of 1920 Takagi gave class field theory the decisive new turn by starting from a new definition of the notion of “class field”. His new definition was more suitable than Weber’s because it allows one to envisage the important completeness theorem right from the beginning.

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HISTORY OF CLASS FIELD THEORY 271

For an arbitrary relative field K/k he makes correspond to each integer divisor modulus m in k a congruence divisor class group interpreluted (erkhart) mod m, viz. the smallest quotient group A,,,/&,, whose principal class I& contains all relative norms N(N) of divisors ‘3 in K prime to m and hence consists of all divisors a in k prime to m with

a N i’V(‘%) mod m, i.e., $& E a = 1 mod m (a E k).

For m sufficiently high, more precisely for all multiples m of a lowest f (the conductor of K/k) the class groups A,,,/& turn out to be “equal” and hence constitute a congruence divisor class group A/H in Weber’s sense with conductor f. According to Weber there holds the already mentioned tist fundamental inequality

[A:Hj=h<n=[K:k],

proved by analytic means. Takagi’s class field definition then runs: K/k is called class field to the congruence divisor class group A/H, if and

only if the equality h = n hoUs. The main theorems of class field theory proved by Takagi, and based on

this definition, may be summarized as follows: The class field relation establishes a one-one correspondence between all

relatively abelian fields K/k and all congruence divisor class groups A/H in k. This statement comprises the existence theorem, uniqueness theorem,

completeness theorem in the older terminology, and also the limitation theorem:

h < n if K/k is not relatively abelian,

which was added later. For “partners” in the said one-one correspondence there hold the follow-

ing further facts:

K E K’-s H z H’ (ordering theorem),

B(K/k) z A/H (isomorphy theorem),

p splits in K into prime divisors of relative degree f and relative order e ifandonly $pf is the least power of p in HP, where A/H, is the maximalfactor group of AIH with conductor prime to p, and where e = [HP : H] (decom- position law).

Hence

?+Pjf, an assertion which was later supplemented by me (1926, 1927, 193Oa, 1930d) to

b = v fp f = M fp x

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272 HELMUT HASSE

(conductor-discriminant theorem) where x runs through the characters of A/H and fx denotes their conductors, while M denotes the least common

multiple taken over all x. x

Whereas the existence theorem is obtained from the uniqueness theorem and completeness theorem by a suitable enumeration, the proof of the completeness theorem comes back to the second? fundamental inequality:

h 2 n.

This is obtained by a far-reaching generalization of the classical theory of genera from Gauss’ Disquisitiones Arithmeticae. Nowadays the theory of cohomology has permitted one to systematize the rather complicated chain of conclusions that leads to this inequality.

5. As to the reciprocity law, the above mentioned further paper of Takagi’s (1922) already gave considerable simplifications of Furtwangler’s rather complicated argument (only 40 instead of 80 pages!). These simplifications were possible because of the availability of the full class field theory instead of that only for the absoZute class field. It was not before Artin, however, that an entirely new idea of fundamental importance was brought to bear on this law. He found the key to its conceptual (as opposed to formal) significance in explicitly giving a canonical isomorphism of the class group A/H onto the Galois group 0(K/k).

Artin (1927) showed that such an isomorphism is obtained by making correspond to each prime divisor p,#, or rather to its class in A/H, the so- called Frobenius automorphism FP of K/k with respect to p. This auto- morphism is defined by

AF’ 3 ABlu’) mod p, for all integers A E K,

where a(p) is the absolute norm of p. Today one writes

and one defines from this the Artin symbol Klk ( > a as a multiplicative function

on the group A, of all divisors a in k prime to b or, what is the same, to f. The isomorphy between A/H and B(K/k) may then be expressed by Artin’s reciprocity law:

=loa~H,.

Artin (1924) had conceived this reciprocity law four years earlier, and had proved it in simple special cases. However, it was not until he came to know

t In modern terminology: first.

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HISTORY OF CLASS FIELD THEORY 273

Tchebotarev’s method of crossing the classes of A/H with the congruence classes corresponding to a relatively cyclotomic field, that he succeeded in giving a general proof. Tchebotarev (1926) had invented this method in order to improve on a theorem of Frobenius (1896), viz. : The prime divisors ‘$ of a normal K/k with Frobenius automorphism F, in a given division (Abteilung) S - ‘F; S (v prime to the order of FV, S running through @(K/k)) have a density, namely the relative frequency of the division in the whole group (li(K/k). The method in question enabled Tchebotarev to generalize this theorem to a single conjugacy class S - ‘F+i’.

For a Kummer field K = k(qb) ( w h ere k contains the nth roots of unity) Klk the Frobenius automorphism p changes J/b into

( > 0 i ;lb, as is clear

from the definition of the power residue symbol b ’

0 - Pn

by Euler’s criterion.

Hence Artin’s reciprocity law yields the assertion: b

0 p depends only on the class to which p belongs in the congruence class

group kod fb corresponding to k(qb).

Here fb denotes the conductor of k(Jfb).

On the other hand, from the definition i 0 Pn

depends only on the residue

class to which b belongs mod p. From this it is easy to deduce the reciprocity in its classical shape and also in the shape of Hilbert’s product formula. Thus it becomes understandable and justified that Artin called the above isomorphy assertion the “general law of reciprocity”.

With the help of this law, Artin (1930) could also reduce the principal divisor theorem, enunciated by Hilbert and not yet proved by Takagi, to a pure group-theoretical proposition, which then was proved by Furtwangler (1930). Further proofs of this proposition were given by Magnus (1934), Iyanaga (1934), Witt (1936, 1954) and Schumann and Franz (1938), whereas Taussky and Scholz (1932, 1934) investigated more closely the process of “capitulation”, i.e. of becoming a principal divisor in the subfields of the absolute class field. Exhaustive results concerning this latter problem, however, have not been achieved up to now.

b 6. Analogously to the process of basing the power residue symbol -

0 on

0”

the more general Artin symbol , I succeeded (1926,1927 and 1930a, b) in

basing the Hilbert norm residue symbol contains the nth

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274 HELMUT HASSE

roots of unity) on a symbol a, Klk

( > -- over an arbitrary k (which need not

contain the nth roots of unity). P

At first, for all prime spots p of k, my defini- tion followed the round-about way Hilbert was forced to take for the prime divisors pin. As a consequence of this, the connection of the symbol with the norm residues became apparent only on the strength of Artin’s reciprocity law. Shortly afterwards, however, I was able to give (1933) a new definition from which this connection was immediately clear. My new definition proceeded on a conceptual basis from the theory of algebras.

I had shown previously (193lb) that every central simple algebra of degree n over a local field k, with prime element rt possesses an unramified (and hence cyclic) splitting field Z,. Consequently such an algebra has a canonical generation

un = xv*

(F, the Frobenius automirphism). % -‘zvup = z,“p

On the strength of this, the residue class

y_p mod+ 1 is invariantly associated with the algebra. For a global relatively n cyclic field K/k I put then

when the cyclic algebra over k, generated by

24” = a, u-‘Ku = KS,

after extension to the completion k,, has the invariant : mod+ 1. In par-

ticular, for a Kummer field K = k(db) ( w h ere k contains the nth roots of

unity) the automorphism changes vb . into

From this definition one infers immediately the local property:

= 1 G a norm from KP/k,,

where KP denotes the isomorphy type of the completions K, for the prime

divisors Cplp. Hence the symbol is now called simply the norm

symbol. Globally I could prove (1926, 1927, 1930a, 193Oc) the norm theorem:

= 1 for all pea norm from K/k,

anticipated by Hilbert for his symbol, as mentioned before. So Hilbert’s

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HISTORY OF CLASS FIELD THEORY 275

product formula for his symbol became the product formula for the norm symbol:

l--$9)=1.

This is equivalent to my sum theorem for central simple aIgebras (1933):

F:zO mod+ 1.

Whereas the definition of the norm symbol generalizes from relatively cyclic to arbitrary relatively abelian fields K/k by formal composition, I showed (1931a) that the norm theorem does not remain true in this generality.

In connection with the local property of the norm symbol I was led (193Oc) to establish the main theorems of a class field theory over local fields kP. Here one has a one-one correspondence between all relatively abelian fields KP/kP and all congruence number class groups A,/H, in k,, such that

( > a, Kplkp

P gives a canonical isomorphism of Ap/Hp onto Q(Kp/k,). The

connection with class field theory over global fields k is given by the relations:

(i) KP/kP represents the isomorphy type of the completion of K/k for the ‘PIP;

(ii) B(K+‘/k,) is the decomposition group of K/k for the !Q[p.

Subsequently, Schmidt (1930) and Chevalley (1933a) gave a systematic development of local class field theory without making reference, as I had done, to that connection with global class field theory. Here I should also mention the essential contributions made by Herbrand (1931 and 1932) to the group theoretical mechanism of certain proofs in local as well as global class field theory, and to the higher ramification theory of normal extensions.

7. In Takagi’s class field theory the characterization of the relatively abelian fields K/k by means of the corresponding congruence divisor class groups A/H in k has a disadvantageous fault of beauty. This fault arises from the approximations A,,,/H, to A/H by a kind of limit process with ascending divisor moduli tn. After p-adic concepts and methods had been brought to bear on class field theory in the manner described before, Chevalley (1933b) had further the happy idea of replacing also the Weber-Takagi characteriza- tion in terms of the congruence divisor class groups A/H by a smoother p-adic characterization. In this he succeeded by introducing his ideal elements, or briefly idsles, namely vectors

a = (. . .,uy,. . .)

with components ap from the single completions k, satisfying a suitable

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276 HELMUT HASSE

finiteness condition, viz.,

up r 1 for almost all p.

He replaced the Weber-Takagi congruence divisor class groups A/H by factor groups A/H of the absolute id2le class group A/k*, where k* denotes the group of principal idbles (. . ., a,. . .) corresponding to the numbers a # 0 from k. He proved that the symbol

(y) = j--j (%y>,

named after him the Chevalley symbol, gives an isomorphism between an idele class group A/H and the Galois group B(K/k). Here, the principal class H consists of all those absolute idble classes in k that contain idele norms from K. In Chevalley’s class field theory this idble class group takes the role of Takagi’s congruence divisor class group A/H. To the set of all relatively abelian fields K/k corresponds in this way the set of all idele class groups A/H with open principal class H in a suitable topology of A, namely that in which the idele units s 1 mod m for all divisor moduli m form a complete system of neighbourhoods of 1.

Thus one can say that by Chevalley’s ideas (not to say: idrYes) the local- global principle has taken root in class field theory.

There is a further fault of beauty in Takagi’s class field theory that was obviated by Chevalley (1940). This is the recourse to analytic means (Dirichlet’s L-series) for the proof of the first fundamental inequality h < n. Chevalley succeeded in proving this inequality in a purely arithmetical manner.

8. There still remains much to be said about further developments arising from the class field theory delineated up to this point. For instance, what lies particularly close to my heart, the explicit reciprocity formulae (determination

( > a, b of the norm symbol p for prime divisors pin); further inclusion of

n infinite algebraic extensions K/k, class field theory over congruence function fields (algebraic function fields with finite constant field), Artin’s L-series and conductors, and so on. I must, however, refrain from talking about those subjects here, because that would exceed the frame of this lecture.

I cannot touch either on the further development of class field theory after the war, but suppose I must conclude my survey of the historical development at this point.

If I have understood rightly, it was my task here to delineate for the mathematicians of the post-war generation a vivid and lively picture of the great and beautiful edifice of class field theory erected by the pre-war generations. For the sharply profiled lines and individual features of this

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HISTORY OF CLASS FIELD THEORY 277

magnificent edifice seem to me to have lost somewhat of their original splendour and plasticity by the penetration of class field theory with cohomological concepts and methods, which set in so powerfully after the war.

I should like to think that I have succeeded to some extent in this task.

REFERENCES?

Artin, E. (1924). I%er eine neue Art von L-Reihen. Abh. Math. Semin. Univ. Hamburg, 3, 89-108. (Collected Papers (1965), 105-124. Addison Wesley.)

Artin, E. (1927). Beweis des allgemeinen Reziprozitltsgesetzes. Abh. Math. Semin. Univ. Hamburg, 5, 353-363. (Collected Papers (1965), 131-141. Addison Wesley.)

Artin, E. (1930). Idealklassen in Oberkorpem und allgemeines Reziprozitatsgesetz. Abh. Math. &mitt. Univ. Hamburg, 7, 46-51. (Collected Papers (1965), 159-164. Addison Wesley.)

Chevalley, C. (1933a). La theorie du symbole de restes normiques. J. reine angew. Math. 169, 140-157.

Chevalley, C. (19336). Sur la theorie du corps de classes dans les corps finis et les corps locaux. J. Fat. Sci. Tokyo Univ. 2, 365-476.

Chevalley, C. (1940). La theorie du corps de classes. Ann. Math. 41, 394-417. Delaunay, B. (1923). Zur Bestimmung algebraischer Zahlkijrper durch Kongruen-

zen; eine Anwendung auf die Abelschen Gleichungen. J. reine angew. Math. 152, 120-123.

Frobenius, G. (1896). Uber die Beziehungen zwischen den Primidealen eines algebraischen Korpers und den Substitutionen seiner Gruppe. Sber. preuss. Akad. Wiss., 689-703.

Fueter, R. (1914). Abel’sche Gleichungen in quadratisch-imaginlren ZahlkGrpern. Math. Annln. 75, 177-255.

Fueter, R. (1927). “Vorlesungen tiber die singularen Moduln und die komplexe Multiplikation der elliptischen Funktionen II” (unter Mitwirkung von M. Gut). Leipzig-Berlin.

Furtwangler, Ph. (1902). Uber die Reziprozitatsgesetze zwischen lten Potenzresten in algebraischen Zablkiirpem, wenn 1 eine ungerade Primzahl bedeutet. Abh. K. Ges. Wiss. Giittingen. (Neue Folge), 2, Nr. 3, l-82.=Math. Annbt. 58 (1904), l-50.

Furtwlngler, Ph. (1907). Allgemeiner Existenzbeweis fur den Klassenkorper eines beliebigen algebraischen Zahlkdrpers. Math. Annln. 63, l-37.

Furtwlngler, Ph. Reziprozitltsgesetze ftir Potenzreste mit Primzahlexponenten in algebraischen Zahlkijrpern I, II, III. Math. An&. (1909), 67, 1-31; (1912), 72, 346-386; (1913), 74,413-429.

Furtwangler, Ph. (1928). I%er die Reziprozitltsgesetze ftir ungerade Primzahl- exponenten. Math. Ant&t. 98, 539-543.

Furtwangler, Ph. (1930). Beweis des Hauptidealsatzes ftir Klassenkijrper alge- braischer Zahlkiirper. Abh. Math. Semin. Univ. Hamburg, 7, 14-36.

t This list of publications from the years 1853-1940 (with one exception from 1954) does not pretend to be a complete list of publications about class field theory and related subjects for this period. It contains only those publications which appeared important as references for what has been said in the lecture.

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278 HELMUT HASSE

Hasse, H. Bericht tiber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkiirper I, Ia, II. Jber. dt. MutVerein. (1926), 35, l-55; (1927), 36,233-311; Exg. Bd. (193Ou), 6, l-204.

Hasse, H. (19306). Neue Begrtindung und Verallgemeinerung der Theorie des Normenrestsymbols. J. reine angew. Math. 162, 134-144.

Hasse, H. (193Oc). Die Normenresttheorie relativ-Abelscher Zahlkorper als Klassen- korpertheorie im Kleinen. J. reine ungew. Math. 162, 145-154.

Hasse, H. (193Od). Ftihrer, Diskriminante und Verzweigungskiirper relativ-Abelscher Zahlkorper. J. reine angew. Math. 162, 169-184.

Hasse, H. (1931u). Beweis eines Satzes und Widerlegung einer Vermutung tiber das allgemeine Normenrestsymbol. Nuchr. Ges. Wiss. Giittingen, 64-69.

Hasse, H. (1931b). Uber p-adische Schiefkorper und ihre Bedeutung ftir die Arith- metik hyperkomplexer Zahlsysteme. Math. An&. 104,495-534.

Hasse, H. (1933). Die Struktur der R. Brauerschen Algebrenklassengruppe tiber einem algebraischen Zahlkorper. Math. Annln. 107, 731-760.

Herbrand, J. (1931). Sur la theorie des groupes de decomposition, d’inertie et de ramification. J. Math. put-es uppl. 10,481-498.

Herbrand, J. (1932). Sur les theortmes du genre principal et des ideaux principaux. Abh. Math. Semin. Univ. Hamburg, 9, 84-92.

Hilbert, D. (1896). Neuer Beweis des Kronecker’schen Fundamentalsatzes tiber Abel’sche Zahlkorper. Nuchr. Ges. Wiss. Giittingen, 29-39.

Hilbert, D. (1897). Bericht: Die Theorie der algebraischen Zahlkijrper. Jber. dt. MutVerein. 4, 175-546.

Hilbert, D. (1898). Uber die Theorie der relativ-Abel’schen Zahlkorper. Nuchr. Ges. Wiss. Giittingen, 377-399= Actu math. Stockh. (1902), 26, 99-132.

Hilbert, D. (1899u). Uber die Theorie der relativquadratischen Zahlkorper. Jber. dt. Mat Verein. 6, 88-94.

Hilbert, D. (18996). Ubcr die Theorie des relativquadratischen Zahlkiirpers. Math. Annln. 51, 1-127.

Hilbert, D. (19OOu). Theorie der algebraischen Zahlkiirper. Enzykl. math. Wiss. I C 4a, 675-698.

Hilbert, D. (19OOb). Theorie des Kreiskorpers. Enzykl. math. Wiss. I C 4b, 699-732. Hilbert, D. (19OOc). Mathematische Probleme. Vortrag auf internat. Math. Kongr.

Paris 1900. Nuchr. Ges. Wiss. Giittingen, 253-297 (reprinted in “A Collection of Modern Mathematical Classics, Analysis” (ed. R. Bellman), Dover, 1961).

Iyanaga, S. (1934). Zum Beweis des Hauptidealsatzes. Abh. Math. Semin. Univ. Hamburg, 10,349-357.

Kronecker, L. (1853). Uber die algebraisch auflijsbaren Gleichungen I. Sber. preuss. Akud. Wiss. 365-374= Werke IV, l-1 1.

Kronecker, L. (1877). Uber Abel’sche Gleichungen. Sber. preuss. Akud. Wiss. 845- 851 = Werke IV, 63-72.

Kronecker, L. (1882). Grundziige einer arithmetischen Theorie der algebraischen Grossen. J. reine ungew. Math. 92, l-122=Werke II, 237-388. See there 0 19, 65-68 = 321-324.

Kronecker, L. (1883-90). Zur Theorie der elliptischen Funktionen I-XXII. Sber. preuss. Akud. Wiss. =Werke IV, 345-496 = V, 1-132.

Kronecker, L. Auszug aus Brief an R. Dedekind vom 15 M&-z 1880. Werke V, 453-458. See also my detailed “Zusutz”, ibid., 510-515.

Magnus, W. (1934). Uber den Beweis des Hauptidealsatzes. J. reine ungew. Math. 170,235-240.

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HISTORY OF CLASS FIELD THEORY 279

S&mid, H. L. (1936). Zyklische algebraische Funktionenkorper vom Grade pm fiber endlichem Konstantenkorper der Charakteristik p. J. reine angew. Marh. 175, 108-123.

Schmidt, F. K. (1930). Zur Klassenkorpertheorie im Kleinen. J. reine angew. Math. 162, 155-168.

Schumann, H. G. (1938). Zum Beweis des Hauptidealsatzes (unter Mitwirkung von W. Franz). Abh. Math. Semin. Univ. Hamburg, 12, 42-47.

Speiser, A. (1919). Die Zerlegungsgruppe. J. reine angew. Math. 149, 174-188. Takagi, T. (1903). Ober die im Bereich der rationalen komplexen Zahlen Abel’schen

Zahlkorper. J. COIL Sci. imp. Univ. Tokyo, 19, Nr. 5, 142. Takagi, T. (1920). Uber eine Theorie des relativ-Abel’schen Zahlkorpers. J. CON.

Sci. imp. Univ. Tokyo, 41, Nr. 9, 1-133. Takagi, T. (1922). Uber das Reziprozitltsgesetz in einem beliebigen algebraischen

Zahlkorper. J. COIL Sci. imp. Univ. Tokyo, 44, Nr. 5, l-50. Taussky, 0. (1932). Uber eine VerschPrfung des Hauptidealsatzes fti algebraische

Zahlkiirper. J. reine angew. Math. 168, 193-210. Taussky, 0. and Scholz, A. (1934). Die Hauptideale der kubischen Klassenkijrper

imagintiquadratischer Zahlkiirper : ihre rechnerische Bestimmung und ihr EinfluD auf den Klassenkorperturm. J. reine angew. Math. 171,19-41.

Tchebotarev, N. (1926). Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehoren. Math. Annln. 95,191-228.

Weber, H. Theorie der Abel’schen Zahlkijrper I, II. Acfu math. Sfockh. (1886), 8, 193-263; (1887), 9, 105-130.

Weber, H. uber Zahlengruppen in algebraischen K&pen I, II, III. Math. Ann/n. (1897), 48,433-473; (1897), 49, 83-100; (1898), 50, l-26.

Weber, H. (1891). “Elliptische Funktionen und algebraischezahlen”. Braunschweig. Weber, H. (1908). “Lehrbuch der Algebra III”.? Braunschweig. Weber, H. (1909). Zur Theorie der zyklischen Zahlkorper. Math. An&n. 67, 32-60. Witt, E. (1936). Bemerkungen zum Beweis des Hauptidealsatzes von S. Iyanaga.

Abh. Math. Semin. Univ. Hamburg, 11, 221. Witt, E. (1937). Zyklische K&-per und Algebren der Charakteristikp vom Gradep”.

J. reine angew. Math. 176, 126-140. Witt, E. (1954). Verlagerung von Gruppen und Hauptidealsatz. Proc. Internat. Math.

Congress II, Amsterdam 1954, 71-73.

t Second edition of “Elliptische Funktionen und Algebra&he Zahlen”.

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CHAPTER XII

An Application of Computing to Class Field Theory

H. P. F. SWINNERTON-DYER

Let G be a connected algebraic group. Chevalley has shown that G contains a normal subgroup R such that R is a connected linear group and GJR is an Abelian variety?; and these properties determine R uniquely. The general theory of algebraic groups therefore depends on theories for linear groups and for Abelian varieties. In Chapter X of this book Kneser has described the known number-theoretical properties of linear groups, and the same thing will be done here for Abelian varieties. But there is a fundamental difference between these two topics. Kneser has expounded a theory which is reasonably complete and satisfactory-the major results have been or are on the point of being proved, and there is no reason to think that there are important theorems as yet undiscovered. For Abelian varieties, on the other hand, very few number-theoretical theorems have yet been proved. The most interesting results are conjectures, based on numerical computation of special cases, and the fact that one has no idea how to attack these conjectures suggests that there must be important theorems which have not yet even been stated.

The original calculations are largely due to Birch and Swinnerton-Dyer ; and their results can be expressed in respectable language through the efforts of Cassels and of Tate. One object of this chapter is to describe the numerical evidence, to show how far the conjectures are directly based on it, and to explain why it seems right to state the conjectures in very general terms although all the evidence refers to cases of one particular kind.

With the advent of electronic computers, calculations which would previously have been out of the question are now quite practicable, provided that they are sufficiently repetitive. There must be many topics in number theory in which intelligently directed calculation would yield valuable results. (At the moment, most number-theoretical calculations merely pile up lists of integers in the manner of a magpie; they are neither designed to produce valuable results nor capable of doing so.) The second object of this chapter

7 An Abelian variety is a connected algebraic group which is also a complere algebraic variety; the group operation on an Abelian variety must necessarily be commutative. The simplest example of an Abelian variety is an non-singular cubic curve in the projective plane, together with a point on it which is the identity for the group operation.

280

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AN APPLICATION OF COMPUTING TO CLASS FIELD THEORY 281

is to show what can reasonably be done on a computer, and what the number- theorist must consider before he can ask someone to do his calculations for him.

When a modern algebraist defines something, he usually begins by taking the quotient of one very infinite object by another. A process of this kind can seldom be carried out on a computer even in theory; and it never can in practice. Thus an object which can only be defined in this sort of way is not effectively computable. Conversely, if something has to be computed it must be defined in a way which is more down-to-earth, even if less canonical. This applies not only to the final result of the calculation, but to anything which turns up in the course of it. This may be difficult or even impossible: for example the Tate-SafareviE group (defined below) has at the moment no constructive definition.

Once a number-theorist has phrased his problem in computable terms, he can estimate how much machine time it will take. This depends on the number of operations involved-an operation for this purpose being an addition, subtraction, multiplication or division. (This is a crude method of estimation, and ignores for example the time spent in organizational parts of the program; but it will give the right order of magnitude.) A calculation which takes lo6 operations is trivial, and is worth doing even if its results will probably be useless. A calculation which takes lo9 operations is sub- stantial but not unreasonable. It is worth doing in pursuit of any serious idea, but not just in the hope that something may turn up; moreover, the method of calculation should now be reasonably efficient, whereas for a smaller problem one chooses the simplest possible method in order to minimize the effort of writing the program. Finally, a calculation which takes 1Ol2 operations is close to the limit of what is physically possible; it can only be justified by a major scientific advance such as landing a man on the moon.

When a calculation is complete, one must still ask whether the results are reliable. A typical program, in machine-coded form, is a string of several thousand symbols; and a slip of the pen in any one of them will change the calculation being carried out by the computer. Most errors have disastrous results, and can therefore be detected and removed; but some errors are analogous to mistakes in a formula, and will merely cause the production of wrong answers. Some calculations are self-checking: one example is given below, in which an integer is obtained as the value of a complicated analytic expression-if this had been wrongly evaluated, the result would not always have come out to be approximately an integer. There are other calculations in which once the answer has been found it can be checked with relatively little effort: for example in searching for a solution of a diophantine equation. But in general it is desirable either to work out some typical results by hand, for comparison with those obtained from the machine, or to have the whole

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282 H. P. P. SWINNERTON-DYER

calculation repeated ab initio by a different programmer on a different machine.

If I is a curve defined over Q, the field of rationals, under what conditions can we say that r contains rational points? This problem is completely solved for curves of genus 0, for I? is then birationally equivalent over Q to a straight line or a conic; so it is natural to consider the case when r has genus 1. This is equivalent to saying that if PI, . . , P., Q,, . ., Qm-, are any points on r then there is a unique Q, such that the P, are the poles and the Qi the zeros of some function on I’. It follows that for any given point 0 on I’ we can give I? the structure of a group with 0 as the identity; for if PI, Pz are any points on I? we define Q = PI + Pz by the property that PI, Pz are the poles and 0, Q the zeros of some function on r. Moreover if 0 and I are defined over Q, then so is the group law; thus the set of rational points on r form a group, which we call To.

We can associate with l? another curve J, the Jacobian of r, as follows. On I? x l? we define an equivalence relation by writing Pi x Q, h) Pz x Q, if and only if PI, Q, are the poles and Pz, Q, the zeros of a function on I’. J is defined (up to birational equivalence) as the curve, each point of which corresponds to an equivalence class on I x I. J is defined over Q and certainly contains a rational point, corresponding to the class of points P x P on I? x I’; and we can write the equation of J in the form

y2z = x3 - Axz2 - Bz3. (1)

We say that r is a principal homogeneous space for J. Clearly r is birationally equivalent to J over Q if and only if it contains a rational point; and because J has a canonical rational point it has a canonical group structure.

The problem of rational points on r now breaks up into two parts: (i) is there a rational point on I, and (ii) what is the structure of the group of rational points on J? Of these, the second is more attractive because there is more structure associated with it. Mordell in 1922 proved the following theorem.

MORDELL'S THEOREM. The group of rational points on J is finitely generated.

It is easy to find the elements of finite order in this group, in any particular case. Thus it is natural to ask for a means of finding g, the number of independent generators of infinite order, and if possible for an actual set of generators. Mordell’s argument is semi-constructive, in that it gives in any numerical case an upper bound for g which is not absurdly large; and by using the ideas underlying his proof one can usually find g and an actual set of generators. But the process can be very laborious and there is no guarantee that it will always work. Moreover g is not connected with anything else in the theory. What should it be connected with?

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AN APPLICATION OF COMPUTING TO CLASS FIELD THEORY 283

We can regard a solution of the equation (1) as the result of fitting together a compatible set of solutions of the congruences

y*z G x3-Axz2-Bz3 (modp”). (2)

One can hope that if these congruences have a lot of solutions it will be relatively easy to find compatible sets, and so (1) will have a lot of solutions and g will be large. Hence we look for a measure of the density of solutions of the congruences (2). Let NP” be the number of essentially distinct primitive solutions of (2). Except at finitely many bad primesp it follows from Hensel’s Lemma that

Npm = p” - ‘Np

and we are therefore led to consider the infinite product

fl (N,lp). (3)

We can also define N,, as the number of points on (1) considered as a curve over the finite field of p elements. For finitely many primes we expect that the factor in (3) is wrong, but it is not yet clear what to put in its place.

There is a more respectable reason for considering the product (3), though in the end it will turn out to be misleading. We have

Np = p-cr,-z*+1

where laPI = p112, * and the local zeta-function of l? is defined as

~r,,W = (l -cr,P-?u--B,P-? (1 -p-3(1 -pi-y *

The numerator of the right-hand side has the value NJp at s = 1; if we write

Ws) = n Kl -~~P-w -spi,p-31-’ (4) then

and formally

L-u)1-’ = fl (N,/P).

The product (4) is only known to converge in Ws > 312, though it is conjectured (and in special cases known) that L,(s) can be analytically continued over the whole plane. Moreover one can at best hope that the product (3) is finitely oscillatory, because one expects that L,(s) will have complex zeros on WS = 1. However, it is easy to calculate the finite products

0’) = I1 (N,lp)

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284 H. P. F. SWINNERTON-DYER

taken over all p 5 P for values of P up to several thousandt. To a sympathetic eye, the results suggested thatf(P) oscillates finitely when g = 0, and tends to infinity when g > 0; and this led to the linked pair of conjectures

f(P) lies between constant multiples of (log P)g,

h(s) has a zero of order g at s = 1.

(Of course the constants in the first conjecture depend on I.) By looking at the behaviour of f(P), Birch and Swinnerton-Dyer were able to predict the value of g for individual curves, and the predictions were right in about 90% of the cases tried; these were curves for which g = 0, 1, 2 or 3. But there seems to be no objective numerical method for estimating g accurately in this way.

To do better than this, we have to know more about L,(s). It is therefore natural to consider curves J which admit complex multiplication, since there is then an explicit formula for ZVp and b(s) is a product of Hecke L-series. The simplest such curves have the form

y2z = x3 - Dxz”, (5)

in which we can assume that D is an integer and is fourth-power-free; and if we write (. .)4 for the biquadratic residue symbol in Q(i) we have the following formulae.

THEOREM (DAVENPORT-HASSE). For the curve y2z = x3 - Dxz2,

(p+l forp G 3 mod 4,

forp E 1 mod 4,

where in the second casep = n?i in Q(i) with TC,?~: E 1 mod (2 -t 2i). There is no easy proof of the full theorem. That ap/n is a power of i goes

back to Jacobsthal, who gave an elementary proof which makes it look like a piece of good luck. The reason why such formulae exist in this case is that Q(i) is here the ring of endomorphisms of J in characteristic 0. Thus if p G 1 mod 4, Q(i) must be the whole ring of endomorphisms of J in characteristic p ; for the known structure theorems rule out any strictly larger ring. Hence ap, being the image of the Frobenius endomorphism, must lie in Q(i); and since CC,&, = p we find that up/z is a unit in Q(i).

t The naive way of calculating N,, is to set z = 1 in (1) and test all possible pairs x, y; this would have taken O(pa) operations, and used more machine time than was justifiable. But because (1) can be written as

y’ = w = x= - Ax - B

it is only necessary to tabulate for each value of w the number of ways in which it is a square; for each value of x one can then read off the number of solutions. In this way, N, can be found after only O(p) operations.

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AN APPLICATION OF COMPUTING TO CLASS FIELD THEORY

Using the values above for the NP we have

285

L,(s) = L,(s) = n (1 +p’-23-l p= 3mod4

where the product is taken over all Gaussian primes R E 1 mod (2 + 29, the sum is over all Gaussian integers u t 1 mod (2 + 2i) and N is the norm for Q(i)/Q. Writing c = 1601 + p, where 1 runs over all Gaussian integers and p over a suitably chosen finite set of values, we have

L,(s) = ; $ 4 ; Lf(Nu)-“. 0

We cannot yet write s = 1, for the inner series would not converge; thus we write

Il(cc,d-Ia12” v+. if+q+ 1-q+ -.L+ c -

1 - - [

E(l -S) 1 I> 9

which converges for 9% > 3, and after some reduction we obtain

Here we can at last let s tend to 1; and since $ (a, 1) = e(a), the Weierstrass zeta function, we obtain

Jw) = i&~(~)4t(i5) - &=@; @) This is an explicit finite formula from which the value of L,(l) can be computed.

In the interests of simplicity, I have derived a relatively clumsy formula for L,(l). In fact, if A is the product of the distinct odd primes dividing D, then L,(l) can be expressed as a sum of O(A’) terms instead of the O(L)‘) given above, and can therefore be calculated in 0(A2) operations. The constant implied in this formula is quite large because each term is com- plicated; but it was possible to evaluate L,(l) for all 1 A 1 < 108, which gave some very large values of D. Moreover, provided that 1 A 1 > 1 the second term in (6) vanishes for a suitable choice of the p and L,(l) can be written as a sum of terms involving Weierstrass p-functions. In this way it can be shown that

D”‘LD(l)/o is a rational integer if D > 0

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286 H. P. I’. SWINNF!RTON-DYER

where CD is the real period of the Weierstrass p-function defined by

P’2 =4p3-4p.

The proof is in two parts: 23’4A &,(1)/o is an algebraic integer, because of the addition formula for p ; and D ‘I4 &,(1)/w is rational, because of the Kronecker Jugendtraum. There is a similar result for D < 0. From this, and the approximate calculations of L,(l), which are accurate to one part in lo’, we can obtain the exact values of L,(l) for a large number of particular curves; and all these support the conjecture that

L,(l) = 0 ifand only ifg > 0.

However, the numerical results still leave something to be explained away or fitted into the theory, to wit, the rational integer D”4LD(l)/w when g = 0. The possible non-trivial factors from which this can be built up are (i) q(D), the number of rational points of finite order on J; (ii) m (D), the order of the Tate-SafareviE group of J; (iii) “fudge factors” corresponding to the finitely many “bad” primes. For the curve (5), q(D) is 4 if D is a square and 2 otherwise. “Bad” primes are those for which J has a bad reduction modulo p; that is, those which divide 20. One has also the right to deem “infinity” to be “bad” if this is expedient.

We have next to define the Tate-SafareviE group. There is an equivalence relation among the principal homogeneous spaces r for a fixed Jacobian J, defined by birational equivalence over Q. The equivalence classes under this relation form the Weil-ChGtelet group,t which is a commutative torsion group of infinite order. Those r which contain points defined over each p-adic field and over the reals, fill a certain number of equivalence classes; and these classes form the Tate-SafareviE group. It is conjectured that this group is always finite, though there is no case in which this has yet been proved; however Cassels has shown that when it is finite its order is a perfect square. The whole group is not constructively defined, and is not computable; but it is theoretically possible to compute those of its elements which have any given order (of which there can only be finitely many), and for the curve (5) it is practicable to find the elements of orders 2 and 4 at least.

Fudge factors must be purely local, and there are now two plausible alternatives: (i) fill in the missing factors of G(S) by means of the functional equation-this is possible whenever there is complex multiplication; (ii) supply the missing factors of the product n(N,/p) by considering the Tamagawa measure of the curve J.

t This can also be defined as the cohomology group W(Q,J)-see Chapter X for the details of a parallel case. The group law can be defined geometrically as follows. There is a map I’ x l-’ + J which can be written P x Q + (P - Q) in an obvious notation. On rl x I?, we define an equivalence relation by writing PI x Pa N Q1 x Ql if and only if PI - Q1 = QP -P,. Now r1 + I?, is defined as I’,. the curve whose points are in one- one correspondence with the equivalence classes on lY1 x r,.

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AN APPLICATION OF COMPUTING TO CLASS FIELD THEORY 287

It must be emphasized that these two alternatives do lead to numerical different results, and that the second one appears to work while the first definitely does not. To define the Tamagawa measure, let o be a differential of the first kind on J; this induces a measure op on J over each p-adic field, and the Tamagawa measure of J is defined to be

I-ISJ%

taken over all primes including infinity. Here o is defined up to multiplication by an arbitrary constant, and the contributions of this constant to the infinite product evidently cancel out.

LEMMA. Zf p is a ‘good”finite prime then SJ o,, = N,/p. Here “good” means that both J and w have a good reduction mod p. We

can take w to be dx dy

2y=-- 3x2-D (8)

possibly multiplied by a p-adic unit. Let (xO,yO) be any solution of

y2 E x3-Dxmodp

with y0 + 0 mod p. By Hensel’s Lemma, to each p-adic x = x0 mod p there corresponds just one y 3 y0 mod p satisfying y2 = x3 - Dx; all these solutions together contribute to J op an amount equal to the p-adic measure of the set of x t x0 mod p, which is p- ‘. A similar argument works for the solutions withy,, = 0 or co. This proves the Lemma.

If we define o to be (a), which is in fact the canonical form for the differential on J, then the bad primes are just those which divide 20 03. It can be shown that

I @co =2oD-*ifD>O

with a similar result for D < 0. Thus the Tamagawa measure of J, defined by (7), is formally equal to

2~00~* [L,(l)]-’ fll op = z(D), say, (9)

where the product is taken over the Unite bad primes p. This is a rational number. The contributions from the bad primes can be found by arguments similar to those of the Lemma, though more tedius; and it can be shown that s e+, for a bad prime p is just the number of components into which J decomposes under reduction modulo p in the sense of N&on (Modtles minimaux . . , Publ. IHES, 21).

The numerical results lead to the conjecture that the Tamagawa measure (9) is equal to

MD>12/dD)- (10) They cannot be said to support this conjecture directly since m(D) cannot

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288 H. P. F. SWINNERTON-DYER

be calculated ; but if we write

Y(D) = MQl”/@>~ the conjectural value of m(D), it has the following properties in every case in which it has been calculated. (i) y(D) is a perfect square, and usually contains no prime factor other than 2; in most cases j(D) = 1. (ii) Whenever the exact power of 2 which divides m(D) has been calculated, this is also the exact power of 2 which divides y(D). In the outstanding cases, m(D) must be and y(D) is divisible by a substantial power of 2. Moreover, a comparison of the conjectures for the isogenous curves y2 = x3 - Dx and Y2 = x3 + 4Dx yields an explicit formula for m(- 40)/1u(D); this has sub- sequently been proved by Cassels.

The surprising part of this conjecture is the presence of a square in (10). We expect to measure a union of representatives of the elements of the Tate-SafareviE group modulo the group of rational points on this union; and the measure turns out to be q(D) rather than a constant.

To describe what happens for g > 0, or for more general curves I, it is convenient to define the modified L-series

LX4 = h-M/n J q# (11)

in which the product is taken over all bad primes including infinity, and L&) is defined by (4) in which the product is taken over all good primes. If J admits complex multiplication, we can give explicit formulae for the coefficients of the power series expansion of L,(s) about s = 1 by arguments similar to those above; the only change is that instead of Weierstrass zeta functions the formulae involve series which converge but have no interesting theoretical properties.

Nelson Stephens, a student of Birch, has carried out these calculations for curves of the form

J: y3 = x3- Dz’, (12)

which also admit complex multiplication. His results are consistent with the formula

LX4 = w- llgK %9/MJJ12 (13) in which g is the number of independent generators of JQ of infinite order, q(J) is the order of the torsion subgroup of JQ, m(J) is the order of the Tate-SafareviE group of J, and K measures the size of the generators of infinite order for J, in the following way. Let P = (x, y, z) be a rational point on J, where x, y, z are integers with no common factor, and write

W) = log IMax tl~~~l~l~~z~>l. Tate has defined the canonical height of P to be

R*(P) = lim n-2R(nP);

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AN APPLICATION OF COMPUTING TO CLASS FIELD THEORY 289

he has shown that this limit exists and behaves like a quadratic form on the additive group JQ. In particular it gives rise to the bilinear form

<Pi, Pz> = 3 [R*(P, +P2)--R*(P1)-R*(P2)].

IfP,, .., PR are the elements of infinite order in a base for JQ, then

K = det (< Pi,Pj>)

which clearly does not depend on the particular base chosen. The conjecture (13) is meaningful for any elliptic curve, although all the

evidence so far given for it relates to curves with complex multiplication. Shimura has shown that L,(s) can be analytically continued for those elliptic curves which can be parametized by modular functions, and has suggested that it should be possible to verify the conjecture for some of them. The few such curves that appear in the literature all have g = 0 and can be easily shown to have L,(l) + 0. Birch has been able to calculate L*,(l) for one of them, and to show that it has the value which the conjecture demands. But we have not yet been able to mechanize the process of finding such curves-though they appear to be extremely common-nor that of calculating L*r( 1).

It is natural to generalize these conjectures to apply to other varieties. Assuming the Weil conjectures, let V be a complete non-singular variety of dimension n; then its local zeta-function has the form

cy,p(s) = h.p(~ 2’4-;;“fP;;;y

p,l ***

where L,,,(S) = n (1 -Cr,jP-3-l. (19)

Here IOlmjl = pmi2, and the product (14) has B,,, factors, where B,,, is the mth Betti number of V. Moreover, we can write

LG) = I-I &n(4 where the product is taken over all primes for which V has a good reduction modulo p. In general we do not know how to define the factors corresponding to the bad primes, and so cannot go on to obtain L:(s); the one exceptional case is when V is an Abelian variety A and m = 1 or 2 n - 1. In this case we can proceed as we did for J; the only novelty is we must now distinguish between A and its dual A, whereas for an elliptic curve J = .f. Since the proper generalization of (13) should follow from (13) when A is a product of elliptic curves, it ought to be

L;nn-l(s+n- 1) = L:(s) z 2”(s- l)“m~~(A)/~(A)tj+i). (15) The form of the denominator on the right has been obtained from the role which duality plays in the proof of Cassels’ relation between m(Ji) and III(J,) for two isogenous elliptic curves J1 and J2.

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290 H. P. I’. SWINNERTON-DYER

Except in this special case, we can only try to generalize the weaker conjecture that L,(S) has a zero of order g at s = 1. In these terms, there is a natural analogue of (15) for general V:

L,,-,(s) has a zero of order g at s = n, where g is the rank of the group of rational points on the Picard variety of V.

But the extra generality is spurious, for one believes that L,,-,(s) depends only on the Picard variety of V and not on V itself’.

For even suffices, Tate in his Wood’s Hole talkt produced a remarkable new conjecture :

L,,(s) has a pole of order v at s = r + 1, where u is the rank of the group of r-cycles on V defined over Q, module algebraic equivalence.

Note that Lz,(s) converges in Ws > r + 1, so that this conjecture can be given a meaning even if L,,(s) cannot be analytically continued; by contrast, all the previous conjectures have refered to a point which is a distance 3 from the region of known convergence. Tate’s conjecture is known to hold for rational surfaces, and he has proved it for other varieties of certain special types. Moreover, by applying it to the n-fold product of an elliptic curve F with itself, he has been able to state the distribution of the characteristic roots a,, and 2,, of I’ as p varies; and these predictions have been verified numerically.

Is there a comparable conjecture for L,,-,(s)? One expects to associate L,,(s) with r-cycles modulo algebraic equivalence, and L,,-,(s) with (r- l)- cycles algebraically equivalent to zero. Moreover, any conjecture must generalize the one stated above for L zn-l(~), and must take into account the duality between L,,,(s) and L,,-,(s). B earing these facts in mind, there is one plausible form that such a conjecture could take. Let ai and a2 be two algebraically equivalent r-cycles on V; this means that there is a non-singular variety W whose points correspond to r-cycles on V, and there are points P, and P2 on W which correspond to a, and a, respectively. We shall say that ai and a2 are Abelian equivalent if it is possible to choose W,P, and P2 in such a way that the image of PI x P2 in the Albanese variety of W is the identity. It is known that this gives an equivalence relation, though it is one about which very little has been proved. The most natural conjecture now is as follows:

L,,-,(s) has a zero of order at least v at s = r, where v is the rank of the group of (r - 1) - cycles on V defined over Q and algebraically equivalent to zero, module Abelian equivalence.

t Published paradoxically in “ Arithmetical Algebraic Geometry. Proceedings of a Con- ference held in Purdue University, December 5-7, 1963.” (Ed. 0. F. G. Schilling.) Harper & Row.

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AN APPLICATION OF COMPUTING TO CLASS FIELD THEORY 291

This is supported by unpublished results of Bombieri and Swinnerton-Dyer for the cubic threefold, and for the intersection of two quadrics in 2n + 1 dimensions. In both these cases the conjecture appears to hold with actual equality. However, numerical calculations show that for the n-fold product of an elliptic curve with itself we can have either inequality or -equality, depending on the particular curve.

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CHAPTER XIII

Complex Multiplication

J.-P. SERRE

Introduction ............... 292 1. The Theorems .............. 292 2. The Proofs ............... 293 3. Maximal Abelian Extension ........... 295

References ............... 296

Introduction

A central problem of algebraic number theory is to give an explicit con- struction for the abelian extensions of a given field K. For instance, if K is Q, the field of rationals, the theorem of Kronecker-Weber tells us that the maximal abelian extension Qb of Q is precisely Qcyc’, the union of all cyclotomic extensions. There is a canonical isomorphism

so we have an explicit class-field theory eve: Q (Chapter VII, 3 5.7). If K is an imaginary quadratic field, complex multiplication does essentially

the same. We get the extensions Kab 3 R 3 K (g is the absolute class field the maximum unramified abelian extension) essentially by adjoining points of finite order on an elliptic curve with the right complex multiplication.

1. The Theorems

Let E be an elliptic curve over C (the complex field); in normal form, its equation may be taken as y2 = 4x3 -g2 x-g,. We know that E z C/T where r is a lattice in C. An endomorphism of E is a multiplication by an element z E C with zF c I?. In general, End(E) = Z; if End(E) is larger, then End(E) @Q = K must be an imaginary quadratic field.

Let R be the ring of integers of such a field K; then End(E) is a subring of R of finite index. Every such subring of R has the form Rf = Z +fR (f > 1 is the “conductor”); so, if E has complex multiplication, there is a complex quadratic field K with integers R, and an integer f, such that End(E) g R,. Conversely, given RI, there are corresponding elliptic curves.

THEOREM. The elliptic curves with given endomorphism ring Rf correspond one-one (up to isomorphism) with the class group Cl(R/).

292

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COMPLEX MULTIPLICATION 293

[Cl(R,) is the same as &RI), the group of projective modules of rank 1 over R,. Iff = 1, it is simply the group of ideal classes of R.]

Sketch of Proof. Such a curve E determines a lattice F by E r C/T (so essentially r is n,(E)). r is an R/-module of rank 1, whose endomorphism ring is exactly R,; this implies that r is an Rf-projective module of rank 1. Conversely, given such a F, C/l? is an elliptic curve with End(C/r) = R,

COROLLARY. Up to isomorphism, there are onlyfinitely many curves E with End(E) = R/; in fact, there are preciseZy h, = Card(Cl(R,)).

To each E, associate its invariant j(E); so there are h, such numbers associated with Rp For simplicity, take f = 1.

THEOREM (Weber-Fueter). (i) The j(E) are algebraic integers. (ii) Let a = j(E) be one of them. Thefield K(u) is the absolute classfield of K.

(iii) Gal(K(a)/K) permutes the j(E)‘s associated with R transitively.

There are analogous results for f > 1; in particular, the j(E) are still algebraic integers. In fact, we can be more explicit. Call Ell(R,) the set of elliptic curves with endomorphism ring RI; if E g C/r E Ell(R,), write j(r) for its invariant; r is an inversible ideal of R/, and j(I) depends only on its class. Hasse (1927, 1931) has proved the following theorem.

THEOREM. Let p be a good prime of K, with (p,f) = (1); let p, = p n R, be the corresponding ideal of R,. Then the Frobenius element F(p) acts on i(r) by

Mr)>p(p) = j(r-p;l).

COROLLARY (f= 1). If Ette E Cl(R), and the Artin map takes c E Cl(R) to b, E Gal(K(a)/K), then a, (e) = e - c.

Briefly, Cl(R) acts on EIl(R) by translation with a minus sign.

2. The Proofs

We describe Deuring’s algebraic proof of these theorems (Deuring, 1949, 1952); for generalizations, see the tract of Shimura and Taniyama (1961).

First, we must make sure that everything we are using is algebraically defined. So, as before, let K be a complex quadratic field with integers R. A curve E defined over an algebraic closure K by y2 = 4x3 - g2 x -gs has an invariant j = 1728gs/A where A = gz - 27g:; and End(E) is well-defined algebraically. We consider the set Ell(RI) of (classes of) curves with endomorphism ring R/.

Our correspondence Ell(R,) c* Cl(R,) was obtained using the topology of C; so it must be replaced by something algebraic. We assert that EII(RJ is an affine space over Cl(R,) ( in other words, given x E El1 and y E Cl we can define x-y E Ell). In fact, if E is a curve and M is a projective module of

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294 J.-P. SERRE

rank 1 over R,, we may define M*E = Hom(M, E). More explicitly, take a resolution

4 Ry+R;!-,M+O;

then Hom(R/, E) = E, so Hom(M, E) is the kernel of ‘4 : E” + Em. Proof of the Theorem. Certainly j(E) is an algebraic number; for if it were

transcendental, there would be infinitely many curves with complex multiplica- tion by R,, but Cl(R,) is finite. In fact, thej(E) are algebraic integers, but we will not prove this (one needs other methods; for instance, show there is a model of E over a finite extension of K with a “good reduction” everywhere).

G = Gal(R/K) operates on the set ElI(R/) and preserves its structure of Cl(R/)-affine space; hence G acts by translations. This means that there is a homomorphism #J : G + CI(R/) such that the action of d E G on Ell(R,) is translation by 4(a).

We make the following assertions.

(9 C$ is onto. If p is a good prime, let Fp E CI(R/) be the image of the Frobenius element

by 4. (ii) Fv = WI. Clearly, (ii) implies (i); for, given (ii), Cl(p) E 4(G) for all good p and

trivially Cl(R,) is generated by the Cl(p)‘s. The assertion (i) tells us that the Galois group permutes the j(E) transitively, and (i) and (ii) taken together tell us that @j(E)) is the absolute class field (for f = 1). Note that it is actually enough to know (ii) for almost all primes p of first degree; class field theory takes care of the others.

Proof of (ii). Let E be an elliptic curve defined over L, where L/K is abelian. If ‘$ is a prime of L, not in a finite set S’ of bad primes, E has a good reduction J&, modulo ‘$. Suppose that ‘$Ip, p a prime of K. If Np is the absolute norm of p, (J!?~)~~ is got by raising all coefficients to the Np-th power. We assert

(iii) UQNv s p*Ep E (@&,. Now (iii) implies (ii), for by (iii), j(E)Np E j(p*E) (modulo 5@), so the

Frobenius mapj(E) +tj(E)Np is the translation j(E) ~+j(p*E).

Proof of (iii). The inclusion map p + RI induces p*E c R/*E = E. This map E + p*E is an isogeny of elliptic curves, and we see easily that its

degree is Np. Taking reductions modulo $3, we have an isogeny E -+ z Case 1. p of degree 1, Np = p. The map E + z has degree p, and may

be seen to be inseparable (look at the tangent space). Hence, it can only be the map xwxP of E -+ EP. For our application, this is actually enough. But we had better sort out the other case:

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COMPLEX MULTIPLICATION 295

Case 2. p of degree 2, so Np = p2 where p is inert in K/Q. Then one can show that B has Hasse invariant zero; that is, it has no point of order p.

The map E-z accordingly has trivial kernel, so again it is purely inseparable, so it is E + f?*‘.

Example. There are either 13 or 14 complex quadratic RI’s with class number 1, namely Q(,/-d) withf= 1 for d = 1,2,3,7,11,19,43,67,163, and possibly ?, and also Q(,/- l), Q(,/- 3) and Q(,/- 7) with f= 2 and Q(J-3) with f = 3. Hence there are 13 or 14 curves with complex multi- plication with i E Q. Their invariants are: (see end of chapter) j = 2633, 2653, 0, - 3353, -215, -21533, -2l*3353 ,

-2153353113 -2183353233293 -m3, 2333113 , 24;553, 3353173 . -j121553.

3. Maximal Abelian Extension

We want the maximal abelian extension Kab of K. As a first shot, try K’, the union of the fields given by the j’s of the elliptic curves of Eli(+), for f= 1,2,...; this field is generated by the i(z>‘s where Im(@ > 0 and T E K. Now enlarge K’ by adding the roots of unity; we get a field K’? = QcyC’ . K’ which is very near Kab. The following theorem states this result more precisely :

THEOREM. Gal(Kab/K “) is a product of groups of order 2.

This follows easily from class field theory and the results of Section 2.

Let now k? be the absolute class field of K, and let E be an elliptic curve defined over R, with End(E) = R. Let L, be the extension of k? generated by the coordinates of the points of finite order of E. This is an abelian extension of K, whose Galois group is embedded in a natural way in the group U(K) = fl U,(K), where U,(K) means the group of units of K at the finite prime V. By class field theory, L, is then described by a homomorphism

8, : I= + U(K)

where IR is the idele group of R. Let U be the group of (global) units of K, so that U = { f 1) unless K is Q(,/- 1) or Q(,/-3). One can prove without much difficulty that the restriction of dE to the group U(R) is:

‘-G(x) = Nr&- ‘1. P&) where ps is an homomorphism of U(R) into U.

The extension L,, and the homomorphism pE, depend on the choice of E. To get rid of this, let X = E/U be the quotient of E by U (X is a projective line), and let L be the extension of R generated by the coordinates of the images in X of the points of finite order of E. This extension is independent of the choice of E.

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296 J.-P. SEIRRE

THEOREM. L is the maximal abelian extension of K.

This follows by class field theory from the properties of 0, given above.

Remark. If U is of order 2 (resp. 4, 6) the map E + X is given by the coordinate x (resp. by x2, x3). Hence Kab is generated by j(E) and the coordinates x (resp. x2,x3) of the points of finite order on E; this has an obvious translation in analytical terms, using thej and g functions.

[For further and deeper results, analogous to those of Kummer in the cyclotomic case, see a recent paper of Ramachandra (1964).]

REFERENCES

Deuring, M. (1949). Algebraische Begrtindung der komplexen Multiplikation. Abh. Math. Sem. Univ. Hamburg, 16, 32-47.

Deuring, M. (1952). Die Struktur der elliptischen Funktionenkorper und die Klassenkiirper der imaginiiren quadratischen Zahlkorper. Math. Ann. 124,393-426.

Deuring, M. Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins. Nachr. Akad. Wiss. Giittingen, Math-Phys. KI. (1953) 85-94; (1955) 13-42; (1956) 37-76; (1957) 55-80.

Deuring, M. Die Klassenkorper der komplexen Multiplikation. Enz. Math. Wiss. Band Iz, Heft 10, Teil II (60 pp.).

Fueter, R. “Vorlesungen tiher die singularen Moduhr und die komplexe Multi- plikation der elliptischen Funktionen”, (1924) Vol. I; (1927) Vol. II (Teubner).

Hasse, H. Neue Begrtindung der komplexen Multiplikation. J. reine angew. Math. (1927) 157,115-139; (1931) U&64-88.

Ramachandra, K. (1964). Some applications of Kronecker’s limit formulas. Ann. Math. 80, 104-148.

Shimura, G. and Taniyama, Y. (1961). Complex multiplication of abelian varieties Publ. Math. Sot. Japan, 6.

Weber, H. (1908). “Algebra”, Vol. III (2nd edition). Braunschweig. Weil, A. (1955). On the theory of complex multiplication. Proc. Znt. Symp. Algebraic

Number Theory, Tokyo-Nikko, pp. 9-22.

STOP PFms

H. M. Stark has shown (Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 216-221) that there is no tenth imaginary quadratic field with class-number 1 (the ? of the Example at the end of $2). Practically simultaneously A. Baker (Mathematika, 13 (1966), 204-216) proved an important general theorem which reduces the problem of its existence to a finite amount of computation.

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CHAPTER XIV

l-Extensions

K. HOECHSMANN

Introduction ........... 297 1. TwoLemmas .......... 297 2. Local Fields .......... 298 3. Global Fields .......... 300 Appendix: Restricted Ramification ....... 302 References ........... 303

Introduction

Let I be a rational prime, R a group of order I (multiplicative). For a pro- finite group G, we shall denote by H’(G) the cohomology group H’(G, Q), G operating trivially on R. If G is the Galois group of a field extension K/k, we shall sometimes write H’(K/k). We shall be interested in these groups particularly for i = 1,2 in the case where k is a local or global field and K its maximal I-extension, i.e. the maximal normal extension whose Galois group is a pro-Z-group.

1. Two Lemmas

LEMMA 1. Consider a family (93 of morphisms of exact sequences of pro-l-groups:

l+R+F+G+l Q” t t t t

l-+R,+F,+G,-+l

where (a) F, F, are free? (H’(F) = H2(FV) = 1). (b) F, F, are built from a minimal system of generators.

G, G, resp. (H’(G) + H’(F), etc., are surjective).

Then R is generated (as a closed normal subgroup of F) by the totality of images 4pU(RV) - the map

H2(G) + n H2GJ 0

is injective. Proof. In analogy to Prop. 26 of Serre (1964) we have: R generated by

the cp,(R,) -+ H’(R)’ + n H’(R,) injective. On the other hand, the trans- v

t Cf. Serre (1964) for the meaning of freedom, etc., in this context. 297

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298 K.HOECHSMANN

gressions (Tg) in the diagram

H’(R)’ 2 H’(G) 1 1

H’(RU)Gv 2 If’(G,)

are isomorphisms by the exactness of the Hochschild-Serre sequence and our hypotheses (a) and (b).

LEMMA 2. Given field extensions as shown in the diagram (everything

K’ Galois over k), where K/k and K’/k’ are l-extensions,

I E = Gal (k/k) is finite with order prime to I, consider the

KAL obvious map 8: Gal (K’/k’) + Gal (K/k). Suppose

I ‘f (a) H’(K’/K) = 1 and

k _A’ k (b) Res: H’(K’/k) + H2(K’/L) to be trivial. I: Then 8 induces an isomorphism

8* : H’(K/k) + H2(K’/k’)‘.

Proof. O* is the composition of Inf: H2(K/k) + H’(K’/k)

and Res: H2(K’/k) + iY’(K’/k’)‘.

The bijectivity of Res follows from the fact that I is prime to the order of E. To see it, one could use dimension-shifting on finite subextensions and the Hochschild-Serre sequence. Hypothesis (a) yields the exactness of the pertinent Inf-Res sequence in dimension 2, hence the injectivity of Inf. Its surjectivity then follows from hypothesis (b) and the injectivity of the restriction from (K’/K) to (K’/L).

2. Local Fields Let k be a finite extension of the rational p-adic field Qp, K the maximal Z-extension of k, and G = Gal (K/k).

Put 6 = rank of Z-torsion part of k*,

ifl-P otherwise.

We are interested in the numbers d1 and dZ of generators and defining relations for G: i.e. the ranks of H’(G) and H’(G).

THEOREM 1.

d’ =n+6+1, d’=b.

Proof. By the Bumside Basis Theorem, it sufhces to count generators of G/G’[G, G], which by local class field theory amounts to looking at k*/k*‘.

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hXENSIONS 299

The 1 in the formula for d’ comes from the infinite cyclic group generated by a prime element, the n+6 from the group U of units. For U is the direct product of a finite cyclic group and a free Z,-module of rank [k: Q,] (cf. Hasse (1962) II. 15. 5). So much for d ‘.

To find d2, we distinguish two cases.

(1) 6 = 1. Inject R into K obtaining an exact sequence I 1

l+n-+K*+K*-+l,

where I means raising to the lth power. Because of Hilbert 90, we have an isomorphism

i : H2(G) + H’(G, K*),,

lower I meaning Z-torsion part. These are the Brauer classes of order I, by local class field theory a group of order 1.

(2) 6 = 0. Adjoin Ith roots of unity obtaining a field k’ and, by construc- tion of its maximal I-extension K’, a diagram as in Lemma 2. The hypotheses of Lemma 2 are readily verified.

(a) H’(K’/K) = Horn (Gal @C/K), a) = 1, since a nontrivial element would yield a proper l-extension of K.

(b) Already the restriction from H’(K’/k’) to H2(K’/L) is trivial, since elements of these groups can, as in (l), be identified with Brauer classes of order I; these die in the vastness of L/k’.

By Lemma 2,

H2(G) = H2(K’/ky.

E is cyclic; let E be a generator. Our injection

i:R+K’*

is not an s-map. Indeed, for w E fi, we have

i(w) = i(0) = [si(o)]“,

where m E Z such that a-‘6 = cm for Zth roots [ of unity. Accordingly, we get a commutative diagram

If’(K’/k’) 1, Br(k’) 81 1 s.m

H’(K’/k’) 1, Br (k’)

i.e. for a E Hz&‘/k’) we have: inv [i(c?)] = m . inv [(ia>“] = m . inv [ia],

which shows that, for a E H2(K’/k’)E, ia = 1 and hence a = 1. Remark. The invariants d’ and d2 and the exponent 1’ of the f-torsion

part of k* determine G completely except in the case 6 = 1, P = 2. If

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300 K. HOECHSMANN

6 = 0, we are dealing with a free group, and the remark is trivial. If 6 = 1, G has the properties:

(1) H2(G) N R. (2) The pairing H’(G) x H’(G) -P n defined by cup-product and (1)

is non-degenerate.

(Via the Kummer isomorphism H’(G) N k*Jk*‘, this pairing corresponds to Hilbert’s Z-power residue symbol.) Pro-Z-groups having a finite number d of generators and satisfying (1) and (2) are called DemzGkin groups. If G is any Demu&in group such that the exponent P of the torsion part of G/[G, G] is greater than 2, generators xl, . . ., xd of G can be found such that the relation deting G takes on the form

431,x,] . . . [xd-l,xd] = 1,

square brackets denoting commutators, If P = 2, there are several group types depending on a more subtle invariant. For details, see Labute (1965) and Serre (1964).

In the present context, all this machinery is necessary only if 1 = p. If Z # p, the cardinality q of the residue class field of k must be = 1 (mod Z) in order for 6 to be = 1. Let Z’(q- 1, s maximal; C be a primitive I”-th root of unity in k. Then G has two generators u and r obtained by arbitrary exten- sion to K of the norm residue symbols for x (prime element) and C. Every finite subextension L/k is metabelian with the unramiCed part T/k left fixed by r. Therefore 7 = ({, LIT), where [ E T may be chosen to be a root of unity. Hence CRC-~ = (a& L/T) = (Cq, L/T) = 7q, this relation holding on every finite L/k and hence on K/k.

3. Global Fields

Let k be a finite extension of Q, and K the maximal Z-extension of k with Galois group G.

For each prime ZJ of k consider the completion k, and its maximal Eexten- sion K, (here we deviate from standard notation!) with Galois group G,.

We flx an extension w of t, to K, obtaining an injection

q,,:G,+G

whose image is the decomposition group of w. Let 6,6, be the Z-torsion rank of k*, k: resp.

THEOREM 2. The (pv induce a monomorphism

p* : H’(G) + u H’(G,,) ”

whose cokernel has rank 6.

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I-EXTENSIONS 301

Proof. We proceed as in the proof of Theorem 1.

(1) 6 = 1. An injection i: R + K* yields corresponding injections iv: R -+ K,* to make commutative diagrams

H’(G) + H’(G,K*), 1 1

H2(Gv) --) H2(G,K:), bijectivity of the horizontal arrows being furnished by Hilbert (Theorem 90) via the exact sequence

1 1

l+SI+KK*-+P-,l

and its local counterpart. We use the right-hand sides of these diagrams and Hasse’s characterization of Brauer classes by their local invariants to obtain the exact sequence

1+ H’(G) 2 u H’(G,) : Cl + 1. ”

where x denotes the sum of invariants written multiplicatively. (2) 6 = 0. Again we adjoin Zth roots of unity obtaining a field k’. We

extend each place u’ of k’ to a place w’ of the maximal Z-extension K’, so that w’ agrees with the already chosen w on K. We now have a diagram as in Lemma 2 for the global fields and, for each v’, a similar local one, showing Ku/k, and Kj/k:# with Galois groups G, and GLS resp. The commutation rule: cpV 0 8, = 8 0 (py. (0’1~) yields a commutative diagram

9.W’

H2(G’) + H’(G:,) 0.1 pv*

H’(G) ‘2 H2(Gv)

for each u’. Taking all these diagrams together and noting the injectivity of 8* (since

H’(K’/K) = 1, again by maximality of K), we deduce the injectivity of ‘p* = n cp: from that of n cp$, which was established in (1).

To ;rove surjectivity, Ge note that, by Lemma 2, the image of 8* is Hz(G)“, hypothesis (b) being verified as before by identifying H2(K’/k’), H’(K’/L) with the I-torsion of Brauer groups over k’ and L resp. and watching the former die (locally everywhere) as we restrict them to L. Next we observe that H’(G,) # 1 e 6, = 1 o u splits completely in k’, and that we can limit our attention to the set Se of these places. We now have a diagram

H;!;“E -, g ~2WtJ

H’(G) 2 ~~.Hf”GJ

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302 K. HOECHSMANN

where 8* is a bijection, and propose to prove the surjectivity of ‘p* by showing that the map

H*(G) -+L&~*(G.)

is no longer injective, if we leave out any b E So from the product on the right. This amounts to finding a non-trivial CI E H2(G’)E such that (p,‘(a) = 1 for all u’ not lying over b.

Arguing as at the end of the proof of Theorem 1 and using the notation introduced there, we find

inv,,, [ i(a”)] = m . inv,. [ icr] (N.B. this time E operates on the places, too.) Therefore

a E H2(G)E e inv#,, [ia] = m. inv,e [ia]. We define the desired a by prescribing the invariants of ia:

(0 if v’J’b

nv,. [iu] = mi

i if')'= db (j=O,...,r-1)

where b’ is a fixed place above b, and r = [k’ : k]. Since r-l

Cm' z 0 (modI), j=O

this does define a Brauer class ia of order I. Remark. By Lemma 1, this result can be translated into a theorem of

H. Koch (1965) about relations defining the group G. If we present G, G. as factor groups of free groups F, I;, and extend cpV to a morphism of the corresponding exact sequences as in Lemma 1, it follows from Theorem 2 that R is generated by the images p>,(R”). Now, R, is generated by a single relation r,, which is trivial if S, = 0 (cf. Theorem 1). Theorem 2 also asserts that the cp,(r,) form a minimal system of relations if 6 = 0 and that, if 6 = 1, a minimal system is obtained by leaving one of them out.

Restricted Ramification APPENDIX

Let S be a set of places on the field k considered in the preceding paragraph, and let K(S) be the maximal Z-extension of k unramified outside of S. So far, we have been dealing with the case S = all places; the other extreme, S = 0, is the topic of Roquette’s Chapter IX. We wish to conclude with some remarks about the general situation.

In Koch (1965), G(S) = Gal (K(s)/k) is studied as a factor group of G. Generators of G are obtained from the norm residue symbols for generators of I/k*I’ (I= ideles of k). More precisely, one considers the exact sequence

1 + U/k*I’ n U --) I/k*l’ + Cl/Cl’ --, 1

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I-EXTENSIONS 303

(U = idele units; Cl = ideal classes), and chooses for generators: (1) pre- images of a basis of the finite cokernel and (2) for each place u a basis B. of UJJL. The natural map

G-+%9 is then given by setting z = 1 whenever r E B,, v # S (the elements of B, lie in the inertia group of u).

If we had a description of G in terms of these generators and some defining relations, we should get a corresponding description of G(S) by simply cross- ing out the unwanted r’s everywhere. But Theorem 2, gives relations for G only if we start with a minimal system of generators; and our system is not minimal! Indeed, a certain finite number of elements of u B, has to be

eliminated to make it minimal, and the outlined procedure Gorks only if S is large enough to include the D’S corresponding to these elements. In gensral this approach is still strong enough, to yield a fundamental inequality of SafareviE (1963, Theorem 5), and in favourable special cases it leads to a satisfactory description of G(S). For details, see (Koch, 1965).

In Brumer (196?), the methods used by us to prove Theorem 2 (case (1)) are directly applied to the more general problem. It is assumed that & = 1 and that S contains all primes lying above I. Then the extraction of Zth roots of S-units leads to extensions unramified outside of S and thus to an exact sequence

142~E&E(s)+1.

where E(S) denotes S-units of K(S). Passing to cohomology, we have

1+ ,H’WS), E(S)) --) ~2(W9~ f, ~2(G(s>, JW)z --, 1. where, for abelian groups X, ,X means X/X’. It is not difficult, to identify the last term with a subgroup of Br (k)l, more precisely: with those Brauer classes of order I whose localizations are trivial outside of S. Furthermore, H’(G(S), E(S)) turns out to be isomorphic to ideal classes of k modulo classes containing elements of S; we shall call this group A(S). We obtain an exact sequence

1 + J(S) -+ H2(G(S)) -+yL6sH2(GJ 1: Q + 1

analogous to the one in the proof of Theorem 1 (part (1)). The more difbcult case (8 = 0) has not yet been successfully dealt with

by this method.

REFERENCES

Brumer, A. (1961). Galois groups of extensions of algebraic number fields with given ramification. (Unpublished.)

Hasse, H. (1962). “Zahlentheorie”, Akademie-Verlag, Berlin.

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304 K. HOECHSMANN /

Koch, H. (1965). I-Erweiterungen mit vorgegebenen Verzweigungsstellen. J. reine angew. Math. 219, 30-61.

Labute, J. (1965). Classification des groupes de Demu&in. Comptes rendues, 260, 1043-1046.

Serre, J.-P. (1963). Structure de certains pro-p-groupes. Skminaire Bourbaki, exp. 252.

Serre, J.-P. (1964). Cohomologie galoisienne. Lecture notes in Mathematics, 5. Springer-Verlag.

SafareviE, I. R. (1963). Extensions with given ramification. Publs. math. Inst. ht. Etud. scient. 18, 71-95. (Russian with French summary.)

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-xv

Fourier Analysis in Number Fields and

Hecke’s Zeta-Functionst

J. T. TATE

1. Introduction . . . . . . . 1.1. Relevant History . . . . . 1.2. This Thesis . . . . . . 1.3. “Prerequisites” . . . . .

2. The Local Theory . . . . . . 2.1. Introduction . . . . . . 2.2. Additive Characters and Measure . . 2.3. Multiplicative Characters and Measure . 2.4. The Local C-function; Functional Equation . 2.5. Computation of p(c) by Special C-functions .

3. Abstract Restricted Direct Product . . . 3.1. Introduction . . . . . . 3.2. Characters . . . . . . 3.3. Measure . . . . .

4. The Theory in the Large . . . . . 4.1. Additive Theory . 4.2. Riemann-Roth Theorem 1 1 1 1 4.3. Multiplicative Theory . . . . 4.4. The C-functions; Functional Equation .

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4.5. Comparison with the Classical Theory A Few Comments on Recent Related Literature (Jan: 1967) References . . . . . . . . .

ABSTRACT

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: 313 316 . 322 . 322 . 324 . 325 . 327 . 327 . 331 : 338 334

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* . Z18

We lay the foundations for abstract analysis in the groups of valuation vectors and ideles associated with a number field. This allows us to replace the classical notion of C-function, as the sum over integral ideals of a certain type of ideal character, by the corresponding notion for ideles, namely, the integral over the idele group of a rather general weight function times an idele character which is trivial on field elements. The role of Hecke’s complicated theta-formulas for theta functions formed over a lattice in the n-dimensional space of classical number theory can be played by a simple Poisson formula

t This is an unaltered reproduction of Tate’s doctoral thesis (Princeton, May 1950). The editors were urged by several of the participants of the Conference to include this thesis in the published proceedings because although it has been widely quoted it has never been published. The editors are very grateful to Tate for his permission to do this and for the comments on recent literature (p. 346).

305

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306 J. T. TATE

for general functions of valuation vectors, summed over the discrete subgroup of field elements. With this Poisson formula, which is of great importance in itself, inasmuch as it is the number theoretic analogue of the Riemann-Roth theorem, an analytic continuation can be given at one stroke for all of the generalized C-functions, and an elegant functional equation can be established for them. Translating these results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming from the archimedean primes and the primes of the conductor. The notion of local C-function has been introduced to give local definition of these factors, and a table of them has been computed.

1. Introduction

1.1. ReIevant History

Hecke was the first to prove that the Dedekind c-function of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for the Dedekind c-function and L-series, but also for a c-function formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number modulo the “conductor”, but also on the position of the conjugates of the number in the complex field. Overcoming rather extraordinary technical complications, he showed (19 18 and 1920) that these “Hecke” c-functions satisfied the same type of func- tional equation as the Dedekind c-function, but with a much more compli- cated factor.

In a work (Chevalley, 1940) the main purpose of which was to take analysis out of class field theory, Chevalley introduced the excellent notion of the idble group, as a refinement of the ideal group. In idbles Chevalley had not only found the best approach to class field theory, but to algebraic number theory generally. This is shown by Artin and Whaples (1945). They defined valuation vectors as the additive counterpart of ideles, and used these notions to derive from simple axioms all of the basic statements of algebraic number theory.

Matchett, a student of Artin’s, made a first attempt (1946) to continue this program and do analytic number theory by means of ideles and vectors. She succeeded in redefining the classical c-functions in terms of integrals over the idUe group, and in interpreting the characters of Hecke as exactly those characters of the ideal group which can be derived from idele characters. But in proving the functional equation she followed Hecke.

1.2. This Thesis

Artin suggested to me the possibility of generalizing the notion of c-function, and simplifying the proof of the analytic continuation and functional

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 307

equation for it, by making fuller use of analysis in the spaces of valuation vectors and ideles themselves than Matchett had done. This thesis is the result of my work on his suggestion. I replace the classical notion of [-func- tion, as the sum over integral ideals of a certain type of ideal character, by the corresponding notion for ideles, namely, the integral over the idele group of, a rather general weight function times an idele character which is trivial on field elements. The role of Hecke’s complicated theta-formulas for theta functions formed over a lattice in the n-dimensional space of classical number theory can be played by a simple Poisson Formula for general functions of valuation vectors, summed over the discrete subgroup of field elements. With this Poisson Formula, which is of great importance in itself, inasmuch as it is the number theoretic analogue of the Riemann-Roth theorem, an analytic continuation can be given at one stroke for all of the generalized c-functions, and an elegant functional equation can be estab- lished for them. Translating these results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming from the archimedean primes and the primes of the conductor. The notion of local c-function has been introduced to give a local definition of these factors, and a table of them has been computed.

I wish to express to Artin my great appreciation for his suggestion of this topic and for the continued encouragement he has given me in my work.

1.3. “Prerequisites”

In number theory we assume only the knowledge of the classical algebraic number theory, and its relation to the local theory. No knowledge of the idele and valuation vector point of view is required, because, in order to introduce abstract analysis on the idele and vector groups we redefine them and discuss their structure in detail.

Concerning analysis, we assume only the most elementary facts and definitions in the theory of analytic functions of a complex variable. No knowledge whatsoever of classical analytic number theory is required. Instead, the reader must know the basic facts of abstract Fourier analysis in a locally compact abelian group G: (1) The existence and uniqueness of a Haar measure on such a group, and its equivalence with a positive invariant functional on the space L(G) of continuous functions on G which vanish outside a compact. (2) The duality between G and its character group, G, and between subgroups of G and factor groups of G. (3) The del?nition of the Fourier transform, j, of a function f E L,(G), together with the fact that, if we choose in G the measure which is dual to the measure in G, the Fourier Inversion Formula holds (in the ntive sense) for all functions for which it could be expected to hold; namely, for functions f E L,(G)

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308 J. T. TATE

such that f is continuous and f~ L,(e>. (This class of functions we denote by ‘Z&(G).) An elegant account of this theory can be found for example in Cartan and Godement (1947).

2. The Lmal Theory

2.1. Introduction

Throughout this section, k denotes the completion of an algebraic number field at a prime divisor p. Accordingly, k is either the real or complex field if p is archimedean, while k is a “p-adic” field if p is discrete. In the latter case k contains a ring of integers o having a single prime ideal p with a finite residue class field o/p of Np elements. In both cases k is a complete topological field in the topology associated with the prime divisor p.

From the infinity of equivalent valuations of k belonging to p we select the normed valuation defined by:

a = ordinary absolute value if k is real. 1: 9 a = s uare of ordinary absolute value if k is complex. Ial = (A+)-‘, where Y is the ordinal number of a, if k is p-adic.

We know that k is locally compact. The more exact statement which one can prove is: a subset B c k is relatively compact (has a compact closure) if and only if it is bounded in absolute value. Indeed, this is a well known fact for subsets of the line or plane if k is the real or complex field; and one can prove it in a similar manner in case k is p-adic by using a “Schubfach- schluss” involving the finiteness of the residue class field.

2.2. Additive Characters and Measure

Denote by k+ the additive group of k, as a locally compact commutative group, and by t its general element. We wish to determine the character group of k+, and are happy to see that this task is essentially accomplished by the following:

LEMMA 2.2.1. If e + X(r) is one non-trivial character of k+, then for each rl E k+, t + Wtl is also a character. The correspondence q c) X(q<) is an isomorphism, both topological and algebraic, between k’ and its character group.

Proqf. (1) X(qr) is a character for any fixed q because the map r + qr is a continuous homomorphism of k+ into itself.

(2) X(h +rlN = Xh 5 +v2 0 = WI W(tl, 0 shows that the map v + X($) is an algebraic homomorphism of k+ into its character group.

(3) X(qt) = 1, all l =z= qk+ # k+ sj tf = 0. Hence it is an algebraic isomorphism into.

(4) X(qr) = 1, all q * k+r # k+ => t = 0. Therefore the characters of the form X(q[) are everywhere dense in the character group.

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FOURIER ANALYSIS IN NUMBER FIELDS AND HJXKE’S ZETA-mCTIONS 309

(5) Denote by B the (compact) set of all r E k+ with ItI < Mfor a large M. Then: q close to 0 in k+ * qB close to 0 in k+ + X(qB) close to 1 in complex plane * X(q<) close to the identity character in the character group. On the other hand, if &, is a fixed element with X(&J # 1, then: X(@ close to identity character =z. X(vB) close to 1, closer, say, than X(&,) =E- &., 4 qB =S q close to 0 in k+. Therefore the correspondence q c, X(qr) is bicontinuous.

(6) Hence the characters of the form X(qa comprise a locally compact subgroup of the character group. Local compactness implies completeness and therefore closure, which together with (4) shows that the mapping is onto.

To fix the identification of k’ with its character group promised by the preceding lemma, we must construct a special non-trivial character. Let p be the rational prime divisor which p divides, and R the completion of the rational field at p. Define a map x + A(x) of R into the reals mod 1 as follows :

Case 1. p archimedean, and therefore R the real numbers.

A(x) = - x (mod 1)

(Note the minus sign!)

Case 2. p discrete, R the field ofp-adic numbers. A(x) shall be determined by the properties:

(a) A(x) is a rational number with only a p-power in the denominator. (b) A(x) - x is a p-adic integer.

(‘IO tid such a A(x), let p’x be integral, and choose an ordinary integer n such that n = p’x (modpy. Then put A(x) = n/p’; A(x) is obviously uniquely determined modulo 1.)

LEMMA 2.2.2. x + A(x) is a non-trivial, continuous additive map of R into the group of reals (mod 1).

Proof. In case 1 this is trivial. In case 2 we check that the number A(x)+@) satisfies properties (a) and (b) for x+y, so the map is additive. It is continuous at 0, yet non-trivial because of the obvious property: A(x) = 0 o x is p-adic integer.

Define now for r E k+, A(r) = A(S,,,t). Recalling that ,!& is an additive continuous map of k onto R, we see that t: + e2nfh(C) is a non-trivial character of k’. We have proved:

THEOREM 2.2.1. k’ is naturally its own character group if we identifr the character e -+ eZniAtqC) with the element q E k+.

LEMMA 2.2.3. In case p is discrete, the character eZniACqC) associated with qistrivialonoifandonlyifq~b -l, b denoting the absolute d@Terent of k.

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310 J. T. TATE

Proof. h(tjo) = 0 * &,(~‘,)) = 0 e &,R(~o) c OR * q E b- ‘.

Let now ,u be a Haar measure for k+.

LEMMA 2.2.4. If we define ,uu,(M) = p(arM) for a # 0 E k, and M a measur- able set in k’, then p1 is a Haar measure, and consequently there exists a number q(a) > 0 such that p, = cp(a),u.

Proof. 5 + a5 is an automorphism of k+, both topological and algebraic. Haar measure is determined, up to a positive constant, by the topological and algebraic structure of k+.

LEMMA 2.2.5. The constant &a) of the preceding lemma is lal, i.e. we have Pww = I4AW-

Pro@I If k is the real field, this is obvious. If k is complex, it is just as obvious since in that case we chose 1~1 to be the square of the ordinary absolute value. If k is p-adic, we notice that since o is both compact and open, 0 < p(o) c co, and it therefore suffices to compare the size of o with that of ao. For u integral, there are N(oro) cosets of ao in o, hence ~(CIO) = (N(ao))-‘p(o) = It+(o). For non-integral CI, replace a by a-l.

We have now another reason for calling the normed valuation the natural one. lcrl may be interpreted as the factor by which the additive group k+ is “stretched” under the transformation < + at.

For the integral, the meaning of the preceding lemma is clearly:

444 = I4 445); or more fully:

J-f<0 40 = I4 .I-f(d) 440. So much for a general Haar measure ,u. Let us now select a fixed Haar

measure for our additive group k’. Theorem 2.2.1 enables us to do this in an invariant way by selecting the measure which is its own Fourier transform under the interpretation of k+ as its own character group estab- lished in that theorem. We state the choice of measure which does this, writing d5 instead of dp(l), for simplicity:

dr = ordinary Lebesgue measure on real line if k is real. d5 = twice ordinary Lebesgue measure in the plane if k is complex. d< = that measure for which o gets measure (A%)-* if k is p-adic.

THEOREM 2.2.2. If we define the Fourier transform f of a function f e L,(k+) by:

f(q) = 1 f(Q e-2niA(qC) de,

then with our choice of measure, the inversion formula

f(5) = If(q) e2”“‘(cq) dq = f( - 0

holdr for f e ?&(k+).

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 311

Proof. We need only establish the inversion formula for one non-trivial function, since from abstract Fourier analysis we know it is true, save possibly for a constant factor. For k real we can take f(c) = e-‘lcl’, for k complex, f(5) = e-2n151; and for k p-adic, f(t) = the characteristic function of o, for instance. For the details of the computations, the reader isreferred to Section 2.6 below.

2.3. Multiplicative Characters and Measure

Our Grst insight into the structure of the multiplicative group k* of k is given by the continuous homomorphism a + Ial of k* into the multiplicative group of positive real numbers. The kernel of this homomorphism, the subgroup of all a with jcrl = 1 will obviously play an important role. Let us denote it by u. u is compact in all cases, and in case k is p-adic, u is also open.

Concerning the characters of k*, the situation is different from that of k+. First of all, we are interested in all continuous multiplicative maps a -P c(a) of k* into the complex numbers, not only in the bounded ones, and shall call such a map a quasi-character, reserving the word “character” for the con- ventional character of absolute value 1. Secondly, we shall find no model for the group of quasi-characters, or even for the group of characters, though such a model would be of the utmost importance.

We call a quasi-character unramified if it is trivial on U, and first determine the unramified quasi-characters.

LEMMA 2.3.1. The unramified quasi-characters are the maps of the form c(a) = Ial” E eslOglul, where s is any complex number, s is determined by c if p is archimedean, while for discrete p, s is determined only mod 2nijlog Np.

Proof. For any s, Ial” is obviously an unramified quasi-character. On the other hand, any unramified quasi-character will depend only on Ial, and as function of Ial will be a quasi-character of the value group of k. This value group is the multiplicative group of all positive real numbers, or of all powers of Np, according to whether p is archimedean or discrete; it is well known that the quasi-characters of these groups are those described.

If p is archimedean, we may write the general element a E k* uniquely in the form a = @, with d E u, p > 0. For discrete p, we must select a fixed element II of ordinal number 1 in order to write, again uniquely, a = &, with a” E u and, this time, p a power of K. In either case the map a + d is a continuous homomorphism of k* onto u which is identity on u.

THEOREM 2.3.1. The quasi-characters of k* are the maps of the form a + c(a) = Z’(E)lal”, where c” is any character of u. Z is uniqueZy determined by c. s is determined as in the preceding lemma.

Proof. A map of the given type is obviously a quasi-character. Con- versely, if c is a given quasi-character and we define c” to be the restriction

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312 J. T. TATE

of c to U, then E is a quasi-character of u and is therefore a character of u since u is compact. a -P c(a)/E(C) is an umamiGed quasi-character, and therefore is of the form Ial” according to the preceding lemma.

The problem of quasi-characters c of k* therefore boils down to that of the characters E of U. If k is the real field, u = { 1, - 1) and the characters are t((a) = B, n = 0, 1. If k is complex, I( is the unit-circle, and the characters are F(C) = d”, n any integer. In case k is p-adic, the subgroups 1 +p”, v > 0, of u form a fundamental system of neighborhoods of 1 in U. We must have therefore c”(1 + p”) = 1 for sufficiently large Y. Selecting v minimal (v = 0 if c’ = l), we call the ideal i = p’ the conductor of Z. Then E is a character of the finite factor group u/(1 +f) and may be described by a finite table of data.

From the expression c(a) = Z(Z)ja[s for the general quasi-character given in Theorem 2.3.1, we see that [c(a)1 = lalb, where c = Re (s) is uniquely determined by c(a). It will be convenient to call cr the exponent of c. A quasi-character is a character if and only if its exponent is 0.

We will be able to select a Haar measure da on k* by relating it to the measure dt on k+. If g(a) EL(k*), then g(<)lSl-’ EL(k+-0). So we may define on L(k*) a functional

Wd = J sco[epe. k+-0

If h(a) = g(pa) (B E k*, fixed) is a multiplicative translation of g(a), then

Wd = 1 sW>ltl-’ dt = Wd, k+-0

as we see by the substitution t -+ /3-‘r; dr + I/31-’ d{ discussed in Lemma 2.2.5. Therefore our functional @ which is obviously non-trivial and positive, is also invariant under translation. lt must therefore come from a Haar measure on k+. Denoting this measure by d 1 a, we may write

~doc)4 a = 1 dOltI-’ &. k+-0

Obviously, the correspondence g(a) t, g(t)l{l-’ is a l-l correspondence between L(k*) and L(k’ - 0). Viewing the functions of L,(k*) and Ll(k+ - 0 as limits of these basic functions we obtain:

LEMMA 2.3.2. g(a) E L,(k*) o g({)lrlml E L,(k+-0), and for these functions

s g(aNa = k+~odt~ltl-ld~~

For later use, we need a multiplicative measure which will in general give the subgroup u the measure 1. To this effect we choose as our standard

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Haar measure on k* :

du=d,a=i, if p is archimedean. a

da = - Np dla=-- Np da Np-1 Np - 1 Ial’

if p is discrete.

LEMMA 2.3.3. In case p is discrete,

ProoJ

I d a = (Nb)-*. ”

Therefore

s Y Y

2.4. The Local C-function; Functional Equation

In this section f(c) will denote a complex valued function defined on k+ ; f(a) its restriction to k*. We let 3 denote the class of all these functions which satisfy the two conditions:

3J f(C), a&CT(C) continuous, E &(k+); i.e.f(t) E 23,(k+)

3J j(a)lal” and~(a)jal” E L,(k*) for Q > 0.

A c-function of k will be what one might call a multiplicative quasi-Fourier transform of a function f o 3. Precisely what we mean is stated in

DEFINITION 2.4.1. Corresponding to each f E 3, we introduce a function Ccf, c) of quasi-characters c, defined for all quasi-characters of exponent greater than 0 by

U 4 = J’ fWc(aYa,

and call such a function a c-function of k. Let us call two quasi-characters equivalent if their quotient is an unrami-

fied quasi-character. According to Lemma 2.3.1, an equivalence class of quasi-characters consists of all quasi-characters of the form c(a) = q,(a)[aI’, where c,(a) is a tied representative of the class, s a complex variable. It is apparent that by introducing the complex parameter s we may view an equivalence class of quasi-characters as a Riemann surface. In case p is archimedean, s is uniquely determined by c, and the surface will be isomor- phic to the complex plane. In case p is discrete, s is determined only mod 2xi/log JTp, so the surface is isomorphic to a complex plane in which

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314 J. T. TATE

points differing by an integral multiple of Zai/log Np are identified-the type of surface on which singly periodic functions are really defined. Looking at the set of all quasi-characters as a collection of Riemann surfaces, it becomes clear what we mean when we talk of the regularity of a function of quasi-characters at a point or in a region, or of singularities. We may also consider the question of analytic continuation of such a function, though this must of course be carried out on each surface (equivalence class of quasi-characters) separately.

LEMMA 2.4.1. A ~-function is regular in the “domain” of all quasi-characters of exponent greater than 0.

Proof. We must show that for each c of exponent > 0 the integral jfb4c(alal”d P a re resents a regular function of s for s near 0. Using the fact that the integral is absolutely convergent for s near 0 to make estimates, it is a routine matter to show that the function has a derivative for s near 0. The derivative can in fact be computed by “differentiating under the integral sign”.

It is our aim to show that the c-functions have a single-valued meromorphic analytic continuation to the domain of all quasi-characters by means of a simple functional equation. We start out from

LEMMA 2.4.2. For c in the domain 0 < exponent c < 1 and 9(a) = Ialc-‘(a) we have

C(f, m?9 21 = r<fi ws, 4 * for any two functionsf, g E 3.

Proof. [cf, c)&& E) = jf(a)c(a) da.j @(B)c-‘@IpI dp with both integrals absolutely convergent for c in the region we are considering. We may write this as an absolutely convergent “double integral” over the direct product, k* x k*, of k* with itself:

I I fW(P>c(aF ‘)[Pld(a, P).

Subjecting k* x k* to the “shearing” automorphism (a, /I) + (a, ap), under which the measure d(a, b) is invariant we obtain

If f(a)~(aP>c(P-‘>[aPld(a, LO.

According to Fubini this is equal to the repeated integral

1 (sf(a)P(aB)lalda)c~-l)[~~d~.

To prove our contention it suffices to show that the inner integral Jf(a)&afi)lal da is symmetric in f and g. This we do by writing down the obviously symmetric additive double integral

I I f(tkh) e-2niA(CBs) d(t, ~9,

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 315

changing it with the Fubini theorem into

f (I f(t) B(V) e- znrA(CBs) dv) d5 = ~.fW(5B) 8,

and observing that according to Lemma 2.3.2 this last expression is equal to the multiplicative integral

I f(c#(c@)[aldl a = constant. /f(a)d(@) laida

We can now announce the Main Theorem of the local theory.

THEOREM 2.4.1. A c-function has an analytic continuation to the domain of all quasi-characters given by a functional equation of the type

WY 4 = P(CN3Y 2). The factor p(c), which is independent of the function f, is a meromorphic function of quasi-characters defined in the domain 0 < exponent c < 1 by the functional equation itself, and for all quasi-characters by analytic continuation.

Proof In the next section we will exhibit for each equivalence class C of quasi-characters an explicit function fc E 3 such that the function

P(C) = C(fc9 4/lltfc, 2) is defined (i.e. has denominator not identically 0) for c in the strip 0 < expo- nent c < 1 on C. The function p(c) defined in this manner will turn out to be a familiar meromorphic function of the parameter s with which we describe the surface C, and therefore will have an analytic continuation over all of C.

From these facts, which will be proved in $2.5, the theorem follows directly. For since C was in any equivalence class, p(c) is defined for all quasi-characters. And if f (r) is any function of 3 we have according to the preceding lemma

WY WC‘,, 21 = KfY wcfc~ 49 therefore

uf, 4 = PWUY 219 if c is any quasi-character in the domain of 0 < exponent c < 1, where Cl.6 4 and Kf’, 2) are originally both defined, and C is the equivalence class of c.

Before going on to the computations of the next section which will put this theory on a sound basis, we can prove some simple properties of the factor p(c) in the functional equation which follow directly from the functional equation itself.

LEMMA 2.4.3.

(1) p(2) = c$ft -

(2) P(E) = cc- l)P(C)*

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316 J. T.-TATE

Proof.

(1) cu 4 = PM.t a = Pw.wdT, 6 = c(- W(fa 4 because](a) = f(-a) and &a) = c(a). Therefore p(c)p(Q = c(- 1).

(2) CUP 4 = ccf, c3 = PGW 8

= PM- 1N.l Q = P(M- w39 Q because f(a) = I( - a) and &a) = E(a). On the other hand,

UC) =po am.

Therefore p(E) = c(- 1)s).

COROLLARY 2.4.1. Ip( = 1 for c of exponent 3. Proof. (exponent c) = 3 * c(a)E(a) = Ic(a)12 = Ial = c(a)t(a) * E(a) = e(a).

Equating the two expressions for p(E) and p(e) given in the preceding lemma yields p(c)p(c) = 1.

2.5. Computation of p(c) by Special C-functions This section contains the computations promised in the proof of theorem 2.4.1. For each equivalence class C of quasi-characters we give an especially simple function fc E 3 with which it is easy to compute p(c) on the surface C. Carrying out this computation we obtain a table which gives the analytic expression for p(c) in terms of the parameters on each surface C. It will be necessary to treat the cases k real; k complex; and k p-adic separately.

k Real t is a real variable. a is a non-zero real variable.

II((f) =-t. Ial is the ordinary absolute value.

& means ordinary Lebesgue measure. da = m. ah

The Equivalence Classes of Quasi-Characters. The quasi-characters of the form Ial’, which we denote simply by IIS, comprise one equivalence class. Those of the form (sign a)laI’, which we denote by f I I”, comprise the other.

The Corresponding Functions of 3. We put f(r) = e-“C’ and f,(e) = r e-‘@.

Their Fourier Transforms. We contend

f(t) -f(C) and f&l = if&). Indeed, these are simply the two identities

03 co

I e -4+2+d,, = e-“C’ and J tte -l=+2~&ld,, = ite-“t=

familiar-fyom classical Fourier analysis. iie first of these can be established

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directly by completing the square in the exponent, making the complex substitution q -P 0 +i& which is allowed by Cauchy’s integral theorem,

and replacing the definite integral 1 e-“C’ dt by its well-known value 1 -CO

1 d The second identity is obtained by applying the operation - - to the

2ni d< first.

The (;-functions. We readily compute:

C(f, 113 = j)(c+$ dcr = j Pa2]#; -00

m

Explicit Expressions for p(c):

n-:r s P(ll3 0 2 = 1-, -Tr 12

( >

= 21-%-*cos ; l-(s) 0 n

2 J+l

s+l -3-r -

d*ll? = i ,: )+1 ( >

2 -I ~c’-~+~~ =-i2’3-‘sin(~)~(s). n 2

Here the quotient expressions for p come directly from the definition of p as quotient of suitable C-functions; the second form follows from elementary r-function identities.

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318 J. T. TATE

k CompIex

t: = x + & is a complex variable. a = rele is a non-zero complex variable.

M) = -2Re(o =-2x. I4 = r2 is the square of the ordi- nary absolute value.

dr = 21dxdyj is twice the ordinary Lebesgue measure.

da = & 2rldrdOl 2 ,a, = -p-- = ; Idr dOI.

Equivalence Classes of Quasi-Characters. The characters c,(a) defined by c,(re”) = ein’, n any integer, represent the different equivalence classes. The nth class consists of the characters c,(cr)[a[‘, which we denote by c.11’.

The Corresponding Functions of 3. We put (x- jy)lnl e-2rW+Y’)

f,(O = {(x+iy)l”l e-z’w+Y9; :: : ;.

Their Fourier Transforms. We contend

f”(r) = i’“y-,(r), for all n.

Let us first establish this formula for n 2 0 by induction. For n = 0, the contention is simply that fO(r) = e -2n(x’+yf) is its own Fourier transform. This can be shown by breaking up the Fourier integral over the complex plane into a product of two reals and using again the classical formula

+m

s e -nu=+2nixu du = eVnx2.

(The factor 2 in the expiient of our function fo(T) just compensates the factor 2 in d5 and in A(<).)

Assume now we have proved the contention for some n 2 0. This means we have established the formula

I f.GW 2nra(C")dq = j"f-.(c),

which, written out, becomes

II w 7 u-jit)).e-2"(U1'~')+4~1(XY-Y~)2du dv = j"(X+iy)ne-2x(%'+Y').

-co --m

Applying the operator

D=$-.(&+i$

to both sides (a simple task in view of the fact that since z” is analytic,

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 3 19

qx+iyy = 0), we obtain ce 01

II (u+iu)“+‘e -2n(u’+~‘)+4nl(xu-y~)2dudv = jn+l(X+iy)"+le-2"(~'+y').

-02 -03

This is the contention for n+ 1. The induction step is carried out. To handle the case n < 0, put a roof on the formula j’-,,(e) = il”x(l)

which we have already proved, and remember that

Lto =f-A- 5) = (- wL.“(0.

The C-Functions. For a = r eie we have

fn(a) = ,I4 e-inee-2nr2 IuI” = rzS

c,(a) = eine 2r dr dO

dcr=rl.

Therefore m 2%

C(f.,c”Il”> = J’fn(ct)c,,(a)[a~da = Is r2(s-1)‘I”le-2nrz2rdrde 00

= zn ~(r2~-1)+‘~1e-z”zd(r2) = (2,90’-“+$$- s , 0

and

Explicit Expressions for p(c).

e = a p-adic variable.

NO = wc3>-

k p-a&c

dt is chosen so that o gets measure (JV~)-~.

a = CC’, non-zero p-adic variable, n a fixed element of ordinal number 1, v an integer.

Ial = (Jp)“.

./yp da

da = Np- 1 - - - so that u gets multiplicative measure (~0)~*.

Ial’

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320 J. T. TATE

The Equivalence Classes of Quasi-Characters. c,(a), for n 2 0 shall denote any character of k* with conductor exactly p”, such that c.(n) = 1. These characters represent the different equivalence classes of quasi-characters.

The Corresponding Functions of 3. We put

j-(t) = {;sfA(c)’ ,

;; ; ; -1 .

Their Fourier Transforms. We contend

f”<r) = {m*wPY, ,

;:; ; mm& .

Proof.

3”(c) = j-j-(q) ,-2=ih(h) drt = j- e-2nih((t- lh) drl. b-$-”

This is the integral, over the compact subgroup b-‘p-* c k+, of the additive character q -+ e-2n*h((C-1)“). If t: z 1 (mod p”), this character is trivial on the subgroup, and the integral is simply the measure of the sub- group: (Nb)*(Nn)“. In case { $ 1 (mod p”), this character is not trivial on the subgroup and the integral is 0.

The C-Functions. First we treat the unramified case: n = 0. The only character of type ca is the identity character, and f. is the characteristic function of the set b-l. We shall therefore compute

CCr,,l]“> = I IaI”da. b-1

Denote by A, the “annulus” of elements of order v, and let b = pd. Then b-l = uz -,, A,, a disjoint union, and

3e is JYb* times the characteristic function of o, so we have, similarly,

C(30,t? = ~cT~,ll’-~> = JQ*.l Ial’-“da 0

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In the ramified case, n > 0,

cu&lll”) = 1 eZ+iA(a) c,(a) IaI’ da b-$-”

= vaf-mNpmvJ~ e2xi”(a) c,,(a) da. A”

We assert that all terms in this sum after the first are 0. In other words that

I e2*“‘@) c,(a) da = 0 for Y > - d-n.

AV Proof. Case 1. v 2 -d. Then A, c b-‘, so e2*‘*@) = 1 on A,, and

the integral is

1 c,(a) da = 1 c,(an”) da = j c,(a) da = 0 A” Y Y

since c,(a) is ramified and therefore non-trivial on the subgroup U. Case 2. -d > Y > -d-n. (Occurs only if there is “higher ramification”;

i.e. if n > 1.) To handle this case we break up A, into disjoint sets of the type a,+b-’ = a,, +pSd = ao(l +pBd-y). On such a set, A is constant = A(a,) and

I e2”‘A(“) c,(a) da = e2~lAk7~)

I c,(a) da.

cltJ+b-’ cao+ b-1

This is 0 because

I c,(a) da =

I c,(a) da =

I c&ad da ao+tl-’ ao(l+p-d-‘) l+p-d-”

= c,W 1 44 da, l+p-d-”

and this last integral is the integral over a multiplicative subgroup 1 + pmdVv of a character c,(a) which is not trivial on the subgroup. Namely, -d > v =S &J-~-’ =E- 1 +pmd-* is a subgroup of k*, and v > -d-n * the conductor P”&I-~-* * c,(a) not trivial on it.

We have now shown

rV,, %I13 = m+d+n)r I eZniA@) c,(a) da.

A-4-n

To write this in a better form, let {E} be a set of representatives of the elements of the factor group u/(1 +p”), so that u = us(l +p”), a disjoint union. Then

A -d-n = un-d-n = U.&X -d-“(l+p”) = ~.(&n-d-“+b-l).

On each of these sets into which we have dissected Awdmn, c,, is constant

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322 J. T. TATE

= C,(&K +-“) = c,(s), and A is constant = A(sr~-~-“). We therefore have

((f,, cnlls) = .Xp(d+nf(~ C,(E) e2’+‘@lnd+“)) j du, * 1 +p”

a form which will be convenient enough. The pay-off comes in computing

r<.L $ = r<xv c,‘ll’-“>. For 3” is A%*JVp” times the characteristic function of the set 1 +p”, a set on which c; ‘(a)laj’ V-S = 1. Therefore

c(J~,~? = JQ+J~” 1 da, a constant! 1+&-n

ExpIicit Expressions for p(c)

AC113 = ~@D”-*PcM ‘f 1 c is a ramified character with conductor f,

such that c(x) = 1.

is a so-called root number and has absolute value 1. {E) is a set of repre- sentatives of the cosets of 1 +f in U.

Taking the quotients of the C-functions we have worked out yields these expressions directly if we remember that b = pd and, in the ramified case, that the conductor of c, was f = p”. The fact that constant p,(c) has absolute value 1 follows from Corollary 2.4.1. Namely, since c is a character, cl/* has exponent 3, so we must have jp(~lj~)l = Ip,,(c)l = 1.

3. Abstract Restricted Direct Product

3.1. Introduction. Let {p] be a set of indices. Suppose we are given for each p a locally compact abelian group GP, and for almost all p (meaning for all but a finite number of p), a tied subgroup HP c GP which is open and compact.

We may then form a new abstract group G whose elements a = ( . . . , up,. . . ) are “vectors” having one component aP E G,, for each p, with a, E HP for almost ail p. Multiplication is defined component-wise.

Let S be a finite set of indices p, including at least all those p for which HP is not defined. The elements a E G such that aP E HP for p 4 S comprise a subgroup of G which we denote by Gs. Gs is naturally isomorphic to a direct product n G, x n HP of locally compact groups, almost all of

PSS oes which are compact, and is therefore a locally compact group in the product

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topology. We define a topology in G by taking as a fundamental system of neighborhoods of 1 in G, the set of neighborhoods of 1 in G,. The resulting topology in G does not depend on the set of indices, S, which we selected. This can be seen from

LEMMA 3.1.1. The totality of all “~arallelotopes” of the form N = n N,,

where N+, is a neighborhood of 1 in GQ for all p, and NQ = HP for almost all p-remember the HP are open by hypothesis-is a fundamental system of neighborhoods of 1 in G.

Proof. By the definition of product topology a neighborhood of 1 in Gs contains a parallelotope of the type described. On the other hand, since N, = HP for almost all p, the intersection

P ) Np n Gs = n Np x Q$‘“p n 4)

Q=s

is a neighborhood of’1 in Gs. It is obvious that Gs is open in G and that the topology induced in Gs

as a subspace of G is the same as the product topology we imposed on Gs to begin with. Therefore a compact neighborhood of 1 in Gs is a compact neighborhood of 1 in G. It follows that G is locally compact.

DEFINITION 3.1.1. We call G (as locally compact abelian group) the restricted direct product of the groups GQ relative to the subgroups Hp.

It will, of course, be convenient to identify the basic group G, with the subgroup of G consisting of the elements “0 = (1, 1,. . . , a,, . . .) having all components but the p-th equal to 1. For that subgroup of G is naturally isomorphic, both topologically and algebraically to GQ.

Since the components, aQ, of any element a of G lie in HP for almost all p, G is the union of the subgroups of the type G,. This fact will allow us to reduce our investigations of G to a study of the subgroups G,.

These Gs in turn may be effectively analyzed by introducing the subgroup GS c Gs consisting of all elements a E G such that a, = 1 for p E S; a, E H,, p 4 S. GS is compact since it is naturally isomorphic to a direct product n H, of compact groups. Gs can be considered as the direct product

PCS Gs = n G, x GS of a finite number of our basic groups G, and the

( > QeS compact group Gs.

We close our introduction of the restricted direct product with LEMMA 3.1.2. A subset C c G is relatively compact (has a compact closure)

if, and only if, it is contained in a parallelotope of the type fl BP, where BP

is a compact subset of GQ for all p, and BP = HP for almost a;1 p.

Proof: Any compact subset of G is contained in some G,, because the Gs are open sets covering G, and the union of a finite number of subgroups G, is again a Gs. Any compact subset of a G, is contained in a parallelotope

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324 J. T. TATE

of the type described, for it is contained in the Cartesian product of its “projections” onto the component groups G+,. These projections are com- pact since they are continuous images, and are contained in HP for p 4 S.

On the other hand, any parallelotope n B,, is obviously a compact subset P

of some Gs; therefore of G.

3.2. Characters Let c(a) be a quasi-character of G, i.e. a continuous multiplicative mapping of G into the complex numbers. We denote by cP the restriction of c to G,: (cJap) = ~(a,,) = ~$1, 1,. . ., a*,. . .) for a, E G,). cP is obviously a quasi-character of G,,.

LEMMA 3.2.1. c,, is trivial on H,, for almost all p, and we have for any tlEG

44 = II c&J P

almost all factors of the product being 1. Proof. Let U be a neighbourhood of 1 in the complex numbers containing

no multiplicative subgroup except (1). Let N = n NP be a neighborhood

of 1 in G such that c(N) c U. Select an S containinlall p for which NP # H,. Then GScN~~GS)cUe,c(GS)=l~C(Hp)=l for p#S. Ifa is a fixed element of G we impose on S the further condition that a E Gs and write a

-+!As- as with as ta G’. Then

c(a) =Vvsc(aP). c(a”) =P~scP(u~) = II %(Ca)* P

since for p # S, cP(ap) = 1. LEMMA 3.2.2. Let c,, be a given quasi-character of Gp for each p, with cp

trivial on HP for almost all p. Then if we define c(a) = n c,,(u,,) we obtain P

a quasi-character of G. Proof. c(a) is obviously multiplicative. To see that it is continuous select

an S containing all p for which cJH+,) # 1. Let s be the number of p in S. Given a neighborhood, U, of 1 in the complex numbers, choose a neigh- borhood V such that V” c U. Let N,, be a neighbourhood of 1 in G, such that c,,(N,,) c V for p E S, and let NP = HP for p # S. Then

c IIN, ( )

c vs c u. P

Restricting our consideration to characters, we notice first of all that c(a) = n c,(a+,) is a character if, and only if, all cP are characters. Denote

by G, tie character group of GP, for all p ; for the p where HP is defined let HP+ c GP be the subgroup of all c+, E GP which are trivial on Hp. Then

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HP compact * 8, = ~ &/HP* discrete - H,* open, and HP open * GJH,

discrete * Gz z H,* compact.

THEOREM 3.2.1. The restricted direct product of the groups eP relative to the subgroups H,* is naturally isomorphic, both topologically and algebraically, to the character group 8 of G.

Proof. Of course we mean to identify c = (. . ., cP,. . .) with the character c(a) = n cP(aP). The two preceding lemmas, applied to characters, show

that this’ is an algebraic isomorphism between the two groups. We have only to check that the topology is the same. To this effect we reason as follows: c = (. .., CP,. ..> is close to 1 as a character o c(B) close to 1 for a large compact B c G o c n BP

( ) close to 1 for B,, c GP, compact,

4 = HP for almost all p o c’,(&,) close to 1 wherever BP # HP and q,(BJ = c&Q = 1 at the remaining p (since HP is a subgroup, c,,(H& can be close to 1 only if cP(HP) = 1) o cP close to 1 in GP for a finite number of p and cP E H,* at the other p o c close to 1 in the restricted direct product of the G,.

3.3. Measure

Assume now that we have chosen a Haar measure da, on each GP such thatsds = 1 for almost all p. We wish to deCne a Haar measure da on G

for &ch, in some sense, da = n da,. To do this, we select an S; then

consider Ga as the finite direct iroduct Gr = P )

G, x G”, in order to PES

define on Gr a measure das = (dh,“(a) .da’, where da’ is that measure on

the compact group Gs for which j da’ = Since Gs is an open G‘ pIsls(il&?v)*

subgroup of G, a Haar measure da on G is now determined by the require- ment that da = da, on Gr. To see that the da we have just chosen is really independent of the set S, let T 3 S be a larger set of indices. Then Gs c Gr, and we have only to check that the daT constructed with T coincides on Gs with the da, constructed with S. Now one sees from the decomposition Gs = (, Eg-&) x GT that da’ = (P ,l$$~~). daT; for the measure on the

right-hand side gives to the compact group Gs the required measure. Therefore

da, =prlIsda,.das =PJJsda+, .P ejJsdad. da’ = da,.

We have therefore determined a unique Haar measure da on G which we may denote symbolically by da = n da,.

P

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326 J. T. TATE

If (p(S) is any function of the finite sets of indices S, with values in a topological space, we shall mean by the expression lim (p(S) = cpO the

statement: “given any neighborhood V of qo, there exisi a set S(v) such that S 3 S(V) * q(S) E V”. Intuitively, lim (p(S) means the limit of q(S)

S

as S becomes larger and larger.

LEMMA 3.3.1. Iff(a) is a function on G,

I f(a) da = lim f(a) da,

S I GS

if either (1) f(a) measurable, f(a) 2 0, in which case + co is allowed as value of the integrals; or (2) f(a) E&(G), in which case the values of the integrals are complex numbers.

Proof. In either case (1) or (2) j f ( ) d a a is the limit of j f (a) da for larger B

and larger compacts B c G. Since any compact, B, is contained in some G,, the statement follows.

LEMMA 3.3.2. Assume we are given for each p a continuous function f, E&(G& such thatf,(aJ = 1 on HP for almost all p. We define on G the function f(a) = nf,(a,), (th is is really a finite product), and contend:

P (1) f(a) is continuous on G. (2) For any set S containing at Zeast those p for which either f,(H,) # 1,

or Ida, # 1, wehave =v

/fbMa =~s[~(avW,]. GS

Proof. (1) f(a) is obviously continuous on any G,; therefore on G.

(2) For a E Gs, f(a) =p~fp(d,). Hence

J‘ f(a) da = f(a) da, =

G.9 I GS

j (Jp&J) (p% . daS) GS

=vlJ [pwds] * J &Is =,rjs [f.fv’~v)d~v]. GS

THEOREM 3.3.1. If fp(a,) and f(a) are the functions of the preceding lemma and iffurthermore

then f(a) E L,(G) and,

I f(a) da = v [j f,(a,) da,].

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Proof. Combine the two preceding lemmas; tist for the function IfW = l-I I&<Ql t o see that f(a) E L,(G), then for f(a) itself to evaluate

j f(a) da.’ We close this chapter with some remarks about Fourier analysis in a

restricted direct product. As we have seen G the character group of G is the direct product of the character groups GP of GP, relative to the sub- groups H,* orthogonal to H,,. Denote by c = (. . . , c,,, . . .) the general element of G. (In this paragraph, c, and cP are characters, not quasi-charac- ters). Let dc, be the measure in G+, dual to the measure da,, in GP. Notice that if f,(a,) is the characteristic function of HP, its Fourier transform

&J = j f,Wc&,) d aP is j da, times the characteristic function of H,*.

A consequence of this fact a% the inversion formula is that

(p%)(J’dc,) = 1. HP f%

Therefore j dc, = 1 for almost all p, and we may put dc = n dc,. H; V

LEMMA 3.3.3. If f,(a,) E B,(G+,) for all p and f,(a,) is the characteristic function of HP for almost all p, then the function f(a) = n f,(s) has the

P Fourier transform f(c) = n jb(c,), and f(a) E Bz,(@.

Proof. Apply Theorem 3.3.1 to the function f(a)c(a> = n &(a,)c+,(a,)

to see that the Fourier transform of the product is the product Jf the Fourier transforms. Since f,(a,) E B,(G,), we have J)‘(cJ E&(G& for all p. For almost all p, &(c,) is the characteristic function of H,* according to the remark above. From this we see thatf(c) E &(G), hence f(a) E Bl(G).

COROLLARY 3.3.1. The measure dc = n dc, is dual to da = n da,.

Proof. Applying the preceding lemma :o the group G with the’measure dc, we obtain for our “product” functions the inversion formula

f(a) = ~f~c)c(a) dc

from the component-wise inversion formulas.

4. The Theory in the Large

4.1. Additive Theory

In this chapter, k denotes a finite algebraic number field, p is the generic prime divisor of k. The completion of k at the prime divisor p shall from now on be denoted by k,, and all the symbols o, A, b, 11, c, etc., defined in Chapter 2 for this local field kp shall also receive the subscript p: oP, A,, b,, I&, cp, . . . , etc.

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328 J. T. TATJ3

DFSINITION 4.1.1. The additive group V of valuation vectors of k is the restricted direct sum, over all prime divisors p, of the groups k,’ relative to the subgroups op.

We shall denote the generic element of V (= valuation vector) by X= ( . . . . xp ,... ). F rom Theorems 3.2.1. and 2.2.1 and Lemma 2.2.3 we see that the character group of V is naturally the restricted direct sum of the groups k,’ relative to the subgroups (b;l. Since bP = o, for almost all p this sum is simply V again! Looking more closely at the identifications set up in these theorems we see that the element t) = (. . . , up, . . .) E V is to be identified with the character

x = (. . . , xp, . . .) + v exp (2niA,(9& = exp $i 1 AJs,s)] P

of V. This suggests that we define the additive function

A(x) = CA, (%I) P

on V, and introduce component-wise multiplication

qx = (. . ., q),. . .)(. . .,x,,. . .) = (. . ., t)&, . . .)

of elements of V in order to be able to assert neatly:

THEOREM 4.1.1. V is naturally its own character group if we identifv the element t) E V with the character x + e2n’A(9*) of V.

On V we shall, of course, take the measure dx = n dx, described in

Section 3.3, dx, being the local additive measure defined i:Section 2.2. Since these local measures dx, were chosen to be self-dual, the same is true of dx, according to Corollary 3.3.1. We state this fact formally in

THEOREM 4.1.2. If for a function f (x) E L,(V) we define the Fourier transform

f(q) = s j(x) e-2n’A(9x)dx,

then for f (x) E B,(V) the inversion formzda

f(x) = I f(q) e2n”‘(x9)dt) holds.

What is the analogue in the large of the local Lemma 2.2.4 and 2.2.5, that is, of the statement d(a<) = ial dC: for a E k* ? In that local considera- tion, a played the role of an automorphism of kz, namely the automorphism c + a& This leads us to investigate the question: for what a E V is x + ax an automorphism of V? We first observe that for any a E V, x + ax is a continuous homomorphism of V into V. A necessary condition for it to be an automorphism is the existence of a b E V such that ab = 1 = (1,1, . . .). But this is also suflticient, for with this b we obtain an inverse map x + bx of the same form. Now for such a b to exist at all as an “unrestricted” vector, we need op # 0 for all p, and then b, = a; l. The further condition

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bo Vmeans apl E eP for almost all p, therefore jaJ, = 1 for almost all x. These two conditions mean simply that a is an id&le in the sense of Chevalley. We have proved

LEMMA 4.1.1. The map x + ax is an automorphism of V if and on& if a is an idtile.

At present we shall consider ideles only in this role. Later we shall study the multiplicative group of idMes as a group in its own right, with its own topology, as the restricted direct product of the groups k,* relative to the subgroups up.

To answer the original question concerning the transformation of the measure under these automorphisms we state

LEMMA 4.1.2. For an idPIe, a,

d(ax) = Ia/ dx, where

I4 = I-I I4 Q (really a finite product).

Proof. If N = n NP is a compact neighborhood of 0 in V, then by

Theorem 3.3.1 and ‘Lemma 2.2.5

The last, and most important thing we must do in our preliminary dis- cussion of V is to see how the field k is imbedded in V. We identify the element t E k with the valuation vector e = (& c, . . . , e, . . .) having all components equal to I$, and view k as subgroup of V. What hind of sub- group is it?

LEMMA 4.1.3. If S, denotes the set of archimedean primes of k, then (1) k n k = o, the ring of algebraic integers in k, and (2) k+ V& = V.

Proof. (1) This is simply the statement that an element C E k is an algebraic integer if and only if it is an integer at all finite primes.

(2) k+Jcs, = V means: given any x E V, there exists a { E k approxi- mating it in the sense that e-x, E D, for all finite p. Such a t can be found by solving simultaneous congruences in o. The existence of a solution is guaranteed by the Chinese Remainder theorem.

Let now ? denote the “infinite part” of V, i.e. the Cartesian product n k+, of the archimedean completions of k. If a generating equation for k

QSS,

over the rational field has rl real roots and r, pairs of conjugate complex

roots, then t is the product of ri real lines and r, complex planes. As such

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330 J. T. TATE

it is naturally a vector space over the real numbers of dimension n = rl + 2rz

= absolute degree of k. For any x E V denote by f the projection

x”= (..., rp ,... +);s, 1 ofx on ?.

LEMMA 4.1.4. If@,, C02,. . . , o,,) is a minimal basis for the ring of integers o

of k over the rational integers, then “0 1 OD

l, 02, . , ., & is a basis for the vector 1

space “v over the real numbers. The parallelotope 5 spanned by this basis

6 = set of all x” =il xv& with 0 I; x, < 1) has the volume ,/Id] (where

d = (det (OF)))” = absolute discriminant of k) of measured in the measure

di =p g dx, which is natural in our set-up. 00

Proof. The projection r + T of k into “v is just the classical imbedding of a number field into n-space. The reader will remember the classical argument which runs:

k separable => d = (det (u#)))~ # 0 3 {&, G,, . . . , &}

linearly independent, and (with a simple determinant computation) E has volume 2-“,//ldl. For us the volume is 2ra times as much because we have chosen for complex p a measure which is twice the ordinary measure in the complex plane.

DEFINITION 4.1.2. The additive fundamental domain D c V is the set of

allxsuchthatxEVs_and~E&

THEOREM 4.1.3. (1) D deserves its name because any vector x E V is con- gruent to one and only one vector of D module the field elements <. In other words, V .s,U~({+ D), a disjoint union.

(2) D has (measure 1. Proof (1) Starting with an arbitrary x E V we can bring it into Vs- by

the addition of a field element which is unique mod o (Lemma 4.1.3). Once in Vs- we can find a unique element of o, by the addition of which we can

stay in V,, and adjust the infinite components so that they lie in 5 (Lemma 4.1.4).

(2) To compute the measure of D, notice that D c Vs- and D = 5 x Vsm. Therefore

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FOURIER ANALYSIS INNUMBER FIELDS AND HECKE'S ZETA-FUNCTIONS 331

Now since the discriminant d (as ideal) is the norm of the absolute different b of k, and since b is the product of the local differents b,, we have IdI = n (XPbp). Therefore the measure which we have computed is 1.

P6SoD COROLLARY 4.1.1. k is a discrete subgroup of V. The factor group V mod k

is compact. ProoJ k is discrete, since D has an interior. V mod k is compact, since

D is relatively compact. LEMMA 4.1.5. A(c) = 0 for all c E k. ProoJ

NO = c A,(t) = c ~p,<q3> = c 1, (gp s,(O) = c I,) P P P

because “the trace is the sum of the local traces”. SincePS(D is a rational number, the problem is reduced to proving that c I,(x) E 0 (mod 1) for

rational X. This we do by observing that the ratikral number 1 n,(x) is P

integral with respect to each fixed rational prime q. Namely

C Apt4 = (, +Ip,lzpw) + 464 + n,bj = (, +Ip,sb)) + (4(x) - 4 P

expresses 2(x) as sum of q-adic integers.

THEOREMS 4.1.4. k* = k, that is A(xr) = 0 for all < e x E k.

Proof. Since k* is the character group of the compact factor group V mod k, k* is discrete. k* contains k according to the preceding lemma, and therefore we may consider the factor group k* mod k. As discrete sub- group of the compact group V mod k, k* mod k is a finite group. But since it is a priori clear that k* is a vector space over k, and since k is not a finite field, the index (k* : k) cannot be finite unless it is 1.

4.2. Riemann-Roth Theorem

We shall call a function C&W) periodic if cp(x+ <) = (p(x) for all 5: E k. The periodic functions represent in a natural way all functions on the compact factor group V mod k. cp(x) represents a continuous function on V mod k if and only if it is itself continuous on V.

LEMMA 4.2.1. If q(x) is continuous and periodic, then j cp(x) dx is equal to: D

the integral over the factor group V mod k of the function on that group which C&S) represents with respect to that Haar measure on V mod k which gives the whole group V mod k the measure 1.

Proof. Define I(q) = 1 cp( x ) d x and consider it as functional on L( V mod k).

Observe that it has the properties characterizing the Haar integral. (To

t Here k* is the dual of k, not the multiplicative group of k (!).

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332 J. T. TATI

check invariance under translation merely requires breaking D up into a disjoint sum of its intersections with a translation of itself.) The functional is normed to 1 because dx = 1.

d k is naturally the character group of V mod k in view of Theorem 4.1.4.

The Fourier transform, e(t), of the continuous function on V mod k which is represented by cp(x) is

e(e) = J q(x) e-zn*A(Cr)dx. D

LEMMA 4.2.2. If q(x) is continuous and periodic and c I@(01 < 00, then <ok

q(x) =exk@(o eznfA(‘c? a

Proof. The hypothesis c j@(t)1 < co means that the Fourier transform

e(C) is summable on k, $:&teeing that the inversion formula holds. The asserted equality is simply the inversion formula explicitly written out.

LEMMA 4.2.3. If f(x) is continuous, c L,( V), and c f(x + q) is uniformly vok

convergent for x 5 D (convergence means absolute convergence because k is nor ordered in any way), then for the resulting continuous periodic function cp(4 = C f(x+ rt) we have 4X0 = 3X0.

risk Proof.

cj3(&) = 1 q(x) e-ZniA(Cx) dx D

=

IP

~ E kj(zt+q) e-2*mclc))dx

=~kll(l+n)e-2x~A(x”d.

(The interchange is justified because we assumed the convergence to be uniform on D, and D has finite measure.)

= zk jDj(x) e-2”fA(sC-qC) dx

=~k:i~(~)e-2xiA(~~)d~

(since A(q<) = 0)

= f(x)e I - 2nf4=0 dx

= 3m. Combining the last two Lemmas 4.2.2. and 4.2.3, and putting x = 0 in

the assertion of Lemma 4.2.2 we obtain

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FOURIER ANALYSIS IN NUMBER mELDSAND HECKB'S ZETA-FUNCTIONS 333

LEMMA 4.2.4. (Poisson Formula.) Iff(r) satisfies the conditions: (1) f(x) continuous, E L,(V); (2) 1 f(x + r) uniformly convergent for x E D;

C=k (3) C If<01 convergent;

C=k then

g(t) =~&m.

If we replacef(x) by f(ax) (a an idele) we obtain a theorem which may be looked upon as the number theoretic analogue of the Riemann-Roth theorem.

THEOREM 4.2.1. (Riemann-Roth Theorem.) Iff(x) satisfis the conditions:

(1) f(x) continuous, B L,( v); (2) c~kfb@ + 0) convergent for all ids/es a and valuation vectors z,

uniformly for x E D; (3) C 1 3(4 convergent for aN hfdes a

<ok then

Proof: The function g(z) = f(ax) satisfies the conditions of the preceding lemma because

gj(x) = 1 f(at)) e-2ani”(rp) dq

(Under the transformation r) -+ n/a, dq + (l/[al) dtp)

We may therefore conclude

that is,

It is amusing to remark that, had we never bothered to compute the exact measure of D, we would now know it is 1. For we could have carried out all the arguments of this section with an unknown measure, say p(D), of D.

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334 J. T. TATE

The only change would be that in order to have the inversion formula of Lemma 4.2.2 we would have to have given each element of k the weight l/p(D). The Poisson Formula would then have read,

$-jgo~=,Ip.

Iteration of this would yield @(D))’ = 1, ;herefore p(D) = 1 I

4.3. Multiplicative Theory In this section we shall discuss the basic features of the multiplicative group of ideles.

DEFINITION 4.3.1. The multiplicative group, I, of id2les is the restricted direct product of the groups k,* relative to the subgroups u,,.

We shall denote the generic idble by a = (. . . , a,, . . .). The name idele is explained (at least partly explained!) by the fact that the idele group may be considered as a refinement of the ideal group of k. For if we associate with an idtle a the ideal p(a) = n pord@*, then the map a + (p(a) is

DC.% obviously a continuous homomorphism of the idele group onto the discrete group of ideals of k. Since the kernel of this homomorphism is Is*, we may say that an idele is a refinement of an ideal in two ways. First, the archi- medean primes figure in its make-up, and second, it takes into account the units at the discrete primes.

Concerning quasi-characters of I, we can only state, according to Section 3.2, that the general quasi-character c(a) is of the form c(a) = n c,(a,), where

c,(a,) is a local quasi-character (described in Section 2.3) and c,(i+,)isunramified at almost all p.

For a measure, da, on I we shall of course choose da = n da,, the da, P

being the local multiplicative measure defined in Section 2.3. We can do nothing really significant with the idele group until we imbed

the multiplicative group k* of k in it, by identifying the element CI E k* with the idele a = (a, a,. . ., a, . . . ). Throughout the remainder of this section our discussion will center about the structure of I relative to the subgroup k*. The first fact to notice is that the ideal q(a) associated with an idtle a E k* is the principal ideal ao generated by a, as it should be. Next we have the “product formula” for elements a E k*. Though this is well known, we state it formally in a theorem in order to present an amusing proof.

THEOREM 4.3.1. lal(= l-j Ia&) = 1 fir aE k*.

Proof. According to Lemma 4.1.2 the (additive) measure of aD is Ial times the measure of D. Since ak’ = k+, aD would serve as additive fundamental domain just as well as D. From this it is intuitively clear that aD has the

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKFi’S ZETA-FUNCTIONS 335

same measure as D and therefore Ial = 1. To make a formal proof one has simply to chop up D and aD into congruent pieces of the form D n (t +aD) and (- {+ D) n aD respectively, < running through k.

This theorem reminds us to mention explicitly the continuous homo- morphism a + Ial = n Ialp of Z onto th e multiplicative group of positive

real numbers. The k&nel is a closed subgroup of Z which will play an important role. We denote this subgroup by J, and its generic element (idele of absolute value 1) by b.

It will be convenient (although it is aesthetically disturbing and not really necessary) to select arbitrarily a subgroup T of Z with which we can write Z = T x J (direct product). To this effect we choose at random one of the archimedean primes of k-call it p,-and let T be the subgroup of all ideles a such that aPO > 0 and aP = 1 for p # p,,. Such an idele is obviously uniquely determined by its absolute value; indeed the map a + lal, restricted to T, is an isomorphism between T and the multiplicative group of positive real numbers, and it will cause no confusion if we denote an idele of T simply by the real number which is its absolute value. Thus a real number t > 0 also stands either for the idble (t, 1, 1, . . .) or for the idele (Jr, 1, 1, . . .>, according to whether p,, is real or complex, if we write the no-component first. Since we can write any idble a uniquely in the form a = Ial .b with la/ E T and b = alai-’ E J, t i is clear that Z = TX J (direct product).

In order to select a fixed measure db on J we take on T the measure dt = dtlt and require da = d t. db. Then for computational purposes we have (in the sense of Fubini) the formulas

BOWL =I [/kWb] t =! [b$] db

for a summable idele functionf(a). The product formula means that k* c J, and we wish now to describe a

“fundamental domain” for J mod k*. The mapping of ideles onto ideals allows us to descend to the subgroup Js, = J n Ia-. To study Js, we map the ideles b E Js, onto vectors Z(b) = (. . ., log lb!,,, . . .)peSb, having one component, log lbl, for each archimedean prime except no. (This set of r = rl +r,- 1 primes is denoted by S’,.) It is obvious that the map b -P Z(b) is a continuous homomorphism of Jsm onto the additive group of the Euclidean r-space. The onto-ness results from the fact that although the infinite components of an idMe b E Js, are constrained by the condition n 1% =p rJ)lP = 1, they are completely free in the set Sb, since we can

ldjust the p. component. k* n Js, is the group of all elements E E k* which are units at all finite

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336 J. T. TATE

primes; that is, which are units of the ring o. The units C for which Z(g) = 0 are the roots of unity in k and form a ftnite cyclic group. It is proved classic- ally that the group of units e, modulo the group of roots of unity C, is a free abelian group on r generators. This proof is effected by showing that the images Z(s) of units form a lattice of highest dimension in the r-space.

If, tl=refore, (&sSls;r is a basis for the group of units modulo roots of unity, the vectors Z(sJ are a basis for the r-space over the real numbers and we may write for any 6 E Js,,

l(b) = &v KG

with unique real numbers x,. Call P the parallelotope in the R-space spanned

by the vectors Z(eJ; that is, the set of all vectors i x,Z(eJ with 0 5 x, < 1. VP1

Call Q the “unit cube” in the r-space; that is the set of all vectors ( . . ..x.,... 1 peS$ with0 5 xP < 1.

LEMMA 4.3.1.

I db _ ww* R

I- u-9 44 ’ where Z”(P) is the set of all b E Js, such that Z(b) E P, and

R = f det (log I~ilp)plw+r e

is the regulator of k.

Proof. Because Z is a homomorphism,

measure of Z-‘(P) volume of P measure of Z”(Q) = volume of Q

= kdet (log Is&) = R,

and we have only to show

I

& _ 2”Cw*

I- (0) 44 l

Z-‘(Q) is the set of all bcsJs, with 1 S 161, < e for no&. Let Q* be the set of all a E IS, with 1 5 lal, -c e for p ES,. Then

because tb E Q* - b E Z-‘(Q) and 1 5 Itbl,, < e. We have therefore only to show that

I da 2”(27rp

P =Jldl*

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 337

Write Q* as the cartesian product Q* = ,g_e= x Is-s where Q: is the

setofalla,ok~suchthatI<la,&<e,forpES,. Then

because for p real,

for p complex,

and

s daSm =,g Is-

1 da, =vg - U?

(JY,b,)-* = -&. w

DEFINITION 4.3.2. Let h be the class number of k, and select idsles b(l) , . . . , bCh’ E J such that the corresponding ideals (p(b(‘)), . . . , cp(@)) repre- sent the diflerent ideal classes. Let w be the number of roots of unity in k. Let E. be the subset of all b E I -l(P) (see preceding lemma) such that

2n 0 s argbpo < ;. We define the multiplicative fundamental domain, E, for

Jmod k* to be

E=Eob”‘~Eob’2’~...~Eob’h’.

THEOREM 4.3.2. (1) J = U, aE, a disjoint union.

2”(21c)“hR (2) j-db = &lw l

B Proof. (1) Starting with any idele b E J we can change it into an idele

which represents a principal ideal by dividing it by a uniquely determined b(‘). If this principal ideal is ao (a uniquely determined modulo units), multiplication by et-l brings us to an idble of J representing the ideal o- therefore into Js,. Once in Js, we can find a unique power product of the fundamental units e, which lands us in J-‘(P), with only a root of unity, i, at our disposal. This C is exactly what we need to adjust the argument of

the p. component to be in the interval 2n

( > 0, w . Lo and behold we are in E. !

For our original id&le b we have found a unique /I E k* and a unique b(‘) such that b E fib(‘)E,.

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338 J. T. TATE

(2) (measure E) = h . (measure E,) = t . (measure I-‘(P)) = 2”(2x)“hR

JW according to the two disjoint decompositions

E = lJ;a 1 b”‘E,, P(P) = u CE,

and the preceding lemma.

COROLLARY 4.3.1. k* is a discrete subgroup of J (therefore of I). J mod k* is compact.

Proof. One sees easily that E has an interior in J. On the other hand, E is contained in a compact.

We shall really be interested not in all quasi-characters of 1, but only in those which are trivial on k*. From now on when we use the word quasi- character we mean one of this type. Let us close our introduction to the id&e group with a few remarks about these quasi-characters.

The first thing to notice is that on the subgroup J, a quasi-character is a character; i.e. lc(b)l = 1 for all b E J, because Jmod k* is compact.

Next we mention that the quasi-characters which are trivial on J are exactly those of the form c(a) = lal”, where s is a complex number uniquely determined by c(a). For if c(a) is trivial on J, then c(a) depends only on 1 al, and in this dependence is a continuous multiplicative map of the positive real numbers into the complex numbers. Such a map is of the form t + tS as is well known.

To each quasi-character c(a) there exists a unique real number CJ such that Ic(a)l = lal”. Namely, lc(a)[ is a quasi-character which is trivial on J. Therefore [c(a)1 = Ial’, for some complex s. Since [c(a)] > 0, s is real. We call c the exponent of c. A quasi-character is a character if and only if its exponent is 0.

4.4. The [-Functions; Functional Equation

In this section f(x) will denote a complex-valued function of valuation vectors; f(a) its restriction to ideles. We let 3 denote the class of all functions f(x) satisfying the three conditions:

(31) f(x>, andf( x are continuous, E L,(V); i.e. f(x) E 5Ba,(V). ) (32) <Zkf (a(% + 0) and @a(” + 0) are both convergent for each idtle a

an> vector x, the coivergence being uniform in the pair (a, x) for x ranging over D and a ranging over any fixed compact subset of I.

(3s) f(a).lal” andf(a)la[“ELr(I) for c > 1.

(Notice that if f(x) is continuous on V, then, a fortiori, f(a) is continuous on I, since the topology we have adopted in I is stronger than that which I would get as subspace of V.)

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 339

In view of (3r) and (3J, the Riemann-Roth theorem is valid for functions of 3. The purpose of (3J is to enable us to define c-functions with them:

DEFINITION 4.4.1. We associate with each f E 3 a function C(f, c) of quasi- characters, defined for all quasi-characters c of exponent greater than 1 by

KC 4 = / f(aMa> da.

We call such a function a c-function of k. Remember that we are now considering only those quasi-characters which

are trivial on k*. These were discussed at the end of the preceding section, where the notion “exponent” is explained. If we call two quasi-characters which coincide on J equivalent, then an equivalence class of quasi-characters consists of all quasi-characters of the form c(a) = c,(a)lals, where c,,(a) is a fixed representative of the class and s is a complex number uniquely determined by c. Such a parametrization by the complex variable s allows us to view an equivalence class of quasi-characters as a Riemann surface, just as we did in the local theory (cf. Section 2.4). It is obvious from their definition as an integral that the C-functions are regular in the domain of all quasi-characters of exponent greater than 1 (see the corresponding local lemma). What about analytic continuation???

MAIN THEOREM 4.4.1. (Analytic Continuation and Functional Equation of the [-Functions.) By analytic continuation we may extend the dejinition of any c-function ccf, c) to the domain of all quasi-characters. The extended function is single valued and regular, except at c(a) = 1 and c(a) = Ial where it has simple poles with residues -of and + r&O), respectively (K = 2”(2z)“hR/(,/ldlw) = vo I ume of the multiplicative fundamental domain). [cf, c) satisfies the functional equation

w-5 4 = r<.L a,

where e(a) = lalc-‘(a) as in the local theory.

Proof. For c of exponent greater than 1 we have

[(f, c) =/f(a)c(a) da = i Ij(tb)c(tb) db d_f = 7 &(f, c):, say. .I It.

Here

C,(J 4 = j- WMb) db J

is absolutely convergent for c of any exponent, at least for almost all t, because it is convergent for some c, and Ic(tb)l = t exponent c is constant for b E J. The essential step in our proof consists in using the Riemano-Roth theorem to establish a functional equation for [,cf c):

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340 J. T. TATB

LEMMA A. For all quasi-characters c we have

(,CT.c)+~(O)~c(tb)db=Clil~4+~(O)S~(~b)db. E E

Proof

C,(f, c) +f(o) j c(tb) db = C 1 f(tb)c(tb) db +f(O) I c(rb) db E crek*aE 6

(Because J = u, aE, a disjoint union)

= C I f(atb)c(tb) db +f(O) J’ c(tb) db aek*E E

(d(ab) = db; c(atb) = c(tb))

= I‘ E

[ C f(otrb)]c(tb)db + /.f(O)c(fb)db. ook* E

(By hypothesis (3J forf, the sum is uniformly convergent for b in the rela- tively compact subset E)

=/ [&M9] cW)db

= J [&$)I @i, c(tb)db E

(Riemann-Roth theorem 4.2.1)

= / [&3(+q $b)db E

(b -+ l/b; db + db). Reversing the steps completes the proof.

LEMMA B.

1 c(tb)db = Id, if c(a) = Ial

k o

, if c(a) is non trivial on J.

Proof.

j c(tb) db = c(t) 1 c(b) db. E E

Here f c(b) db is the integral over the factor group J mod k* of the character E

of this group which c(b) represents. Therefore it is either K (= measure of E), or 0, according to whether c(a) is trivial on J or not. In the former case we must notice that c(t) = jtr = t’.

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 341

To prove the theorem write, for c of exponent greater than 1,

UC) =i,,,,s =jctu4~ +Jim$ 0 0 1

The f is no problem. For it is equal to the integral of Aa) c(a) da over . that half of I where Ial 2 1. Therefore it converges the better, the less the exponent of c is; and since it converges for c of exponent greater than 1, it must converge for all c. Now, the point is that we can use Lemma A

(and the auxiliary Lemma B) to transform the d into an $ thereby obtaining

an analytic expression for CV; c) which will be good for all c. Namely:

where the expression { { . . . >} is to be included only if c is trivial on J, in which case we assume c(a) = Ial”. W e are still looking only at c of exponent greater than 1. If c(a) = 1 al” this means Re (s) > 1, which is just what is needed for the auxiliary integrals under the double bracket to make sense. Evaluating them and making the substitution t + l/t in the main part of the expression we obtain

and therefore

The two integrals are analytic for all c. This expression gives therefore the analytic continuation of [V; c) to the domain of all quasi-characters. From it we can read off the poles and residues directly. Noticing that for c(a) = Ial: e(a) = la]‘-“, we see that even the form of the expression is unchanged by the substitution Cr; c) + (3, Q. Therefore the functional equation

holds. The Main Theorem is proved!

4.5. Comparison with the Classical Theory We will now show that our theory is not without content, inasmuch as there do exist non-trivial c-functions. In fact we shall exhibit for each

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342 J. T. TATE

equivalence class C of quasi-characters an explicit function f E 3 such that the corresponding [-function c(J c) is non-trivial on C. These special C-functions will turn out to be, essentially, the classical c-functions and L-series. The analytic continuation and the functional equation for our I-functions will yield the same for the classical functions.

We can pattern our discussion after the computation of the special local c-functions in Section 2.5. There we treated the cases k real, k complex, and k p-adic. Now we treat the case

k in the large !

The Equivalence Classes of Quasi-Characters. According to a remark at the end of Section 4.3, each class of quasi-characters can be represented by a character. To describe the characters in detail, we will take an arbitrary. but fixed, finite set of primes, S (containing at least all archimedean primes) and discuss the characters which are unramified outside S. A character of this type is nothing more nor less than a product

c(a) = n c&h,) P of local characters, c+,, satisfying the two conditions

(1) cP unramified outside S. (2) n c&x) = 1, for a E k*.

P To construct such characters and express them in more concrete terms,

we write for p fz S:

CJQp) = qQlapI$ E, being a character of up, fP a real number (cf. Theorem 2.3.1). For p $ S, we throw all the local characters together into a single character, say

c*(a) =v(jsqxap).

and interpret c* as coming from an ideal character. Namely: The map

Q + (P,(a) =vJJsPorde=

is a homomorphism of the idele group onto the multiplicative group of ideals prime to S. Its kernel is I,. c*(a) is identity on 1,. We have therefore

c*(a) = ideW), where x is some character of the group of ideals prime to S. Our character c(a) is now written in the form

44 =v~s~v(~v) .vIjJal~~. x(cP&)).

To construct such characters we must select our 2,, tP and x such that c(a) = 1, for a E k*. For this purpose we first look at the S-units, E, of k,

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FOURIW ANALYSIS IN NUMBER FIFJLDS AND HBCKE’S ZETA-FUNCTIONS 343

i.e. the elements of k* n I,, for which (ps(s) = o. Assume S contains m+ 1 primes; let e, be a primitive root of unity in k, and let (sr, .Q, . . . , E,} be a basis for the free abelian group of S-units modulo roots of unity. For c(a) to be trivial on the S-units it is then necessary and sufficient that c(e,) = 1, 0 % v 5 m. The requirement C(Q) = 1 is simply a condition on the ZP:

We therefore fust select a set of i?,, for p E S, which satisfies A. The require- ments c(s,) = 1, 1 5 v 5 m, give conditions on the tP:

JJe,I~ =v~sE[l&J, 1 5 v < m

which will be satisfied if and only if the numbers rP solve the real linear equations

(B> v~SiPw4v =ilog flE(E” ) (pes P vv)) 1 s v 5 m

for some value of the logarithms on the right-hand side. We now select a set of values for those logarithms and a set of numbers rP solving the resulting equations B. Since, as is well known, the rank of the matrix (log le&) is m, there always exist solutions rp. And since c log l.Q, = 0 for all Y, the most

V6S

general solution is then rp = rP+ r, for any r. While we are on the subject of existence and uniqueness of the rp we may remind the reader that if p is archimedean, different rP give different local characters c+, = ZJF; but if p is discrete, those rP which are congruent mod 2n/log Jyp give the same local cP.

Having selected the f, and r+,, how much freedom is left for the ideal character x ? Not much. The requirement c(a) = 1 for all a E k* means that 1 must satisfy the condition

(C)

for all ideals of the form rps(a), the ideals obtained from principal ideals by cancelling the powers of primes in S from their factorization. These ideals form a subgroup of finite index hs (less than or equal to the class number h; hs = 1 if S is large enough) in the group of all ideals prime to S. Since the multiplicative function of a on the right-hand side of condition (C) has been fixed up to be trivial on the S-units, it amounts to a character of this sub- group of ideals of the form &a). We must select x to be one of the finite number h, of extensions of this character to the group of all ideals prime to S.

The Corresponding Functions of 3. Having selected a character

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344 J. T. TATB

unramifled outside S, we wish to find a simple function f(x) E a whose C-function is non-trivial on the surface on which c(a) lies. To this effect we choose for each p E S some function f&J E 3P whose (local) C-function is non-trivial on the surface on which cP lies (for instance select f to be the function used to compute pP(cP 11:) in Section 2.5). For p 4 S, we let f,(x,) be the characteristic function of the set 0,. We then put

f(x) = n fp@p>. P

(We will show in the course of our computations that the f(r) is in the class 3.) Their Fourier Transforms. According to Lemma 3.3.3,

Ad = l-y&J,

and moreover, f E ‘z$(V), i.e. f satisfies axiom (3J. Notice that f(x) is the same type of function as f(x), except for the fact that at those x # S where b, # 1, f&J equals Nb,* times the characteristic function of bpr, rather than the characteristic function of 0,.

The C-Functions. Since 1 f(a)] la]” = n &(a,)1 lal% is a product of local

functions, almost all of which are 1 onPUp, we may use Theorem 3.3.1 to check the summability of If(a)1 la]” for Q > 1. A simple computation shows, for p 4 S,

J’ Ifp(a,>lla,l; da, = l fz+lo. kS

The summability follows therefore from the well-known fact that the product

l-I l P$S, l-4-” is convergent for Q > 1. Well known as this fact is, it should be stressed that it is a keystone of the whole theory. The existence of our C-functions just as that of the classical functions, depends on it. It is proved by descend- ing directly to the basic field of rational numbers (see, for example, E. Landau, “Algebraische Zahlen”, 2nd edition, pages 55 and 56). Because!(x) is the same type of function as f(x), we see that If(a)1 lal” is also summable for Q > 1. Therefore f(x) satisfies axiom (33).

Having established the .summability, we can also use Theorem 3.3.1 to express the C-function as a product of local C-functions. Namely

C(fa 4 = !J cpv,, c&l)

for any quasi-character c = n cP of exponent greater than 1. If c now denotes P

our special character,

44 = l-l cp(ap) =pl&e,(ap). xh(a)h P

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 345

we can compute explicitly the local factors of [(f, c 113 for p $ S. Indeed

CJfp ~~11:) = J cv(ap)laplY~, *v

dn,+ = 1 -x(p)./y^;s’

because, for p 4: S, cJa,> = x(p ord*a+$. If therefore we introduce the classical C-function C(s, x), defined for Re (s) > 1 by the Euler product

we can write

KfY cl13 =vlJs(&, c,ll:> .,l&J%*. C(s, xl-

We see that our ccf ~11’) is, essentially, the classical function C(s, x). It may be remarked here that we could have obtained directly the additive expression for C(s, x),

x(4 C(s,x) = c -

Q htz$Ldp Jva”’

had we computed ccf, c 11’) by breaking up I into the cosets of I,, integrating over each coset, and summing the results, rather than by using Theorem 3.3.1 to express the c-function integral as a product of local integrals.

Treating the C-function off in the same way we Cnd

u.f,~ =pI;l$fp9 c~hrJlsx@,w~;s 41 -s, x-9,

for Re (8) < 0. Before discussing the resulting analytic continuation and functional

equation for [(s, x), we should set our minds completely at rest by checking that ourf(x) satisfies axiom (3J; that is, that the sum

is uniformly convergent for a in a compact subset of I and Y E D. We can do this easily under the assumption that, for p E S, the local functions & we chose in constructing f do not differ too much from the standard local functions which we wrote down in Section 2.5. Namely, we assume for discrete p E S, that f, vanishes outside a compact; and for archimedean p, that f,(x,) goes exponentially to zero as xP tends to inGnity. Under these assumptions one sees first that there is an ideal, a, of k such that f(a(x + t)) = 0 if t $ a,

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346 J. T. TATE

for all a in the compact and x in D. The sum may then be viewed as a sum over a lattice in the n-dimensional space which is the i&rite part of V, of the values of a function which goes exponentially to zero with the distance from the origin. The lattice depends on a and X, to be sure, but the restriction of a to a compact means that a certain tied small cube will always fit into the fundamental parallelotope of the lattice. The uniform convergence of the sum is then obvious.

Analytic Continuation and Functional Equation for c(s, x). The analytic continuation which we have established for our C-functions, both in the large and locally, now gives directly the analytic continuation of &, x) into the whole plane. Our functional equations

C<f: ?I= i(L 4 and SJ.& cp> = P,(c,)~,(& tip)

yield for [(s, x) the functional equation

3(1--w?) =J-p,c%ll;+i”> ; Q “Jq-*X-‘@,) * &,x).

The explicit expressions for the local functions p,, are tabulated in Section 2.5. The meaning of the fi,, and tp, and their relationship to the ideal character x, is discussed in the first paragraph of this section.

These ideal characters, x, which we have constructed out of idele charac- ters, are exactly the characters which Hecke introduced in order to define his “new type of c-function”. {(s, x) is that C-function; and the functional equation we have just written down is the functional equation Hecke proved for it.

A Few Comments on Recent Related Literature

[Added January 19671

In Lang’s book Algebraic Numbers (Addison-Wesley, 1964) there is a review of the theory above, in which the global results are renormalized in a way which corresponds more closely to the classical theory and which is more suitable for applications. The applications to prime densities, the Brauer- Siegel theorem, and the “explicit formulas” are also treated there. Lang has urged me to point out that in the main theorem on equidistribution of primes (Theorem 6, p. 130 lot. cit.) one must assume that a(Jp) = G, not only that a(J,) = G; thus the first application following that theorem, concerning the equidistribution of log p, is not valid.

For a reinterpretation of the proof of the functional equation given above, see Weil, Fonction z&a et distributions, S&ninaire Bourbaki, No. 312, June 1966.

Siegel’s work on quadratic forms has inspired much ad&c (addle is the modern term for valuation-vector) analysis in recent years. For some

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FOURIER ANALYSIS IN NUMBER FIELDS AND HECKE’S ZETA-FUNCTIONS 347

crucial results and bibliography in that direction, see Weil’s papers in Acta Mathematics, 1964 and 1965.

REFERENCES

Artin, E. and Whaples, G. (1945). Axiomatic characterization of fields by the product formula for valuations. Bull. Am. math. Sot. 51,469-492.

Cartan, H. and Godement, R. (1947). Thkorie de la dualitC et analyse harmonique dans les groupes abCliens localement compacts. Ads. scient. EC. norm. sup., Paris (3), 64, 79-99.

Chevalley, C. (1940). La thkorie du corps de classes. Ann. Math. 41, 394-418. Hecke, E. Uber eine neue Art von Zetafunktionen und ihre Beziehungen zur

Verteilung der Primzahlen. Math. Z. 1(1918), 357-376; 4 (1920), 11-21. Reprinted in “Mathematische Werke”, Vandenhoeck und Ruprecht, GGttingen, 1959.

Matchett, M. Thesis, Indiana University (1946), unpublished.

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Exercises?

Exercise 1: The power residue symbol . . . . . . 348 Exercise 2: The norm residue symbol . . . . . . 351 Exercise 3: The Hilbert class field . . . . . . . 355 Exercise 4: Numbers represented by quadratic forms . . . 357 Exercise 5: Local norms not global norms. . . . . : 360 Exercise 6: On decomposition of primes . . . . . . 361 Exercise 7: A lemma on admissible maps . . . . . . 363 Exercise 8: Norms from non-abelian extensions . . . . . 364

Exercise 1: The Power Residue Symbol (Legendre, Gauss, et al.)

This exercise is based on Chapter VII, § 3, plus Kummer theory (Chapter III, 3 2). Let m be a tied natural number and K a lixed global field containing the group h of mth roots of unity. Let S denote the set of primes of Kcon- sisting of the archimedean ones and those dividing m. If al, . . . , a, are elements of K*, we let S(a,, . . ., a,) denote the set of primes in S, together with the primes v such that [Q& # 1 for some i. For a E K* and b E Is(‘)

the symbol f is defined by the equation 0

where L is the field K(l;fla).

EXERCISE 1.1. Show z 0

is an mth root of 1, independent of the choice

of “Ja.

EXERCISE 1.2. Working in the field L’ = K(?/a, “Ja’) and using Chapter VII, $ 3.2 with K’ = K and L = K(mJa>, show

($) =(;) (fz) if b E I%“* 0’).

EXERCISE 1.3. Show

(ii) =@ (i) if b E Ia(‘).

t These “exercises” refer primarily to Chapter VII, “Global class field theory*‘, and were prepared after the Conference by Tate with the connivance of Serre. They adumbrate some of the important results and interesting applications for which unfortunately there was not enough time in the Conference itself.

348

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EXERCTISES 349

Hence,

EXERCISE 1.4. (Generalized Euler criterion.) If’ u 4 S(u) then mj(Nv- l),

where NV = [k(v)], and 0

% is the unique mth root of 1 such that

0 No-l a --ii- EU

V (mod P,).

EXERCISE 1.5. (Explanation of the name “power residue symbol”.) For v $ S(u) the following statements are equivalent:

a

() c-1 i =l.

V

(ii) The congruence a?’ = a (mod p,,) is solvable with x E 0,. (iii) The equation x” = a is solvable with x E &.

(Use the fact that k(o)* is cyclic of order (No-l), and Hensel’s lemma, Chapter II, App. C.)

EXERCISE 1.6. If b is an integral ideal prime to m, then m-1

(3 - i =[" force&.

(Do this first, using Exercise 1.4, in case b = 1) is prime. Then for general b = C n,v, note that, putting Nu = 1 +mr,,, we have

Nb = n (1 +mr,,p s 1 + m C n, r,, (mod m’).)

EXERCISE 1.7. If Q and b E Iz(“) are integral, and if a’ E u (mod b),

then6) =(i).

EXERCXE 1.8. Show that Artin’s reciprocity law (Chapter VII, § 3.3) for a simple Kummer extension L = K(“Ja) implies the following statement: rf b and 6’ E Is(“), and b’ 6-l = (c) is the principal ideal of an element c E K*

such that c E (K,*)” for aN u E S(u), then G)=(t). Note that for v # S,

the condition c E (II;*)” will certainly be satisfied if c = 1 (mod p,).

EXERCLSE 1.9. Specialize now to the case K = Q, m = 2. Let u, b, . . . denote arbitrary non-zero rational integers, and let P, Q, . . . denote positive,

odd rational integers. For (a, P) = 1, the symbol (;) = (&) =*1 is

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350 EXERCISES

defined, is multiplicative in each argument separately, and satisfies

(g=(i) if a = b (mod I’).

Artin’s reciprocity law for Q(Ja)/Q implies

6) =(;) if P 3 Q (mod 84,

where a, denotes the “odd part of a”, i.e. a = 2’ao, with a, odd. (Use the fact that numbers = 1 (mod 8) are 2-adic squares.)

EXERCISE 1.10. From Exercise 1.9 it is easy to derive the classical law of quadratic reciprocity, namely

P-l

q-p-, f

0

p*-1 P-l Q-l

= (-1) 8 P Q , and Q p =(-1) 00

2 ‘T-

Indeed the formula (*) above allows one to calculate g 0

as function of P

for any tied a in a finite number of steps, and taking a = - 1 and 2 one proves the first two assertions easily. For the last, define

<P, Q> = (5) (z), for (p, Q> = 1.

Then check first that if P E Q (mod 8) we have

<P,Q) = (2)

and the given formula is correct. (Writing Q = P+8a one finds using Exercise 1.9 that, indeed,

Now, given arbitrary relatively prime P and Q, one can find R such that RP = Q (mod 8) and (R, Q) = 1 (even R E 1 (mod Q)), and then, by what we have seen,

<P, Q> CR, Q> = <PR, Q> = .

Fixing R and varying P, keeping (P, Q) = 1, we see that <P, Q) depends only on P (mod 8). By symmetry (and the fact that the odd residue classes (mod 8) can be represented by numbers prime to any given number), we see that <P, Q) depends only on Q (mod 8). We are therefore reduced to a small finite number of cases, which we leave to the reader to check. The next exercise gives a general procedure by which these last manoeuvres can be replaced.

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EXERCISES 351

Exercise 2: The Norm Residue Symbol (Hilbert, Hasse) We assume the reciprocity law for Kummer extensions, and use Chapter VII, $ 6. The symbols m, K, S, and S(ur, . . . , a,) have the same significance as in Exercise 1. For a and b E K* and an arbitrary prime u of K we define (a, b), by the equation

(~u)~*(*) = (a, b),,ju,

where $, : K: + G” is the local Artin map associated with the Kummer extension K(‘“Ja)/K.

EXERCISE 2.1. Show that (a, b), is an mth root of 1 which is independent of the choice of vu.

EXERCISE 2.2. Show (a, b),(u, b’), =(a, bb’), and (a, b),(u’, b), = (au’, b),. Thus, for each prime v of K, we have a bilinear map of K* x K* into the

group p,,, of mth roots of unity.

EXERCISE 2.3. Show that (a, b), = 1 if either a or b E (K,*)m, and hence that there is a unique bilinear extension of (a, b), to K,* x K:.

This extension is continuous in the v-adic topology, and can be described by a finite table of values, because Kz/(K:)‘” is a finite group (of order m2/lmjo, where lrnlu is the normed absolute value of m at v). Moreover, the extended function on K: x K,* can be described purely locally, i.e. is inde- pendent of the field K of which K, is the completion (because the same is true of $3, and induces a non-degenerate pairing of K,*/(K:)” with itself into p,,,; however we will not use these local class field theoretic facts in most of this exercise. For a general discussion of (a, b),, and also for some explicit formulas for it in special cases, see Hasse’s “Bericht”, Part II, pp. 53-123, Serre’s “Corps Locaux”, pp. 212-221, and the Artin-Tate notes, Ch. 12. The symbol (a, b), defined here coincides with that of Hasse and Serre, but is the opposite of that defined in Artin-Tate. While we are on the subject, our local Artin maps $, coincide with those in Serre and in Artin-Tate, but are the opposite of Hasse’s.

EXERCISE 2.4. Show that (a, b), = 1 if b is a norm for the extension K,(vu)/K,. (See Chapter VII, 5 6.2; the converse is true also, by local class field theory, but this does not follow directly from the global reciprocity law.)

EXERCXE 2.5. We have (a, b), = 1 if a+ b E (Kz)“‘; in particular, (a, -a), = 1 = (a, 1 - a),. (This follows from the purely algebraic lemma: Let F be afield containing the group p,,, of mth roots of unity, and let a E FL. Then for every x E F the element ~“‘-a is a norm from F(vu). Indeed, let am = a. The map d H au/a is an isomorphism of the Galois group onto a subgroup ,u~ of p,,, and is independent of the choice of a. Hence if (CJ is a

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352 EIERcIsm

system of representatives of the COSets of & in &,,, we have for each x E F

Q.E.D.)

EXERCISE 2.6. Show that (a, b),(b, a), = 1. (Just use bilinear@ on 1 = (ab, -ab),.)

EXERCISE 2.7. If u is archimedean, we have (a, b), = 1 unless K, is real, both a < 0 and b < 0 in &,, and m = 2. (In the latter case we do in fact have (a, b), = - 1; see the remark in Exercise 2.4. Note that m > 2 implies that K,, is complex for every archimedean a.)

EXERCISE 2.8. (Relation between norm-residue and power-residue symbols.)

If t, # S(u), then (a, b), = a o(b)

0 ; ; in particular, (a, b), = 1 for a 4 S(a, b).

(See ,the fist lines of Exercise 1 for the definition of S and S(u), etc. The result follows from the description of the local Artin map in terms of the Frobenius automorphism in the tutramified case. More generally,

0 # S =z- (a, b), x 0 g , where c = (- l)o(a)u(b)&b)b-~a~

is a unit in & which depends bilinearly on a and b. To prove this, just write a = &%z,, and b - nuCb)bo where v(x) = 1, and work out (a, b), by the previous rules; for the geometric analog discussed in remark 3.6 of Chapter VII, see Serre, lot. cit., Ch. III, Section 4.)

EXERCISE 2.9. (Product Formuh.) For a, b E K* we have n (a, b), = 1, the product being taken over all primes u of K

Exnncrsn 2.10. (i%e general power-reciprocity law.) For arbitrary a and b in K* we define

where (b)” is defined in Chapter VII, 3 3.2.

Warning: With % 0

defined in this generality the rule

does not always hold, but it does hold if S(b) A S(a, a’) = S, and especially

if b is relatively prime to a and a’. The other rule,

in general.

($=(;)($holds

Using Exercises 2.6, 2.8 and 2.9, prove that

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EXERCISES

In particular

353

(*I and

a b-l

00 iJ a =c~s(h a),, if S(a) n S(b) = S,

c**> A 0 b =v~s(l, b),, if S(1) = S.

EXERCISE 2.11. If K= Q and m = 2, then S = (2, co}, and for P > 0 as in Exercise 1.10, we have (x, P), = 1. Hence the results of Exercise 1.10 are equivalent with

(-l,P)z= (-l)y PZ-1

w%= (-UT, P-l Q-l

and (P, Q))z = (- l)y ‘2,

for odd P and Q. On the other hand, these formulas are easily established working locally in Qz. In particular, the fact that (1+4c, b), = (- l)ol(*)‘, from which the value of (a, b), is easily derived for all a, b using Exercises 2.2,2.5 and 2.6, is a special case of the next exercise.

EXERCISE 2.12. is unramified at u.

An element a E K is called u-primary (for m) if K(“Ju)/K For u 4 S, there is no problem: an element a is *primary

if and only if u(a) I 0 (mod m). Suppose now u divides m and m = p is a prime number. Let C be a generator of q, and put 1 = 1 -C. Check that Apzp-‘/p is a unit at u, and more precisely, that rZp-’ E -p (mod PA), so that Ap-l/p z - 1 (mod p,). Let u be such that a = 1 (modpllo,), so that we have a = 1 +Kc, with c E 0,. Prove that a is u-primary, and that for all b,

(a, b), = C-S@)“(b),

where S denotes the trace from k(u) to the prime field and E is the u-residue oft. Also,ifu = 1 (mod p&), then u is u-hyperprimary, i.e. u E (Kz)“.

(Let ap = a, and write a = 1 +ti. Check that x is a root of a polynomial f(X) E o,[ X] such that j’(X) = Xp- X- c (mod p,). Thus f’(x) E - 1 $0 (mod P,), so Ud = &c”J u is indeed unratied. > And if c 3 0 (mod pJ then f(X) splits by Hensel’s lemma, so &(“Ja) = &. Now xp = x+ c (mod p,), so if% = p*, then

On the other hand, if a’ = [a = 1 +W, then x’ E x- 1 (mod p,). Com- bining these facts gives the formula for (a, b),.)

EXERCSE 2.13. Let p be an odd prime, [ a primitive pth root of unity, K = Q(& and m = p. Then p is totally ramified in K, and 1 = 1 -C generates the prime ideal corresponding to the unique prime u of K lying over p. Let U, denote the group of units = 1 (mod 2’) in Kf, for i = 1,2, . . . . Then the image of vi = 1 - rZ’ generates tJ,IU,, r, which is cyclic of order p, and the image of L generates K:/(K:)PIJ,. By the preceding exercise,

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354 EXERCISES

u p+l = uap. Hence the elements L, 5 = qr, l-l2 = tf2,. . ., l-L’= qp generate (Kf)/(K:)P. But that group is of order p2/1pI, = ~l+~, so these generators are independent mod pth powers. Show that

(a> (Vi3 rljh = (Vi, Vi+j)u(Vi+j, ?j>v(Vi+ jt J)i’, for all kJ 2 1.

(b) Ifi+jrp+l,then(a,b),= 1foralla~U~andb~U~.

Cc) (Vi, n>o = 1

1, forl<i<p-1 (, for i = p.

(d) (a, b), is the unique skew-symmetric pairing K: x K,* -+ pp satisfying (a) and (c).

(For (a), note qj+l$, = If,+j, divide through by v~+~, and use Exercise 2.5 and bilinearity; the oddness of p, which implies (a, b) = (a, -b) in general and (a, a) = 1 in particular, is used here. The rest all follows easily, except for (c) which is a consequence of the preceding exercise; but note that the first (p- 1) cases of (c) are trivialities, because

h 4 f, = (1 -A’, Ai), = 1 * (vi, A), = 1 for 1 5 i I; p- 1.)

EXERCI~B 2.14. (Cubic reciprocity law.) Specialize to p = 3 in the pre- ceding exercise. The ring of integers R = 2 + ZC is a principal ideal domain, whose non-zero elements can be written in the form lv[pa, with a E f 1 (mod 3R). Prove

(*) (;) = ($ for relatively prime a and b, each = + 1 (mod 3R),

and also

(**I , for a = f(l+ 3(m+ rtc)).

As an application, prove: If q is a rational prime E 1 (mod 3), then 2 is a cubic residue (mod q) if and only if q is of the form x2 +27y2 with X, y E Z. (Write q = nR with rr E + 1 (mod 3R). Then Z/qZ z RjlrR, so 2 is a cubic

residue (mod q) if and only if 0

2 7T

= 1. Now use (*), and translate f = 1 0

into a statement about q.)

EXERIXE 2.15. Let L be the splitting field over Q of the polynomial X3 - 2. The Galois group of L/Q is the symmetric group on three letters. Using the preceding exercise, show that for p # 2,3 the Frobenius auto- morphism is given by the rules:

FL&p) = (l), ifp = 1 (mod 3) and p of the form x2 +27y2, %&4 = 3-cycle, if p E 1 (mod 3) and p not of the form x2 + 27u2, h/p(P) = 2-cycle, if p E - 1 (mod 3).

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ExERclsEs 355

Hence, by Tchebotarov’s theorem, the densities of these sets of primes are l/6, l/3 and l/2, respectively.

EXERCISE 2.16. Consider again an arbitrary K and m. Let a,, . . ., Us be a finite family of elements of K*, and let L be the Kummer extension generated by, the mth roots of those elements. Let T be a finite set of primes of K containing S(a,, . . . , a,), and big enough so that both JK = K*JR,=, and Jt = L*J=,=s, where T’ is the set of primes of L lying over T. Suppose we are given elements 5v,i E c(m, for u E T and 1 5 i 5 r, such that

(i) For each i, we have fl & = 1, and “ST

(ii) For each u E T, there exists an x, E K: such that (x,, a,)” = [y,l for all i.

Show then that there exists a T-unit x E Kr such that (x, a,)” = cu,i for all ooTandall1 ~ilr.

The additional condition on T, involving T’, is necessary, as is shown by the example K = Q, m = 2, T = (00,2,7}, r = 1, a, = - 14, co3,1 = - 1, 52,l =-I, L,l = 1. To prove the statement, consider the group X = l’-&~V*)/WV*)“‘~ th e subgroup A generated by the image of KT, and the smaller subgroup A, generated by the images of the elements a,, 1 5 i I r. The form (x, JJ) = nvsT (x,, y,), gives a non-degenerate pairing of .X with itself to p,,,, under which A is self orthogonal, and indeed exactly so, because [X] = mzt and [A] = m’, where t = [T]. (See step 4 in the proof of the second inequality in Chapter VII, 5 9, the notations S, n, and s there being replaced by T, m, and t here.) Thus X/A = Horn (A, pm) (note by the way that both groups are isomorphic to Gal (K(vKr)/K), by class field theory and Kummer theory, respectively), and, vice versa, A RJ Horn (X/A, /A,,,). So far, we have not used the condition that JL = L*JLsT.. Use it to show that if a E A and n,(a) E q(A,) for all v, where nv is the projection of X onto K,*/(K,*)“, then a E A,, i.e. vu EL. Now show that, in view of the dualities and orthogonalities discussed above, this last fact is equivalent to the statement to be proved.

Exercise 3: The Hilbert Class Field Let L/K be a global abelian extension, u a prime of K, and iv : K: + JK the canonical injection. Show that v splits completely in L if and only if i,(K:) c K*NLIKJL, and, for non-archimedean v, that v is tutramified in L if and only if i,(U,) c K*N,IKJL, where U,, is the group of units in K,,. (See Chapter VII, $ 5.1, 5 6.3.) Hence, the maximal abelian extension of K which is unramified at all non-archimedean primes and is split com- pletely at all archimedean ones is the class field to the group K*J,,,, where S now denotes the set of archimedean primes. (Use the Main Theorem

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356 EXJZRCISES

(Chapter VII, 5 5.1) and the fact that K*NLIKJL is closed.) This extension is called the Hilbert class field of K; we will denote it by K’. Show that the Frobenius homomorphism FEIK induces an isomorphism of the ideal class group Hx = Ig[Px of K onto the Galois group G(K’/K). (Use the Main Theorem and the isomorphism JRIJR,s L$ I&) Thus the degree [K’ : K] is equal to the class number h, = [Hg] of K. The prime ideals in K decompose in K’ according to their ideal class, and, in particular, the ones which split completely are exactly the principal prime ideals. An arbitrary ideal a of K is principal if and only if F,,,,(a) = 1.

The “class field tower”, K c K’ c K” = (K’)‘c . . . can be infinite (see Chapter IX). Using the first two steps of it, and the commutative diagram (see (11.3), diagram (13))

I K m?!5, G(K’I’K) con

1 I V

I Ke FK”x’ + G(K”/K’),

Artin realized that Hilbert’s conjecture, to the effect that every ideal in K becomes principal in K’, was equivalent to the statement that the Ver- lagerungt V was the zero map in this situation. Now G(K”/K’) is the com- mutator subgroup of G(K”/K) (Why?), and so Artin conjectured the “Principal ideal theorem” of group theory: If G is a finite group and G’ its commutator subgroup, then the map V: (G/G’) + G’/(Gy is the zero map. This theorem, and therewith Hilbert’s conjecture, was then proved by Furtwiingler. For a simple proof, see Witt, Proc. Intern. Conf. Math., Amsterdam, 1954, Vol. 2, pp. 71-73.

The tist five imaginary quadratic fields with class number # 1 are those with discriminants - 15, -20, - 23, - 24, and - 31, which have class numbers 2, 2, 3, 2, 3, respectively. Show that their Hilbert class fields are obtained by adjoining the roots of the equations X2 + 3, X2 + 1, X3 - X- 1, X2 + 3, and X3 + X- 1, respectively. In general, if K is an imaginary quadratic field, its Hilbert class field K’ is generated over K by thej-invariants of the elliptic curves which have the ring of integers of K as ring of endo- morphisms; see Chapter XIII.

Let Jz denote the group of ideles which are positive at the real primes of K and are units at the non-archimedean primes. The class field over K with norm group K*J& is the maximal abelian extension which is unramified at all non-archimedean primes, but with no condition at the archimedean primes; let us denote it by Ki. Let Pg denote the group of principal ideals of the form (a), where a is a totally positive element of K. Show that F,,,, gives an isomorphism: &/Pi % G(K,/K). Thus, G(K,/K’) is an elementary

t Called the transfer in Chapter IV, 5 6, Note after Prop. 7.

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EXERCISES 357

abelian 2-group, isomorphic to P=/Pz. Show that (PK : Pi)(Ks : Kg) = 2”, where Ki = K* n J& is the group of totally positive units in K, and rl is the number of real primes of K.

We have Q1 = Q, clearly, but this is a poor result in view of Minkowski’s theorem, to the effect that Q has no non-trivial extension, abelian or not, which is unramified at all non-archimedean primes (Minkowski, “Geometrie der Zahlen”, p. 130, or “Diophantische Approximationen” p. 127). Consider now the case in which Kis real quadratic, [K: Q] = 2, and rl = 2. Show that [Kl : K’] = 1 or 2, according to whether Ne = - 1 or Ne = 1, where E is a fundamental unit in K, and N = Nx,o. For example, in case K = Q(,/2) or Q(J5) we have K’ = K, because the class number is 1, and consequently also Kl = K, because the units E = 1 + ,/2 and E = +(l + ,/5) have norm - 1. On the other hand, if K = Q&/3), then again K’ = K, but Kl # K, because E = 2+,/3 has norm 1; show that K, = K(J- 1). In general, when - 1 is not a local norm everywhere (as in the case K = Q(J3) just considered), then Ns = 1, and Kl # K’. However, when - 1 is a local norm everywhere, and is therefore the norm of a number in K, there is still no general rule for predicting whether or not it is the norm of a unit.

Exercise 4. Numbers Represented by Quadratic Forms Let K be a field of characteristic different from 2, and

a non-degenerate quadratic form in n variables with coefficients in K. We say that f represents an element c in K if the equation f(X) = c has a solution X = x E K” such that not all xi are zero. Zff represents 0 in K, then f represents all elements in K. Indeed, we have

(tX+ Y) = t’f(x)+m(x, Y)+,f(Y). Iff(x)=Obutx#(O,O,..., 0), then by the non-degeneracy there is a y E K” such that B(x, JJ) # 0, so that f(tx +JJ) is a non-constant linear function of I and takes all values in K as I runs through K.

A linear change of coordinates does not affect questions of representability, and by such a change we can always bring f to diagonal form: f = c a, A’: with all a, # 0. If f = cX: - g(X,, . . ., X,) then f represents 0 if and only if g represents c, because if g represents 0 then it represents c. Hence, the question of representability of non-zero c’s by forms g in n - 1 variables is equivalent to that of the representability of 0 by forms f in n variables. The latter question is not affected by multiplication off by a non-zero constant; hence we can suppose f in diagonal form with a1 = 1 in treating it:

EXERCISE 4.1. The form f = X2 does not represent 0.

EXERCISE 4.2. The form f = X*-b Y* represents 0 if and only if b E (K*)*.

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EXERCISE 4.3. The form f = X2-b Y2-cZ2 represents 0 if and only if c is a norm from the extension field K(Jb).

EXERCISE 4.4. The following statements are equivalent: (i) Theformf= X2-bY2 - cZ2 + acT2 represents 0 in K. (ii) c is a product of a norm from K&/a) and a norm from K(Jb),

(iii) c, as element of K(,/ab), is a norm from the field L = K(Ja, ,/b). (iv) The form 9 = X2-bY2 -cZ2 represents 0 in the field K(,/ab).

(We may obviously assume neither a nor b is a square in K. Then the equivalence of(i) and (ii) is clear because the reciprocal of a norm is a norm, and the equivalence of (iii) and (iv) follows from Exercise 4.3 with K replaced therein by K(dab). It remains to prove (ii) o (iii), and we can assume ab 4 (K*)2, for otherwise the equivalence is obvious. Then Gal (L/K) is a four-group, consisting of elements 1, p, rr, r such that p, Q, and r leave fixed, respectively, Jab, ,/a, and ,/b, say. Now (ii) * (ii’): 3 x, y E L such that x” = x, y’ = y, and a?+“yl +p = c; and (iii) e (iii’) 3 z EL such that zI+~ = c. Hence (ii) * (iii) trivially. Therefore assume (iii’), put u - c-lzb+l and check that ub = u, i.e. u E K(Ja), and up+l = 1. Hence by Hilbert’s iheorem 90 (Chapter V, 5 2.7) for the extension K(Ja)/K, there exists x # 0 such that x0 = x and xp-’ = u. Now put y = zp/x, and check that (ii’) is satisfied.)

So far, we have done algebra, not arithmetic. From now on, we suppose K is a globalfield of characteristic # 2.

EXERCISE 4.5. The form f of Exercise 4.3 represents 0 in a local field K,, if and only if the quadratic norm residue symbol (b, c), = 1. Hencefrepre- sents 0 in K, for all but a finite number of o, and the number of n’s for which it does not is even. Moreover, these last two statements are invariant under multiplication off by a scalar and consequently hold for an arbitrary non- degenerate form in three variables over K.

EXERCISE 4.6. Let f be as in Exercise 4.4. Show that iff does not represent 0 in a local field K,, then a 4 (K:)2, and b 4 (Ki)‘, but ab E (Kz)2, and c is noi a norm from the quadratic extension K&/a) = K&/b). (Just use the fact that the norm groups from the different quadratic extensions of K,, are subgroups of index 2 in K:, no two of which coincide.) Now suppose conversely that those conditions are satisfied. Show that the set of elements in K,, which are represented by f is N- cN, where N is the group of non-zero norms from K,(,/ a , ) and in particular, that f does not represent 0 in K,. Show, furthermore, that if N-cN # K:, then - 1 $ N, and N+ N c N. Hence f represents every non-zero element of K, unless K, NN R and f is positive definite.

EXERCISE 4.7. A form f in n 2 5 variables over a local field K, represents 0 unless K, is real and f definite.

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EUIRCISES 359

EXERCWI 4.8. Theorem: Let K be a global jield and f a non-degenerate quadratic form in n variables over K which represents 0 in &for each prime v of K. Then f represents 0 in K. (For n = 1, trivial; n = 2, cf. Chapter VII, 5 8.8; n = 3, cf. Chapter VII, $ 9.6 and Exercise 4.3; n = 4, use Exercise 4.4 to reduce to the case n = 3; Anally, for n 2 5, proceed by induction: Let

f(X) = ax: + bX2 -9(X3,. . ., XJ, where g has n-2 2 3 variables. From Exercise 4.5 we know that g repre- sents 0 and hence every number in K, for all v outside a kite set S. Now (K,*)’ is open in K:. Hence, by the approximation theorem there exist elements x1 and x2 in K, such that the element c = ax: + bxj: # 0 is repre- sented by g in K, for all v in S, and hence for all v. By induction, the form cY2-g(&. . . ., XJ in n- 1 variables represents 0 in K. Hence f does.)

EXERCISE 4.9. Corollary: If n 2 5, then f represents 0 in K unless there is a real prime v at which f is definite.

EXERCISE 4.10. A rational number c is the sum of three rational squares if and only if c = 4”r where r is a rational number > 0 and $ 7 (mod 8); every rational number is the sum of four rational squares.

EXERCISE 4.11. The statements in the preceding exercise are true if we replace “rational” by “rational integral” throughout. (The 4 squares one is an immediate consequence of the 3 squares one, so we will discuss only the latter, although there are more elementary proofs of the four square statement not involving the “deeper” three square one. Let c be a positive integer as in 4.10, so that the sphere IX]’ = X:+Xf+Xz = c has a point x = (x,, x2, xs) with rational coordinates. We must show it has a point with integral coordinates. Assuming x itself not integral, let z be an integral point in 3-space which is as close as possible to x, so that x = z+a, with 0 < lal2 I 3/4 < 1. The line I joining x to z is not tangent to the sphere; if it were then we would have lal2 = lzi2- 1x1’ = jzl’-c, an integer, contra- diction. Hence the line I meets the sphere in a rational point x’ # x. Now show that if the coordinate of x can be written with the common denominator d > 0, then those of x’ can be written with the common denominator d = lal’d < d, so that the sequence x, x’, (x’)‘, . . . must lead eventually to an integral point. Note that d’ is in fact an integer, because

d’= jal’d = Ix-zj2d = (1x1’-2(x,z)+Iz12)d = cd-2(dx,z)+lzl’d.)

EXERCISE 4.12. Let f be a form in three variables over K. Show that if f does not represent 0 locally in K,, then the other numbers in K, not represented by f constitute one coset of (K,*)2 in K:. (Clearly one can assume f = X2-b Y2- ~2’; now use Exercise 4.6.) Using this, show that if K = Q and f is positive definite, then f does not represent all positive integers. (Note the last sentence in Exercise 4.5.)

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For further developments and related work see 0. T. O’Meara: “Intro- duction to Quadratic Forms” (Springer, 1963) or Z. I. Borevii? and I. R. SafareviE, “Teorija Cisel” (“Nauka”, Moskva, 1964). [English translation, Z. I. Borevich and I. R. Shafarevich, “Number Theory”, Academic Press, New York: German translation, S. I. Borevicz and I. R. SafareviE, “Zahlentheorie”, Birkhluser Verlag, Basel.]

Exercise 5: Local Norms Not Global Norms, etc.

Let L/K be Galois with group G = (1, p, cr, z) z (Z/ZZ)*, and let K,, K,, and K3 be the three quadratic intermediate fields left fixed by p, 6, and z, respectively. Let N, = Nz,,,&‘) for i = 1, 2, 3, and let N = N,,,(L*).

EXERCISE 5.1. Show that N1 N2N3 = (X E K*]x2 E N}. (This is pure algebra, not arithmetic; one inclusion is trivial, and the other can be proved by the methods used in Exercise 4.3.)

EXERCLSE 5.2. Now assume K is a global field. Show that if the local degree of L over K is 4 for some prime, then N1 N, N3 = K* (cf. Chapter VII, 0 11.4). Suppose now that all local degrees are 1 or 2. For simplicity, suppose K of characteristic # 2, and let Kl = K(,/uJ for i = 1,2,3. For each i, let S, be the (infinite) set of primes of K which split in 4, and for x e K* put

=~g~1A =orIy2JLa = fL

where (x, y),, is the quadratic norm residue symbol. Show that Ni. N2 N3 = Ker 9~ and is a subgroup of index 2 in K*. (The inclusion Nl N2 N3 c Ker Q is trivial. From Exercise 5.1 above and Chapter VII, 5 11.4 one sees that the index of N1 N2N3 in K* is at most 2. But there exists an x with (p(x) = - 1 by Exercise 2.16.)

EXERCISE 5.3. Let K = Q and L = Q(,/13, ,/17). Show that if x is a

product of primes p such that 0

fi = - 1 (e.g. p = 2, 5, 7, 11, . . .), then

&I = F7 * 0

Hence S2, 72, 102, 112, 142,. . . are some examples of numbers

which are local norms everywhere from Q(,/13, ,/17) but are not global norms. Of course, not every such number is a square; for example, -142 is the global norm of 3(7 + 2&3 +,/17), and comparing with the above we see that - 1 is a local norm everywhere but not a global norm.

EXERCISE 5.4. Suppose now that our global Cgroup extension L/K has the property that there is exactly one prime v of K where the local degree is 4: Let w be the prime of L above t’ and prove that fi -‘(G, L*) = 0, but

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ExERcms 361

fi -‘(G, Lz) z Z/22. (Use the exact sequence near the beginning of paragraph 11.4. The map g is surjective, as always when the 1.c.m. of the local degrees is the global degree. And the map g: R -‘(G, Jd --+ R(G, Cd is also injective, because of our assumption that the local degree is 4 for only one prime.)

Let A, resp. A,, be the group of elements in L*, resp L$ whose norm to K (resp. to K,,) is 1, and let A be the closure of A in L$ It follows from the above that

A = (L*)p--(L*)yL*)y

and that

A = (L*w)p-‘(L~y-‘(L*,))‘-’

is of index 2 in A,,,. Now, as is well known, there is an algebraic group T defined over K (the twisted torus of dimension 3 defined by the equation %/r&Q = 1) such that T(K) = A and T(K,) = A,. Hence we get examples which show that the group of rational points on a torus T is not necessarily dense in the group of v-adic points (see last paragraph below). However, it is not hard to show that if T is a torus over K split by a Galois extension L/K, then T(K) is dense in T(K,) for every prime v of K such that there exists a prime v’ # v with the same decomposition group as v; in particular, whenever the decomposition group of v is cyclic, and more particularly, whenever v is archimedean.

As a concrete illustration, take K = Q and L = Q(,/- 1, ,/2) = Q(t), where c4 = - 1. Then L is unramified except at 2, but totally ramified at 2, and consequently there is just one prime, 2, with local degree 4. Let M = Q(i) where i = C2 = J-1, and let L, and M,, denote the completions at the primes above 2. It is easy to give an ad-hoc proof without cohomology that the elements of L with norm 1 are not dense in those of Lz: just check that the element z = (2+i)/(2-i) 8 M,, is a norm from L, to L,,, but that z(M,*)~ contains no element y E M such that y is a global norm from L to A4 and such that NM,o(y) = 1.

Exercise 6 : On Decomposition of Primes

Let L/K be a finite global extension and let S be a finite set of primes of K. We will denote by Spl, (L/K) the set of primes v 4 S such that v splits com- pletely in L (i.e. such that L @ K, z KckKJ), and by Splk (L/K) the set of

primes v #S which have a silit factor in L (i.e. such that there exists a K-isomorphism L 3 G). Thus Spl, (L/K) c Spli (L/K) always, and equality holds if K is Galois, in which case Spl& (L/K) has density [L : K ]- ’ by the Tchebotarov density theorem. (Enunciated near end of Chapter VIII, $ 3.)

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362 EXERCISES

EXERCISE 6.1. Show that if L and M are Galois over K, then

L = M-2 SPJ, (M) c SPl, VA

(Indeed, we have

SPL (LWK) = Spb W/K) n SPL W/K),

SO

L c M + Spl,(M) t Spl&) =r Spl&M/K) = Spl,(M/K) =s-[LM:K]=[M:I+LcM;

where was Galoisness used?) Hence

L = Mo Spl,(L) = Spl, (M).

Application: If a separable polynomial f(X) E K[XJ splits into linear factors mod p for all but a finite number of prime ideals p of K, then f splits into linear factors in K. (Take L = splitting field of f(X), and M = K, and S large enough so that f has integral coefficients and unit discriminant outside S.) Finally, note that everything in this exercise goes through if we replace “all primes v 4 s” and “all but a finite number of primes u” by “all tr in a set of density 1”.

EXERCISE 6.2. Let L/K be Galois with group G, let H be a subgroup of G, and let E be the fixed field of H. For each prime tt of K, let G” denote a decomposition group of 0. Show that v splits completely in E if and only if all of the conjugates of G” are contained in H, whereas v has a split factor in E if and only if at least one conjugate of G” is contained in H. Hence, show that the set of primes Spli (E/K) has density $P&-~]/[G]. Now

prove the lemma on finite groups which states that the union of the conju- gates of a proper subgroup is not the whole group (because they overlap a bit at the identity!) and conclude that if Splk (E/K) has density 1, then E = K. Application: If an irreducible polynomial f(X) E K[X] has a root (mod p) for all but a finite number of primes p, or even for a set of primes p of density 1, then it has a root in K. This statement is false for reducible polynomials ; consider for example f (X) = (X2 -a) (X2 - b) (X2 - ab),where a, b, and ab are non-squares in K. Also, the set Spl’ (E/K) does not in general determine E up to an isomorphism over K; cf. Exercise 6.4 below.

EXERCISE 6.3. Let Hand H’ be subgroups of a finite group G. Show that the permutation representations of G corresponding to H and H’ are iso- morphic, as linear representations, if and only if each conjugacy class of G meets H and H’ in the same number of elements. Note that if H is a normal subgroup then this cannot happen unless H’ = H. However, there are examples of subgroups H and H’ satisfying the above condition which are not conjugate; check the following one, due to F. Gassmann (Math. Zeit., 25, 1926): Take for G the symmetric group on 6 letters (xi) and put

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EXERCISES 363

H = (11 (Xl X,)(X, X4), (Xl x,>w, X,)9 (Xl X,)(X, x3>> H ‘= (1, (Xl x,>w, X‘s), (Xl X,)(X, X,)9 (X3 &)(X5 X,>~

(H leaves X5 and X6 fixed, where H’ leaves nothing fixed; but all elements # 1 of H and H’ are conjugate in G.) Note that there exist Galois extensions of Q with the symmetric group on 6 letters as Galois group.

EXERCISE 6.4. Let L be a finite Galois extension of Q, let G = G(L/Q), and let E and E’ be subfields of L corresponding to the subgroups H and H’ of G respectively. Show that the following conditions are equivalent:

(a) H and H’ satisfy the equivalent conditions of Exercise 6.3. (b) The same primes p are ramified in E as in E’, and for the non-

ramified p the decomposition of p in E and E’ is the same, in the sense that the collection of degrees of the factors of p in E is identical with the collection of degrees of the factors of p in E’, or equivalently, in the sense that A/pA zz A’IpA’, where A and A’ denote the rings of integers in E and E’ respectively.

(c) The zeta-function of E and E’ are the same (including the factors at the ramified primes and at co.)

Moreover, if these conditions hold, then E and E’ have the same discriminant. If H and H’ are not conjugate in G, then E and E’ are not isomorphic. Hence, by Exercise 6.3, there exist non-isomorphic extensions of Q with the same decomposition laws and same zeta functions. However, such examples do not exist if one of the fields is Galois over Q.

Exercise 7: A Lemma on Admissible Maps

Let K be a global field, S a finite set of primes of K including the archimedean ones, H a finite abelian group, and cp: Is + H a homomorphism which is admissible in the sense of paragraph 3.7 of the Notes. We will consider “pairs” (L, a) consisting of a finite abelian extension L of K and an injective homomorphism a: G(L/K) + H.

EXERCISE 7.1. Show that there exists a pair (L, a) such that L/K is unrami- fied outside S and q(a) = a(F,,,(a)) for all a E Is, where FLIR is as in Section 3 of the Notes. (Use Proposition 4.1 and Theorem 5.1.)

EXERCISE 7.2. Show that if q(u) = 1 for all primes tr in a set of density 1 (e.g. for all but a finite number of the primes of degree 1 over Q), then cp is identically 1. (Use the Tschebotarov density theorem and Exercise 7.1.) Consequently, if two admissible maps of ideal groups into the same finite group coincide on a set of primes of density 1, they coincide wherever they are both defined.

EXERCISE 7.3. Suppose we are given a pair (L’, a’) such that a’(FrIK(u)) = q(u) for all o in a set of density 1. Show that (L’, a’) has

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364 EXERCISES

the same properties as the pair (L, a) constructed inExercise 7.1; in fact, show that if L' and L are contained in a common extension M, then L’ = L and a’ = a. (Clearly we may suppose M/K finite abelian. Let 8, resp. O’, be the canonical projection of G(it4IK) onto G(L/K), resp. G(L'/K). By Exercise 7.2 and Chapter VII, § 3.2 we have a O 0 O FMIEL = a' o 8' o FM,=. Since a and a' are injective, and FM,, surjective, we conclude Ker 8 = Ker 8’, hence L = L’, and finally a = a’.)

Exercise 8: Norms from Non-abelian Extensions

Let E/K be a global extension, not necessarily Galois, and let M be the maximal abelian subextension. Prove that NE,, C, = NM,,&,, and note that this result simplifies a bit the proof of the existence theorem, as remarked during the proof of the Lemma in Chapter VII, 3 12. [Let L be a Galois extension of K containing E, with group G, let H be the subgroup corre- sponding to E, and consider the following commutative diagram (cf. Chapter VII, 3 11.3):

8-2(H,Z)# Hub2;C,/N,,&, z! l?'(H,C,) car 1 *1 1 NE/K 1 car

fi -2(G, Z) # Gab ? CK/NLIRCL x fi’(G, Cll).

Since G“"/bJ(Ho"> z G(M/Q this gives the result.]

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Author Index

Numbers in italics indicate the pages on which the references are listed.

A Artin, E., 45, 56, 70, 82, 91, 143, 173,

176, 220, 225, 230, 272, 273, 277, 306,347,356

B Bore], A., 262, 263, 264 BoreviE, Z., I., 360 Bourbaki, N., 51, 67, 119, 127, 138,

143, 161 Brauer, R., 160, 225,230 Bruhat, F., 256, 264 Brumer, A., 248, 303, 303

C Cartan, H., 139, 308, 347 Cebotarev, see Tchebotarev Chevalley, C., 180, 205, 206, 230, 250,

252, 253, 264, 265, 275, 276, 277, 306,347

D Dedekind, R., 210,230 Delaunay, B., 267,277 Demazure, M., 253, 265 Deuring, M., 293, 296

E Eilenberg, S., 127, 139 Estermann, T., 213, 230

F Franz, W., 273, 279 Frobenius, G., 273, 277 Frohlich, A., 248, 249 Fueter, R., 267, 277 296, Furtwiingler, Ph., 270, 273, 277, 356

G Gaschiitz, W., 241 Gassmann, F., 362 Godement, R., 51,67,262,265,308,347 Golod, E., S., 232,241, 249 Grothendieck, A., 253,265

H Hall, M., Jr., 219, 230 Harder, G., 258, 265 Harish-Chandra, 262,263,264 Hasse, H., 204, 211, 230, 232, 249,

269, 271, 273, 274, 275, 278, 293, 296, 299,303, 351

Hecke, E., 205, 210, 211, 214, 218, 230,306,347

Herbrand, J., 275, 278 Hilbert, D., 267,268,269,270,278,351,

355,356 Hochschild, G., 197

I Iwahori, N., 256,265 Iwasawa, K., 249 Iyanaga, S., 273,278

K Kneser, M., 255,264,265 Koch, H., 249, 302, 303,304 Kronecker, L., 266,267,278

L Labute, J., 300, 304 Landau, E., 210, 213,214,230 Lang, S., 51, 168, 193, 214, 225, 230,

346 Lubin, J., 146

365

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366 AUTHOR INDEX

M Magnus, W., 273,278 Mahler, K., 68 Mars, J. M. G., 262,265 Matchett, M., 306, 347 Matsumoto, H., 256, 265 Minkowski, H., 357 Montgomery, D., 127

N Nakayama, T., 197, 199 Noether, E., 22

0 O’Meara, 0. T., 360 Ono, T., 261, 262,265

R Ramachandra, K., 296,296 Raynaud, M., 161 Rosen, M., 248 Rosenlicht, M., 250, 251;26.5

S SafareviE, I. R., 200, 232,

303,304, 360 Samuel, P., 4’ S&mid, H. d9, .?79 Schmidt, F (I> ?75, 279 Scholz, A., 249, 273 Schumann, H. G., 273,279 Serre, J.-P., 41, 87, 89, 90,

168, 187, 198, 249, 253, 265,297, 300, 304, 351

241, 249,

Shimura, G., 165, 168, 293,296 Siegel, C. L., 264, 265 Speiser, A., 225, 230, 267, 279 Stalin, J. V., 366 Steenrod, N., 127 Steinberg, R., 255, 265 Swan, R. S., 22, 161

T Takagi, T., 267, 270, 272,279 Taniyama, Y., 168, 293, 296 Tam, J.T., 51,70,143,146,173,176,199 Taussky, O., 249,273,279 Tchebotarev, N., 273,279,355, 363 Titchmarsh, E. C., 211,213, 230 Tits, J., 256, 264

V Vinberg, E. B., 241

W Wang, S., 176 Web&, H., 210, 230, 266, 267, 268,

279. 296 Weil,’ A., 51, 67, 70, 173, 200, 253,

258, 262, 264,265,296, 346, 347 Weiss, E., 86, 229, 230

Z 255, 257, Zariski, O., 41

Zippin, L., 127