Zeta-funct ions of Modula Diagonal Quotient SurfacesIntroduction A rnodular diagonal quotient...
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Zeta-funct ions of Modula Diagonal Quotient Surfaces by Satyagraha Mohit A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen's University Kingston, Ontario. Canada Decernber, 2001 copyright @ Satya Mohit. 2001
Transcript of Zeta-funct ions of Modula Diagonal Quotient SurfacesIntroduction A rnodular diagonal quotient...
Zeta-funct ions of Modula Diagonal Quotient Surfaces
by
Satyagraha Mohit
A thesis submitted to the Department of Mathematics and Statistics
in conformity with the requirements for
the degree of Doctor of Philosophy
Queen's University
Kingston, Ontario. Canada
Decernber, 2001
copyright @ Satya Mohit. 2001
Acquisitions and Acquisitions et BiMiographic Senrices services bibliographiques
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Abstract
This thesis develops an expression for the zeta-function of the modular diagonal quotient
surface &, of prime level p. in terms of its space of cusp forms.
In so doing, it presents an anaiysis of the space of cusp forms and (1, 1)-cohomology
spaces of the (disconnecteci, singular) surface Z, and t h a e of its reductions and shows that
each of these spaces can be embedded hinctoriaiiy in the correspondhg (1, 1)-cohomology
space of the (smooth, geometrically connecteci) product surface X(p) x Xb), where X(p)
is the principal modular cume of level p.
Using the resuits of Carlton about "Atkin-Leher theory for MDQS'sn, we associate to
each of the ( 1.1 )-cohornology spaces above an isomorphic " H d e shiRn . This technique
t hen enables the application of Eider-Shimura theory, with the aid of which we show that
the main factor of the zeta-function of Zp can be expresseci as a product of Rankin-Selberg
Lfunctions associated to a certain canonical basis of its space of cusp forms.
Acknowledgements
The completion of this thesis and of my graduate education wodd never have been
possible without the efforts of many people, at Queen's and in Montreal. Throughout the
course of my graduate studies and before, 1 have been fortunate to have encountered and
have had the opportunity to l e m from many dedicated, generous and uispired teachers
and mathematicians. 1 can only hope that my education wiU prove worthy of the efforts
that such people have expended.
Unfortunately. it is impossible to thank every specific person who has contributed to
my graduate education in some way. However. 1 have trieci to mention the people who 1
feel have made the most significant contributions which have aided me during the course of
my t hesis research.
Firstly, 1 would Lüce to t hank my advisor, Professor Ernst Kani, who has devoted a huge
amount of time anc. effort to teaching me and training me during the past three years. 1 have
greatly benefited from his vast knowledge of arithmetic geometry and number theory, as
weil his scrupulous and conscientious approach to mathematical research and education. In
particular. 1 would Like to thank him for giving two excellent specialized courses which helped
a great deai with my research (and t hat of othen). for carehilly reading and commenting
on everything I gave him. for always having t h e for my problems and for many stimulating
mathematical discussions. Finaily, 1 would like to thank him for his extreme patience with
the production of this thesis.
1 would like to thank Professor Ram Murty. who has been a teacher, mentor and close
fiiend to me for the past six years. In the time that 1 have been at Queen's, he has advanceci
a thriving number theory group and seminar series and given many wonderful courses on
diverse topics. Moreover, his ent husiasm for mat hematics and his attitude toward research
iii
have helped to inspire many students, induding myself. I would like to thank him for the
extreme generosity and kindness which he has always shown me and the innumerable things
he has taught me (both about mathematics and othenvise) in the past years.
1 would like to thank Eriends and coileagues Adam Van Tuyl, A h a Cojocaru, Srinath
Baba, Remus Floricel, Yu-ru Liu, Wentang Kuo for many helpful discussions as weil as for
t heir jo kes and camaraderie.
1 would iike to t hank Jennifer Read, whose cornpetence, kindness, helpfulness and sunny
disposition has made life easier for many of the inhabitants of the bmment of Jeffrey Hd.
I would like to thank m y family and friends for moral support and encouragement during
my long and lean years of graduate school.
Special thanks to my girbiend Joanna Meadows. who has waited patiently for me and
stood by me, supported and encouraged me throughout my doctoral studies. 1 never would
have made it without her.
I would iike to thank NSERC. Queen's University and the CRM for the generous han-
cial support that I have received for the past five years. without which my thesis research
would not have been possible.
S t atement of Originality
The parts of this thesis which constitute original research are the contents of Ch.1 54.
Ch.2 56, Ch. 3 $4, Ch.4 51.3.4.5. Ch. 5 53,5,7.
The main original contribution of this thesis is the expression for the zetahction of
the surface Zp that is contained in Theorems 2.24 and Coroilary 5.21.
Contents
Abstract
Acknow ledgements
S tatement of Originality
Chapter O. Introduction
Chapter 1. Modular Diagonal Quotient Surfaces
1. Introduction
2. Igusa's mode1 of X( N) over L[k]
3. The quotient of a quasi-projective scheme by a finite group
4. Models of modular diagonal quotient surfaces
Chapter 2. The Zeta-function of a Modular Diagonal Quotient Surface
1. The zeta-function of an arithmetic scheme
2. Reductions of the quotient of an arithmetic scheme
3. The étale cohomology spaces of a quotient variety
4. The Künneth formulaand the C(I,ll-function of aDiagonal quotient surface
5. Basic theorems of étale cohomology
6. The zeta-function of Z N , 8 Fq
Chapter 3. Moduiar Correspondences and the (1, 1)-cohomology of ZNTT
1. Correspondences on X x X and endomorphisms of the Jacobian of X
2. The Hodge decomposition for c u v e s and their Jacobians
3. The dgebra of moduiar correspondences and the Hl-cohomology of X ( N )
4. The (1.1)- singdar cohomology of ZNT7
2
Chapter 4. The space of cusp f o m on & and its (1. 1)-cohomology
1. Introduction
2. Atkin-Lehner theory on MDQS's
3. The (1. 1)-cohomology of & 4. The Hecke shift of a 'Tb)-module
5 . The Hecke shift of the (1. 1)-étaie cohomology of Z, 8 Fq
Chapter 5. Eider-Shimura theory and the L~l,l)-function of Z,
1. The Eichler-S himura congruence relation
2. The modified Weil pairing
3. Min-Selberg L-funct ions
4. The semi-simplicity of TQ.
5. Full submoduies and reIative characteristic polynomials
6. The Morita Theorem
7. The L(I,l)-funetion of Z, 8 Fq
Bibliography
Introduction
A rnodular diagonal quotient surface is defined as a certain quotient of the product
surface X(N) x X (N) by a "twisted diagonal subgroupn of its automorphism group, where
X(N) denotes the principal modular c w e of level N (we give a precise definition in 51
of Chapter 1). The study of such surfaces originates in the work of Hermann, [Hr], Kani-
Schanz [KSl], [KS2] and Kani-Rizzo [Kq.
The main reason why moduhr diagonal quotient surfaces are of interest to number the-
orists is that these surfaces have a moduiar interpretation which turns out to have powerful
diophanthe applications. In this preliminary chapter, we give a cursory idea as to the
nature of such applications. refening the interested reader to F'rey's article [Fr], and to
Süverman's book [Sil] for basic facts about eiliptic c w e s .
We denote by GQ the absolute Galois group ~al@/Q).
In a famous 1978 paper, Mazu [Ma] asked the following question
QUESTION OF MAZUR. Let N 2 7. Do there exist two non-isogenous elliptic curves
E, Er /Q such t hat t here is a GQisomorphiam of their N-torsion subgroups
Recall t hat
E[N] := { P E E(Q) : [NIP = O},
(where O is the point at infinity on the elliptic curve). We write E - E' to denote the
relation of isogeny.
REMARK. Mazu's original question actudy asks about the existence of a symplectzc
isomorphism between the 1'-torsion subgroups of non-isogenous eiliptic curves over Q. This
notion is dehed in terms of the Weil pir ing, which is a certain GQ-compatible pairing on
4
O. INTRODUCTION 5
E[N] x E [ q . Using this pairing, one can dehe the notion of the determinant of a GQ-
isomorphism y3 : E[N] 1 E1[N] (namely, $ is said to have determinant 7 E ( Z / N Z ) X if it
raises the Weil pairing to the rth power ) . A symplectic isomorphism is t hen d e h e d to be
a GQ isomorphism of determinant 1.
Subsequently, several authors have provided an afhmative answer to Maur's question.
In fact, infinitely many examples have been provided for N = 7.11 (by Kraus-Oesterlé.
Frey, Halberstadt-Kraus. Kani-Rizzo) , but one expects that, for large N, this phenornenon
is extremely rare.
In fact, Frey has made the foiiowing conjecture.
CONJECTURE (Frey, 1988). (t) If E/Q is a heà elliptic curve, the set
SE,^ := {Et /Q, Et + E such that E1[W 1 E[N]} = 0,
for all N sufficiently large.
Using the deep work of Wiles. Frey [fi] has shown that his Conjecture (t) is equivaient
to:
ASYMPTOTIC FERMAT CONJECTURE . Fix a, b.c E Z with abc # O. The famdy of
equat ions
d " ' b y n + c r n = O , n z 4
has only finitely many primitive integer solutions.
The Asymptotic Fermat Conjecture is a consequence of the famous ABC-Conjecture
of Masser and Oesterlé. Although Frey's Conjecture s e e m much weaker and more precise
than the ABC-Conjecture. it is generaily considered to be completely intractable by current
met hods.
More recently. Darmon has proposed the fouowing variants of F'rey's Conjecture.
CONJECTURES (Darmon. 1994). 1) (Strengthening of (t)) There is a constant M such
that the set
for ail primes N M .
2) (Weakening of 1) ) There is a constant M such that
for all primes N 2 M .
REMARKS. 1) In fact. Frey's and Darmon's Conjectures are formuiated for ail number
fields K . We have stated them above only the special case K = Q In part idar . the general
version of Darmon's Conjecture 2) asserts that the constant M is independent of the field
K .
2) For an eiiiptic c w e E/Q and a squarekee integer D, let ED denote its LIth quadratic
twist. Then given E [ N ] E f [ N ] then we also have ED[N] &, ED[N] for aii D and
thus, Conjecture 2) is a "useful weakening" of 1).
3) Kani further conjectures that Darmon's Conjecture 2) holds for the precise constant
M = 23 and gives a geometric interpretation of this conjecture in terms of rational points
in the f d y of moddar diagonal quotient surfaces (henceforth abbreviated to MDQS).
MDQS's are intimately connected to the above problems in the foiiowing way: there is
a (disconnected) MDQS SN which has an open afnne subset ZN such that
each Gp-isomorphism S, : E[N] E f [ N ] detemines, in a natuml way, a Qmtional
point PE,Er,iU E SN(()) and conuersely, every mtional point arises in this way.
More precisely, Kani-Rizzo [KRi] show t hat there is an "almost biject iver correspon-
dence
SN/{twists) s SN(Qnt,
where ZL(Qm denotes a set of suitably dehed "non-trivial pointsn. Thus, Conj. 2) of
Darmon is equivalent to the finiteness of the set
(for some constant M).
Now, the arena of Modular Diagonal Quotient Sudaces offers some powerful and con-
structive methods to the study of the Asymptotic Fermat Conjecture and other related
diophantine questions. For example, since such a surface is, in a precise way, a modular ob-
ject , t here are many explicit constructions which do not exist for ot her surfaces (or general
varie t ies ) . For exarnple. for these surfaces. it is possible tu explicitly construct their Néron-Severi
groups (and this has been done in certain cases by E. Kani) as weli as to construct expiicit
canonical bases of their spaces of cusp forms (that such a canonical basis ez+sts is proven in
this thesis).
The goal of my thesis is to develop an expression for the zeta-funetion of a modular
diagonal quotient surface of prime level (see Chapter 2 51 for a precise definition of zeta-
function).
A zeta-function is a certain (Gvalued) function which one can associate to an arithmetic
object and which is believed to encode a great ded of specific (arithmetic) information about
the object and. in addition, to have good analytic properties. The most familiar exarnple
of a zeta-function is the Riemann zeta-function, which is associated to the integers.
In the part icular case of MDQS's (because of t heir modular interpretation) , one expects
that the coefficients of the zeta-bction of a good reduction ZN @ Fq of ZN should yield a
formula for the number of solutions to the relevant moduli probiem (mod q). This gives a
particularly interest ing application of t his study.
Due to the well-developed and extensive Eramework of modularity, the zeta-function of
a modular curve is weli-undentood and, in fact, there is an "explicit" expression for the
zeta-funetion of any modular curve defmed over Q. Using standard techniques, it is easy to
extend this understanding to a product of two modular curves.
In this thesis, we use the framework of modularity to derive an expression for the zeta-
function of an MDQS (of prime level). While this expression is not made explicit, the main
result of the thesis has the immediate consequence that the zeta-tùnction of an MDQS has
meromorphic continuation.
O. INTRODUCTION 8
The main result of this thesis (Coroilary 5.21) is that the "main factor" (the L(I , l ) -
function) of the zeta-function of the MDQS of prime level Zp can be expresseci as a product
of Rankin-Selberg Gfunctions attached to a certain canonical basis of its space of cusp
forms. which in turn. c m be identified wit h a set of tensor products of H d e eigenforms in
the space of weight 2 cusp forms &(r(p)), up to Euler iactors at primes dividing p3 - p.
In order to prove this theorem. we show (Theorem 4.28) that, for a prime q { p3 - p.
the (1. 1)-étale cohomology space of the reduction (of the mode1 of &/Z[$] constructed
in Chapter 1) Z, 8 Fq can be identified with a certain subspace (Hedresubmodule) of
the (l,l)-étale cohomology of the (smooth, geometricdy connecteci) surface Y@),[;I @ Fq, where := X(P),IjI x X ( J I ) , ~ ~ ] and denotes Igusa's model of X@) over
P P P
q 1 . We then proceed (in Chapter 5) to compute the characteristic polynomial of the re-
striction of the Frobenius endomorphism of Y @)Z[;l @ Fq to the subspace above, using the
Eichler-S himura congruence relation (as the main ingrdient ) . Before presenting and proving the main resuit, it is necessary to develop a fairly large
amount of "background materiale. of which some is known (but not in the literature) and
some is new.
First. in Chapter 1. we construct a model of ZN over Z[k] (Theorern 1.14),
generalizing the mode1 of ZN over Q constructed in [KRi]. The remainder of the thesis is
then devoted to determining the zeta-functioo of the model ZNtk' in the case that N = p
is prime.
In Chapter 2. we reduce the determination of the zeta-fundion of Z, to the computation
of its L(I,Il-funftion (Theorem 2.24). that is. we define the L(I,I)-hinction and compute the
other -trivial factors" of the zeta-hinction of Sp.
In Chapter 3. we summarize some (known) results about the action of modular wm-
spondences on the HL-cohomology of the moduiar m e X(N) and proceed to use these
results in order to understand the (1, 1)-singular cohomology of ZN in terms of its space of
cusp forms (Theorem 3.32).
Finally, in Chaptea 4 and 5 (after explaining Carlton's results about Atkin-Lehuer
theory on MDQS's) . we state and prove the main theorem(s) (Theorem 4.3 and Coroliary
5.21).
CHAPTER 1
Modular Diagonal Quotient Surfaces
1. Introduction
Let N be a positive integer and let X ( N ) c = ï(N)\d8 be the principal modular curve
of level N . Here. 9' = 3 u {s} U Q is the "compactifiedn upper half-plane and
r ( N ) := ker (SLz(Z) + SL~(Z/NZ))
is the principal congruence subgroup of level N.
Recall that the space of holomorphic ditferentials C 2 x ( N ) , can naturdy be identined
with the space of weight 2 cusp forms on r ( N ) ,
(see for example [DDT], 91.2).
The natural map
m w c - X(Uc
is a Galois cover of c w e s with group
(see [La2], Ch. IV 52) and thus. G acts on X ( N ) C as a group of automorphisms. Thus, G
ais0 acts on RX(N)c
Let Y ( N ) c be the product surface
We have G x G Ç Aut Y (N)ê. We consider the *twisted diagonal subgroups"
1. INTRODUCTION
where Q, = ( O <:) E G L & Z / N Z ) / { f 1). for r E ( Z / N Z ) X .
For each 7 E ( Z I N Z ) X ! d e h e the modular diagonal quotient surface
and let
be the (disconnecteci) surface with the G,,'s as its connected components.
The curve X ( N ) C has a canonical mode1 X ( N ) Q mer Q with the property that its
space of holomorphic dinereotials RX(NIQ can be identifieci with the subspace of S2 ( I ' (N))
consist ing of forms wit h rat ional Fourier coefficients,
(see [DDT]. Theorem 1.33). The m e X ( N ) Q is cailed "Shimura's canonicai modeln of
X(N).
The morphism X ( N ) c - X ( l ) e descends to a morphism
on Shimura2s Qmodels but this cover is not Galois. Thus. G does not act on X(N),Q, but
oniy on
X ( N ) a C , ) := X ( N ) Q @ QCN 1.
Now let Y ( N ) q = X ( N ) Q x X ( N ) Q and put
It is weU-known that, if VI K is a quasi-projective variety over a field K and A E Aut V
is a finite group. then the quotient A\V exists as a quasi-projective Mnety (see for example,
[Mul], Theorem on p. 66 and Remark on p. 69). Moreover, if V / K is proper. then so is
A\V (Ex. II.4.4 in [Ha]).
Thus. for each AN,. we have a proper suIfaee
2. IGVSA'S SIODEI, OF .Y(hr) OVER Z&] 12
In fact, ZN,? has a model over Q Kani and Rizzo [KRi] construct such a model as a
quotient of a geometrically disconnected surface by a finite group.
THEOREM 1.1. (Kani-Rizzo) The surface ZN,? h a a rnodel ',, over Q and further,
there is a morphisrn
@Q : Y ( N ) Q + @., such that the aCN)-base change of @Q is the natuml quotient map
In particulut, Gal(Q(cN )/QI normalizes A N , in Autq Y (N& 1.
PROOF. See [KRi] Prop. 14 and Remark 15.
The goai of this chapter is to explain that (using "Igusa's modeP of X ( N ) o w Z[+]),
one can extend the Kani-Rizzo t!teorem to construct a model of the surface ZN,? over Z[+].
The goal of this thesis is to develop an expression for the zeta-function of the modular
diagonal quotient surface of prime level
where 2;:' is the rnodel which we construct in this chapter.
2. Igusa's model of X(N) over Z[+]
Of central importance in the theory of modular curves is the foiiowing hindamental
theorem. due to Igusa and Deligne-Rapoport.
THEOREM 1.2. The cume X(N) has a smooth proper model X(N)Z[k ,C ,vI over Z[k.cN], whose fibres are geometrically irreducible.
PROOF. By the work of Igusa. X ( N ) has a model over Z[+,cN] which has an open
affine subset which represents the functor classifying "elliptic curves with full level N-
structure" (see 52. of the Introduction of [DR]). By the work of DeiîgneRapoport, the
compaetification of this f i e open subset is a proper smooth cuve over Z[+.CN] with
geometridy irreducible fibres over Z[+.cN]. This is proven in [KMa], Cor. 10.9.2. O
2. IGUSX'S MODEL OF X(N) OVER Z[+l
.v ' as a scheme over Z[+], via the naturd morphism Now, consider X ( N ) , p cLvi
PROOF- wN)q+,S,vI is smooth o v e Spec (Z[k ,~~] ) , by Theorem 1.2, which is smooth
over Spec(%[+]). since Q(CN )/Q is unmmified outside N. Thus, X(N)Z(k,c,vl is smooth
over Z[+], since the composition of smooth morphisms is smooth (see [Ha], Prop. 111.10.1).
To see that X ( N ) = jlN ,C,vl/Z[+] is proper, note that
1) a composition of proper morphisms is proper ([Ha], Cor. 11.4.8) and
2) a finite morphism is proper ([Ha]. Exercise U.4.1).
Thus, by 2),
2.1. Minimal models of cunres
Let K be a number field and let R be a Dedekind domain with quotient field K.
Let X / K be an absolutely irreducible smooth projective curve.
By a weil-known result of Abhyankar, X has a pmper regular model over R, that is,
t here is a reguiar (relative) c w e X / R. which is proper over R such t hat X 8 K 5 X . This
tact is proven in Artin's article [ArZ].
DEFINITION. ([Lic]) Let X be a regular proper curve over R. A minimal model of X/R
is a pair (Y, x ) , where Y is a regular proper eurve over R,
is an R-morphism that is birational such that, for any birational map
(where Xr is a reguiar proper c w e over R)? the induced birationd map
2. IGUSA'S MODEL OF S ( N ) OVER Z[;b]
PROPOSITION 1.4. If a curve X/R has a minimal model (r, Y ) , then it is unique up
to unique isornorphzsrn. That is. if (6. Y') is another such model, then there as a unique
R-isomorphism q : Y % Y' sueh that 7 o K = n'.
PROOF. Since Y and Y' are each birational to X, they are birational to each other
and thus, there is a unique bimtional map r ) : Y - -, Y' such that q o n = r'. Since
Y' is a minimal model. the birational map 7 must be an R-morphism. Since Y is also
a minimal model, the inverse bimtional rnap rl-l is &O an R-morphism. Thus. r) is an
R- isomorp hism. O
PROPOSITION 1.5. Let X;'" be a minimal model X/R. Then any automorphism
PROOF. An automorphism
a E A u t K X @ K
defines a birational map a : XF'"- + XF'" and thus, by dennition of minimal model. an
R-morphism a : XP" + X?'". In addition, the inverse a-1 defmes a birational rnap and
so defines an R-endomorphism of ~ 2 " . Thus. a E AutR xRrnin. O
Thus, we have
A U ~ R xPn = AutK XR 8 K.
The following fundamentai theorem is due independently to Lichtenbaum and Shafare-
vich.
THEOREM 1.6. (Minimal Models Theorem) Let X / R be a pmper regular curue whose
generic fibre is a smwth projective absolutely i d u c i b l e cume of genus 2 1. Then X hos
a minimal model over R-
PROOF. See [Lic]. Theorem 4.4.
2. IGL'SA'S XIODEL OF X(N) OVER Z[+I 15
From now on. given a c w e X/R satisfying the hypotheses of Theorem 1.6, we denote
by A'Rrnin its minimal model over R.
REMARK. If. in the situation of Theorem 1.6, X/R is proper and smooth over R and
g ( X K ) 2 1, then X z xgin. Indeed. X is then reguiar and the canonical morphism
is given by a sequence of blowdowns of (-1)-rational cuves in the fibres of X / R (cf. [Ch].
Theorem 1.2). However. since the fibres of X/R are irreducible c w e s of genus >_ 1. no
such (-1)-curves exist on X. so n is an isomorphism.
Now. the curve X ( N ) has genus 2 1 for all N 2 6. (This is apparent fiom the usual
formula for the genus of X (N) for N 2 3. See for example [KSZ] Prop. 1 .l .)
Thus, in part icular, for any N >_ 6, we have t hat Igusa's model X (N)= ,C,v is isomor-
phic to X(N)",(,~].
COROLLARY 1.7. Let N be an integer 2 6 . Then the minimal model
as smooth and pmper over Z[$] and has geometricully imducibie fibres.
PROOF. For 1V 2 6, the c w e X(N) has genus 1 1. Thus, by Theorem 1.6 (and
Abhyankar's Theorem)! X(N)q has a minimal model X(N)Z[+l over Z[+]. Now since
Z[+,c,v]/Z[+] "tale, X(N)q+] 8 Z[+&J] is a minimal model of X(N)WC,v). By the
above remark and Theorem 1.2. we have that X(N)Z(,gl €3 Z[+,c,v] - X(N)zIiB,cNcNI Thus,
X(N)Z,hl 8 Z[+.cLv] is smooth over Z[+.C,V] and hence X(N)z(+l/Z[;%] is alw smooth (by
faithfully Bat base change). In addition. its fibres are geometrically irreducible, by Theorem
1.2. 0
2.2. The action of the group
Let XN = X ( N)Q @ Q(cN )' considered as a scheme mer Q. It is a fact that the funaion
field of XN is the field
FN := Q(fr : r)
3. THE QUOTIEXT OF A QUASI-PROJECTIVE SCHEME BY A FINITE GROUP 16
generated by the f icke functions f, for r E (z/Nz)*\ {O) (see [La2], Chapter 6, 92), which
has a natural jaithful action of the group
Thus, Ç c A u ~ Q X ~ .
Since XN = X ( N ) q @ Q ( C N ) . each o E Gal(Q(C,v)/Q) U s to an automorphisrn 1 @a E
Autq XN. Explicitly. if' O, : Ç cr C' (where C is a primitive Nth root of unity), then its lift
is 1 8 a, = ( 8 :) E Ç (cf. [KRi], p. 6-7. in particular, Remark 12). Thus. the subgroup
can be identified with the Galois group. Note that we have = HG, where
COROLLARY 1.8. Fiz an integer N 2 6 . The group Ç acts natumlIy on x ( N ) " , ~ ~ ~ ,
PROOF. Since x ( N ) $ , ~ ~ ~ is the minimal mode1 of XN/Q(CN) and G C Autac, X,
by Prop. 1.5. the elements of G naturdiy induce elements of AU^^^^,^^^ x ( N ) " , ~ ~ ~ .
Furthemore. via X (N)Z(+l @ Z[+.cx] = X ( Q [ + ,cNI, the group H is the iift of
to X(N)Z[h,C,Vl and thus. H A u ~ ~ ( I ~ Xw h'
We therefore have
G = GH A u t q , ~ ~ XN.
3. The quotient of a quasi-projective acheme by a fmite group
In this section, we explain that a stdicient condition for the existence of the quotient
of a scheme X over a base S by a finite group A Auts X is that X/S is quasi-pmjectzve.
3. THE QUOTIENT OF -4 QEASI-PROJECTIVE SCHEME BY A FINITE CROUP 17
DEFINITION. Let !ü be a category.
Suppose that X is an object in 'B and A is a finite group of automorphisms of X.
A categorical quotient of X by A is a pair (YJ), where Y is an object in b and
is a A-invariant morphism sat isfying the universai property that
for any 4-invariant morphism f : X + 2. there is a unique morphisrn g : Y + Z such
that f = g 0 r.
We wili use the following theorem.
THEOREM 1.9. Let A be a finite group acting on a scheme X svch that !or any point
P E X , the A-orbit of P is contatned in an afine open subset.
Then there is a pair (Y ,a) , where Y is a scheme and a : X + Y is a A-invariant
morphism such that
1) as a topological space. Y = A\X
2) the natuml homomorphisrn Ou + ( ~ ~ 0 ~ ) ~ is an isomophism
3) (Y? n) is a categoRcal quotient of X by A, hence is unique up to unique isornorphism.
PROOF. See [Mul], Theorem 1(A) on p. 11 1. O
Note that, if X is a scheme over a base S and A is a group of S-automorphisms, then
i f the quotient A\X exists. it has a naturai structure as an S-scheme.
Now, let X/S be a quasi-projective scheme over S (in the sense of [Ha], p. 77) and let
A ç Auts X. We wiU prove the following.
COROLLARY 1.10. The quotient A\X exists as an S-scheme. Moreouer, i f X is proper
over S , then so is A\Xo pmuided that the base S is noethen'an.
By Theorem 1.9. in order to show that the quotient of X by A exists. it sufnces to show
that each A-orbit A(P) in X is contained in an afüne open subset. For this we appiy the
foiiowing result bom EGA ([Gril).
4. MODELS OF MODULAR DIAGONAL QUOTIENT SURFACES 18
LEMMA 1.11. For any finite set of points 2 C X , the- is an tafine open subset O C X
containing 2.
PROOF. In this case, the definition of "quasi-projective" given above is equivaient to
t hat of Grothendieck in EGA (see [Ha], remark foiiowing the definition on p. 103). We can
therefore apply Cor. 4.5.4 of [Grl]. O
PROOF OF COROLLARY 1.10. By Lemma 1.11. a quasi-projective scheme X/S satisfies
the hypothesis of Theorem 1.9 and t hus, the quotient A\X exists. For the second assertion,
see [Ha], Exercise 4.4. O
COROLLARY 1.12. Suppose that X/S is separated. Then X is couered by open afine
A-stable sets and any finite set of points is contained in an open, afine A-stable set.
PROOF. Let P E X. Thcn by Lemma 1.11, A(P) 2 O for an atfine open subset O C X.
Nowo for any two open affine subsets O. 0, On O' is also &ne, since X/S is separated,
by Exercise 4.4 of [Ha].
Thus, nbEA06 is an affine open A-stabie set containing P.
To prove the second assertion, consider a finite set of points Pi, . . . , P,. By Lemma
1.11. t here is an affine open set O C X which contains each of the A-orbits of t hese points,
Then the set nseaOd is an afnne. open A-stable set containhg Pl, . . . , P,. O
REMARK. Note that, if one develops the theory "frorn scratch", Cor. 1.10 is not really
a "corollary" of Theorem 1.9, since one needs this fact in order to protie the Theorem. In
his book, Mumford proves these two facts simultaneously.
4. Mcdels of modular diagonal quotient surfaces
In this section. we extend the construction of Kani-Rizzo [KRi] to show that the surface
ZN,? has a model <!fi over Z [f 1.
4.1. The construction of the model of Z N , over a[+]
4. MODELS OF MODt'LAR DWGONrU, QUOTENT SURFACES
Let N 2 6 so that the genus of X(N) is 2 1. The minimal mode1 X ( N ) ~ ~ =: X, q & v l
is smooth and proper over Z[+] by Cor. 1.3. By Cor. 1.8. we have
Since the restriction map
is an isomorphism. the image of G~(Q((CN)/Q) in Ç is quai to that of AU^^^+^ Z[+,o].
which is the subgroup
Let a, be the elexnent of AutZlkl Z[k.(N] such that a, : CN c, ~ 5 . Then u, c, h, E 4.
Let Y (N)Z!+I be the product surface
and let
For each 7 E (Z /NE) '. let
be the fibre product of A!:''' with itself, taken with respect to the structure morphism
on the second factor.
PROPOSITION 1.13. a) Each Y:,:+' is imducible and thw, is notumlly an imdunble z [+ j
component of YN .
4. MODELS OF LIODL'LXR DIAGONAL QUOTIENT SURFACES
b) The stabilizer of Y:?' in E x Ç is
independent of y.
In fael. on element g = (gi,g2) E E x G permutes the cornponents of Y$;+' ocmrding to
the rule
where
d ) For euch 7, we have an isomorphism
PROOF. a) There is a naturd embedding
1
whidi is a smooth. projective surface over Z[+]. Thus, $,FI is a smooth and therefore
Y'[+' . the surface Bat. quasi-projective surface over 2[+]. The generic fibre of N,,
Y$, is irreducible. since it is geometricaüy irreducible over Q(CN ) i. e.,
is irteducible (cf. proof of Prop. 13 of [KW).
Now. any scheme V that is flat and quasi-projective over a Dedekind domain R whose
generic fibre V 8 K is irreducible h necessarily irreducible itself (it suftices to prove this for
4. MODELS OF hIODULAR DIAGONAL QUOTIENT SUWACES 2 1
affine varieties, for which it is obvious fiom the fact that if AIR is a 0at R-algebra, then
A L-) A @ K, so that if A @ K is a domain, then so is A). zr+i Thus, Y'[+' N,r is irredueible and thus, is an irreducible cornponent of Y , .
b) Let g = (g1,g2) E Ç x Ç. Then g E 6~,, if and only if g factors over the unique
morphism
J : K v , ~ YN
such that Pry,,T of = P~Y,., , .~* where pr, denotes projection onto the it"actor of these
products for i = 1.2. Equivalently, g E e N , if and only if
Thus. eNn = r. (cf. proof of Prop. 13 and Remark 15 of [KRi]).
C) Since Gd(()( jN ) /Q) acts on X ~ J via
it acts on Yhr via
In addition. the image of the automorphism u7 : ( ++ in Ç is the same, whether one
considers it as an element of
or of
WQ(h /QI AU~Q XN.
Thus. the result foilows koom [KRi].
d) By definition of $,$" as a base-change of Y(N)ZIkl, we have a finite 'H-equivariant
map
4. bf0DEI.S OF MODL'LAR DIAGONAL QUOTIENT SURFACES
and hence, by the universal property of quotients. an induced morphism
n : .w\Y:~\~~ -, Y ( N ) ,
The morphism rT is birationd (cf. [Km]), finite and hence an isomorphism, since both
schemes are cormal.
(thus. &, = A N n aS in 81).
Notice that ANn n 'F1 = (1) and that further, H n o m a h e s A N , in X A N - ~ , that is,
A u l t lA~,?.
NOTATION. Let
The inclusion of subgroups î i C AN, induces a morphism
The a h of this section is to prove the following theorem. which shows t hat is, in
fact. a mode1 for ZN,, /Z [+] . This generaüzes the construction of [KRi].
THEOREM 1.14. The Z[+,(s]/Z[+]-bose change <Pt of 3 is the quotient map
Before proving Theorem 1.14. we first develop some preparatory results about base
change.
4.2. Base change results
Let R be a Dedekind domain wit h quotient field K. Let V / R be a normal scheme which
is quasi-projective and flat over R. Suppose that K f / K is a Galois extension with group H
4. MODELS OF SIODULAR DNGONAL QUOTIENT SUFLF.4CES 23
and let R' be the integral closure of R in Kt. Suppose hvther that VRt := V @ R' is integral
and that
A AutR~ VRl
is a finite group such that H n A = {1) and H normalizes A in AH. We will prove the
foilow hg .
PROPOSITION 1.15. Let W := (AH)\VR~. Then W is a mode1 of A\V', over R. That
is, the Rt/R-base change of the natuml rnap
is the natuml quotient map VR1 + A\VRt.
En order to prove Prop. 1.15. we first establish the special case where V is affine, which
is the content of the following lemma.
LEMMA 1.16. Let R be a Dedekind domaàn and let A be an R-algebm that is an integrully
closed domain containing R. Let K. Kt. H be as above and suppose further that A' :=
A 8 Rf is a domain. Let A .4utRr A' be a finite gmup such that H normalizea A in A H .
Then the natuml rnap
1 AH ( A ) 8 R' +
as an isomorphism.
PROOF. We have (A') = A 8 (R') = A. since R' is integrally closed. Tbus, we have
a natural inclusion
since R' is R-torsion-free and thus. flat (because R is Dedekind),
Now, since A, R' are integrdy closed and At = A 8 R' is a domain, A' is integraily
closed (see [BoZ], Chapter V. 51.7). Thus. ( A ' ) ~ . and ( A ' ) ~ ~ are each integrdy closed
ring. as is ( A ' ) ~ ~ @ R'. so that
4. SIODELS OF SIODULAR DIAGONAL QUOTIENT SURFACES 24
is an integral extension of integrdy closed domains. To prove the result, it therefore sufEces
to establish that these domoins have the same quotient fields, i. e., that the result is true in
the case t hat R = K. R' = K' and A is a field.
By hypothesis then. K' and A are disjoint extensions of K. via their natural embeddings
in the field A Kt and we have the field inclusions
Since the h s t extension has degree # H. the second containment is an equality, which proves
the daim. U
Now, let : Spec(Rt) -. Spec(R) denote the basechange map, which induces (V-
iinearly) a basechange map D R t I R y : VRf IR< V . Since this rnap is H = A U ~ R R'-invariant,
it factors through a unique morphism H\VRt + V which is an isomorphism. since V/ R is
flat (in [KMa] Prop. A7.1.1. the authors prove this foc affine schemes which easily implies
the quasi-pro jective case).
The base change of the morphism
is a A-invariant map
w hich t herefore factors through a unique morphism
e : a\(v 8 El') 4 (AH\VRt) 8 R'.
We will show that 8 is an isomorphism. By Lemma 1.16, 8 is an isomorphism if V is
affine, nomai and flat over RI
Now. since VRt/R is quasi-projective. VRf has an afline open cover consisting of AH-
stable sets (0:) and for each such O:, the restriction
4. MODELS OF MODULAR DIAGONAL QL'OTTENT SURFACES
is an isomorphism.
Thus. it suffices to show t hat 0 is injective. Since any two (say A-inequident) points
of Vtr are containeci in an f i e . open AH-stable subset, this is clear from the previous
PROOF OF THEOREM 1.14. We check that the hypotheses of Prop. 1.15 are satisfied:
1) Y:?' is integal. since it is irreducible. by Prop. 1.13 a) and is smooth (and therefore
pot) over Z[+].
2) It is easy to see that AN,, ri 31 = ( 1} and that î f normahes A N , in ANV7H E 6N,.r.
3) Z[),(N]/Z[$] is an integral extension of Dedekind domains. O
In the next chapter. we discuss the reta-function of the mode1 and reduce its
determination to the computation of its "L( I,l)-function".
CHAPTER 2
The Zeta-function of a Modula. Diagonal Quotient Surface
1. The zeta-function of an arithmetic scheme
An arithmetic scheme is a scheme of finite type over Spec(Z). In this section. we recaii
the definition and basic properties of the zeta-function of an arithmetic scheme. We then
recaii t hat the zeta-function of a proper scheme over a finite field h a a factorization as a
product of L-functions associated to its étaie cohomoIogy spaces.
DEFINITION. (See [Sel]) Let X be an arithmetic scheme. The zeta-function of X is the
forma1 Dirichlet series
where x denotes the set
field at x.
The zeta-function of
of ciosed points of X and Nz := # K ( x ) is the order of the residue
any arit hmet ic scheme sat isfies the following basic properties.
PROPOSITION 2.1. We have
a) <(X. s) converges absolutely for Re(s ) > dim X .
b ) If X is the disjoint union of (ciosed) subschemes, X = U X i , then
c ) I f f : X + Y is a morphism, then
where X, = f is the fibre over y.
26
i . THE ZETA-Ft'SCTION OF Al ARITHMETIC SCHEME 27
d) C(X, s) = 3(X, (I-~), where 3 ( X , z) E Q( (2)) is the classical zeta-hinction of X/Fq
PROOF. See [Sel]. Theorem 1 for a), 31.5 for b),c) and 81.6 for d).
Thus. if X is a scheme over Z, then
since the closed points of SpecZ are exactly the non-zero prime ideais ( p ) for primes p.
EXAMPLES. I) Let X = Spec(Z). Then
is the Riemann zeta-function.
2) Let V = Spec(IFp[x]). Then
(see [Lo] Example VIIL5.2) and thus, C(K s) = (1 - &)-'.
3) Let X = Spec(Z(x1). Then the surjective morphism
gives the factorization
P
and X 8 IFp = Spec(Fp [z]). so
4) Let X = Ci. Then
and
1. THE ZETA-FUSCTlON OF AS ARITHMETIC SCEEME
(see [Lo] Example VIII.5.5). so ((ph, s) = ((s)((s - 1).
We now recall the factorization of the zeta-function of a proper variety over a finite
field. In this thesis. an integral variety is an integral scheme of h i t e type over a field and
a varietg is an integml uanety that is geometrically irreducible.
Let V/Fq be a sepamted variety. One can attach to V 63 pq a set of étale cohomology
spaces
% , ( V B F ~ , Q ) := l im&t(Va - F q , ~ / ~ n ~ ) ~ ~ , @ 1 # q n
as weH as étale cohomology spaces ~ i t h compact support"
H ~ ( V @ ~ ~ , Q ) : = ~ ~ ~ ~ ( V O ~ ~ , Z / ~ " Z ) @ ~ , Q C I # q n
(étaie cohomology groups with compact support are defined in [Mil], p. 93) which are
(possibly) non-zero only for O < i 5 2 dim V.
By definition of these spaces. we have the following.
LEMMA 2.2. ljV/Fq 2s a proper uoriety, then
Now. let ipq E EndV denote the qth power Frobenius morphism, which extends to a
morphism gq = rpq 8 1 E End( V 8 Fq ) . which in tum, induces a morphism
on each of the étale cohomology spaces of V 8 P',, as well as the compact support spaces
(since t hese spaces defme functors) . Since each of these spaces is finite-dimensionai, each
TKEOREM 2.3. Let V/Fq be a sepamted uatiety. Then
2. REDt'CTIOXS OF THE QUOTENT OF AK MUTHMETIC SCHEME
and thw, 3(V,z) 2s a national function of x.
PROOF. See [Mil], Theorem 13.1.
COROLLARK 2.4. Let V/Fq be n proper uanety. Then
In the case that V/Fq is smooth and projective, the assertion of Cor. 2.4 is the first
of the famous "Weil Conjectures' and is a well-known consequence of the Lelschetz Fued
point formula.
2. Reductions of the quotient of an arithmetic scheme
If X is a scheme of finite type over a ring of S-integers OKtS of a number field K, then
X is an arithmetic scheme. since s p e ~ ( O ~ , ~ ) is of finite type over Spec(Z) ( O K , ~ is a hitely
generated L-dgebra) . Let S = S p e ~ ( 0 ~ , ~ ) for fked K. S and let X/S be a qua3i-pmjective scheme of hite
type. Suppose that A Ç Auts X.
Then the quotient map X - A\X exists (by Cor. 1.10) and induces a morphism of
each of the fibres over S,
X 8 FQ - ( A i x ) 8 FQ,
for e E SP~C(OK.S) (Say Q # {O)).
The goal of this section is to prove the foiiowing extension of the results of [KMa].
PROPOSITION 2.5. Let Q be a prime of O , , ly2ng ouer a national prime q not diuidang
#A. The; the natuml rnap
PROOF. In [KMa], the authors prove this for a f i e schemes. See Prop. A7.1.3.
2. REDUCTIONS OF THE QUOTIEIVT OF AN ARITHMETIC SCHEME 30
By Cor. 1.8. .Y has a cover consisting of open, affine A-stable sets (Oi}. For each such
subset Di, the natural map
is an isomorphism. by the Katz-Mazur result. Thus, it suflices to show that 11 is injective.
Let Pl, Pz E X @ IFQ with Pl A(P2). the A-orbit of P2. There is an open &ne
A-stable set 0 8 FQ C 8 FQ containhg Pl,P2, by Cor.l.8. Then $(P) # $(Q), since i1>
restricts to an isomorphism on
Thus, Q is injective.
Application to MDQS's.
Proposition 2.5 shows that there is a natural isomorphism
Let Q be a prime of Z[+,c,v] king over a rational prime q not dividing N $ ( N ) $ ( N ) .
Then by the above. the FQ/Fq-base change of the Fq-morphism
is the natural quotient map.
Thus we have an isomorphism
that is Robenius-compatible.
In the next section. we show that
3. THE ÉTALE COHOMOLOGY SPACES OF X QUOTIENT VARfETY
(compatibly with Frobenius) for ail i 2 2.
3. The étaie cohomology spaces of a quotient Mviety
Let V/Fq be a smoot h projective variety and let A be a h i t e group acting (h -linearly)
on V 8 Fq that has finitely many fized points. Note that we do not assume that A acts as
a group of automorphisms of V.
In this section, we explain that if *an Fq-quotient of V by A" eùsts (in the sense defined
below). one can identify the (higher) étale cohomology spaces of the quotient ( A\V) 8 Fq with the A-invariant subspaces of the cohomology of V @ Fq ,
for i 3 2. compatibly with F'robenius in the sense that the isomorphism identifies the action
of the Frobenius endomorphism of this quotient A\V with the restriction of the Frobenius
endornorphism of V to the A-invariant subspace.
DEFINITION. Let V/P$ be a smooth, projective variety and let A C Autpq (V 8 Fq) be
a finite subgroup.
a) A quotient of V by A is a pair ( Z , K ) , where Z/Fq is a variety and n : V + Z is a
morphism such that the Fq-base change of a is the natural quotient map
b) An étale quotient of V by A is a pair (2. K) as in a), where the morphism n is étale.
REMARK. 1 ) A necessary and sufficient condition for the quotient morphism V @ Pq -. h\V 8 pq to be étale is that A has no fted points. (see [Sel], 52.1 or [Mul], Theorem A
on p. 66)
2) A sufficient condition for an IFq-quotient of V by A to exist is that GFq normalizes
A in AutFq V @ Fq, so that G F ~ cornmutes with the quotient morphism
3. THE ETALE COHOMOLOGY SPXCES OF A QUOTIENT VARIETY 32
Assuming this hypothesis, it is easy to see that a quotient A\V exists and is unique
up to unique isomorphism. For suppose that $1 : V -. WL and & : V - W2 are two such
quotients. Then the Fq-base change of each #* 8 1 is a '%me quotient" and thus. factors
though a unique isomorphism
by the universal property of quotients. Then there is a unique isomorphism a : Wl 3 W2
such that >12 = a 0 $1 or equivalently. a = $9 o
When a quotient A\ V is étole. the situation is much simpler. The following is a standard
consequence of Poincaré duality.
TBEOREM 2.6. Let V/Fq be a smooth pmjectaue uanety and let A C Auttq (V 8 Rq) be
a finite group.
Let r : V - W = A\V be an étale quotient. Then n induces a canonical isomorphism
that is Fro benius-compatible for al1 a .
PROOF. By hypothesis. there is a finite étale map ~r : V - W of constant degree
d := #A such that
is the naturai quotient map. Since n is a morphism of Fq-schemes, Q cornmutes with
geometric F'robenius.
Thus, by (the proof of) Lemma 1.12 of [Mil], there is a (push-forward) tmce map
such that
1) tr, oo: = [q = multiplication by d and
2) 0 tr, = x6EA 6'.
3. THE ETALE COHOhlOLOCY SPACES OF A QUOTIENT VARiETY
Thus, the pd-back map
is injective. since tr, om: is an isomorphism. and is &O surjective, since o: 0 tr, surjects
onto H;,(v 8 F q 7 ~ ) - \ . Moreover. since w commutes with Frobenius, w? commutes with the Robenius endo-
morphisrn on cohomology. LI
We now retum to the general situation, where 4 has finitely many fixed points in V@Fq.
By this hypothesis, the quotient A\(V 8 Fq) has finitely many singularities.
In this situation, we can still establish the assertion of Theorem 2.6 for the ich cohomol-
ogy spaces of the quotient A\ V 8 Fq for i 2 2. by applying Poincaré duality for cohomology
uith compct support to the open subvariety of A\(v@F~) obtained by removing the (finitely
many ) singularities.
The treatment in the remainder of this section is loosely based on some private notes
of E. Kani [K4].
THEOREM 2.7. Let i~ : V - W = A\V be on IFq-quotient of V b t ~ A. Then n induces o
canonical isomorphism
thut is Frobenius-compatible for each i 2 2.
We will prove Theorem 2.7 at the end of this section.
Recaii that , by Lemma 2.2. if VIFq is a pmper variety, then one cari identify its étale
cohomology spaces with its (étale) cohomology spaces uith compact support,
The foiIowing result asserts that. if the complement of the open set U C V is finite,
then the (higher) cohomology with compact support spaces of U 8 Fq are e p a l to those of
voQ.
3. THE ETALE COHOhlOLOCY SPACES OF A QUOTIENT VARIETY 34
PROPOSITION 2.8. Let VIFq be a projective variety, let Z be a jinite set of closed points
and let U = V\Z. Then we have
Proposition 2.8 follows kom the foilowing two lemmas.
LEMMA 2.9. Let V/Fq be a sepamted variety, let Z C X be a closed subscheme and let
U = V\Z. Then we have a long ezact sequence of cohomoiogg gmups
- - -+ g ( U . Z I N Z ) + If,(V. Z / N Z ) + IT,(Z, Z / N Z ) + C L ( u , Z / N Z ) + - .
PROOF. See [Mil], Remark 111.1.30.
LEMMA 2.10. Let C V be a finite set of closed points. Then
H&(Z. z / N Z ) = &(Z, Z I N Z ) = {O)
for all 2 2 1 .
PROOF. The first equality foiiows from the fact that Z/Fq is proper.
Since Z is a finite set of points. Z = U,(P,} , we have
The cohomology groups of a closed point P can be identified with the Galois cohomology
groups of the residue field K ( P ) ,
which are O for i 2 1 (see [FK] p. 14).
PROOF OF PROP. 2.8. By Lemrna 2.9. we have a long exact sequence
with G,(z. Z / N Z ) = {O} for i 2 1: by Lemma 2.10.
3. THE ETALE COHOSIOLOCY SPACES OF A QUOTEKT VMUETY
Thus, we have that
is surjectiue for a.ii n and
Hc(U. Z/lnZ) 1 He, ( X . Z / l n Z )
for al1 n, for i 2 2. Thus. we have
H ; ( U + Q ) - H ~ ~ ( x . Q ) and f f , (Li .Q) H;,(X.Q) for i 2 2.
cl Theorem 2.7 foilows from Prop. 2.8 and the following consequence of Poincaré duality
for cohomology with compact support (which is analogous to Theorem 2.6).
PROPOSITION 2.11. Let n : V + W be an Fq -quotient of V by A, where V is a projectiue
variety. Suppose that there is an open subset O C W such thot
is étale. Then ~r induces a canünical isomorphism
W' := (m 1). : ~ ~ , ( o ~ N F ~ , Q ) 1 r r , ( ~ ' e F ~ , ~ > ~
for each i that is compatible with Frobenius.
PROOF. W = A\V is proper over Fq (by Exercise 4.4 of [Ha]) and
n lo r : 0' + O
is finite (of constant degree d = #A) and étale and further,
is the natural quotient map. Thus, by [Mil]? p. 171, there is a trace map
for each à such that
1) tr, ow: = [dl and
2) & 0 tïm = xdEa 6: (P. 168 of [Mil]).
3. THE ÉTALE COHOMOLOGY SPXCES OF X QUOTIENT VARIETY 36
Thus, as in the proof of Theorem 2.6, it foilows that n: is injective and surjects onto
EP,(O', Q O
PROOF OF THEOREM 2.7 Let Z V be the set of fixed points of A, whïch is finite. by
hypothesis. Let O C W be an open subset which has finite complement such that
for i 2 2.
Thus, by Prop. 2.11, we have
for i 2 2. Since a is a morphism of Fq-schemes, (n @ 1 ) g cornmutes with the Robenius
endomorphism. O
Application to MDQS's.
In order to apply Theorem 2.7 to our situation, we must verify that the group
has finitely many fued points.
This is easily accomplished. using the following observation of Kani-Schanz (cf. [KSI],
Let a, denote the automorphism of G obtained by conjugating with the element Q, =
4. THE KÜNNETH FORhfüLX -4XD THE C(Iml,-FUIICTION OF A DIAGONAL QUOTIENT SURFACE37
PROPOSITION 2.12. Let y = (xi, x z ) E Y ( N ) @ &. Then the AN,,-stabilizer of y is
PROOF. Clearly, (9, a, (g)) (xi. 4 = (xi, zz) if and only if g E Gzt and a, (g) E G,, or
equivalent ly. g E a; ' (Gz2 ) . O
COROLLARY 2.13. The gmup AN,, C Aut (Y (N) @ Pq) has finitefy many f i ed points.
PROOF. A 6nite group acting on a eunie has fmitely many fked points (since a surjective
morphism of curves is u n r d e d outside 6nitely many points).
By Prop. 2.12. the number of 6xed points of AN,, is then less than or equal to the
square of #{ h e d points of G E Aut(X(N) @ F q ) } ,
which is finite.
Thus, we- can apply Theorem 2.7 to our situation to conclude the foilowing.
COROLLARY 2.14. We haue
compatibly with Frobeniw, for i >_ 2.
4. The Künneth formula and the C( l,l)-function of a Diagonal quotient surface
In the previous section. we showed that, if VIFq is a smooth projective surface and
4 C h t F q V 8 Fq is a finite group with finitely many Fuced points such that an Fq-quotient
W of V exists? then we have a canonical isomorphism
which commutes with Frobenius for each i 2 2.
We now specialize to the case where V is a product sudace,
4. THE KUNNETH FORMCLA AND THE C~L,I,-FUSCTION OF X DIAGONAL QUOTIENT SURFACE38
where X/Fq is a smooth projective geometrically irreducible curve.
In this case, the Robenius morphism (pq E End Y is the product
where t,bq E End X is the Frobenius morphism on X.
By Cor. 2.4. the zeta-function of Z/Fq has the factorization
where
By Theorem 2.7 we have. for ê 2 2, that
For our purposes, the "main factor" of Z is its &-factor, In this section, we show that,
by the Künneth formula for smooth product surfaces, one can express this C2-factor as a
product of a main factor (the C(l,l)-huietion) associated to the A-invariant subspace
and two 'trivial3 factors (the L(0,2) and t(2,0,-fmon) which each have degree at most one.
Let VI. V2 be smooth projective varieties over IFq. For each i, j, let
The following result is known as the Künneth formula.
THEOREM 2.15. For a positive integer r . the- is a functoriol isomorphism
zs a functorial isomorphism.
In particulor, if E Endpq V , 8 Fp, Q = 1,2, then tue have
5. BASIC THEOREMS OF ÉTALE COHOMOLOCY 39
PROOP. See Remark IV.8.24 of [Mil]. O
COROLLARY 2.16. Let Z be a quotient of a product surface Y = X x X/Fq by a finite
group A E Autpq (Y @ Fp). Then we have an isomorphism of Robenius-modules
which cornmutes with the action of the Frobenius endomorphism.
PROOF. The first isomorphism is given by Theorem 2.7. Since the Kiinneth decompe
eition çommutes with elements of EndEp (Y 8 a,) and we have that the group A and the
geometric Frobenius morphism are Fq -linear, we have the second isomorphism. U
NOTATION. For each i, j with i + j = 2. let
P ROOF. Since the isomorphism in Cor. 2.16 is Fkobenius-compatible. the (reciprocal)
characterifitic polynornial of Robenius on the H2-spze factors as the product of its char-
acteristic polynomials on each the (0,2), (1.1) and (2,O) spaces. O
Note that
H::**)(Y 8 IF~,Q) = HO(X B Q) ag H*(X O F ~ ) ,
which is 1-dimensional (since X is an absoiutely irreducible c w e ) . Thus, the factors
C(03) (2, x) and f (2,0) (2. x) each have degree at most 1.
5. Basic theorems of étale cohomology
In this section. we develop some basic theorems of étaie cohomology, which will be used
in the next section to compute the itriviai factors" of the zeta-hinction of Z N , 8 Fq .
5.1. The reductions of a smooth variety over a number field
Let X be a smooth projective Mnety over a number field K.
S. BASIC THEOREMS OF ÉTALE COHOMOLOGY 40
One c m associate to X/ K a set of étale cohomology spaces Hi, (X 8 K, Q ) on which
the Galois group GK = al( KIK) acts. as well as a set of singular cohomology spaces
0 @Q) (on which complex conjugation acts). These are zero uniess O 5 i 5 2d.
where d := dimX.
For each finite place v of K. let K u denote the completiùn of K at v , let O,, denote its
ring of integers. M , its maximal ideal and let IFv = O , / M , denote the reaidue field.
Let Ch. denote the set of finite places of K. There is an embedding K L, and thus.
an inclusion Du = GK, L, GK. which is well-defined up to conjugation in GK.
Moreover, there is a canonical map
- (wbere denotes the integral closure of O. in Ku) whose kemel IV is caiied the inertia
s u ~ r o u p of D,.
DEFINITION. Let u f CK. X is said to have good teduction ut v if there is a smooth
projective scheme X, /O, such t hat
If X has good reduction at u. t hen the reduction of the mode1 Xv,
is a smoot h projective variety over IFu.
Ln addition, note that, in the case of good reduction, the inertia subgroup I, C D, acts
trivially on
K , ( X @ K ~ Q )
(see [Seal, 92.3 Remark 3).
In general. there may be several non-isomorphic models Xv/Ou and reductions. How-
ever, the following standard consequence of the "smooth base change theorem" (see [Mil],
Theorem VI.4.1) shows that, if u is a prime of good reduction for X, then the étale coho-
mology spaces of the reduction X, 8 F',, are independent of the srnwth model Xu/Ov.
5. BASIC THEOREMS OF ÉTUE COHOMOLOGY 4 1
THEOREM 2.18. Let u E CK be a prime of goad A c t i o n for X . Then then is an
isomorphism (depending on K L, K.}
that is compatible with the Du-action.
PROOF. See [Ta]. p. 108 or [Mil]. Cor. VI.4.2. a
An easy consequence of the smooth base change t heorem is that if K K. is an inclusion
of olgebmically eloaed fieldp. then the étale cohomology spaces of X 8 K are the same as
those of X @ K ( s e [Mil]. Cor. VI.4.3). Thus, in particular, we have
and
% , ( X @ K , Q ) 2 P&WCQ).
Thus, Theorem 2.18 (combined with the above f a t ) proves the foliowing.
COROLLARY 2.19. Suppose that X/K 2s a smooth ptujective varietg and v is a prime
of good reduction. These is a canonicul isornorphism
As we explain below. there is a hinctorial isomo~hjsm between the étale and singular
cohomology spaces of a smooth projective variety over @
5.2. Cornparison behreen singular and étale cohomology
Let X/@ be a proper variety. Then any étale covering of X can be "refinedn by a
covering for the complex topology and, from this, one obtains canonicd maps
(X* A) = IT,(X* A) + Hh;,,,(X. A) = GiDg(x, A)
for =y abelian group A ( s e 5 1.6 of [Mi21 or [F'K]? Chapter 1 511).
Here the subscript c indicates cohomology with compact support. In either case, the
fact that X is pmper impiies by definition that the compact support space is the same as
the cohomology space in question.
THEOREM 2.20. For any Gnite abelian gmup A, the cananical map
qt (X* A) + q i n g ( x ? A)
i s an isomorphism.
PROOF. See [FK]. Theorem 1.6.
COROLLARY 2.21. There is a canonical isomorphism
Kt W? Q 1 qi& QI BQ Q *
PROOF. By Theorem 2.20, each canonical ma;,
g, ( X . z / ln%) + HQx, z / l n Z )
is an isomorphism.
Now, for singular cohomology, one has
by the Universai Coefficient Theorem (see for example pi], Theorem 3.6).
5.3. The Lo-function of a variety over Fq
Let X/Fq be a (geometricaiiy irreducible) proper variety and for each i , let
G(X/Fqyx) := det(l - x @ ~ I H / l t ( . Y @ ~ q , Q ) ) w
In this subsection. we establish the fouowing.
PROPOSITIO~~ 2.22. For any geornetrically irreducible vanety X/&, we have
Co(X/Fq, x) = l - 2.
Prop. 2.22 follows £rom the foilowing lemma.
LEMMA 2.23. Let X/Fq be a geornetrically connected scheme. Then
H:,.(x 8 IF~.z /Nz) 2 Z ~ N Z .
PROOF. See [FK], p. 21.
6. THE ZETA-FUNCTION OF ZN,, 8 Fq
PROOF OF PROP. 2.22 By Lemma 2.23, we have that each
Thus. taking inverse limits and tensoring with Q . we have H:(X @ Eq, Q ) 2 Q . Also note tbat the Frobenius endomorphism 4: acts triuially on this space (since )q
acts as the identity on the underlying topological space of X 8 lÏ$, it acts trivially on the
constant shed). w hence
6. The zeta-hinetion of ZNn 8 IFq
In this section. we compute the "trivial factorsn of the zeta-huiction of the surface
where q is a prime not dividzng #AN,? = #G (except for the f l-factor. which is only
computed conditionally for al1 primes except those in a finite exceptional set ).
We will prove the following, using the results of Kani & Schanz [KSI], P(S2) about the
singular cohomoiogy of the surface ZN,? @ @ and the cohomological facts developed thus
far (in 53,4,5).
THEOREM 2.24. a) For any prime q ( N $ ( N ) # ( N ) , we have & ( z ~ ~ , ~ , z ) = 1. Abo,
6. THE ZETA-Ft3CIlOK OF Zs,, 3 Fq
We wili prove Theorem 2.24 later in this section. In addition, we will show that Ci (& @
iFq , z ) = 1 for all but finitely many primes q, assuming a certain ( kasonable" ) hypot hesis.
For my Dirichlet series D ( s ) wit h an Euler product D(s) = np D&) and any f i t e set
of primes S. define the *away from S Dirichlet sene D(~)(S) = ripas D,,(S).
COROLLARY 2.25. Fu a prime p and assume that Li(& @ iFqo x) = 1 for ail q f P3 - p
and al2 7 E (ZlpZ) '. Let S = {qlp3 - p } . Then we have
and
dP3-P
and L(l,l) (Zp 8 IFq. q-') i s the reciprocal chamctefistic polynomial of 4; acting (diagondly)
In order to prove Theorem 2.24. we record the foiiowing results of Kani & Schanz.
THEOREM 2.26. a ) For each i,
bJ The Betli numbers b,(ZN,?) = dime: H&,(ZN,T, C) of ZN,? are:
and
4
6- THE ZETA-FUNCTIOK OF ZN,, @ Fq
PROOF. a) See Prop. 2.7 of [KSI].
b) By Prop. 2.8 of [KSl], we have
where := g(G\X(N)) . Here. ij = g(X(l)) = O, which proves the claim.
c) We have
(H::~)~J~-T 2 (Hsing @ &ag,, )A".L t
where HSingi7 is the G-module whkh is equol to Hsing as a ve~tor space and the G-action is
obtained by twisting tbat on Hsing by the automorphism aT.
NOW, G z AN.I = A G x G is the diagond subgroup. The character of IIsing @ HSing,?
is the pmduct hh,. where h, is the G-character of Hsing,T.
We then have
dim(Hsing O H S ~ ~ , ~ ) ~ N * . = ( h b , lG) = (h, &).
By Prop. 2.8 of [KSI], we have
and since h, is mal-ualued (see [KS 11, Remark 1.1 l), we have h, = &.
PROOF OF THEOREM 2.24a). For i >_ 2, we have
by Theorem 2.7.
By Cor. 2.19 combined with Cor. 2.21, we have
for i > 2.
We proceed to deduce the f-factors in descending order.
6- THE ZETA-FUNCTIOh' OF Z,v,, 8 F.
First, since b4 (ZN,?) = 1 by T h r e m 2.26 b), we have that
is a 1-dimensional subspace of a 1-dimensional space. the containment is an equality and
(For any smooth projective variety V/Fq of dimension d . H~~ (V @ Fq Y Q ) is 1-dimensional
and &(v/&. x) = 1 - qdx. by [Ha], top of p. 456.)
Since b3(ZNVt) = 0 by Theorem 2.26, we have L ~ ( & , ~ , x) = 1.
Now, by Cor. 2.16, we have
where
Set t ing
we have (by the Künneth f o d a for singulcrr cohomology) that
Since X(N) @ @ is connected, its Ho and ~ * - s ~ a c e s are 1-dimensional, so b(0,2)(ZN,î.) and
b(Z,O) (ZN,7) are each <_ 1. By Theorem 2.26. we have
By cohomologicaI cornparison and base change. we then have that
6. THE ZETA-FUNC'MON OF ZN,, 8 Fq 47
is 1-dimensional and thus, the above containment ia an equality (and the same gws for the
(%O)-~bspace 1- Thus,
- Finally, ne have by Prop. 2.22 of 55 that Lo(ZN,?, z) = 1, since Z N , ~ ~ is geometrically
irreducible (In Chapter 1 54, we showed that the surface cl' is geometricaliy irreducible
is geometrically Muc ib le over Z[#, ,&], ovet Z [h ,(NI, so t han the quotient surface N,7
so its mode1 over Z [k] is geometricaliy irreducible over Z [hl). b) Putting together the factors in a) yielde the assertion, by the nretorization of the
zeta-funtion. O. - We now give a wnditiond proof that Ll(Z~,,, x) = 1 for alma& al1 primes. The
foliowing diseussion is based on private notes of E. Kani [KSI. A
Let ZN,, denote the minimal desingularization of the suhce ZN,?. Then by defmition, n
there is a birational morphiem Z N , ~ + GtT.
REMARK. In faa, E. Kani h a suggested a proof of hypothesis (H), ming the the fact 1
that the singularities of &F1 @ Fq are "the same and of the same type as" those of &.
The remahder of thie section is devoteci to the proof of Prop. 2.27. Foilowhg PSI, we prove the assertion, using facts about birational morphisms of n o m d , intogml demes
and a further mult of Kaai & Schanz.
PROOF. The first assertion foiiows hom PSI] Prop. 3.1 combined with the cornparbon
theorem (Cor. 2.21). The second is a standard coneeguence of the smooth apeciaiiwrtion
Theorem (Theorem 2.16).
LE- 2.29. If f : X + Y is a bimtional m o r p h h of nomd, intagml achemes, then
for any constant aheuf M , the induced mop
is injective.
PROOF SKETCH. For a given normal integral scheme X with generic point gx : r) 4 X,
we have
g*Mq =MX
for any constant sheaf M (where (gx).M,, denotes the pwh-~omurd da, by [Mil], Ex-
ercise 3.7.
Thus, if f : X + Y is birational, then f 0 gx is the generic point of Y and thus,
by the above fact.
The Leray Spectral sequence (se [FK], p. 28) giws rise to a short exact sequenœ
(cf. [CE], p. 329).
Thus, since f,Mx = My, we have that
f : H'(Y, My) + H'(x, M x )
iB injective.
PROOF OF PROP. 2.27. By hypothesis (H), there is a birationai morphism - f : (ZN,~)Z @ I F q ' @ Fq
We have seen in Chapter 1 that z",fkl is a normal, integral scheme, with geometRcaUy
irredueible fibres over Z[+]. The m i W desinguiarization is also integral and normal A
(since smooth). For a prime q of good reduction, ZN,? @ Fp is smooth, thus n o r d
Thus, we can apply Lemma 2.29 to condude that
is injective. By Lemma 2.28, the right hand side is {O) for any prime q that is of good Cc,
reduction for ZN,?. Thus, for any such prime q, we have &(ZN @ &,z) = 1. O
Modular Correspondences and the (1,l)-cohomology of ZN,t
The goal of this chapter is to study the reiationship between the (1, 1)-cohomology of
ZN and its space of cusp forms.
In order to proceed towards this g d , we nret present some background material about
the action of correspondences on the Hl-cohomology of a eurve. We then give an exposition
of some more specific results about the action of the 'algebra of moduler correspondenees"
of X(N) x X(N) on H&(x(N), C) . In 54, we WU apply these resuits to prove a p r h theorem rehting the (1,l)-&uplar
cohomology of ZNI, to the apace of cusp forme on ZN.
1. Correspondences on X x X and endomorpbms of the Jacobian of X
Let X / K be a smooth projective curve over a field K such tbat X(K) # 0 and let Jx
be its Jacobian.
In this section, we define the dgebm of cowwpondencea on X x X and show that it is
canonically isomorphk to the endomorphism algebra of .lx.
We t hen define an action of this algebra on W1 (X) := HA (X O K, Q ) that is compatible
with the action of the endomorphism Wbra ~nda Jx on H 1 ( J x ) := H&(Jx @ K , Q ) via
the canonid isomorphiwi H1(X) = H1(Jx). We will define the p u p of correspondencea on X x X as a certain quotient of the
Néron-Severi gmup of X x X , whose definition we first r d .
Let Y/ K be a smooth projective surface.
DEFINXTION. Let Div(Y) be the fke abelian group generated by aii integral curveu
1. CORRESPONDENCES ON X x X AND ENDOMORPHXSMS OF THE JACOBIAN OF X 51
More generaiiy? for any variety V / K , let Div(Y) be the fiee abelian group generated by
ail integral subvarieties of d i m e m i o n 1 on V .
An element of Div(Y) is called a (Weil') diMsor on V.
Let Div.(Y) denote the subset of Divisors on Y which are dgebmically cpuiuuierat tu O
(in the se- of ml, Exercise V.1.7),
PROPOSITION 3.1. Div,(Y) is a subgroup of Div(Y) and thtu, H definea an epuivdmce
relation on Div(Y) (that is compatible wàth i b grmcp rtmctun).
PROOF. See [Ha], Exercise V.1.7.
The Néron-Severi group is the quotient NS(Y) = Div(Y)/ Diva(Y).
Now, let Y = X x X , where as before, X / K b a smooth projective curve. We make the
futther hypothesis that K is perfeet This ensures that the nonnakation c of every curve
C / K is mooth ([Hi], Exercise 111.5.8 asserts that the normalbation of C @ K ie smooth. --
Since K is perfect, c @ K is a no& cunie and thus, C @ K = c @ K by the universai
property of the notmaliaation given in [Ha] Ex. II.3.8, so c b ahPo smooth).
NOTATION. For a nvve C / K , let u : c -r C denote its (smooth) normalization.
Let D(Y) denote the subgroup of NS(Y) which is generated by the images of curves of
the fotm
X x { P ) , { Q } x K W E X ,
whete P, Q are close- poanb of X .
D E F ~ O N . Dehe the p u p of corre~pondence clas9es on Y = X x X to be the
quotient
C(Y) := NS(Y)/D(Y).
W, let C X x X be an absolutely irreducib1e curve.
1. CORRESPONDENCES ON X x X .%ND ENDOMORPHTSMS OF THE JACOBUN OF X 52
We have a pair of morphisms
where pri denotes projection onto the a* façtor.
One can associate to C a canonid homomomorphism
(and extend Zheady), where C.({P) x X) denotes the intersection O-cycle of the two
elements of Pic(X x X ) (as in b], 52.4) and (prz). denotes the push-forsrard map on
cycles (see [Fu], p. 11-14). Then Tc preserves the subgroup of Div(X) conkt h g of principal
divisors (see [KI] equation (1.1.9)) and thus, induces an element Tc E EndPic(X).
Let v : c + C be the (smooth) normalization of C and consider the morphisms
For each i , we have (functorial) pull-back and push-forward maps
S U C ~ that ( f i ) . 0 fi' = [deg f i ] .
Thw, the asmciation C c, Tc is compatible with &use change.
PROOB. See [KI], equation 1.1.8. for the ûrst assertion. The second then foiiow8 h m
the fact that the pull-back and push-forward maps on Pic(X) are fuactorial. O
NOW, for any surjective morphism f : X + Y of smooth projective curves, f m and
fa each preserve the subpup consisting of d e g m zen, divisors and also, the subgroup of
principal divisors (aee [Sil], Remark II.3.7).
L. CORRESPONDENCES ON X x X AND ENM)MORPEIISMS OF THE JACOBUN OF X 53
Thus, ne have induced homomorphisms
Since the Jacobian mpresents the pico-functor, f' is represented by a homomorphh
of abelian vanettes and, by the autodualiv of the Jacobian, so is f,,
Thus, Prop. 3.2 shows that Tc natwally induces an dement TC f End Jx.
REMARK. Note that we can 868ociate a homomorphism TB to an arbitmry curve B
(which is not necessarily absolutely irreducible). For supposing that B @ K = UiBi, the
homomorphism
Tg := Tg, E End Pic(X @ K) i
cornmutes with GK and therefore defines an element of End Pic(X).
NOTATION. Rom now on, given D E Div(X x X), dewte by TV the indu& element
End Jx. Thus, notice that if Tc H TC, then TC = O means only that TClpiP(X) = 0.
THEOREM 3.3. (CM elnuovo)
a) If D E Div.(X x X ) , then TD = 0.
6) If the image of C in NS(X x X ) is in D(X x X ) , then = 0.
c) In fact, the terne1 of
NS(X x X ) - End Jx
is ezoctiy the arbgroup of degenemte davisor closses (under aigebmic equavulence). Thw,
the- is a natud p u p isomorphism
PROOF. See [MU], Cor. 6.3.. Alternstely for a), see pi] p. 244.
1. CORRESPONDENCES ON X x X' AND ENDOMORPIfISMS OF THE JACOBIAN OF % 54
We cm use the isomorphism in Theo- 3.3 to define the composition of tro m m
spondence classes Ci, C2 E C(X x X ) to be the unique comapondence daas Ci 0 C2 for
w hich
TC^ o q = Tc, 0 Tc2 E End Jx.
Then, by definition, we have a nng Qomorphinn
C(X x X) 1 End Jx.
DEFINITION. Define the algebm of co~~esponde~xs on Y = X x X to be the Qalgebra
and simikrly, dehe the endomorphism ufgebm to be the Qalgebra
~ndO( J') := End Jx @z Q
1.1 The action of comrpondenceii on H1(X)
Let X/K be a smooth projective curve (where K is a perfect field) and let
W e first explain that there is a funaorial isomorphism
and then define the action of the ring C(X x X ) on H'(X) sa that it is compatible with
the action of End Jx on
H'(J*) := H ; ~ ( J K K ,Q)
via this isomorpbm. In this way, we a n invoke the theory of Jambians without ha* to
check any compatibilities.
A choice of a rational point Po E X(K) determines an inclusion
i, : X ( t ) ~f JX(L) =pic0(x@L)
P ct cl(P - Po)
1. CORRESPONDENCES ON X x X AND ENDOMORPEIISMS OF THE JACOBIAN OF X 55
for every extension LIK. This data deterrnines a (basechange compatible) embedding
i pb : X LS J'
PROPOSITION 3.4. The map i i defnies an isomorphism that is independent of Po.
Monover, it is finctoriol in the seme fhat if f : X + Y is a surjective m o t p h h of
smooth curves, then we have
where i x = iXVh : X ~f Jx and i y = iY,f(q) : Y v JY.
PROOF. See [Mi31 Cor. 9.6 for the h t assertion and Mo] p. 43 for the second. For
the last assertion, it suffices to show that f. 0 ix = iy 0 f. For t h , aee [K7], p. 48. O
COROLLARY 3.5. We have an isomosphiam
that is independent of Po.
PROOF. By the cornparison theorem (Cor. 2.21), we have an identification irg 8 1 =
iypb Thus,
and, in particuiar then, iv 8 1 is injective, ahich implies that iy is injective and thus
an isomorphism. O
REMARK. Of course, Corollary 3.5 can be proven in a much simpier and more natu-
r d way and was, in faet, h m for at least nRy years before the development of étde
cohomology theory.
1. CORRESPONDENCES ON X x X AND ENDOMORPMSMS OF TRE JACOBUN OF X 56
Let H i ( X ) = HA(X,Q) O+ H L , ( X , Q ) .
Ta emphasize the independence of the isomorphism i; of y we nRte
Now, End Jx acts naturaily on HL ( J ~ ) ,
End Jx + (End H' ( J ~ ) ) "
f * fd
and thus, we can use the isomorphisms
( as in Theorem 3.3) and
: H I ( & ) 2 H ~ ( x )
(Prop. 3.4 and Cor. 3.5) to dehe an action of C(X x X) on tll(x).
DEFINITION. Define the action of C(X x X) on H I (X) as foiiows: Let B E C(X x X)
and let B@ be the element
W e have thus defineci a ring homomorphism
such that the fo11owing diagram cornmuta
C(X x X) 1 End Jx
RERUARK. In faa, using Prop. 3.2, one can show that, if C / K is an irreducible cuwe
and fi = pri Ic o u and u : c -t C denotes its smooth wrmalieation, then 0 = f: 0
1. CORRESPONDENCES ON X x X AND ENDOMORPHISMS OF THE JACOBIAN OF X 57
For an abelian variety A, let %(A) denotes its 1-adic Tate-moddc Recall that the Tate
apo=
T P ( 4 := 9(4 82 , Q
is bctorialiy the dual of H:t (A @ K, Q ) , t hat is, there is a canonical isomorphiam
H ~ J A 8 K,Q) 1 e(~)" := H O ~ ( ~ ( A ) , Q )
such t hat , via this isomorphhm, the puii-bael h! of h E End A becornes the dud (hl) t , where
h' is the endomorphhm of (A) induced by h (see [MJ], Remark 11 -5). The foilowing is
a basic tact in the theory of abelian varieties.
PROPOSITION 3.6. Let A, B be abelian varietics over a field K. Then for any 1 # char K,
the nutuml map
Hom(A14 + Homz, (%(A), W B ) )
is injective. In fact,
is injective.
PROOF. See [Mi4], Theorem 12.5.
We ean use Prop. 3.6 to prove the analogous injectivity statement for C(X x X) acting
on H1 ( X ) .
COROLLARY 3.7. The map
is injective.
PROOF. By Prop. 3.6, we have End Jx @ Zi + Endz(Jx), ao that
(End J x ) @ Q r ~ n d c ( ~ x ) * (End Jx) 8 Q v (End~i(~~)~)'~,
since duality of -or spaces is a perfect duaiity. Thus, by the functorial Womorpbms
H ~ ( X @ K, q) 1 H;~(JX GD K, Q) 1 ~ ( J X ) ~
1. CORRESPONDENCES ON X x X' AND ENDOMORPHISMS OF THE JACOBIAN OF X 58
(see Prop. 3.4 for the Guet and [Mil] Remark 11.5 for the second), ne have
(End Jx) @ Q .-, ( ~ n d H& (x, Q )) OP.
Then by the commutative diagram (t) abow, we have
Let X/C be a (smooth projective) c m . Recall from Chapter 2 (Cor. 2.16) that t h e
is a cornparison isomorphhm
which is fuactorial with respect to motphisms.
COROLLARY 3.8. The isomorphism
commutes with the action of C(X x X ) and is, thw, an isomorphh of C ( X x X) 8 Q - modules.
PROOF. By the commutative diagram (t), it suffices to show that
H ~ J J X ~ Q ) @ Q 1 H&(JX,Q)
as modules over End J x , which is true by the usual cornparison theomm. O
We now deduce the d o g u e of Cor. 3.7 for singular cohomology, wing Cor. 3.8, noting
t hat this assertion is actually much easier to prove directly.
COROLLARY 3.9. The n a t u d map
C(X x X) @ Q + ( ~ n d @))OP
is injective.
1. CORRESPONDENCES ON X x X AND ENDOMORPHISMS OF TEfE JACOBIAN OF X 59
PROOF. By Cor. 3.8, thia foiions from the fact that
which follows hom Cor. 3.7.
1.2. Reductions of Correspondences
Let X / K be a smooth projective curve over a number ficld K with X(K) # 0 and let
Jx be its Jacobian.
If C Ç X x X is a curve, then a direct discussion of what coustitutes a "reductionn of C
(with respeet to compatible mod& of X and C over OK) ia somewhat tridry. Fortmately,
for our purposes, one can circumvent this diffiiculty through the use of Némn modeb of
Jacobians.
Let R be a Dedekind domain wïth quotient field K and Let AIK be an abeiian varieW.
D E F ~ T I O N . ([Arl], 51) A Nkon mudel for A over R is a smooth group scheme A R / R
whoae generic fibre is iaomorphic to A and whid, satisfies the folloaring u n i d property:
For any smwth group rcheme X/R and mtiond map OR : X- + AR, q5 cztends
uniquely fo a m o r p h h X + AR
The universal property above characterizes Néron modela uniquely.
TBEOREM 3.10. (NQon) The h n mode1 AR/R ezista and U of finite tlvp wcr R.
PROOF. See [Arl], Theorem 1.2.
PROPOSITION 3.11. Any endomorphiam a E E ~ ~ K A eztends to an endomorphiam in
E R ~ R A R . Thus, we have an isomorphiam
End A 1 End AR.
PROOF. By definition of the Néron model, we have a natutal injective map End A L,
End AR. This map is also surjective. since any e n d o m o r p b of AR d c t s to an ande
morphiwi of each of the fibres. O
1. CORRESPONDENCES OEi .Y x S AND ENDOMORPHISMS OF THE JACOBIAN OF X 60
Now, for any prime P of good reduction for AR, the reduction AR @ Fp is an abelian
variety over the residue field Fp and, by Prop. 3.1 1, my element u E End A extends to a
morphism AR -t AR. Since t his induces a morphism of each of the fibres, in particular. if
AR has good reduction at P. then the fibre of a at P gives an endomorphism
We therefore have a ring homomorphism
End A + ~ n d A.
PROPOSITION 3.12. Let À = AR 8 Fp, where P is a prime of good neduetion for AR.
a) The naturd map EndA + End ts injective.
b ) For any rationaf prime 1 of good reduction for .-AR, P f f , thete as a natuna1 isomor-
phisrn A[Ln] 1 A[ln] for of1 n and thw, x(A) 2 z(A).
PROOF. a) See [La4], Theorem 3.2.
b) By[SeT], Lemma 2, the reduction map defines an isomorphism 4 (A)'P 1 q ( A ) , where I p C GK, denotes the ineaia subgroup. For a prime P of good reduction, Ip acts
trividiy on G(A), so this proves the claim. CJ
The goal of this subsection is to deduce the analogue of Prop. 3.12 for the ring of cor-
respondences. In order to do this, we wiU use the foiiowing result about relative Jacobians.
THEOREM 3.13. Let X / R be a smwth cume. Then there is an obelian scheme f i I R / R
(with connected fibres) and a morphism offunctors
(where hzx/ , is the tùnctor of points of Jx and is the pico-funcior m an [Mi3],
58) such that, for any scheme T offinite type over X , the homomorphism
is injective and i s an isomorphàsm whenever X x~ T has a section.
1. CORRESPONDENCES ON S x .Y AND ENDOMORPHISMS OF THE JACOBWN OF X 6 1
PROOF. Çee Theorem 8.1 of [Mis] for the existence of JXlR as a group scheme (under
more general hypotheses) and p. 193 for the fact that is an abelian scheme when
X/R is smooth. I7
The abelian scheme ZXlR is c d e d the relative Jacobian of X / R . - COROLLARY 3.14. If P zs a prime of good reduction for X / R , then JXlR JX,K(pr
In addition, we have that JYlR 63 K I: Jsos.
PROOF. See [Mi3], p. 193, where the author shows that the relative Jacobian cornmutes
with base change. O
The following is a basic fact about Néron models.
PROPOSITION 3.15. Let AIR be an abelian scheme. The A is the Némn model of its
generic fibre.
PROOF. See [Arl], Cor. 1.4. O
COROLLARY 3.16. Let X/R be a srnooth curue. Then the nlatiue Jacobtan J x / R is the
Néron model of the Jacobian of the generic fibre X 8 K.
PROOF. By Prop. 3.15, Jx is the Néron model of Jx @ K, which is the Jacobian of
X @ K, by Cor. 3.14.
Now, let X/K be a smooth projective m e over a number field K and let S denote
the set of primes of bad reduction for X. Let R = OK,S and let xmin = XEin denote the
minimal model of X over R. Then xrnin/~ is smooth and thus,
by Cor. 3.14. Aiso. by Cor. 3.16. for any P E Spec(R), we have - &"n/R = J ~ n , F p -
Via t hese identifications and the injection
End Jx ' E I L ~ ( J ~ ) ~ 8 Fp
1. CORRESPONDENCES 01'; .Y x .Y AND ENDOhfORPHISMS OF THE JACOBWN OF X 62
of Prop. 3.12 a), we have an injection
px ,p : End Jx L, End
CORCLLARY 3.17. Hypotheses as above, for any P E Spec(R), there is a natuml ring
injection
(where- denotes reduction (mod P ) ) that L9 compatible with reductiow of endomorphtsm
in End J,\-.
PROOF. First ly, We have a ring homomorphism
Also. we have
and
xmin - - End J- C(Xmin Xmin
s* in / F,, 1 -
Thus, we have a "natural" injective map
9-& *.Y - X ' " -2
C(X x X) End JA = E n d ( J x ) ~ 9 End (Jx)R = End J- Xmin/F,
+ C(Xm'n )
that is compatible wit h reductions of endomorphisms. U
COROLLARY 3.18. The isomorphism in Cor. 2.19
i s an isomorphism of C ( X x X ) 8 Q -modules, where the right hand stde is wwidered to
be a C(X x X ) -module via the injection in Cor. 3.1 7.
1. CORRESPONDEXCES OiV S x -Y AXD ENDOMORPHISMS OF THE JACOBlAN OF X 63
PROOF. We have an isomorphism
over (End J x ) 8 Q * (End J: 8 Fp ) @ Q (by Cor. 2.19 and Prop. 3.12 a)).
Thus,
over C(X x X) 8 Q. O
1.5. The Weil pairing and the Rosati involution
Let X/R be a smooth c w e , where R = OK,S îs a ring of S-integers (for S a finite set
of primes of OK). Then X has a (relative) Jacobian J x / R which is the Néron mode1 (over
R) of the Jacobian of X 8 K (by Cor. 3.16).
Now, any relative Jacobian &/R has a canonical pfincàpal polatization (see [Mua], p.
118)
Restricting X to any fibre of over R which is an abelian variety gives the principal
poiarizat ion
The polarization A induces a non-degenerate, bilinear Weil pairhg of Galois modules
- where pin denotes the group of fnth roots of unity in the algebraic closure ~ ( z ) (see [Mid],
By Prop. 3.12, there is a naturd ismorphism
Since the canonid polarization is basechange invariant, we have the following huther fact,
which will prove useW in Chapter 5.
LEMMA 3.19. The WeiI pairing is compatible with r p , thut is, if P,Q E z ( J x @ K ) ,
then
p' ( r p ( P ) t Ma) = e : ~ ~ ( p ? QI-
PROOF. We have etSK(p. Q) = ec(P, XQ). where el : q ( J x ) x q(&) denotes the usual
WeiI pairing (as [Mi3], p. 132). The pairing el is compatible with rp
(t his follows hom the proof of Prop. 2.4 on p. 338 of [Gr2], which is much more general) . Thus, we have
as claimed.
Now. the polarization A, also induces an involution * R on ~ n d ' (JXl 8 K (z)) cded
the Rosati involution,
where ii denotes the dual of a.
Recall the following basic facts.
LEMMA 3.20. Let X/K be a smooth projective curue and let a E EndJxlr< Then the
Rosati conjugate cr'R is the adjoint of a with respect to the Weil pairing on (JxIK) .
PROOF. See [Mi4]? p. 137 or [Mulj, p. 189. O
For an irreducible curve B C X x X denote by Bt the tmwpose of B. That is, if
w E End(X x X) is the morphism which interchanges the factors of X x X, t hen Bt := w(B).
The taking of transposes defines an involution of C(X x X ) .
LEMMA 3.21. The zsomorphism iIs : C(X x X) 1 End Jx s a t w e s
2. THE HODGE DECObIPOSITiON FOR CURVES rLYD THEiR JACOBIANS
where C denotes the transpose of the comspondence C .
To prove Lemma 3.21, we extend the definition of the Rosati conjugate as follows.
Let X, Y be smooth projective curves over a field K and let Xx, X y denote the canonical
polarizations of t heir Jacobians. For f E Hom( J,s, Jy ), define
Then one has, for f E Hom(Jdy, Jv),g E Hom(&, Jz) that (g 0 f ) ' R = f m R 0 goR.
PROOF OF LEMMA 3.21. We have Tc = (h). 0 ( fl)*. by Prop. 3.2. Thus.
LEMMA 3.22. Let X/@ be u smwth pmjectiue curve. Then the Rosati conjugate a ' R as
the adjoint of a with respect to the Petersson znner product on RJx = Rx.
PROOF. Eichler shows that C is adjoint to C with respect to the Petersson product on
Rx (see [Ei], p. 274). Since the isomorphism RJx I, OR takes the action of End Jx to the
action of C(X x X) by definition. we then have that *(CL) is Petersson-adjoint to B(C).
Since 9(Ct) = 8(C)*R by Lemma 3.21. the c l a h is established.
2. The Hodge decomposition for c w e s and their Jacobians
Let X/@ be a smooth projective cuve and let Jx be its Jacobian. Let Rx, fix, ClJx, RI, denote the spaces of holomorphic and anti-hornoiornorphic 1-forms on X , Jx,
respectively as in [Fol, 519.1 (and [LB], p. 17).
R e d ([Fol Theorem 19.4) that t here is a canonical Hodge decomposition
which is funetorid with respect to the pull-badrs and push-forwards of holomorphic maps.
The goal of this section is to deduce the stmnger statement t hat, in faet, the Hodge
decomposition is an isomorphism of C(X x X)-modules (with respect to the action of
2. THE HODGE DECOMPOSITION FOR CURVES AND THEIR JACOBIANS 66
C(X x X) on Rx and fix defineci below). We deduce this fiom the Hodge decomposition
for H1(&,@).
Fix a point Po and consider the embedding
We have a pull-badc map on dtflerenttable 1 - j o m
and, since ipo is holomorphic. iPpo presewes 1-forma *of (1,O) and (O, 1)-type" and the
subspaces of these consisting of holomorphic and anti-holomorphic 1-forms. respectively,
and thus, we have maps
PROPOSITION 3.23. a) We have a bnctoriai Hodge dewmposition
H ~ ~ ( J x , @ RJ, $ 0 ~ ~ .
b ) iahoio is an isomorphism and thw,
ts an isomorphism of Hodge structures.
PROOF. See [LB] 54 (Theorem 4.1 and its proof, in particular, p. 17). Altemately for
b). see [Mi31, Prop. 2.2. O
is an isomorphism of End Jx-modules.
As before, defme an action of C(X x X) on RA. and fix, by setting
2. THE HODCE DECOMPOSITION FOR CURVES AND THEXR JACOBIANS
for B E C(X x X), where B : C(X x X) End&, so that the diagram
C ( X x X ) i, End&
1 4 (End Qx)OP 1 (End RJ,)OP
is commutative.
We therefore have:
COROLLARY 3.24. The Hodge decomposition
REMARK. One can also prove Cor. 3.24 directly using the fact that, for an integral c w e
C, we have Cs = fi 0 ( f i )$ , where f, = pr, Ic 0 v and v : c 4 C is the smooth normalization
of C as in 51.1.
From now on. we make the identification
There is a canooical sesquilinear pairing on Hi (X. C) defined by
This pairing restricts to a Hermitian pairhg on Rx x Rx and induces a bilineor pairing
on ft,~ x Rx,
3. THE .UCEBRA OF MODCLAR CORRESPONDENCES AND THE Hl-COHOMOLOGY OF X ( N ) 68
3. The algebra of modular correspondences and the Hl-cohomology of X(N)
In this section, we show that the space V = of anti-holomorphic dinerentials on
X ( N ) is isomorphic to the contragredient module V' of V = RX(N) with respect to the
modula algebra' . We t hen show t hat t his contragredient module is isomorphic to the
module obtained by conjugating with the element Q-i := ( iL y ) . This is a result which is
due to E. Kani (to the author's knowledge) and the treatment herein is boseiy based on
his private notes [K6] on this subject.
In 54. we wili use this result in order to undentand the (1. 1)-cohomology of ZN,,,
in t e m of the space of cusp forms on SN.
3.1. Modular conespondences
Following Shimura, let R(I'(N), A) be the fiee abelian group generated by the double
cosets
r ( N ) w N )
of r ( N ) in
A = (a E M2(Z)Ideta >O}.
Shimura defines a natural way to multiply two such cosets, which makes R(I'(N), A) into
an algebra (see [Shl], p. 54). We write M = R ( î ( N ) , A) to denote this dgebm.
Recall that the usual nth Hecke correspondence T, is dehed as
where the sum is taken over all double cosets of î ( N ) in
(as in [Shl], p. 60).
One then has for a prime q that
3. THE ALGEBRA OF SIODL'LAR CORRESPOSDEKCES .4ND THE HL-COHOMOLOGY OF X ( N ) 69
(see equation 3.3.9 of [Shl]) . One can then define the Hecke algebra
and the Hecke algebra "away fiom the level" t (N) = QT, : gcd(n, N) = 11 C T.
Now. for any a E SL2(L). we have
since ï ( N ) is a normal subgroup. Thus. by [Shl] Prop. 3.7. for all a,@ E S L 2 ( Z ) ,
in the ring R ( ~ ( N ) . A ) .
For an element a E SL2 (Z), identifjr the coset
with the image of a in the goup G' := SL2(Z/NZ). In this way, we have QG'] C M as a
subring.
REMARK. Let M = MQ = (T. G) C M. Then. in fact, one can show that the images
of M and Mi in End V are the same. but we will not use this fact here. See the forthcoming
paper [K2].
Now, r e c d t hat the elements of M induce divisors on X ( N ) x X(N) and thus, algebmic
correspondences,
MI + := C(X(wqG) X(N)q(b)),
and the map M + C is a ring homornorphism (see [Shl], 87.2 and p. 76).
Thus. M acts (contravariantly) on each of
3. THE ALGEBFU OF 5IODUL4R CORRESPONDENCES AND THE Hl-COHOMOLOGY OF X ( N ) 70
As weU, for any prime Q of Q ( C N ) lying over a rational prime q f N, the algebra
acts on
and t hus,
so the Robenius rnorphism ipq acts on the latter.
NOTATION. Let H be a right Mmodule, that is. we have a ring homomorphism
M + (End H)*Y
We use the notation rn: for the image of an element m E M via the above representation
and the notation h@ for the image of the algebra M Alternately, we use the "slash notation" :
In the cases of the Mmodules under consideration above, we use the more specific
notation tet, tfsing, tfho~o. danti when necessary.
Now, we have (a pRon) two actions of the algebra (T, G) on RX(Nl?
(where J ( N ) denotes the Jacobian of X ( N ) ê ) and the clapsical action of M on the spoce of
c w p fowns of wezght 2
Note the above isomorphism is an isomorphism of Mmodules, that isl for any rn E M., we
have mlCmP = mZholo via this map.
Thus, in order to apply results from the (classical) theory of moduiar forms, we must
show that these two actions are the same.
Of course, this is a weil-known fact.
THEOREM 3.25. a) Let tn = $(T,) E EndJ(N) and let T:'~ E End V denote the
classical nth ~ e c k e operator on V . Then the representation
,O : End J ( N ) + ~ n d s2 ( r ( N ) )
Q c, 2 0 ai o (ig)-'
satisj?es p(Q ) = T F - ~ . b ) Furthemore, if g E G C M then p( * (g ) ) = gscmP, where g D C q E End V denotes the
classical operator.
PROOF. a) See [Sh2], proof of Prop. 9 on p. 534 (in particular, the comment following
the commutative diagram) or [Shl], p. 170-171.
b) By Prop. 3.4.
and. by the functoriaiity of the Hodge decomposition, this implies that
9 ~ ~ I O I O = i j h ~ ~ ~ O (* (g).)'ho" 0 (i'hO1O)-L = p ( ~ (g)) .
Then since gzholo = gZCUSP. the claim is proven.
REMARK. Let
I)J1' =@[mg : m € 4 C EndV.
There is a canonical map
but. in general, it is not injective. Thus, V is not in general a faithful M 8 Grnodule. See
[Shi]! p. 85.
3. THE rUGEBRr\ OF MODCWR CORRESPONDENCES AND THE HL-COHOMOLOGY OF X ( N ) ï2
Kowever, we remark by contrast that, if
(where X(N)q denotes Shimura's canonical model over Q) , then we have an injection
CQ v End VQ
and thus. Cq@@ - En& V (see [K3I1 Example 4.5b), where it is shown t hat End J ( N ) P - End VQ) .
In part icdar , t his implies t hat
since the Hecke correspondences T, are dehed on Shimura's model X(IV)Q, x X(N)P (see
Remark 7.7 of [Shl].)
3.2 The Hodge decomposition and contragredient modules
For a E A. dehe the tmnspose of a double coset [a] = r ( N ) d ( N ) E M b y
[a]' := ~ ( N ) & ( N ) .
Then set
[a]' := [ u I ~ ' w - ' 1 = r ( i V ) ( w a t w - ' ) r ( ~ ) ,
where w = ( y ;' ) . Then * defines an involution of hl, which Shimura calls the main
involution ([Shl], p. 72). Extend this Ciinearly to an involution of & := M @ C
REMARK. It is easy to see that we have = T,, for each of the Hecke operators. In
addition, for each g E Gr C M gt E C and wgt w - = g-L. Thus, * restricts to an involution
of M := (T. G) and also of Mg and Mc := M 8 @
DEFINITION. Let A / F be an aigebra over a fieid F, equipped with an involution * and
let W be a right A-module. The contmgredient A-module is the set
3. THE ALGEBRA OF MODULAR CORRESPONDENCES AND THE Hl-COHOMOLOGY OF X ( N ) 73
on which A acts on the right via
f ( ~ ) l a = f (zla* 1.
Thus? W is a right A-module.
In this subsection, we show that V := is the contragrdient Mc-module of V ,
that is, V 2 V' as an Mc-module.
LEMMA 3.26. Suppose that there U a non-degenemte F-bilinear pairing
of right A-modules svch that
( u l o , ~ ) = (u,wla*)
for al1 a E A.
Then W - U' as an A-module.
PROOF. Since ( . ) is non-degenerate, it defines a vector space isomorphism
where Fu is the linear functional defined by Fu : w * (u, w ) .
By hypothesis, we have (ula,w) = (u. wla*)? thus p(&) : w c, (u,a'w), so
p(uIa) = FuIa : w * Ful, ( w ) = Fu (wla* 1 = Fu (w) la-
Thus, p defines an isomorphism U -, W* of A-modules. O
In what foliows. write * for the main involution on M (or M). Recall the following basic
fact from the t heory of modular forms.
PROPOSITION 3.27. a) For any rn E MI, f .g E V, we have
where ( , ) denotes the Petersson pmduct on V .
3. THE ALGEBR4 OF MODG'LAR CORRESPONDENCES AND THE H L - C O ~ 0 M O L O ~ ~ OF X ( N ) 74
b) T,' = wT,w-' /or al1 n and g* = g-' for g E G'.
c ) If p : M + C denotes Shimum 's representotion, then p(m)'R = p(m8).
PROOF. a) See equation (3.4.5) on p. 76 of [Shl].
b) The k t assertion follows fiom the fact that TA = T,, and the second follows imme-
diately from a matrix computation.
c) See [Shl]. p. 171 (equation after (7.2.5) ). 0
Now. we have a Gbilinear pairing on V x V ,
COROLLARY 3.28. We have P 2 Y' as an Mc-module.
PROOF. By Proposition 3.27, the Petersson pairing on V x V satisfies
for al1 m E M a .
Thus, for w f V. E E P. we have
for any m E Mw, where the penultimate equation follows from the faa that the "bar"
R-isomorphism V - V cornmutes with M
Since [ . ] is Gbiiinear, we then have [w(,,e] = [w.cl,-] for aii m E M c . Thus, by
Lemma 3.26. we have that P z V'. I7
3.3. Complex coqjugation and automorphisms of M[
Consider the automorphkm of M defined by conjugation with the element Q-1 =
3. THE A L C E B U OF MODULAR CORRESPONDENCES AND THE HI-COHOMOLOGY OF X ( N ) 75
REMARK. The automorphism K fies the Hecke dgebra "away from the level" T(N),
by [Shl], equation (3.5.19).
Now , for an arbitrary Wmoduie W, let W- 1 denote the MLmodule obtained by truisting
the action of MI by the automorphism n. that is, W-l = W as a vector space. but MI acts
via
~ I r n - i := wlK(m).
Notice that n restricts to an automorphism of M := (T, G') aod thus, of M n and Mc.
The goal of this subsection is to prove the following.
THEOREM 3.29. ( E . Kani) We have = V' as an Mc-moduk.
In order to prove Theorem 3.29, we will define a non-degenerate, Mc-compatible C
bilinear pairing
( , : v x VSl + c
and then invoke Lemma 3.26.
We employ the following lemma.
LEMMA 3.30. (E.Kani) Let U be a finite-dimen3iona1 @-vector s p c e that is a right
module over A := AR. an W-algebra equipped with an involution *. Suppose that there is a non-degenemte Hermitian pairing
such that
(ula? W ) = ( U T ~ I a m )
for al1 a F AR.
Let T denote cornplez conjugation and suppose that there is an Llsnear automorphism
i E AutÂ(U) and an automorphism cr of Ag such that
3. THE XLCEBR4 OF MODULAR CORRESPONDENCES AND THE Hl-COHOMOLOGY OF X ( N ) 76
for ail c E C, u E CI and a E AB.
Then the pairing
is non-degenemte. @-bilinear and satasfies
for al1 a E & := AR@@.
PROOF. Clearly, [ . ] is non-degenerate, Giinear in the fint variable. and klinear in
the second.
Let u , w € V , c € @ Then
(where the second equation foilows hom the fkst condition in ( t ) above), so that [ , ] is
iinear in the second variable.
Now, for any a E Ag! we have
by the second and third conditions in (t). Now, since [ , ] is Gbilinear. this suflices to show
that the relation hoids for ail a &. U
NOTATION. Let Li, be the &-module obtained by twisting the action of & on U by
the automorpbism a. That is. Ua = U as a vector space. but & acts on Ua via
COROLLARY 3.31. Under the hypotheses of Lemma 3.30, we have Ua zz U' as an &-
module.
3. THE ALGEBRA OF MODCLAR CORELESPOXDENCES AND THE Hl-COHOMOLOGY OF X ( N ) 77
PROOF. The pairing
of vector spaces may equaiiy weii be considered to be a pairing on U x U, and by Lemma
3.30, we have
for al1 a E &. Thus by Lemma 3.26, we have an isomorphism U 2 (11.)' and thus,
u" rrL (k)" h un. 0
Finaily, we apply this to our situation in order to prove Theorem 3.29.
We let >r E Aut M be the automorphism induced by conjugation with Q-i.
Rirther. let i be the Klinear automorphism of V defiaed by
(as in [Miy] p. 115. denoted f,. and [Shl] p. 83, denoted f C ) .
PROOF OF TKEOREM 3.29. The Petersson pairing
(is non-degenerate) and satisfies
for al1 m E M (where * is the main involution of M).
In order to apply Cor. 3.31. it suffices to veri@ that the t h conditions (t) in Lemma
3.30 hold for the operators 7. a.
The first is clear: - -
?(cf) =cf(-2) = E f ( - E ) = W?(f
That the second condition holds is proven on p. 83 of [Shl].
It is easy to vetify the identity
4. THE (1.1)- SLiiCt'LAR COHOMOLOGY OF ZN,,
which y ields the ident ity
for a E A.
Thus, for any m = [ r ( N ) d ( N ) ] E M,
(where the pendthnate equation follows fiom the k t that, since ï(N) is a normal subgroup
of SL2(Z) , we thus have d ( N ) = I ' (N)w) .
This verifies the third condition GK equation (t) and thus, by Cor. 3.31, we have
4. The (1.1)- singular cohomology of ZN,?
Let 7 E ( Z I N Z ) ', let Z N , = GV7 and let
We now apply the results of the previous section in order to study the cohomology space
Rz,, and, in particular. to relate this space to the space of cusp forms on ZN.
THEOREM 3.32. There is a T(N)@~-isornorphâsm
such that
4. THE (1.1)- SINGULAR COHOMOLOGY OF Zn.,, 79
PROOF. There is a functorial isomorphism HSsg = V @ Y , by the Hodge decomposition.
By Cor. 3.28, P = V' as an M-module and by Theorem 3.29 then, z as an M-
module.
Thus, t here is an M@'- isomorphism
so that via this isomorphism,
Now, the nat ural isomorphism of uector spuces V = V- given by the identity cornmutes
with T ( N ) (but not with M). since the automorphism a-1 &es T(N).
Furthermore. we have
which takes (V @ V- l ) ' L ~ . y to V-?. Similady, we have
and thus, the identitiy map induces T(N)@~-isomorphisms
v . L @ v ; v'*. v-l@v-l z ve2
which take (V- @ v ) ~ N ~ - to V-? and (V- 8 V- )A".- to V7 , respectively.
Thus, taking the sum of the T(N)@~-isomorphisms of the preceding paragaph, we have
a T(N)@~-isomorphism which has the desireci property. CI
CHAPTER 4
The space of cusp forms on S, and its (1,l)-cohomology
1. Introduction
Fu< a prime q { prlr(p)4(p) = p3 - p.
In Chapter 2 56 (Theorem 2.34). we reduced the determination of the zeta-function of
Zp @ Fq to the computation of the "L(l,l~-huictionn
w here
and (&)@2 deno tes the q'h geometric Robenius morphism acting (diagonaily ) on ( H ; ~ ) P - ~ .
B y the cornparison and speciaüzation t heorems and the Hodge decomposition resulta
of Chapter 2. the space t) is closely reiated to the space of c w p f o m on the disconnected
(where the Hecke operator T, is deûned in 53.1) and let T(p) denote the Hecke algebra
"away fiom the levei" . Tb) =Tc@) = qTn : p t n ] C T.
NOTATION. Note that this use of the symbol ïï codicts with the notation of Chapter
3. where we used T to denote the (infinite-dimensional) Hecke subalgebra of Mi
80
In this chapter, we wiii prove the following theorem, using Carlton's results about
"Atkin-Lehner theory on MDQS's" .
THEOREM 4.1. a) The sum x, V zs direct.
b) The space V := C, V C vS2 zs a module having multiplicity one and, in
fact, zs a b e module of runk one over the restncted algebm T(p)@21v.
c ) V has a baszs of normaluied Tb)@*-eigenfoms.
Theorem 4.1 is prown at the end of §2. By Theorem 4.1, V has a canonical bais
consisting of normalized (in the sense explained at the begining of 92) 'F(p)B2-eigenforms in
Va2. Cal1 this basis V.
At the end of 93, we wiii use the Hodge decomposition and cornparison results to deduce
the following analogue of Theorem 4.1.
THEOREM 4.2. a) The sum get = &Z/pZ)x %: 2s direct.
b ) Ret is a f i e module of mnk 4 over the algebm '&(p)@21g6t, where
fi@) :=Q[T~ : p i n ] C ~ n d &
and Tn denotes the reduction of the correspondence T, (as defined in Chapter 3, 51.2).
Theorems 4.1 and 4.2 show that the space of cüsp forms on the disconnected surface
Zp (respectively, the (1,l) étale cohomology of Z, @ Fq) can be embedded in the space of
cusp forms of the connected (smooth) surface Y @) (respectively, the ( 1, l)-étale space of
Y @ ) @ &). This rather strange fact (which is only true in the case of prime level) shows
that. in fact, the L(I,l)-function of 2, divides the L(l,ll-function of Y@).
In fact. the main remit of this thesis is that the L(l,l) function of Zp c m be expresseci
as the product of Rankin-Seiberg L-functions associateci to the canonical basis V of V (up
to Euler factors at primes q(p3 - p). In Chapter 5 57, we wiil prove the foilowing.
THEOREM 4.3. Let q { p3 - p. Then
(where L,(F,x) zs a certain degree 4 polynomial defined in Chapter 5, 53).
2. ATKIK-LEHNER THEORY ON MDQS'S 82
It turns out that the space of cusp forms V is not preserved by the full Heeke aigebra
T@* which presents an obstacle to the direct computation of &( 1,1) (6, z).
We overcome this obstacle by "aigebraicdyn associating to V ( i e . . by means of an
algebraic correspondence) a certain Hecke submodule of Va2.
2. Atkin-Lehner theory on MDQS's
In a recent paper. Carlton [Ca] studies the spaces of cusp for= on modular diagonal
quotient surfaces. In his paper, Carlton (axzong other things) develops a theory of Hecke
operators and (partially) develops a version of Atkin-Lehner theory in this context.
Carlton's results can be used to deduce important consecluences for the zeta-functions
of MDQS's (of arbitrary level).
The goal of this section is to explain Carlton's main relevant results and, in particular,
to prove Theorem 4.1.
Fbc a positive integer N. a residue class 7 E ( Z J N Z ) and let
where V = VN = S 2 ( r ( N ) ) .
An element 3 E Va2 has a Fourier expansion in 2 variables
LI, 12) = an, ,nl (3)~;' q;'. g, = e2riz~lN.
DEFINITION. A form 3 E va2 is cailed nomalired if al-1 (7) = 1.
The foilowing lemma shows that most of the Fourier coefficients of a fonn 3 E V C VB2
are zero. which (usualiy) makes it more couvenient to consider the product n, V7 (see [Ca],
55).
LEMMA 4.4. Let 3 be an element of P. Then the Fourier eoeficient %,,,,(3) = O if
ni + yn2 f O (mod N).
PROOF. Let T = ( A ) E G. Then we notice that
2. -4TKIX-LEHNER THEORY ON blDQStS
since Q,T(Q,)-L = F.
For any F(z iT z?) = C E va2, we have
Thus, 3 = 31 (T,T. ) implies that %, ,,, = O unless n 1 + 7712 I O (mod N) .
Now. observe that For each dl N. there is a natural injection
where 7 E (ZINZ) and denotes the image of 7 under the natural rnap ( Z I N Z ) - (Z/ (hr /d )Z ) '. The image of any element under t bis map
satisfies a,, ,n2 = O unless dlni (and thw, d also divides 712 since ni + 7722 I O (mod N)).
That the converse holds is Carlton's main Atkin-Lehner theory-type result.
PROPOSITION 4.5. Let 3 E v".' and arsume that, for some dlN, u,,,,,,(F) = O unless
dlnl. Then 3 E vNId~?
PROOF. See [Ca], Prop. 2.3.
In the case of prime level (and weight 2), this gives the following.
COROLLARY 4.6. Let 7 E Vpv7 and assume that a,,,,, (7) = O zj p { ni and p f n2.
Then 3 = 0 .
PROOF. By Prop. 4.5, 3 E vNIN-? = However, VI = S2 (SL2(Z)) = (O), so
this space is O. 0
Now. Carlton defines (on p. 215) a Hecke opemtw tnLqn2 on the space of cusp forms
27 = @,v".v for each pair (nit n2) of positive integers (ni, ne) coprime to N. He then
proves the following.
2- .ATKIF-LEHNER THEORY ON BIDQS'S
PROPOSITION 4.7. The sum map
for al1 ni. n2 copnme to N. whew T-,, ,,, = Tn, 8 T,, E End va2. Thus, yN = x, V N 3 has a simultaneous basis of eigenfowns under the Hecke algebra
"away h m the leveln T(N)@* C En& va2.
PROOF. See [Ca], Prop. 4.3 and 4.4.
Remarks.
1) The sum map C is not generaiiy injective. but this does tum out to be the case when
N = p is prime. As we explain. Carlton deduces this fact from the more general result that
C descends to an injection on the "multipiicity-one quotient" of n, vN". 2) Similarly, Prop. 4.7 does not provide a canonical basis for vN, as the T(N)@*-
eigenspaces are not, in general. 1-dimensional. However, the "mdtiplicity-one quotientn
of vN does have this property and we shali see that this d i c e s to ensure that VP has
multiplicity one.
NOTATION. Now, for each y E ( Z / N Z ) X , consider the subspace ( v ) O L D ç Vr defineci
by
and let = v/(w)OLD. Also. let voLD C V be the subspace
and let v = ri/voLD.
2. .4TKIK-LEHNER TBEORY ON MDQS'S
Now, let vNEW be the subspace
REMARK. Notice that the terms *OLD" and "NEW" here do not correspond to (or even
bear an obvious relation to) the usual notions of "oldn and "new" subspaces in the theory
of rnodular forms.
LEMMA 4.8. We have
v = vNEW $ vOLD
as a decomposition of T ( N ) -modules.
" Then it is shown PROOF. For an element f = a,qE E V, let := &n,N)=i anqN.
in [KMo] 52 (using certain -twisthg operators") that f( E V. Write
where := &N)=l a , q s VVNEW. f ' = C(n,N)>I hqv voLD. T ~ W , every element
f E V can be written uniquely as a sum of elements in the OLD and NEW subspaces, so the
sum is direct. The formulae for the action of the Hecke operators show that these spaces
are preserved by ïï(N). O
We remark that, in the case N = p. we have fb) = f li-,, which is in V , since V is a
G-module.
LEMMA 4.9. We have
V 8 voLD + voLD 8 V = W := {F E ~ @ * l a ~ ~ , ~ , ( 3 ) = O if gcd(ni, N) = gcd(nz, N) = l}.
Thus, the notuml map va2 + vG2 has kernei V @ voLD + voLD 8 V and thw ,
v n (v B voLD + voLD 8 V I = (vyoLD.
PROOF. Clearly, we have the containment
Now. we have the decompostion
P2 = (V 8 vOLD + v O L D @ V) @ (vm)@*.
2. ATKIS-LEHNER THEORY ON MDQS'S
Suppose that 3 E W and mite
with respect to the above decompostion. If rNEW # O. then we have a,, ,,, (FNEw) # O for
some pair of intergers nl, n;! coprime to N. Then
which is a contradiction. Thus. FNEW = O and 3 E V O voLD + voLD €3 V . which completes
the proof. O
By Lemma 4.9. there is a weii-defined. injective map - v@* and thus. a sum map
h
PROPOSITION 4.10. The map C is injective and thus, tue have an zsomorp.phism
PROOF. See [Ca]. Prop. 4.3.
COROLLARY 3.11. If N = p is a prime. then the sum V = C, V7 2s direct.
PROOF. We have (V7)OLD = ( ~ p ~ ~ ) ~ ~ ~ = (O} by Cor. 4.6. Thus, we have
and the latter sum is direct. by Prop. 4.10.
COROLLARY 4.12. If N = p 2s a prime, then for any T(p)@2-eigenform 3 E V, we have
ai,r (3) # 0-
PROOF. For p C nm. let A,., denote the eigenvaiue of 3 with respect to the (n,m)th
Hecke operator. Then by [Ca] Cor. 4.1. we have
2. .%TKIS-LEHNER THEORY ON MDQS'S 87
for aii n and m coprime to p. Thus. if ai, i (F) = O, then a,,,,(F) = O uniess p divides n or
m which implies that 3 = 0, by Cor. 4.6. O
Since the subspace voLD C VN is a T(N)-submoduie. we have that is naturally a
T(N)-module.
- @2 LEMMA 4.13. has rnultiplicity one as a T(N)-module. Thus, VN has multiplicity
one as a T(N)@*-module.
PROOF. Let N(V) denote the set of normalized newforms of all leveis in V. By A t h -
Lehner theory, V = VN = $ CV(Vl Vf where the Vf's are precbely the distinct T(N)-
eigenspaces and t herefore, voLD = $fE~(V, voLD n V! and
where 6 := V,/(K n VI)
To show that each 6 is 1-dimensional. it suffices to show that
dirn voLD . Vf 2 dim Vj- - 1
w hich proves the inequali ty.
Also,
is a multipiicity one T(N)@~-module.
In particular, we have the foiiowing consequence, which, combined with Cor4.11, corn-
pletes the proof of Theowm 4.1.
3. THE (1.1)-COHOMOLOGY OF Zp
COROLLARY 4.14. Fiz N = p a prime. The g p c e
has a bas& conszsting of nonnalired T@)@2-eigenforms and further, hm rnultiplicitg one as
-module.
Thus in particular. V zs free of mnk 1 over ~ ( p ) @ ( ~
PROOF. Since Tb) is a commutative. *-closeci algebra, it is semi-simple and hence. so
is 'II'@)@*. Thus, any T@)@2-module, in particular V , bas a basis of T'@)a2-eigenforrns. By
Cor. 4.12, each such eigenform 3 satisfis ai,i (3) # O and thus, 3 can be normalized.
By Lemma 4.13, vB2 has multiplicity one and, by Cor. 4.11,
so V has multiplicity one.
Thus, V is hee of rank 1 over 1 v .
PROOF OF THEOREM 4.1. See Cor. 4.11 for a) and Cor. 4.14 for b), c).
3. The (1. 1)-cohomology of 2,
In the previous section. we showed that
1) the sum t" is direct and
2) V is a free 'Tb)@* IV-module of rank 1.
In this section. we deduce the analogue of these results for
A) The (1,l)-singular space 'Hsing
B) The (l,l)-étale space &, of 2,8 Pq For Xsinrt this is a direct consequence of Corollary 4.14 and Theorem 3.32. As we show,
the analogue for &, then follows kom the cornparison theorem and specialization results
developed in Chapter 2.
3.1. The (1. 1)-singular cohomology of Z,
3. THE (1.1)-COHOMOLOGY OF Zp
Recaii (Theorem 3.32) t hat t here is a ~@)@~-isornor~hisrn
such that
Now, let
THEOREM 4.15. a) The sum x, 31:& C H:& 2s direct and Ilsng 2 v4 as a
module.
b} RSing is a free module of mnk 4 ovet T @ ) @ * ( ~ ~ ~ ~
PROOF. We have
under the T(p)@2-isomorphism of Theorem 3.32.
Thus. since V is a T(p)@2-rnodule. Rsing is &O a ~ @ ) @ ~ - m o d u k and Using zz v4 as a
T(p)@2-module.
Now. T@)a2 acts diagonally on (V@2)4, so that
By Cor. 4.6. V is free of rank 1 over ~ ( p ) @ ~ I V , so '+fing i~ kee of rank 4 over this
algebra. Cl
We will now prove the andogous fact for TQ@). For this, we use the foilowing Lemma.
LEMMA 4.16. Let A be a commutative, semi-simple, fintte-dimeBst'onal Q-algebna acting
o n a finite-dimensional Q-uector space W . If H C W is a subspace such that H 8 @ i a an
A @ @-submodule of W @ @, then R is an A-submodule of W .
3. THE (1.1)-COHOhfOLOGY OF Z, 90
PROOF. Suppose, on the contrary, that there are eiements a E A, w f W such that
aw 4 W. Then (a @ l ) (w 8 1) = (au @ l) , so that aw W implies that (aw) @ i # W @t C,
so W @ @ is not an A @ Grnodule. O
Now. the action of T(p) = Tc@) on Hsiw cornes fmm the action of
that is. en, @ @ = Gng as a Tc(p)-module.
Thus. the fact that Using = C, U&,g is a T'&)-module implies that
is a TQ(p)@2-moduIe with
Thus, the foiiowing is immediate.
COROLLARY 4.17. a) The sum &'H;: H:& is direct.
6 ) ~2~~ i s ta j k e TQ(p)@21 p -module of mnk 4. %ag
3.2. The (1: 1)-étale cohomology of Z, @ l&,
Recall t hat
by Cor. 3.7 and also,
as a Q-module, by Cor. 3.8.
PROPOSITION 1.18. a ) The sum & Ret HE* zs dinct.
b) ?let = x, ?tet is a free module of mnk 4 over the algebm
3. THE (1.1)-COHOMOLOGY OF Z,
PROOF. We have that each
via the
T hus,
as a 4 @)@2-module. so the sum is direct.
By Cor. 4.17 b) , zzng is hee of rank 4 over TQ (p)02 1 Q , which implies that $'& is 'sing
free of rank 4 over
Then Uet = 312& as a Tl (p)a2-moduIel so ?tet is fiee of rank 4 over 4 @)@2171et Ci
PROOF OF THEOREM 4.2. Fix a prise q and consider the space
We have already seen (in Chapter 3, 91.2) that
and that fi, 2 as a Ci-module.
Thus, the sum
is direct and fiet is a hee module of rank one over Tl @)@2 1 WCt, which completes the proof.
O
4. THE HECKE SHIFT OF A T(p)-MODULE
4. The Hecke shift of a Tb)-module
It tunis out that the space V is not preserved by the full Hecke algebra p 2 , but only
by the Hecke algebra "away 60m the leveln 'II'@)@*.
In this section. we show that one can (algebraicaiiy) associate to V a canonieal p2-
submodule vT C v@* such that V 2 vT as a 'Tb)@2-module. We also deduce the corre-
sponding fact for the (1.1)-étale cohomology space f i c ,
These facts wili prove usehi in the next chapter when we compute the L(i,I)-function
of Zp.
4.1. The Hedce shift of a T(p)-submodule of V .
R e c d that the group G := SL2(Z/pZ)/{Il} acts on X(p) as a subgroup of its auto-
morphism group, and we have Shirnura's representation
Now, the conespondence algebra C acts contravariantly on V, that is, there is an dgebra
homomorp hism
C -+ (End V ) O P .
into "the opposite ring" of En& V.
Thus, @ := C @ C acts on V &o. As previously mentioned, this is not in general a
faithfil action.
The naturd morphism
is a Gaiois cuver of c w e s with group
We therefore have an exact sequence of vector spaces
4. THE HECKE SHIFT OF .A T(p)-MODULE
where ~p is the correspondence
LEMMA 4.19. Let j = a,q; E V . Then
PROOF. We compute
in the notation of 52. Thus. we have
as Tb)-modules.
We now show that the subspace VNEw V has the important property that the pth
Hecke operator T; d s h e s on it.
PROPOSITION 4.20. We houe T~E; = T; on V and thw, T; = O on V-.
PROOF. We have
P I ~ (see 51 of [Li2]) and
PI^ on V (Lemma 4-19]. Clearly then for any f E V .
so that ~ j c ; = T;.
4. THE HECKE SHlFT OF A T(p)-MODULE
Now,
by definition and, by the above. for any v' = (1 - ck)(v) E vNEW, we have
PROPOSITION 4.21. The opemtor c b commutes with the Heeke olgebm away from the
leuel T@)e.
PROOF. We have the decomposition
(into eigenspaçes under the Carkm subgroup) indexed by the characters A on (Z/pZ)%.
We show that Tb) commutes with c k on each
Recaii that, on VA.
T; = u, + A([)&
for any prime 1 # p, where Lii and & are the usual operators on qp-expansions (see (Li21,
51).
It therefore suffuces to show that Ul and 6 commute with CL on V.
Let f E C an$ E V . Then
and
un pli n
Thus, EL commutes with T@): = C@ : 1 # p] and so does (1 - c;).
We use Propositions 4.20 and 4.21 to associate to any Tb)-submodule of V a T-
subrnodt.de.
4. THE HECKE SHIFT OF A T'@)-MODULE
PROPOSITION 4.22. Let W C if be a Tb)-submodule. Then
a) W I p , , ) is a T-submodule of V and
6) if W n voLD = {O}, Mien W I p c p 1 = W os a T(p)-module.
PROOF. By Prop. 4.21. the hear map
. v - v P J E W (1 - ej,) . S V
is a homomorphism of Tb)-modules. Thus, the image WI~l-,,l of W is a Tb)-submodule
of V.
Since T; = O on vNEiV by Prop. 4.20, vNEW is a Grnodule and TlyNEW = P.(p)IVNEW.
Thus, in particular, WI(l-,,) C vKEW is a 7'-submodule.
Evidently, if ker(1 - e;) n W = {O}, then (1 - e;) restricts to an isomorphism
DEFINITION. Let W be a T(p)-submodule of V.
a) The Hecke shift of W is the T-submodule
b) If W n voLD = {O}, then we say that W is quasi-new.
The utility of this notion in this simple context is austrateci by the following k t .
PROPOSITION 4.33. If W is a quasi-new T(p)-submodule of V such that W is j k e over
Tb) 1 of mnk 1. then the Hecke shift W' is free of runk 1 over T w .
PROOF. We have W' C vNEiV and T; = O on vNm by Prop 4.20. thus
and WT h W as a T(p)-module. since it is quasi-new.
4. THE HECKE SHIFT OF A T(p)-MODULE
4.2. The Hecke shift of V
Evidently, we can also apply the Hecke shiR technique to associate to a ~ (~ )@~-submodule
of va2 a T@2-submodule.
Notice that
Thus, if W is any 'Tb)"2-submodule of ve2. then Wl(l- ,pp~ is a ~ 2 - m o d u l e with
Im(1 - c;)@~ = (vXEW)@* and (ker(1 - eL)B2 = V @ voLD + voLD 8 V.
DEFINITION. Let W be a Tb)@*-submodule of va2. a) The Hecke shiJt of W is the 'I'@2-submodule
W' := wI(L-(p).2.
b) If W n (voLD @ V + V @ voLD) = {O): then we Say that W is quusi-neut.
PROPOSITION 4.24. Let W C Va2 be a quasi-new T(p)@2-submodule such that W is
free of mnk 1 over 'R'(p)@*(rv. Then W' = W is free of mnk 1 over P21wr.
PROOF. We have wT C ( v ~ ~ ~ ) @ ~ ! so T; @ 1 = 1 @ T; = O on W? Thus,
COROLLARY 4.25. V C V8' i s quasi-new and thw, V zz v', which is a free TIv-module
of mnk 1.
4. THE HECKE SHIFT OF A F(p)-MODULE
PROOF. We show that V is quasi-new, that is,
Suppose that a E ker(1 - ep)@2 n V. Then we can mite a uniquely as
Then
Since the sum
is direct, this implies that each aT(l - c ~ ) @ ~ = O. But since ker(1 - c ~ ) @ ~ nV7 = (w)OLD =
(O}, this implies that each a, = O and thus, a = 0.
V is kee of r a d 1 over T@)@2(v, by Cor. 4.14. Thus, by Prop. 4.20 above, is kee
of rank 1 over TV.
4.3. The Hecke shift in general
By Prop. 4.20. we have
in En& V.
Now, since the representation C + End V is not faithfui in general, this is not enough
to conclude that (1 - c p ) 0 T, = O in C.
However, this does turn out to be true in this case, since the correspondences c p and
Tp are defieci m e r Q.
From now on. we make the identification
(which is permissible because of the correspouding identification End J(p)Q = En&Q J@)Q
since O/Q.
4. THE HECKE SHfFT OF A Tb)-MODULE
THEOREM 4.26. a) c p E CQ.
b) We have (1 - c p ) o Tp = O in éq.
PROOF. a) As ?reviousIy mentioned. the naturai map
is a Galois cover of c w e s with group P = (T) . Furt hermore, @ descends to a morphism J>p
on Shimura's Qrnodels (since it cornmutes with the Galois action corning h m this choice
of Qrnodels).
Thus, we have (Qrational) puil-back and push-forward maps on Jacobians
It is not hard to show that, via the isomorphism
ep gets identifhi with the endomorphism
(cf. [KRoj, equation (6) on p. 249). Thus, c p E CQ.
b) Since 1 - ( p . Tp E CQ L, (End VQ)OP, this follows fiom the fact that ~ i ( 1 - c p ) = O
on VQ E V& U
By Theorem 4.26, we can associate a Hecke shift to a b@)-submodule of any Q-
module.
In order to apply this terminology to the Tl@)@2 z Tl@)@2-submodule
we develop it in slightly more generality.
Thus. let M be a faithful module over
where F/Q is an extension.
5. THE HECKE SHIFT OF THE (1, I)-~TALE COHOMOLOGY OF Z, @ f ,
Let TF(p) = TQ@) @Q F and identify c p with its image c p @ 1 E C$.
Since (1 - c p ) o Tp = O in CQ, if W C M is any TF@)-submodule, W I p , , ) is a
~F-subrnodule.
DEFINITION. Let W C hf be any TF@)-submodde.
a) The Hecke shzft of W is the Tpmodde WI~I-,,).
b) If W n ker(1 - E ; ) = {O), then W is cailed quai-new.
In order to apply the Hecke shift in the aforementioned context, we generalize the above
definition to the 2-variable case.
DEFINITION. Let W Ç Ma2 be any TF@)@*-submodule.
a) The Hecke s h y of W is the ~2-submodule W' := W l~l-,,p~.
b) If W n ker(1- E;)@~ = {O}. then W is caiied quasi-new.
PROPOSITION 4.27. Let H E Ma2 be a quosi-new ~~(~)@*-subrnodule such thot H is
free of finite rank n over T ~ ( ~ ) @ ~ I H.
Then H~ îs free of mnk n mer TH = T'$'le
PROOF. Again. we have
5. The Heeke shifi of the (1,l)-etale cohomology of Z, 8 pq
Let
7 7
where & = H ~ ( x ( P ) B q , ~ ) . We showed in the proof of Theorem 4.2 (in 53.2) that uet is a fiee module of rank 4
over the algebra $ (p)@2 lGét . IR this section, we will prove the foUowing.
S. THE HECKE SHIFT OF THE (1, 1)-ÉTALE COHOMOLOGY OF Zp @ Fq
THEOREM 4.28. We have
Üet n (1 - O 1)@* = {O}
and thus, ( 1 - Z; @ l)@* : get 1 Hz, which is a free module of mnk 4 over
Furthemore, ( 1 -
We first prove the
Tget := q21gzt. 2p ip 1la2 commutes with the geometric Robenius endomorphism.
analogue of t his result for the ( 1,l )-singuiut cohomology of Z, 8 QI
5.1. The (1, 1)-singular cohomology of 2,@ C
Let Zp = $, let
As before, identify É P with its image ~p 8 1 in
PROPOSITION 4.29. a) We haue
Hsing n ker(1 - É;)@~ = {O}.
Thus, R : ~ ~ is free of mnk 1 over
T R ~ ~ ~ ~ := PZ I R C . sing
b j Thw. H& n ker(1 - c;)@* = {O} and (?&JT is /ree of mnk 4 over
PROOF. a) By Theorem 4.15, there is a T@)@2-isomorphism
SU& that e(Using) = v4. Now. the correspondence c p &O commutes with the isomorphism V 2 since by
the identity Q-lT(Q-i)-l = T-'. we have Q-icP(Q-l)-l = c p .
Usin, n ker(1 - CL)@? 1 (V n ker(1 - c:)@~)~ = {O),
by Cor. 4.25.
Now, Rsing is tree of rank 4 over T(P)@~)%~,,, by Theorem 4.15. Thus, by Prop. 4.27,
~ 2 , ~ is free of r d 4 over lï%ing.
b) We have
31:, n ker(1 - c L ) @ 2 C RSing n ker(1 - E;)@~ = {O},
by 4. S ince
@ C 2 H$"~
as a T ' , Q ( ~ ) ~ ~ 8 Grnodule (by definition) and
via this map, the fact that 'Wsing is a T@)@2-module implies that ~2~~ is a TQ(p)@2-module
and u ! ~ ~ )Rp C z Using as a T,Q@)@~ @ Grnodule.
If 312ng is not kee of rank 4 over the semi-simple dgebra
t hen for some character x E Irr(TQ(p)@2) ,
Q where nx denotes the multipliczty of x in the representation Usin,. Then writing x @ @ =
C, X i as a sum of distinct, Galois-conjugate characters shows that, for any of the characters
xt E I~(T.&)@~) above, we have n,, # 0-4, contradicting the faet that Raing is fiee of radc
4 over Tc(P)@21%i,g (by Theorem 4.15). III
In order to prove Theorem 4.28. we will deduce the analogue of Prop. 4.29 for singuiar
cohomology with Q -coeficients.
First, we establish the following iemma.
5. THE HECKE SHIFT OF THE (1, 1)-ÉTUE COHOMOLOGY OF 2,@ Fq
LEMMA 4.30. Let CI he an abelion gmup, let H be a q q - m o d u l e . Then the map
H%Q - ( H ~ Q ) ~
PROOF. Consider Q and Q as trivial H-modules and let Hi := H 8 Q . Then H" =
H @au] Q and we have
HY = & O ~ [ U , Q = (H@q[tqQ) @O.
Let k e r ~ ( l - denote the kernel of (1 - cp)@2 acting on (eng)@'. PROPOSITION 4.31. We have
ker(1 - I E ~ 0 I ) @ ~ = (kerQ(l - c$)@*) 63 Q
PROOF. This foilows fiom the following more general (obvious) fact. Let M be a vector
space over a field K and f E EndK M. Then for any extension LI K, one has
COROLLARY 4.32. We have
G U,, n ker(1 - CL O 1)@2 = {O}.
PROOF. Mie have
( ~ 2 ~ ~ @ Q) n (ker(1 - @ Q) = ('Hzng n ker(l - E$)@') @ Q = {O},
by Prop. 4.29.
By Prop. 4.27 t hm. we have that (R:!~)~ is a kee module of rank 4 over T2 1 9, . %Ilg
5.2. The proof of Theorem 4.28
PROOF OF THEOREM 4.28. Notice first that we have
H g 2 2 (HJP,J@*
by Cor. 2.9. Thus,
'fi4, n ker( l - cp 8 l)@* 2 .HZ, n ker(1 - rpng @ 1)@2 = {O),
by Cor. 4.32.
By Prop. 4.27. we then have gct ??: via the endomorphism (1 - 2;' @ 11)"~. which
cornmutes with the Robenius morphism. since r p E % v Cpq (and since Frobenius is in
the centre of CF,). O
CHAPTER 5
Eichler-Shimura theory and the L(l,l)-function of Z,
In this chapter, we wiU prove Theorem 4.3.
By Theorem 4.28, this is reduced to the assertion that for each prime q i P3 - p,
det (1 - zNQ) = n &(Fr z), Ff v
where
HP := (#$'217i:t7
4: is the gge~metrk Frobenius endomorphism and Lq(F,z) ia the qîh local l-factor
attached to the T'@2-eigenform 3 (the degree 4 polynomial E o ( 3 , x) is ody defined in 53 of
this chapter) . Since 7?; is a free module of tank 4 over Tg;iet. where
and H, is Tgdl -1inear (since & is TI-iinear), the computation of det (1 -ZN,) (or equivaiently,
ch (Nq,z)) can be done in two steps:
1) compute the
2) compute the
relative characteristic polynomial
41 [t]/Qz[ll (ch (q, 4) - "et
By a w e U - b n hear algebra formula (see [Bol], Coroilary on p. 548), the absolute
characteristic polynomial ch (Np? x ) is equd to the expression in 2).
The key to the computation seems to be to corne up with an "intrinsic" expression for
the relative characteristic polynomid in t e m s of (an appropriate subalgebra or quotient
of) the correspondence algebra itself. We will do this in 57.
104
1. THE EICRLER-SH1,CIURA CONGRUENCE RELATION 105
First, we recaii that, by the Eichier-Shimura congruence relation, there is a simple and
explicit expression for the degree 2 polynomial
and, using tensor products. this yields an expression for
Then in 55 (iising the semi-simplicity of the Hecke algebra), we show that the relative
characterist ic polynomial ch (NT: z) is exactly the polynomial obtained by restncting the
coefficients of Z(x) to the submoduie HL. Finally in 97? we calculate the n o m of this polynomial in te- of the space V, with
the aid of the Morita theorem.
1. The Eichier-Shimura congruence relation
For any prime q, let @q E End J(N)F, denote the gth-power aithmetic Robenius end*
morphism.
The foUowing celebrated theorem is the main tool which is used to express the Li-
hinction of X ( N ) Z [ + I as a product of ("away fiom N") Lfunftions attacheci to normalized
newforms in Sz (T'(N)).
THEOREM 5.1. (Eider-Shimura) For any prime q N, we have
where oq = (PO' 4) E G is the ditzmond opemtor and *R denotes the Rosati involution.
PROOF. See [Ei], equation (34) on p. 312 where he pro- the assertion for aii pRmes
of gmd reductaon for X(N). By Igusa's Theorem, the assertion then holds for aü q f N. O
Recail also the following well-known consequence of Atkin-Lehner theory.
PROPOSITION 5.2. The Tate space ~ ( J ( N ) ) is a j+ee '&-module of mnk 2.
1. THE EICHLER-SHIhlCTRA CONGREENCE RELATION 106
PROOF SKETCH. In [DDT], Lemma 1.39, the authors prove that, for any congnaence
subgroup r with ï o ( N ) 2 r > r 1 (N), the Tate space of the Jacobian of the modular c w e
Xr := ï\fi8 is hee of rank two over the corresponding 1-adic Hecke algebra There is a
natural isomorphism of Riemann surfaces X(N) 1 Xiv = rN\4'. where ïN is a ceratain
subgroup containing r1(N2) and this isomorphism preserves the Hecke structures on these
curves. O
In the next section, we will use the "modified Weil pairingn CO deduce the foiiowing
consequence of Theorem 5.1 and Prop. 5.2.
THEOREM 5.3. The relative chamcteristic polynomtal of (Ilio)$ E Endq(J(N)&) r(
Now, recall from Chapter 3 that there is a natural isomorphism
such that, via this isomorphism. the pull-back h: of an endomorphism h E End J ( N ) h
becomes the dual h!. where h2 is the endomorphism of T: ( J (N)) induced by h (see [Mi3],
Remark 11.5).
Furt hermore. t here is a natural isomorphism
which identifies hr with h2. where h is any endomorphism of X(N)&, by Prop. 3.4.
Thus, if 4, E End X ( N ) F q is the geometric Frobenius endomorphism, then 4; gets
identified with ((4,):) ' = ($& via the natural isomorphism
which takes to (El,. We therefore have the foilowing corollary to Proposition 5.3.
COROLLARY 5.4. The space fie, := Hkt (X(N)Fq, Q) LP a fize Tl-module of mnk 2 and
ch(&/'@) = x2 - (T& + ciq.
2. THE hfODIFIED WEIL PAIRING 107
PROOF. The characteristic polynomial of any square mat& and its dual (transpose)
are equal. Thus. the characteristic polynomiai of $9 is qua1 to that of (1(1,):, which is equd
to that of (&)?. O
2. The moditied Weil pairing
Recail Erom 81.5 of Chapter 3 that there is a canonical Weil paiting
such that its image under the reduction map
is the canonical Weil pairing e l on T, ( J ( N ) % ) for al l primes p, 2 { N, p # 1.
Now, the Weil pairing e? satisfies
for any a E End J ( N ) (by Lemma 3.20). In addition, the restriction of the Rosati-involution
to the image M + End J ( N ) q acts the same as the image of Shimura's "main involutionn
( Prop. 3.27).
Extend + Q-linearly to obtain an involution of
PROPOSITION 5.5. Fiz an isornorphisrn p : Zl 2 Zl(l). Then the Weil piring ef
defines an isomorphism
TP( J(W) TP(JVN' of ( J ( N ) ) with i ts Mi-contmgredient.
2. THE MODIFiED WEfL PrURING
PAOOF. Fix an isomorphism p : &(1) 1 Zl to obtain a map
TP(J(W) -, G'(J(N))" x * 4;,
Q where & : y c, p 0 el ( x , y).
By the above, we have
e p ( W , Y) = e p h (m'):y)
for ali m E M i . Thus. since ep is non-degenerate. the map z c, & defines an isomor-
phism ~ ( J ( N ) ) 1 ~ ( J ( N ) ) * which gives an isomorphism ~ ( J ( N ) ) 1 ~ ( J ( N ) ) ' after
tensoring with Q . O
Now, since there is a naturai isomorphism ( J ( N ) ) " 2 Hit (X(N)q, Q ), it is conve-
nient to modifjr the Weil pairing (as in [DDT] Lemma 1.38 and [Co] Theorem 4.8) in order
to define an isomorphism
Z ( J ( N ) ) 1 T ~ ( J ( N ) ) ~
NOTATION. If u E Aut X ( N ) . then £rom now on, we omit the + in the induced sut*
morphism a. E Aut J ( N ) , in order to keep the notation as unencumbereci as possible.
Now although ( J ( N ) ) is naturaily a right Ml-module, since the subalgebra Ti C Ml
is commutatiue. it is a bimoduie.
P ROPOSITIQN 5.6. The "modified Weil painng "
PROOF. We have
2. THE MODiFLED WEIL PAIRING
for al1 T E Tt (since Tt = T) and thus,
for aUT E Tl.
Composing with p gives an Q-isomorphism
On the other hand, for arbitrary h E End J ( N ) , we have
Now, since the Weil pairings are compatible with good reduction, we have the andogous
relation for h = $.+ E End J(N)Fq and thus,
In order to prove Prop. 5.3, it remains ody to estabhh the foilowing.
LEMMA 5.7. We have
- = o,'~t, 9
~ * ; R W - I = &&iR
in End J ( N ) p q .
PROOF. In ~ n d J~ ( N * ) ~ ~ y one has
where $; denotes the geometric Frobenius endomorphism and w ~ 2 denotes the Fricke in-
volution. See [DDT] pp. 38-39 and [Shl] Cor. 7.10.
Let BN = ( P O y ) . Notice that
2. THE MODIFIED WEIL PAIRING
where ï ,v is the congruence subgroup
{a= ( t d ) E S L 2 ( Z ) : a i d = 1 (mod N ) , c = O (mod N')}.
We therefore have a naturai isomorphism of Riemann surfaces
Since each of the c w e s X ( N ) . X A r has Shimura's canonical Qmodel. by Prop. 7.2 of
Shimura, a morphism of the above type can be dehed on these modeis if and only if it
cornmutes with the (lifteci) Galois action. It is easy to verify that pN does commute with
this action. That is, there is a Qisomorphism
which extends to a morphism of the minimal models of these c u m over Z[+] and thus, to
a morphism of eaeh of the fibres over Z[+],
for each q f N.
Conjugating equatioo (t) with &r gives
Now, we have
B ~ u ~ ~ N ~ = N - W
as matrices and thus, on V,
(@N~fl&l)' = W'
since the two sides of t his equat ion are defineci over iFq .
PROOF OF PROPOSITION 5.3
B y the Eider-S himura congruence relation,
By Lernma 5.7 and the remarks preceding it , the endomorphism
has the same characteristic polynomial over as (@q)n and, in part idar , they have the
same trace. -
tr(Siq)p = t r ( 6 & ; R ) ~ = (T&
By Lemma 19.2 of [Mi4], we have $q$;R = q in End J ( N ) F , . By the Eider-Shimura
relation, we t hen have
and t hus 1
det $ J ~ = - ((tr $,12 - tr(#)) = q(5,):. 2
3. Ranl<in-Selberg Lhuictions
Fix a prime q t N. let
and let @(z) denote its image in pQ[z], where f&[~] denotes the image of TQ under the
representation
TQ c C(X(NIq x X(NIq) &t-
By Cor. 5.4, fio is a £ree '&-module of rank 2 and
Thus,
det(1- zq$/q) = ëd(z).
Now. H z 2 is a free q2-module of r d 4 and
Using the explicit expression for the (relative) characteristic polynomial of 4; and the
following element ary lemma. we compute the relative characterist ic poly nomial of (+i)@2
over ?y.
LEMMA 5.8. Let R be a commutative ring, let V be a j k e R-module of mnk 2 and let
cr, P E EndR V with
ch (a. x) = z2 - A1x + Ag, ch (p ,2 ) = z2 - B12 + BO.
Then the chamcteristic polynomial of a @ @ E Endmz v@* is
PROOF. Fix an R-basis of V. Say {vl , u2) and suppose that
with respect to this basis.
3. RANKIN-SELBERG L-FUNCTIONS
with respect to the basis { v 1 ~ v i , v i 8 v z , y@u1, v2@u2} of P2, where ai := a;@l , B, = 1 8 b i .
Now, we have
Al = al + a 4
A. = a1a4 - a ~ a g
and similarly for the Bi's.
Thus, we have
er = aiSr + ai194 + a4A+ a 4 8 4
and similarly the other ci's can be expressed as ploynomials in the aiBj73.
Computing the characteristic polynomial of the above matrix ushg MAPLE and com-
parhg coefficients gives the resuit. O
Now, let
8 , ( x ) = 1 - e ix + ezx2 - e3s3 + e4z4 E '&&],
w here
PROOF. Combining the Eider-Shimura relation (Theorem 5.3) Mt h Lemma 5.8 gives
the assertion. O
Now by the usual linear aigebra formula (see [Bol] Cor. on p. M8),
(where q!~: denotes the element of ~ n d at induced by the the Frobenius morphism on
X ( N ) F q ) where the last equation foilows from the fact that the n o m is invariant under
tensoring with Qalgebras (see [Bol], Ch. III 59.1).
We therefore &O have
where p:uP denotes the image of Tc under the jaithful representation Tc L, (En* V ) O Y
For two normalized Hecke eigenforms f, g in V := S2 ( ï (~ ) ) ? let
D EFINITION . Let f , g be normalized Hecke eigenforma of some level in V = Sk (I' ( N) ) ,
Associate to ( j, g ) the Rankin-Selberg L -function
where c = Xf Ag? the product of the Nebentypus charaeters of f, g.
REMARK. Our definition cliffers from the Wxmdard" one of [Li] (p. 135), in two insignif-
icant ways- Namely, we have not used the norrnalization s - s + k - 1 and, in addition, o u
definition of L ( f x g , s) agrees wit h t heir definition of L( f x g* , s) (up to the normalization) , where g* := C & ( f )qN is the komplex conjugate" newform of g.
From an analytic perspective. the dennitions of [Li] (for example) are of course, more
convenient .
3. RXXK1,hi-SELBERG L-FUNCTIONS 115
Now. recall that the L-function of a Hecke eigenform f E V has an Euler product of
the form
where, for a prime q f N.
PROPOSITION 5.10. Let f , g be two nonnaizzed newforms (of some leuel) in V . Then
the Rankin-Selberg L-funetion of j x g has an Euler pmduct
zuheie. for a prime q N.
PROOF. See [Mur], p. 55.
Using Prop. 5.10 and Lemma 5.8. we will easily verie the foilowing.
PROPOSITION 5.11. We have
for any newforms f , g E N ( V ) and any q f N.
PROOF. Multiplying out the product in Prop. 5.10 yields
where
3. RAXKIX-SELBERG L-FUNCTIONS 116
LEMMA 5.12. If 3 E is a normalized T(~)@*-ei~enjoonn, then f l i e n is a unique
pair of newforms f , g with
for al1 n. m coprime to N .
PROOF. By Atkin-Lehner theory.
gives a decomposition of V as a T-module and the VI's are precisely the distinct T(N)-
eigenspaces in V. Thus.
and the Vf If Vgos are precisely the distinct T(N)@~-eigenspaees in va2 . Thus. if 3 E v@* is a T(N)@~. then 3 E Vf @ V, for a unique pair of newforms
f, g E N(V) . Now by the proof of Cor. 4.12, we have
for aii n. m copnme to N. for a nonnalàzed eigenform 3 (where A n V m ( 3 ) denotes the (n, m)th
Hecke-eigenvaiue of 3. Thus we have
for ail nt m coprime to N.
Associate to 3 the "away from N" Rankin-Selberg Gfunction
ThmT Prop. 5.11 has the foilowing additional consequence, which we will use to prove
Theorem 4.3 in 56.
4. THE SEbII-SIMPLICITY OF TQ.
COROLLARY 5.13. I fN = p is ta prime, we have
for any prime q + p3 - p.
PROOF. We have
Thus, for any TI^)@* -eigenform E in the same eigenspace as Fr, we have
since the n o m of the polynomial @(z) depends only on T'@)@*sigenvalues.
4. The semi-simplicity of TQ.
In order to proceed towards a proof of Theorem 4.3, we k s t show that the Hecke algebra - @2
TQ E CQ is semi-simple. which, in tum. irnplies that each of the algebras Ili, fi, q2, v2, 4 are also semi-simple.
This fact will aUow us to give a particulary elementary derivation of the expression for
the relative characteristic polynomid of N, .
Since VQ is a fuithhl TQ-module. to show that TQ is semi-simple. it suflices to show
that TQ acts semi-simply on VQ.
4. THE SEMI-SIMPLICXTY OF aQ.
LEMMA 5.14. The jollowing are equivalent.
a) TQ as sema-simple.
6 ) T = TQ 8 C as semz-simple.
c ) i l = VQ 8 @ has a complete basis of T-eigenjorms.
PROOF. Recall that a commutative algebra is semi-simple if and only if it is a direct
sum of fields.
a) a b) : The algebra TQ is hite-dimensional and thus, if it is semi-simple, then it is a
direct sum of number fields. For any number field K, K 8 @ is a direct s u m of fields (copies
of @) and thus, TQ 8 C is &O a direct s u m of fields.
b)=+ a): If TQ is not semi-simple, then it has a ailpotent element a # O. Then a @ 1 is
a non-zero nilpotent element of Tc, so Tc is not semi-simple (since it is commutative).
b ) e c): Since T c is a commutative Gdgebra, a Tc-module is irreducible if and only
if it is 1-dimensional. Since V 2 Tc as a T-module, V is the direct sum of 1-dimensional
Tc-modules if and only if T is the direct sum of 1-dimensional, and therefore irreducihle,
T-modules. This shows that V has a complete b a i s of eigenforms if and only if Tc is
completely reducible or equivalently, semi-simple. Cl
PROPOSIT~ON 5.15. The Hecke algebra TQ is semz-simple.
P ROOF. By Lemma 5.14. it siiffices to show that V has a cornplete basis of T-eigenforms.
By Atkin-Lehner theory. we have a decomposition
where the Vf 's are precisely the distinct T@)-eigenspaces in V. Moreover,
It therefore sufices to show that, when nf = p, Vf has a complete basis of T-eigenforms.
By [Miy] Theorem 4.6.17. the Fourier coefficient a, = %(f) # O in this case.
We have
5. FULL SUBMODULES AND RELATn'E CHARACTERISTIC POLYNOMlALS
and thus, the form
f(p) = fI(1-c;) = / - %f&I
is a T-eigenforrn and. since a, # O, f and f(p) are iinearly independent and {f, fb)} is a
buis for VI. O
5. Full submodules and relative characteristic polynomials
Let T be a sez.&simple commutative algebra over a field F of chatacteristic zero and
let W be a fiee 2'-module of rank n. Consider T to be a subalgebm of EndF W .
DEF~NITION. A T-submodule U of W is called firll if it is free of rank n over the
restriction T f u.
For an irreducible character A E Irr(T), let W(X) denote the T-isotypic component.
T hus ,
and, in addition. since T is semi-simple commutative. for each X there is an idempotent
such that W (A) = WcA. Moreover. the cA's form a set of commuting, orthogonal idempo-
tents.
PROPOSITION 5.16. Let U be a T-submodule of W. Then the foiiowing are equiudent.
i ) LT is a fil l T-submodule
ia) LI = eAES W(X) for some subset S C Irr(T)
iii) U = WC for some idempotent r f T
iu) 11 2 W @T Tlt. as a T-module.
PROOF. i) + ii): the condition that W is fiee of rank n is equivalent to the assertion
that W(X) zz W," for each X E Irr(T). where WA denotes an irreducible T-module with
character A.
If ii) does not hold, then for X E Irr(T). we have
5. FGLL SLBhlODULES XSD RELATDrE CHARACTERISTIC POLYNOMLALS
for some m with O < rn < n. Thus, Li cannot be kee of rank n over Tlu.
ii) iii): The EX'S are a set of cornmuting, orthogonal idempotents. Thus, for a module
of the form
we have
üi) + iv): Since T is commutative. for any idempotent c E TT we have the orthogonal
decomposit ion
iv) i): Clearly. we have
w here
S := (A E Irr(T) : U(X) # O } .
Thus? the module W @T Tlu i U if and only U(X) = W(X) for aU X E S'. O
LEMMA 5.17. Let Tt be a T-algebm and let W' = W & Tt , so W' is a /ree T'-module
of mnk n. Let a E EndT W and extend ai Tt-iànearly to an element a' E EndT* W t . Then
we have
ch (af /T . x) = (ch (a/T. 2)) 8 1.
PROOF. See [Bol], equation (12) on p. 542.
COROLLARY 5.18. Let U Ç W be a full T-submodule and let a E EndT W. Then
where the right hand sàde means that the c o ~ u e n t s of ch (a/T, x) E T[x] are restricted to
U.
5. FULL SCBMODCLES AND RELATTVE CHARACTERISTIC P O L Y N O W S 121
PROOF. By Proposition 5.16, we have Tlu = Te and since T is commutative, this
implies Tlrr is a T-module. Thus, by Lemma 5.17, we have
whi& gives the above expression. El
Evidently. we can apply Prop. 5.18 and the preceding results to the full submodules
(using the fact t hat the Hecke dgebra TQ is semi-simple, ao q2 and v2 are a h ) .
PROPOSITION 5.19. a) The= is an idempotent cv E q2 such that vT = v @ ~ I , , . b ) There is an idempotent c - E q2 such that
%t
c) We have
det(l -ZN:& ét ) = @ ( z ) I ~ ~ = (08(4)2%4t.
PROOF. a) Since GQ normalizes the action of A,, in
each of the spaces ((vQ)e2@Q) h, a GQ-invariant bai s , that is, each V7 has a "Qstructuren
VG with V@@=V7.
Moreover, since vT = C, V l ( l - , , p r is a p2-module and the action of p2 on va2
"cornes &omr that of q2 on vg2, the space
Sioce q2 is semi-simple (we showed in 94 that TQ is semi-simple and thus, so is q2 and V$ is indeed a full $2-module. by Prop. 5.18, there is an idempotent a E q2 such
t hat
6. THE MORITA THEOREM
since va = O C.
b) Since there is a ~2-isomorphism
i t sufnces to show t hat there is an idempotent r E q2 such that
Since u$~, is isotypic (Theorem 4.28), this follovs from Prop. 5.18.
C) The restriction T - is equal to v2 . Since gk is an isotypic v2-submodule of %t "ét
Bg2, this follows h m Lemma 5.8. 17
In the next sections, we show that one can relate the two idempotents in Prop. 5.19
and thereby prove Theorem 4.3.
6. The Morita Theorem
Let C be a semi-simple algebra over Q and let V be a iaithful right Grnodule via
p : é v (End V)OY
DEFINITION. A subspace W C V is called algebraic (with respect to the representation
p ) if there is an a f C such that W = Imp(a) = VI,.
REMARK. Note that the image VI, of an element a E é depends ody on the left ideai
Ca that it generates,
VI4 = Imp(a) = Imp(Ca) .
Thus, a subspace W C V is algebraic if and only if there exist a finite set of elements
a l , . . . ak E SU& that
6. THE MONTA THEOREM
where 2i is the left ideal generated by the ai's,
Let Algc V denote the set of ail algebmic subspaces of V and let Lid(é) denote the set
of al1 Ieft ideals of C.
There are maps
(in a semi-simple algebra, any left ideal is generated by an idempotent) and
We have the following result. which as a generalization (due to E. Kani) of a speciai
case of the Morita Theorem(s).
THEOREM 5.20. The maps w and u are inverses of each other and thw, w defiries a
bijection
Lid(C) cr Algc(V).
In particular, for any finite sets of elements {ai, a*, . . . .at}, { b i t . . . , bi} C C,
PROOF. See [K3], proof of Theorem 3.2 for the case where V is a finitely generated
Grnodule. For the general case, since C is semi-simple and the representation
is faithful. we have that p is fait hfully 0at and t hus, the asertion holds by Theorem 3.1 of
* cl We are now ready to prove Theorem 4.3.
7. The L(l,ll-function of 2, BE',
For each 7 E (ZlpZ) " . let 6, be the element
d7 := cap,,(l - c p l a 2 ,
w here
By definition, we have
By Prop. 5.18. there are idempotents EV. c E q2 such that *tir
By Theorem 5.20, t his implies t hat the left ideals
are equal. as are the id&
Thus, for any faithful representat ion
we have
We can now prove Theorem 4.3.
PROOF OF THEOREM 4.3. By Prop. 5.19 c). we have
Since we now have that
by Theorem 5.20, we have
by Cor. 5.13.
For a nomalized Tb)-eigenform 3. Iet
where S is a set of primes containing those primes dividing N.
COROLLARY 5.21. We have
where S = ( q l p 3 - p } .
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