BOL. SOC. BRAS. MAT. VOL. 9, N. ~ 1 (1978), 89-95

Adele rings of global field of positive characteriste

89

Stuart Turner

1. The zeta function, the adele ring, and the Jacobian variety.

Let k be a global field of characteristic p > O with field of constants grq. Let f~ denote the set of places of k. For v e f~ let k v be the completion of k at v and qv be the cardinality of the residue field of k~. For s s C , Re (s)> 1 the zeta function (k of k is given by the absolutely convergent product (k(S) = [-I (1 -- q - S ) - 1

r E f 2

There exists a complete non-singular curve C defined over ~: such that K(C), the'field of rational functions on C, is isomorphic t o k~ C is unique up to U:q-isomorphism. Let N(C),, denote card C(IFr the number of 6:q,~rational points on C. Let ~c be the zeta function of C and set

r = q:S, tl'ien ~k(s)= ~c(r) and d(ln~c(r)) dT - ~ N(C), T"- t. This power n > _ l

series has radius Of convergence q-1.

Finally, ~c can be written

Pk(T) (1) (c(T) = ( 1 - T ) ( 1 - q T )

with Pk(T) 6 7/[T], deg Pk(T)= 2g where g is the genus of C. Pk satisfies

the functional equation Pk(T)=q y T2YPk(..--.~ ~ and Pk(O) = 1. / - - \

Let k~ deno te the adele ring of k.

Theorem 1. Let k', k" be global fields of positive characteristic such that Pk, = Pk,," Then k~ and k~ are isomorphic topological rings.

ProoJ2 Let P(T) be a polynomial which occurs as the numerator of the zeta function of some global field k of positive characteristic. It suffices to show that one can use P to construct a topological ring isomorphic to k A. First, (deg P)/2 is the genus .q of k. Write P(T)=a2 T 2 y + . . . + 1. From the functional equation for P one sees that the cardinality of the constant field of k is the positive integer q such that qg= a2,. Hence P determines ,q and q.

Recebido em novembro de 1977.

90 S. Turner

Let C be

~c(T) = (1 -

of P. there d(ln ~c( T))

dT lytic function n a positive

any complete non-singular curve defined over IFq such that

P(T) . (The examples in w show that for some choices T ) (1 - qT) can exist non-isomorphic C with this property.) Since

N(C),. T"-I and since this power series defines an ana- n > l in the disc I r l < q-~, the N(C),, are determined by P. For integer define M(C). = card {x e C(D:q.) I x ~ C(IFq.,), n' n,

n':~ n}. The subfields of IFq. containing B:q are of the form IFq., with n' n, so it is easy to see that the set {N(C).,},,<_. determines M(C).. As we have seen, the N(C)., can be determined from P.

For each positive integer n let r be a complete eguicharacteristic discrete valuation ring with residue field I:q.. By the Cohen structure theorem ([17], Chapter VIII, w r is isomorphic to I:q.[[X]]. Let k be the quotient field of r.. Let r,..~, r,..2 . . . . . r,M,. be complete discrete valuation rings isomorphic

to r and k, , , l . . . . . k,,.M,, be their quotient fields. Consider the sets

{k,..i}n~N and {r.i}.~N.

l < i < M n l < i < M n

Form the restricted direct product D of the k,..i with respect to the r,,.r D is clearly isomorphic to k A.

Corollary. Let k', k" be global fields with field of constants $ q such that (k, = (k,," Then k" a and k~ are isomorphic topological rings.

Proof The corollary follows immediately from the theorem and from (1).

Remark. It can be shown that there exist algebraic number fields k' and k" with the same q-function, but with non-isomorphic adele rings ([3], Theorem 1).

Theorem 2. Let k, k' be global fields of positive characteristic such that k A and k' a are isomorphic topological rings. Then (k = ~k, and Pk = Pk,"

Proof To prove the first assertion it suffices .to show that one can deter- mine ffk from k a. Let M be a closed maximal ideal of k a, then there exists a v~t s f~ such that k f fM and kvM are isomorphic topological fields. Conversely, given v ~ f~, there exists a closed maximal ideal M c k a such that k/Mo is isomorphic to k,. This correspondence between the closed maximal ideals of k A and f~ is one-to-one ([2], proof of lemma 7). Let ro be the maximal

Adele rings of global fields 91

compact subring of k o, m o be the maximal ideal of r v, and qv = card (ro/mo). Since (k(s)= I-I ( 1 - q -~)-1, kA determines (k"

r~f~

Let q be the cardinality of the field of constants-of k. Since Pk(q -~) = = ( 1 - q -Z) (1- ql-S)(k(S), to prove the second assertion of the theorem it suffices to show that one can determine q from k a. The topology of kA x, the adele group of k, call be defined directly from the topology of k A, without referring to k or its completions ([-13], Chapter IV, w definition 2). Let !l'kA x--, IR+ ~ be the continuous homomorphism defined by the Haar module. Then q generates the image of II in IR+ x. ([-13], Chapter IV, w proposition 3; Chapter VII, w corollary 6.).

Remark. The same proof shows that two algebraic number fields with isomorphic adele rings have the same zeta funtion.

Let k, k' be global fields with field of constants IFq and C, C' be complete, non-singular curves defined over [Fq with their function fields isomorphic to k and k' respectively. Let A = J(C) and B = J(C') be their Jacobian varieties. A and B are defined over IF. Let ~A ~ End A (resp. rc~ E End B) be the Frobenius endomorphism of A (resp. B) and fA (resp. fn) be the characteristic polynomial of ~a. Then Pk(T)= qgfa(T ). Furthermore, A is F(isogenous to B if and only ifJ~ = f n ([,10), theorem 1). Combining this result with the corollary to theorem 1 and theorem 2 gives.

Proposition. Let k, k' be global fields with field of constants ~:q. the following are equivalent:

i) ~k = ~k'" ii) J(C) and J(C') are ~q-isogenous.

iii) k a and k' a are isomorphic topological rings.

Then

In [,11] it is shown that there exist infinitely many non-isomorphic singular curves of genus two defined and irreducible over 0:pn, n odd, which have the same zeta function.

2. The adelic theta funct ion

It is not possible in general to reconstruct C from J(C). However, if ~g(0) denotes the polarization of J(C) defined by a canonical divisor 0 (or equivalently by a translation 0 a of 0) on J(C), C can be reconstructed from the principally polarized abelian variety (J(C), rg(0)) ([7], [,16]). In the classical case where C is replaced by a compact Riemann surface M and

92 S. Turner

J(C) by an algebrizable complex torus J(M) it is possible to choose the canonical divisor 0 on J (M) so that 0 is "cut out" by the zeros of Riemann's theta function ([5], [6], [8]). We now show that it is possible to recover k from k A in an analogous way.

Recall that k can be canonically identified with a discrete subfield of k a which we also denote by k. Let /~A be the Pontryagin dual of k A. Let T be the group of complex numbers of modulus one and < . > :kAX ka ~ T denote the canonical pairing. Let R be the maximal compact subring of k A and ~ be the characteristic function of R. Define

Ok : kAx ~A --" C

by

Ok(x, x*) = fk r + ~) < 4, x* > d~(~),

where a i s the Haar measure on k which gives each point measure o n e . O k

is the adelic analogue of Riemann's theta function ([15], w167 16, 40). 0 k is continuous and since k is discrete

Ok(x, x*) = y. r + 4) < ~, x* >. {~k

Proposition. Le t= Fq* = { x * 6 l c A l x * ( a ) = 1 for a6[Fq}. Let x 6 k A. I f there exists ~ ~ k such that x + ~ ~ R, then Ok(X, X*) ---- .q < ~, , x * > , for x* ~ Fq. and Ok(x,x*)=O for x * r I f there exists no ~ ~k such that x + ~ ~R, then Ok(X ,x*) = 0 for all x* ~ kA"

Proof Let ~ k be such that x + ~ R . Then x + ~ + a ~ R for all a ~ IFq and ~ ' ~ k has the property x + ~ ' e R if and only if 4 ' = ~ + a for some a~ Fr Hence

ok(x, x*) = y <~ + a, x*> = <~, x*> ~ <a, x*> a @ Fq a E O:

for any x*~[c A. If x*~ [Fq., this sum has the value q<4, x*>. If x r [Fq,, this sum is zero. The last assertion of the proposition is obvious.

Theorem. Let A be a topological ring and k, k" be global f ields such that k A and k' A are isomorphic to A. Let (o : A - , k A ,and ~ : A - , k" A be isomorphisms of topological rings and ~ : ~a ~ "4, (O : ~c A ~ .~ b e the dual isomorphisms. Let O a, k : A x .4 ~ C (resp. O a,k, : A x .,~ --* C) be the function

Adele rings of global fields 93

O:(c~, $-1) (resp. O~ Let F and F' be the fields of constants of k and k" respectively., I f 4)- I(F) = O- l(F') and 0 A k(X, X*) = 0 a k,(X, X*) for all x e ~9- i(k) U O - l( k ) and all x* ~ 49- l(F),, then "~- l(k) = O-'i( k ).

Proof. Let ~q=Cb- i (F)=$- l (F ' ) . By the propositon OA.k(X,X*)= = q( -- X, X*) for each x*e ~:q,, so by the hypothesis OA,k,(X, X*) = q( -- X, X*). Using the proposition one sees that there exists r such that OA,k,(X, X*)= = q ( $ - l(~,), X*) for each x* e )Fq,. So ( - x, x*) = ( ~ - 1(r x*) for each x*e~q, . However, the set of x* in A7 which induce the trivial character on IFq separate the cosets of IFq in A. So ~ - 1(~') = - x + a for some ae IFq. Hence 4>- ~(K) c ~ - t(k'), symmetrically $ - t(k') c ~k- l(k).

3. Examples.

We now give two ways of constructing sets of isogenous, non-isomor- phic cuves of genus one. By the proposition of 1. Such curves have the same zeta function and the adete rings of their function fields are isomorphic.

Let j e I]:qn, n 1, j r ~:q. There exists an elliptic curve /: defined over IFqn and unique up to D:qn-isomorphism with invariant j([4]). Let k be the function field of E and k q. be the image of k under the endomorphism x ~ x q. There is an elliptic c u r v e E (q) defined over ~qn with function field k q. E (q) has invariant ft. Since j r E and E ~q~ are not isomorphic over IF--q or over any other field containing ff:q. Let i : E--+ E ~q) be the morphism determined by the inclusion k q c k. i is a purely inseparable isogeny of degree q.

The existence of non-isomorphic elliptic curves which are separably isogenous can be shown using the general theory of abelian varieties over finite fields. We begin by recalling the principal theorems.

Let A be an abelian xariety defined over [i:r n a be the Frobenius endomorphism of A and fA be the characteristic polynomial of x A. fa is. a monic polynomial degree 2 dim A with integer coeffients. The roots of fA have complex absolute value ql/2 ([15]). Since fa(Xa)=0, nA may be regarded as an algebraic integer determined up to conjugacy. For every embedding p 'Q(xA)--*C, p(xa) has absolute value qi/2. Generally, an algebraic integer n is called a Well number for q if for every embedding p " �9 p(x) has absolute value ql/2 ([9], [12]). There is a one-to-one correspondence between the set of conjugacy classes of Weil numbers for q and the set of isogeny classes ,of elementary abelian varieties defined over Fq (Eli, [9], [12]).

From now on we assume that A is elementary and q = p".-Let End A denote the ring of I:(endomorphisms of A and let E = (End A)�9 E is a

94 s. Turner

division algebra with center F = �9 E has invariant 1/2 at each real place of F, invariant zero at each finite place of F prime to p, and invariant

o rd~ (ha) [ F v . Q p ] at a place v of F lying over p. Finally, 2 d im A = ordv (q) = [ E : F ] I I 2 [ F " ~)] ([9]).

Let I be an ideal in E n d A and H ( I ) = r~ Ker a. I is called a kernel ideal if I = {alaH(1)=O } ([12], 3.2). Let I, J be kernel ideals. Then A / m l ~ is 0 : - i somorphic to A / mj ) if and only if I = Z] for some invertible

2 ~ E ([12], theorem 3.11). Let M be a maximal order in E. Then there is an abelian variety B, 0:~-isogeneous to A, with End B - M ([12]) theorem 3.13).

Every ideal I ~ End B is a kernel ideal and the rank of H(I ) equals the reduced n o r m of I ([12], theorem 3.15); End B/H,~ is also a maximal order ([12], theorem 3.14).

�9 - 1 + ~ - S 3 1 - 1 - ~ / - 3 1 Consider r e = , ~ = . lr and ~ are Weil

2 2 numbers for 8. Let A be an elementary abelian variety in the isogeny class

determined by ~r and ~ .Z [T r ] is the maximal order in � 9 31) and (2) = PIP2, where

(1 ( = -- -- ' - ' P2 = 1 + ~ / - - 3 1 ,

(z~)=p 3 and ( ~ ) = p ~ . Tate 's theorem shows that dim A = 1 and End A x �9 ~_ �9 0. By Waterhouse ' s theorems there exists an elliptic curve E defined over [F 8 and isogeneous to A such that End g = Z[rc]. Since the norm of p~, i = 1, 2, is two, the isogenies i t : E --, E/u(p o and i 2 : E ~ E/n<p~ )

are of degree two. As (1), Pl, and P2 are representatives of the three ideal classes of � 9 31), no two of these curves are i somorph ic over 0: s. The isogenies i I and i 2 may be inseparable, however theorem 5.3 of [12] implies that thare exist separable isogenies f rom any one of these curves to any other.

References

[1] T. Honda , Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20, 1968, 83-95.

[2] K. Iwasawa, On the rings of valuation vectors, Ann. Math 57, 1953, 331-356. [3] K. Komatsu. On the adele rings of algebraic number fields, preprint. [4] S. Lang, Elliptic Functions, Addison-Wesley, Reading, Mass, 1973, Appendix I by J. Tate. [5] J. Lewittes, Riemann Surfaces and the theta functions, Aeta Math. 111, 1964, 37-61. f6| H. Martens, Three lectures on the classical theory of Jacobian varieties, Algebraic geometry,

Oslo, 1970: Wolters-Noodhoff, Groningen, 1972.

Adele rings of global fields 95

[7] T. Matsusaka, On a theorem of Torelli, AJM 80 (1958), 784-800.

[8] A Mayer, Generating curves on abelian varieties and Riemann's theta-function, Ann. Seuola Norm. Sup. Pisa, Ser. III, 19, 1965, 107-111.

[9] J. Tare, Classes d'isog#nie des vari~t#s ab~liennes sur un corps fini, Sem. Bourbaki 21, 1968/69, no. 352.

[10] J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2, 1966, 134-144.

[11] S. Turner, Principal polarizations of abelian surfaces over finite fields, to appear.

[12] W. Waterhouse, Abelian varieties over finite fields, Ann. Scient. l~c. Norm. Sup, 4, t. 2, 1969, 521-560.

[13] A. Weil, Basic Number Theory, Die Grundlehren der math. Wissenschaften, Band 144, Springer-Verlag, Berlin and New York, t967.

[14] A. Well, Courbes alg~briques et variet~s ab~liennes, Hermann, Paris, 1971.

[15] A. Weil, Sur certains groupes d'opdrateurs unitaires, Acta Math., 111, 1964, 143-211.

[16] A. Well, Zum Beweis des Torellischen Satzes, Nachrichten der Akademie der Wissens- chaften in G6ttingen 2 (1957), 33-53.

[17] O. Zariski and P. Samuel, Commutative Algebra, Van Nostrand Princeton, 1960.

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