Number Theory: Séminaire de Théorie des Nombres, Paris, 1992-3 0521559111, 9780521559119

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London Mathematical Society Lecture Note Series. 215

Number Theory Seminaire de Theorie des Nombres de Paris 1992-3

Edited by

Sinnou David Universite Pierre et Marie Curie, Paris

CAMBRIDGE UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge C132 I RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1995 First published 1995 Library of Congress cataloging in publication data available British Library cataloguing in publication data available

ISBN 0 521 55911 1 paperback Transferred to digital printing 2004

Number Theory Paris 1992-93

Table des Matieres K. ALLADI

Decomposition of the integers as a direct sum of two subsets ..................... I Y. ANDRE

Theorie des motifs et interpretation geometrique des valeurs p-adiques de G functions (une introduction) .................................................................37 N. BOSTON

A refinement of the Faltings-Serre method ...............................................61 J. BOXALL

Sous-varietes algebriques de varietes semiabeliennes sur un corps fini..69 P. COHEN Proprietes transcendantes des fonctions automorphes .............................81 E. FOUVRY et M. Ram MURTY

Supersingular primes common to two elliptic curves .................................91 V GRITSENKO

Arithmetical lifting and its applications ..................................................103 G. HARMAN

Towards an arithmetical analysis of the continuum ............................... 127 H. HIDA

On A-adic forms of half integral weight for SL (2) /Q .............................139 J. MARTINET

Structures algebriques sur les reseaux .................................................. 167 H. OUKHABA

Construction of elliptic units in function fields ........................................187 I. PAYS

Arbres, ordres maximaux etformes quadratiques entieres ..................... 209 T.N. SHOREY

On a conjecture that a product of k consecutive positive integers is never equal to a product of mk consecutive positive integers except for8.9.10=6! .......................................................................................231 P. STEVENHAGEN

Redei-matrices and applications ...........................................................245

R. TIJDEMAN

Decomposition of the integers as a direct sum of two subsets .................261 Y.G. ZAHRIN

CM Abelian varieties with almost ordinary reduction .............................. 277

Number Theory Paris 1992-93

Liste des conf6renciers 5 octobre : H. CARAYOL. - Representations galoisiennes et congruences 12 octobre : E. PEYRE. - Points de hauteur donnee sur une surface de Del Pezzo

D. BROWNAWELL. - Transcendance on Drinfeld modules 19 octobre : I. PAYS. - Algebres de quaternions et formes quadratiques entieres

2 novembre : G. HARMAN. - Types of fractions near irrationals 9 novembre : P. RAMBOUR. - Proprietes galoisiennes d'un anneau d'entiers en caracteristique p

16 novembre : T. HALES. - Sphere packings 23 novembre : E. FOUVRY. - Nombres premiers supersinguliers

30 novembre : V. GRITSENKO. - Maass wave functions on four dimensional hyperbolic space

7 decembre : T.N. SHOREY. - Squares in products from a block of consecutive integers 14 decembre : R. TIJDEMAN. - Complementing sets of integers

4 janvier : M. HINDRY. - Hauteurs de Neron sur les varietes abeliennes

11 janvier : Q. LIU. - Modeles stables des courbes de genre deux 18 janvier : G. CHRISTOL. - Indice d'operateurs differentiels 25 janvier : R. COULANGEON. - Reseaux unimodulaires et quaternioniens lei fevrier : N. BOSTON. - Some applications of families of Galois repre-

sentations

8 fevrier : H. OUKHABA. - Unites elliptiques et unites cyclotomiques dons les corps defonctions lei mars : J. BOXALL. - Sous-varietes algebriques de varietes abeliennes sur un corps fini 8 mars : J. MARTINET. - Structures algebriques sur les reseaux 15 mars : E. ULLMO. - Geometrie d'Arakelov des courbes elliptiques

22 mars : J.-M. COUVEIGNES. - Calcul et rationalite desfonctions de Belyi

29 mars : R. MURTY. - The supersolvable reciprocity law

5 avril : M.L. BROWN. - Modules singuliers et modules supersinguliers de modules de Drinfeld 26 avril : L. SCHNEPS. - Groupe de Grothendieck-Teichmiiller et automorphismes de groupes de tresses

3 mai : J.-P. WINTENBERGER. - Relevements de representations adeliques associees awc motifs de modules de Drinfeld

10 mai : D. BLASIUS. - Good reduction of Shimura varieties 17 mai : S. SAITO. - Cohomological Hasse principle for a surface over a number field

24 mai : H. HIDA. - A-adic forms of half-integral weight 7 juin : W. RASKIND. - Applications d'Abel-Jacobi £-adiques superieures Y. ANDRE. - Pour une theorie inconditionnelle des motifs 14 juin : K. ALLADI. - The combinatorics of words and applications to partitions

21 juin : P. COHEN. - Proprietes transcendantes des fonctions automorphes

28 juin : M. KUWATA. - Points rationnels sur les tordues de courbes elliptiques

Y. ZARHIN. - Hodge and Tate cycles on simple abelian fourfolds

Les textes qui suivent sont pour la plupart des versions ecrites de conferences donnees pendant l'annee 1992-93 au Seminaire de Theorie des Nombres de Paris. Ce seminaire est financierement soutenu par le C.N.R.S. et regroupe des arithmeticiens de plusieurs universites et est dotee d'un conseil scientifique et editorial. Ont ete aussi adjoints certains textes

dont la mise a la disposition d'un large public nous a paru interessante. Les articles presentes ici exposent soit des resultats nouveaux, soft des syntheses originales de questions recentes ; ils ont en particulier tous fait l'objet d'un rapport. Ce recuefi doit bien sur beaucoup a tous les participants du seminaire et a ceux qui ont accepte d'en reviser les textes. Il doit surtout a Monique Le Bronnec qui s'est chargee du secretariat et de la mise au point definitive du manuscrit; son efficacite et sa tres agreable collaboration ont ete cruciales dans l'elaboration de ce livre.

Pour le Conseil editorial et scientifique S. DAVID

Number Theory Paris 1992-93

The method of weighted words and applications to partitions Krishnaswami Alladi

ABSTRACr. The study of identities of Rogers-Ramanujan type forms an important part of the theory of partitions and q-series. These identities relate partitions whose parts satisfy certain difference conditions to partitions whose parts satisfy congruence conditions. Lie Algebras have provided a natural setting in which many such identities have arisen. In this paper a new technique called "the method of weighted words" is discussed and various applications illustrated. The method is particularly useful in obtaining generalisations and refinements of various Rogers-Ramanujan type identities. In doing so, new companions to familiar identities emerge. Gordon and I introduced the method a few years ago to obtain generalisations and refinements of the celebrated 1926 partition theorem of Schur. The method has now been improved in collaboration with Andrews and Gordon thereby increasing its applicability. The improved method yielded a generalisation and a strong refinement of a recent partition conjecture of Capparelli which arose in a study of Lie Algebras. Another application is a refinement and generalisation of a deep partition theorem of Gollnitz. A unified approach to these partition identities is presented here by blending the ideas in four of my recent papers with Andrews and Gordon. Proofs of many of the results are given, but for those where the details are complicated, only the main ideas are sketched.

1. - Introduction Identities of Rogers-Ramanujan type form an important part of the theory of partitions and q-series. Generally, one side of these identities is in the form of an infinite series, while the other side is an infinite product.

Usually, the series is the generating function for partitions whose parts satisfy certain difference conditions whereas the product is the generating function for partitions whose parts satisfy congruence conditions. The literature on such identities is vast (see for instance Andrews [101, [11)).

2

K. ALLADI

Andrews' monograph [I I I gives a quick overview of several recent major advances and discusses applications to many areas within mathematics, and even to Physics. The generic name "Identities of Rogers-Ramanujan Type", stems from

the two celebrated identities due independently to L. J. Rogers and S. Ramanujan which are the prototype, namely, 00

CIO

1

q

n=O

(1 - q)(1 - q2)...(1 - qn)

M=0

(1 - q5-+l)(1 - q5m+4)

and 00

qn2+n

(1.2)

n=0

n = H (1-q)(1-q2 )...(1-q) M=0

1

(1 - q5m+2)(1 - q5m+3)'

Indeed, even today, (1.1) and (1.2) are unmatched in simplicity, elegance and depth. These two identities have nice combinatorial interpretations as observed by MacMahon and Schur : THEOREM R. - For i = 1, 2, the number of partitions of an integer n into parts with minimal difference 2 and each part > i, is equal to the number of partitions of n into parts =- ±i (mod 5).

However, no simple combinatorial proof of (1.1) and (1.2) is known. One reason for the difficulty is that we do not know how to refine (1.1) and (1.2) by introducing a free parameter whose power would represent an important statistic in the partitions being counted. The term refinement is explained below.

There are several examples of Rogers-Ramanujan type identities for which refinements are known, the most famous being the 1926 partition theorem of Schur [221 : SCHUR'S THEOREM. - Let S(n) denote the number of partitions of n into

distinct parts - ±1 (mod 3). Let Si (n) denote the number of partitions of n into parts with minimal difference 3, and such that no two consecutive multiples of 3 can occur as parts. Then S(n) = Si(n)-

The equality in Schur's theorem can be refined to (1.3)

S(n; k) = Si (n; k),

TILE METHOD OF WEIGHTED WORDS ANDAPPLICA77ONS TO PARTITIONS

3

for any positive integer k, where S(n; k) and Si (n; k) count partitions of the type counted by S(n) and Si (n), but with the added restriction that there are precisely k parts, and with the convention that parts - 0 (mod 3) are counted twice. Combinatorial proofs can usually be given for partition identities which permit refinements. This is also the case with Schur's theorem; for combinatorial proofs of Schur's theorem, see Bressoud [14] or Alladi-Gordon 111.

Schur had originally stated the equality of three partition functions T (n) = S(n) = S1 (n), where T (n) is the number of partitions of n into parts ± 1 (mod 6). We do not discuss T(n) in this paper because refinements of the type (1.3) are not possible with T(n). The equality T(n) = Si(n) can

be considered as the next case beyond the Rogers-Ramanujan identities because the minimal difference 2 is replaced by 3 and the modulus 5 is replaced by 6. But in doing so, Schur realised that he needed the extra condition that consecutive multiples of 3 should not occur as parts. The purpose of this paper is to discuss a new technique called the method o f weighted words originally due to Alladi and Gordon [ 1 ], [2], and which has recently been improved by Alladi, Andrews and Gordon [3], [4]. This method is particularly useful in the study of Rogers-Ramanujan identities which permit refinements. Indeed, the method provides substantial refinements and generalisations often involving several free parameters which keep track of the number of parts in various residue classes. In many instances these refinements lead to bijective proofs of the partition theorem in question, in a natural way. The basic idea of the method is to consider positive integers which occur in various colours denoted by a, b, c,.... We then form words whose letters are symbols a, b, c.... with subscripts, where the subscript denotes the integer or part of the partition, and the symbols a, b.... will denote the colour

of that part. Special types of words are considered by impossing certain order rules on the symbols and certain gap conditions on the subscripts. These will correspond to the gap conditions on the parts in the partition theorem being discussed. The usefulness in the method lies in the fact that the symbols a, b, c.... play a dual role. On the one hand they represent colours, and on the other, they are free parameters when computing generating functions. The upshot of all this is that the partition theorem in question becomes a special case under certain dilations and translations, and the free parameters a, b, c.... provide the necessary refinements of the partition theorem. By changing the order rules on the symbols a, b, c..... several new companion partition identities are generated. The method was introduced in [11 to obtain refinements and generalisations of Schur's theorem. Subsequently [2], it was noticed that by changing the order rules for the symbols several companions to Schur's

4

K. ALLADI

theorem would be generated. More precisely, the function Si (n) is only one of six partition functions S,, (n),µ = 1, 2, ... , 6, all equal. Generalisations, refinements and companions to Schur's theorem are discussed in § § 4, 5.

In recent years, Lie Algebras have provided a general setting where Rogers-Ramanujan type identities have been discovered (see Lepowsky and Wilson [19], [20], [211). Motivated by a study of Vertex Operators in Lie Algebras, Capparelli [ 151 [16] made the following conjecture : CAPPARELLI'S CONJECTURE. - Let C* (n) denote the number of partitions of

into parts = ±2, ±3 (mod 12).

Let D(n) denote the number of partitions of n into parts > 1 with minimal difference 2, where the difference is > 4 unless consecutive parts are multiples of 3 or add up to a multiple of 6. Then C*(n) = D(n). Capparelli's conjecture was proved by Andrews in 1992 [12] using generating functions. Since there are similarities between Capparelli's conjecture and Schur's theorem, it was natural to see whether the method of weighted words would apply. And indeed it did, but the key idea was to replace C* (n) by the equivalent function C(n) which denotes the number of partitions of n into distinct parts = 2,3,4 or 6 (mod 6). This is for the purpose of refinements. In otherwords, C* (n) is like T (n) in Schur's theorem for which refinements are not possible and C(n) is like S(n) in Schur's theorem which permits refinements. Andrews, Gordon and 1 [3] obtained a three parameter refinement and generalisation (see Theorem 14 in § 7) from which Capparelli's conjecture followed as a special case under suitable dilations and translations. The deepest application of the method of weighted words so far has been a proof of a substantial generalisation and refinement due to Andrews, Gordon and myself [4], of the following formidable theorem of GOllnitz [ 18] : GOLLNITZ'S THEOREM. - Let A(n) denote the number of partitions of n into

parts - 2, 5, 11 (mod 12). Let B(n) denote the number of partitions of n into distinct parts

2, 4,

or5 (mod 6). Let C(n) denote the number of partitions of n in the form ml + m2 + + m,, no part equal to 1 or 3, and such that m1-m1+1 > 6 with strict inequality if mi - 6, 7 or 9 (mod 6).

Then A(n) = B(n) = C(n).

The equality A(n) = B(n) is trivial, and so the real challenge is the equality B(n) = C(n). Moreover, the function A(n) is like C*(n) in Capparelli's conjecture and T(n) in Schur's Theorem, permitting no refinements. So we will not discuss A(n) in this paper.

7HE METHOD OF WEIGHTED WORDS AND APPLICA77ONS TO PAR7TITONS

5

Gbllnitz [18] actually established the refinement

B(n;v) =C(n;v),

(1.4)

where B(n; v) and C(n; v) denote the number of partitions counted by B(n) and C(n) respectively, with the additional restriction that there are precisely v parts, and the convention that parts - 6, 7 or 9(mod 6) are counted twice. His proof is complicated and the details are forbidding. Andrews [8] gave a proof using generating functions, and subsequently gave a second proof ([ 111, § 10) where he used computer algebra to simplify the calculations. He then asked for a proof which lends more insight into the equality (1.4). We believe that our proof by the method of weighted words does provide this insight (see § 9 for a sketch of the main ideas behind this proof).

The main idea is to consider integers in six colours, of which three colours a, b and c are primary, and three colours ab, ac and be are secondary.

We then impose certain order rules on the coloured integers. Gbllnitz' theorem is viewed as emerging from an incredible key identity (see (9. 11) in § 9). The proof of this key identity is deep and difficult and may be found in [4]. It requires not only Watson's q-analog of Whipple's theorem but also the 6q'6 summation of Bailey. This is substantially more than what is required to prove either Schur's theorem or the Rogers-Ramanujan identities.

One advantage in using primary and secondary colours is that this explains the requirement for strict inequalities in Gbllnitz' theorem when mI - 6, 7 or 9(mod 6). As will be seen in § 9, the residues 2, 4 and 5(mod 6) correspond to the primary colours a, b and c whereas the residues 6, 7 and 9(mod 6) correspond to the secondary colours ab, ac and be. Another advantage with our approach is that Schur's theorem falls out as a special case by setting c = 0. For then, colours c, ac and be disappear and we are left only with three colours a, b and ab, which corresponds to Schur's theorem. The paper concludes with § 10 where some further problems going beyond Gollnitz' theorem are described.

2. - Notations We adopt the standard notation n-1

(a)n = (a; q)n =

(1 - aqj) j=0

for any complex number a, and 00

(2.1)

(a),, = n-00 lim (a)n = fl(1 - aqj), for j=0

IqI < 1.

6

K. ALLADI

In fact (2.1) can be used to define (a),,, for all real n by means of the relation (a)n =

(2.2)

.
MQ?k). Pour associer a une cohomologie classique sur Vk, disons la cohomologie cristalline, un foncteur fibre sur M(Vk), it faut faire agir les correspondances motivees modulo - sur les espaces de cohomologie cristalline

en respectant les fonctorialites idoines, c'est-a-dire relever de maniere "coherente" les classes mod. - en de vraies correspondances motivees. C'est possible, du moins lorsque V est "engendree" par un seul schema X de fibre generique XK connexe [A93][A] :

TxfoRkME 7. - II existe un foncteur fibre gradue 4HHri8 sur M(X®) a

valeurs dans les W(k)Q-espaces vectoriels, tel que pour tout m, Hcris(X= /W(k))Q, et tel que l'isomorphisme de BerthelotOgus induise un isomorphisme de foncteurs fibres gradues HHR 0 CP Hcris ®Cp) o sp sur M(-K)

Dans le cas d'une famille projective lisse X -> S avec bonne reduction

en v comme au § 2, on peut appliquer ce resultat en prenant pour X un modele de Xe, pour chaque T; E S(K) p-adiquement proche de 0, separement. La compatibilite des constructions pour plusieurs est 0 etant fixe, j'ignore par exemple si la classe cristalline problematique : de la specialisation de tout dans H,,,,is(Xok) = Hcris(X= ) 15

Voir note (8).

THEORIE DES MOTIFS ET INTERPRtFATION GEOME7RIQUE

57

cycle motive sur une puissance de Xo coincide avec la classe en theorie H,,*ris ; c'est toutefois vrai pour les intersections de classes de diviseurs sans hypothese supplementaire sur Xk, et pour tout cycle motive si Xk est une variete abelienne. Cela permet d'appliquer la construction du § 7 dans de nombreuses situations en se passant de l'hypothese (N), du moans lorsque Q = Q.

c) Citons quelques applications du theoreme 6 dans le cas d'un corps de base k C_ C. Soient S un schema reduit connexe de type fini sur C, f : X -> S un morphisme projectif et lisse, et 0 une section globale du faisceau R2P f,,Q(p). Grothendieck a conjecture que si la fibre 03 est algebrique en un point

s c S(C), it en est de meme en tout point [G661. En s'appuyant sur le "theoreme de la partie fixe" de Deligne [D71 ] § 4, on montre :

Tii9OREME 8. - Si la fibre 0, est motivee en un point s E S(C), it en est de meme en tout point (pour un choix convenable de V). Avec les notations du § 7b, on en deduit : COROLLAIRE. - Lie Gdj f,g est un ideal de Lie G,g ; c'est aussi un.sousespace motive de End HdR(Xe).

Enfin, a l'aide d'un cas particulier du theoreme de deformation 8 et de [A92], une argumentation suivant le fil de celle de Deligne pour prouver que tout cycle de Hodge 1'est absolument sur les varietes abeliennes, permet de montrer : ThIoREME 9. - Tout cycle de Hodge sur une variete abelienne complexe est motive (pour V convenable).

d) Quelques mots sur la demonstration du theoreme 1 [A]

On se ramene a la situation du §2 define sur un certain corps de nombres de base K0, par le changement de variable x t--> 1/x, et en remplacant f par 1/x n o f : X = A --> S (a fibres non geometriquement connexes), n etant choisi de facon a tuer la ramification au-dessus de x = 0 (pour le nouveau choix de x). Les coefficients de la matrice Y(x) sont alors des series de puissances en x a coefficients dans Ko (en fait des G-fonctions). D'autre part, le nombre de places v de K = K0(Z;') divisant exceptionnel est borne par hypothese (par n52). Le principe de la preuve consiste a construire des relations polynomiales 0 de degre borne a coefficients dans une extension de degre borne de K entre les evaluations v-adiques en Z; des coefficients d'une

58

Y. ANDIZt

matrice Y(x) solution de 1'equation de Picard-Fuchs, a multiplier entre elles ces relations (il y en a au plus nb2), avant de conclure par le "principe de Hasse" de Bombieri cite dans l'introduction (compte-tenu de cc que les points de hauteur et degre bornes sont en nombre fini). L'hypothese que les varietes abeliennes dans la fibre x = 0 sont de type CM entraine que la composante neutre de Go = G,,,,ot(Xo) est un tore. Nos motifs seront modeles sur des varietes abeliennes ; nous utiliserons les consequences suivantes du fait d'avoir affaire a des varietes abeliennes : i) Q = Q, car sur toute variete abelienne, l'involution de Lefschetz est algebrique (Lieberman-Grothendieck) ; ii) X. a bonne reduction potentielle partout; de la resulte 1'existen-

ce d'un revetement etale S' -> S, et pour toute place finie w, d'un ouvert U,,, de Zariski de SK_ au-dessus duquel f 79 a bonne reduction (Ogus). 0 se fait en La construction des relations "exceptionnelles" Q(Y; adaptant l'argument d'espaces homogenes de 7b. Enfin, borner le degre de relations exceptionnelles revient a borner le degre de certains tenseurs "exceptionnels" invariants sous G,,,,ot (Xg) ; pour cc faire, on recourt au lemme suivant :

LEMME. - Sott G un groupe algebrique semi-simple complexe. Il n'y a qu'un nombre fini de classes de conjugaison de sous-groupes fermes connexes H C G contenant le centre de la composante neutre de leur normalisateur dans G.

Manuscrit recu le 22 janvier 1994

TH9ORIE DES MOTIFS ET INTERPRETATION GEOME`IRIQUE

59

BIBLIOGRAPHIE

[A86] Y. ANDRE. - Multiplication complexe daps un pinceau de vartetes abeliennes, Sem. de Theorie des Nombres de Paris, 1984-85, (C. Goldstein, ed.) Progress in Math. 63 Birkhauser Boston (1986), 1-22. [A891 Y. ANDRE. - G -Junctions and Geometry, Aspects of Math. vol. E13, Vieweg, Braunschweig/Wiesbaden (1989). [A901 Y. ANDRE. - p-adic Betti lattices, in "p -adic analysis" Proceedings of the Trento conference, (F. Baldassarri, S. Bosch, B. Dwork, eds.) Springer L.N.M. 1454 (1990). (A92] Y. ANDRE. - Une remarque a propos des cycles de Hodge de type CM, Sem. de Theorie des Nombres de Paris, 1989-90, (S. David, ed.) Progress in Math. 102 Birkhauser Boston (1992), 1-7. [A931 Y. ANDRE. - Pour une Teorie inconditionnelle des motifs, soumis a publication, (premiere version prepubliee a 1'Univ. Paris 6). [A] Y. ANDRE. - Realisation de Betti des motifs p-adiques, en preparation, (premiere partie prepubliee a l'I.H.E.S., Avril 1992). [BO831 P. BERTHELOT, A. OGUS. - F-isocrystals and the De Rham cohomology I, Inv. Math. 72 (1983), 159-199. 1Be931 F. BEUKERS. - Algebraic values ofG functions, J. reine angew. Math.

434 (1993), 45-65. [Bo8 1] E. BOMBIERI. - On G -functions, in Recent progress in analytic number theory, Durham 79, Academic Press (1981) vol. 2, 1-67. [D711 P. DELIGNE. - Theorie de Hodge II, Publ. Math. I.H.E.S. 40 (1971),

5-57. [D801 P. DELIGNE. - La conjecture de Weil II, Publ. Math. I.H.E.S. 52 (1980), 137-252. [D901 P.

DELIGNE. - Categories tannakiennes, in The Grothendieck

Festschrift, Birkhauser Boston (1990), vol II, 111-195. [F891 G. FALTINGS. - Crystalline cohomology and p-adic Galois representations, in Algebraic analysis, Geometry and number theory, J.I. Igusa ed., proc. of the JAMI inaug. conf. John Hopkins Univ. (1989) 25-80.

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[Ge86] L. GERRITZEN. - Periods and Gauss-Manin connection for families of p-adic Schottky groups, Math. Ann. 275 (1986) 425-453. [G661 A. GROTHENDIECK. - On the de Rham cohomology of algebraic varieties,

Publ. Math. I.H.E.S. 29 (1966), 93-103. [GM78] H. GILLET, W. MESSING. - Riemann-Roch and cycle classes in crystalline cohomology, Duke Math. J. 45 (1978), 193-211. [J921 U. JANNSEN. - Motives, numerical equivalence, and semi-simplicity, Inv. Math. 107 (1992), 447-452. [KM74] N. KATz, W. MESSING. - Some consequences of the Riemann hypothesis for varieties overfinitefields, Invent. Math. I.H.E.S. 23 (1974), 73-77. [090] A. OGUS. - A p-adic analogue of the Chowla-Selberg formula, in "padic analysis" Proceedings of the Trento conference, (F. Badassarri, S. Bosch, B. Dwork, eds.) Springer L.N.M. 1454 (1990).

Yves Andre UA 763 du C.N.R.S.

Universite Paris 6 College de France 3, rue d'Ulm 75231 PARIS 05

Number Theory Paris 1992-93

A refinement of the Faltings-Serre method Nigel BostonN

1. - Introduction In recent years the classification of elliptic curves over Q of various conductors has been attempted. Many results have shown that elliptic curves of a certain conductor do not exist. Later methods have concentrated

on small conductors, striving to find them all and hence to verify the Shimura-Taniyama-Weil conjecture for those conductors. A typical case is the conductor 11. In [1[, Agrawal, Coates, Hunt, and van der Poorten showed that every elliptic curve over Q of conductor 11 is Q-isogenous to y2 + y = x3 - x2. Their methods involved a lot of computation and the use of Baker's method. In [ 121, Serre subsequently applied Faltings' ideas to reprove this result in a much shorter way. He called this approach "the method of quartic fields". In this paper I first seek to refine this method and to make it possible to classify elliptic curves over Q of conductor N for a large number of N. These N are all prime and so this work is indeed superceded by the result of Wiles that every semistable elliptic curve over Q is modular (if fixed). The advantage of my method is that it provides a much simpler approach (when

it works). Like Wiles, I am using deformations of Galois representations

but in a more elementary way. The second half of the paper indicates how the Faltings-Serre method can be used to describe spaces of Galois representations and gives the first applications of the method to mod p representations with p 2. The main result of the first half is Theorem 1 below. Note that there are extensive tables of class numbers and units of cubic fields due to Angell [2) and that information on quartic fields is not required Partially supported by NSF grant DMS 90-14522. 1 thank God for leading me to these results. I thank J.-P.Serre for generously sending me copies of his unpublished work. 1

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N. BOSTON

THEOREM (1.1). - Let N be a prime - 3 (mod 8) , such that 3 divides neither

h (Q (')) nor h (Q (,/-_N)) . Let M be one of the cubic subfields of the unique cubic cyclic extension K of Q() of conductor 2. Suppose that h(M) is odd and that the minimum polynomial modulo N of a fundamental unit of M has a quadratic residue and a quadratic non-residue root.

Then there is at most one Q-isogeny class of elliptic curves over Q of conductor N with given trace of Frobenius at 2, a2.

Remarks (1) There is a unique such field K because 2 is inert in Q(om) and 3 does not divide h(Q(v/---N)). (2) By Cohen-Lenstra heuristics, 47% of N should satisfy )). Apparently most (but not all, e.g. 571) of these 3 { h(Q(/))h(Q( N have h(M) odd. Some of these satisfy the condition on the fundamental unit (e.g. N = 11, 67,179,...); some don't (e.g. N = 19,43,163,...). (3) The prime N may satisfy the hypotheses of the theorem but there be no elliptic curve over Q of conductor N, (e.g. N = 227, 251, ... see 151).

2. - The Basic Set-up and Elementary Properties

_

Let E be an elliptic curve over Q of conductor N. Let p : Gal(Q/Q) -* GL2(F2) give the action of Galois on the 2-division points of E. The curve E has no rational points of order 2 since its conductor is neither 17 nor of the form u2 + 64 (i.e. it is not a Setzer-Neumann curve) [ 131.

Using work of Brumer and Kramer [5] based on work of Serre [ 11 ], we can deduce various properties of E and p. Firstly, plus or minus the discriminant of a semistable elliptic curve with no rational point of order 2 is never a perfect square. It follows that E has supersingular reduction

at 2, since Q(v) (0 being the discriminant of E) is Q(v) or Q(v) and so has no unramified cyclic cubic extensions by the hypotheses of the theorem. Secondly, since E is supersingular modulo 2, its 2-division field is a cyclic cubic extension of Q(v) unramified outside 2 and totally ramified

at 2, and moreover 2 is inert in

From this it follows that

Q(/) = Q(/), that p is surjective, and that the 2-division field of E (i.e. the fixed field of ker p) is K.

3. - The Faltings-Serre Method [12] Suppose that E' is another elliptic curve over Q with conductor N and the same trace of Frobenius at 2. Assume that E' is not isogenous to E. Let P I P ' : Gal(Q/Q) --> GL2(Z2) give the action of Galois on the Tate modules T2 (E), T2 (E') respectively. By Faltings [7], p and p' are not isomorphic. By section 2, their reductions modulo 2 are isomorphic.

Pick the largest a such that p and p' are isomorphic modulo 2a. Replacing p' by a conjugate if necessary, we can assume that they are equal

A REFINEMENT OF THE FAL77NGS-SERRE METHOD

63

modulo 21. Define o : Gal(Q/Q) -+ M2(IF2)°oGL2(IF2) by

a(x) = ((p'(x) - p(x))/2a(mod 2), P(x)), where M2(IF2)° denotes the 2 x 2 matrices over IF2 of trace zero (mapped to since det p equals det p'). Let k be the fixed field of ker or.

4. - The Proof of the Main Theorem The idea of the Faltings-Serre method is to use it to produce a representation a that can then be shown not to exist by methods of algebraic number theory (in particular tables of number fields). This then shows that there cannot be two non-isogenous curves with the properties stated in the main theorem. PROPOSITION (4.1). - The extension K/K is unramiiled outside N.

Proof : since E has supersingular reduction at 2, the theorem of HondaHill-Cartier [8[ implies that the characteristic polynomial of the formal group

associated to E at 2 is the same as the characteristic polynomial of the system of 2-adic representations at 2. This says that a2 determines the formal group at 2 of E, which determines the 2-adic representation of a decomposition group D2 at 2, i.e. pID2

p'1 D2 If x E D2 (so in particular if

x is in an inertia group at 2), then a(x) = (0, p(x)).

It remains to show that such an extension K/K cannot exist. The key idea is to use two results of Nicole Moser 1101. The first one is :

(1) h(K) = (ah(M)2h(Q(V-__N))/3 (a = 1 or 3) PROPOSITION (4.2). - The class number h(K) is odd.

Proof : this follows from the above formula (1), from our hypothesis that h(M) is odd, and from genus theory, which tells us that h(Q(vl--N)) is odd (since N is prime). Secondly, Moser showed [ 101 that K has a Minkowski unit, i.e. a single

generator of its unit group modulo torsion as a Z[Gal(K/Q)]-module. To apply this, consider by global class field theory the exact sequence of IF2 [Gal (K/Q)]-modules : 0 -+ B -4 U --* (DpjN U p _*

-+ 0,

64

N. BOSTON

where U is the global units of K modulo squares, U. is the local units of K. modulo squares, and P is the Galois group over K of a maximal elementary 2-abelian extension L unramified outside the primes of K above N. Now dimF2 U = 3, dimF2 Up = 1 implying that dimF2 P = dimF2 B. Since K C L, it remains to show that B = 0. The existence of a Minkowski unit implies that U - {±1} G V, where V is an irreducible 2-dimensional F2 [Gal(K/Q)]-module. Sowe just need an element of V which is not in one of the kernels from U - U. The image in V of the unit in the hypotheses of the theorem satisfies this.

5. - Examples (1) N = 11. There is an elliptic curve over Q of conductor 11, namely (11A) y2+y = x3-x2. Let E be another such. Then [51,[131 p is determined,

E has supersingular reduction at 2, and M has odd class number. In fact M is the cubic field of discriminant -44. By [6] a fundamental unit of M has minimum polynomial x3 + x2 + x - 1, which factors modulo 11 as (x + 3)2(x + 6). Since -3 is a quadratic non-residue modulo 11, theorem 1 shows that every elliptic curve over Q of conductor 11 with a2 = -2 is isogenous to (11A). As in Serre's original letter [ 121, this classifies up to isogeny every elliptic

curve over Q of conductor 11, because a similar argument to the above shows that an elliptic curve with good reduction outside 11 and a2 = 2 (respectively a2 = 0) is isogenous to (121A) (respectively (121D))-

(2) N = 67. There is an elliptic curve over Q of conductor 67, namely

(67A) y2 + y = x3 + x2 - 12x - 21. Let E be another such. Then [5], [ 13] p is determined, E has supersingular reduction at 2, and M has odd class number. In fact M is the cubic field of discriminant -268. By [6] a fundamental unit of M has minimum polynomial x3 - 7x2 + 13x - 1, which factors modulo 67 as (x+16)2(x+28). Since -16 is a quadratic non-residue modulo 67, theorem 1.1 shows that every elliptic curve over Q of conductor 67 with a2 = 2 is isogenous to (67A).

The same argument as for N = 11 now applies, because the twist of (67A) by the quadratic character associated to Q( -67) is an elliptic curve of conductor 672 with a2 = -2 and with the same p and the curve of CM type relative to Q( -67) is an elliptic curve of conductor 672 with a2 = 0 and the same T.

6. - Deformation Spaces of Galois Representations The homomorphisms p and p' are lifts of the same p to Z2. They therefore

lie in the deformation space of lifts of p [3]. The Faltings-Serre method constructs from them a third lift a to the dual numbers F2 [E] (E2 = 0). We consider below some applications of this idea.

A REFINEMENT OF THE FALTINGS-SERRE METHOD

65

Let p : Gal(Q/Q) -+ GL2(FP) be an absolutely irreducible representation. Let C denote the category of complete, noetherian local rings

with residue field IFP. Objects of this category are rings of the form Zp[[Ti, ..., T,.]]/I. If R is such a ring, then two representations P1, p2 : Gal(Q/Q) -> GL2 (R) will be called strictly equivalent if conjugate by an element of r2(R) := ker(GL2(R) -+ GL2(]FP)). A strict equivalence class of lifts of p is called a deformation of p. Fix a finite set of rational primes S containing the primes ramified in p. Define a functor 17: C - - --+ Sets by : F(R) = {deformations of p to R unramified outside S} Mazur [91 proved that .F is representable, i.e. that there exists a representation L; : Gal(Q/Q) -+ GL2 (R) (the universal deformation) lifting p and

parametrizing lifts of p to R in C unramified outside S up to strict equivalence via Hom(R, R). The set Hom(R, Z,,) will be called the deformation space of lifts of p.

At the joint AMS-LMS conference in Cambridge, England, in 1992, I suggested that deformation spaces of Galois representations should have some special properties. In particular, it appears that they are often coordinatized by their restrictions to various inertia subgroups It (e E S), namely (i) the restrictions to It V E S - {p}) should indicate which component the lift is on and (ii) the restriction to IP should indicate where the lift is on that component. This idea is now to use the Faltings-Serre method to prove some cases of (ii). The novelty of this approach lies in replacing the prime 2 by more general primes. See [31 for a further discussion of this. Example (1) Let E be the elliptic curve X0(49). This is an elliptic curve over Q of conductor 49. In [4), it is calculated that the universal deformation

ring of the Galois representation given by the 3-division points of E with S = {3, 7} is Z3[[T1, T2i T3, T4]]/((1 + T4)3 - 1). Thus its deformation space

splits into three explicitly given components {T4 = 0}, {T4 = w - 1}, {T4 = w2 - 1}, where w is a primitive 3rd root of unity, and as shown in 141 any lift to Z3 lies on the first component. Also, (i) above holds. In other words, the image of an inertia group at 7 determines on which component a representation over Z3 lies. Now let p and p' be two lifts Of T to Z3 (so lying on the first component). Suppose that they agree on inertia at 3. We shall show that they are actually strictly equivalent (so give the same point in the deformation space). Assume for now they are not strictly equivalent. Since p is absolutely irreducible, two lifts are strictly equivalent if and only if they are isomorphic. For suppose that p' = A-1 pA with A E GL2 (Z7,). Then A centralizes the image of p and so by Schur the image of A in GL2(]FP) is a scalar matrix,

66

N. BOSTON

i.e. A = BC where B is scalar and C E I'2(Zp). But then p' = C-1pC. Since p and p' are not isomorphic, or defines a homomorphism from Gal(Q/Q) into the semidirect product M2 (F3)oGL2 (1F3) (= GL2 (IF3 [e]), e2 =

0) unramified outside S = {3, 7}. Letting K and k denote, as before, the fixed fields of p and a respectively, we get that K/K is unramified outside 7. It is also unramified outside 3 because (i) holds, i.e. p and p' agree on inertia at 3. Such an everywhere unramified field extension of K does not exist, since its Galois group would be a quotient of the ideal class group killed by 3 with Gal(K/Q) acting via the adjoint action. This is excluded as explained in [41 by the work of Coates and Flach since 3 does not divide the numerator of a certain special value of the L-function of the symmetric square of E.

(2) Following [91, let p be a prime number of the form 27 + 4a3 and K be a splitting field over Q for x3 + ax + 1. Embedding Gal(K/Q) = S3 in GL2(Fp), we obtain a representation p : Gal(Q/Q) -+ GL2(1Fp) unramified outside p. Letting S = {p}, Mazur showed that R = Zp[[T1iT2,T3]]. Let p and p' be lifts of p to 7Gp unramified outside S. Suppose that they agree on inertia at p, but are not strictly equivalent. Then they produce, as in (1), an unramified p-extension of the fixed field of ker p, but as Mazur showed in (91, p-h(K), a contradiction thereby proving (ii) in this case. Manuscrit recu le 17 aout 1993

A REFINEMENT OF THE FALTINGS-SERRE METHOD

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REFERENCES

[1] M. AGRAwAL, J. COATES, D. HUNT and A. VAN DER POORTEN. - Elliptic curves

of conductor 11 Math. Comp. 35 (1980), 991-1002. [2] I.O. ANGELL. - A table of complex cubic fields.

[3] N. BOSTON. - Deformations of Galois representations, (a monograph), in preparation . [4] N. BOSTON and S.V. ULLOM. - Representations related to CM elliptic curves,

Math. Proc. Camb. Phil. Soc. 113 (1993), 71-85. [5] A. BRUMER and K. KRAMER. - The rank of elliptic curves, Duke Math. J. 44,

no 4 (1977), 715-742. [6] B.N. DELONE and D.K. FADDEEV. - The theory of irrationalities of the third degree, AMS, Providence, RI, 1964.

[7] G. FALTINGS. - Endlichkeitssatze fur abelsche Varietaten fiber Zahlkorpern,

73 (1983), 349-366. [8] W. HILL. - Formal groups and zeta-functions of elliptic curves, Invent. Math. 12 (1971), 321-336.

[9] B. MAZUR. - Deforming Galois representations, Proceedings of the March 1987 Workshop on "Galois groups over Q" held at MSRI, Berkeley, California.

[10] N. MOSER. - Unites et nombre de classes d'une extension galoisienne diedrale de Q, Abh. Math. Sem. Univ. Hamburg 48 (1979), 54-75.

[I I I J.-P. SERRE. - Proprietes galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. [ 121 J.-P. SERRE. - Letter to Tate, Oct. 26, 1984.

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[ 131 B. SETZER. - Elliptic curves of prime conductor, J. London Math. Soc. 10 (1975), 367-378.

Nigel BOSTON

Department of Mathematics University of Illinois 273 Altgeld Hall, MC 382 1409 West Green street Urbana, IL 61801 U.S.A.

Number Theory Paris 1992-93

Sous-vari@tes algebriques de varibtes semi-abeliennes sur un corps fini John Boxall

RESUME : Dans cet article nous etendons les resultats deja obtenus dans [Boll concernant l'intersection dans une variete abelienne d'une sous-variete avec certains groupes de points de torsion aux varietes semi-abeliennes, le corps de base etant un corps fine.

SUMMARY : In this paper we extend results already proved in [Boll concerning intersections on abelian varieties of subvarities with certain groups of torsion points to semi-abelian varieties, the base field a finite field.

1. - Introduction

_

Soit k un corps fin!, soit k une cloture algebrique de k, soit 1 un nombre premier different de la caracteristique de k et soit G le groupe des racines de 1'unite dans k dont 1'ordre est une puissance de 1. Nous nous proposons

de montrer que 1'ensemble des ( E G tels que 1 - (E G est fini. Plus generalement, soient (a, b) E k2 avec ab 54 0 : nous allons montrer que I'ensemble : J((, 1q) E G2 I a( + br1=1 }

_

est fini.

Dans ce but, designons par r le groupe de Galois de k sur k et remarquons que si ((, 71) E G2 verifie (1)

a(+brj=1,

alors on a egalement a a( + b au = 1 pour tout or E I' et done (2)

a'(+ b'uj = 1 ,

ofi l'on a pose :

a =a ( et b'=b"'

.

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J. BOXALL

Or, les equations (1) et (2) ont au plus une solution ((, rl) si (a(, ark) # ((, rl) ; it suffit donc pour conclure de montrer qu'il existe un sous-groupe fini Go de G ayant la propriete que pour tout ((, 77) E G2 avec ((,,q) Go, ii existe or E F tel que ((a - 1)(, (a - 1)11) soit un element de Go different de (1, 1). En effet, toute solution 7,7) n'appartenant pas a Go satisait a (1) et (2) avec (a'a-1, b'b-1) E Go \ (1, 1). 11 y a done au plus IGo12 + IGoI 1 solutions.

-

Or, it est aise de construire un tel sous-groupe Go. Posons l' = 4 ou 1' = l selon que 1 = 2 ou l est impair. Soit k' 1'extension de k dans k engendree par les racines l'-iemes de l'unite, soit le l'ordre de G fl k'* et soit e : F --4 Zt le caractere cyclotomique. On sait alors l'image de Gal(k/k') par e est (1 + 1eZ1)". Soit done (E G avec (V k' et soit ln, (n > e), l'ordre de (. Si l'on choisit a E Gal(k/k') de telle maniere que e(a) - 1 (mod ln) mais e(a) 0- 1 (mod In-1), alors (a - 1)( appartient a k' et est different de 1. On en tire que Go = G fl k'* convient. Voici une autre interpretation de ce resultat. Designons comme d'habitude par G.,,,, le groupe multiplicatif : le groupe G n'est autre chose que le groupe des points de torsion de Gvm, dont l'ordre est une puissance de 1. Soit X la courbe dans G2 definie par 1'equation ax+by = 1. Selon notre resultat X (k) fl G est un ensemble fini qui peut etre effectivement determine. Pour tout Q = (c, 71) E G2 (k), on designe par TQX le translate de X par le point

-Q =

Si donc P E X(k) fl Get si a E I', alors aP E X(k) et donc P = aP - (aP - P) E Top_ pX (k), d'ou P E X fl Tp_pX (k). (c-1

77-1)

Autrement dit, si P = ((,'rl), alors ((, rl) est une solution des equations (1) et (2) que, on le sait, n'ont qu'un nombre fini de solutions (eventuellement une au plus) lorsque aP P. Pour conclure it ne reste qu'a montrer que pour tout P E X (k) fl G en dehors d'un sous-groupe fini effectif Go on peut choisir a de telle maniere que aP P et aP - P E Go : d'apres 1'alinea precedent, le choix Go = G fl k'* convient. De ce point de vue, ce resultat est capable d'importantes generalisations(*). Soit E une variete semi-abelienne, c'est-a-dire une extension d'une variete abelienne par un tore (pour plus de details, le lecteur se reportera au debut du §2). On suppose que E est definie sur k et on designe par G un sous-groupe de E(k) (puisque k est un corps fini, tout element de E(k) est d'ordre fini). Soit X une sous-variete fermee de E que l'on suppose definie sur k : comme precedemment, on designe par TQX le translate de X par -Q E E(-k). Pour simplifier on supposera (_k) que X est k-irreductible. Nous nous interessons alors a determiner X f1 G. Bien sur, ceci ne sera possible que si G est d'une nature tres particuliere. Comme dans le resultat qui (*l Apres avoir termine ce texte, je me suss apercu que l'argument qui vient d'etre presente se trouve (pour une courbe plongee dans une variete abelienne) dans l'article de Raynaud ([R31, p3-4).

SODS-VARIETE AWEBRIQUES DE VARIETES SEMI-ABELIENNES SUR UN CORPS FINI

71

vient d'etre demontre, on pourrait prendre comme G le groupe des points de torsion dont l'ordre est une puissance de 1. Plus generalement, soit S un ensemble fini de nombres premiers et soit 1(S) le monoide multiplicatif engendre par S. Pour tout n c Q (S), on designe par E[n] le groupe de points de n-torsion de E(k) et l'on pose ES = UfEO(s)E[n]. Nous demontrerons alors au §2 le theoreme suivant : THEOREMS A. - Avec les notations et les hypotheses qui viennent d'etre introduites, it existe un ensemble fini de couples (Pi, Bi)iEI, Oft Pi E Es, Bi est une sous-variete semi-abelienne de E et Tp, Bi C X, tel que :

X (k) n Es = U Tp Bi. iEI

On en tire immediatement que, lorsque G = Es, on a X (l) n G C UiEI Tp Bi(k). Ce dernier resultat a deja ete demontre dans [Boll lorsque E est une variete abelienne. Le resultat analogue lorsque k est un corps de nombres a ete demontre par Bogomolov [Bgl], IBg2l. Ensuite les travaux de Raynaud [R1], [R2J, [R31, Hindry [HI et Faltings IF] ont etabli le meme resultat lorsque G est le groupe de tous les points de torsion sur k, ou le groupe E(k), ou meme lorsque G est 1'enveloppe divisible d'un sous-groupe de type fini de E(k). Le cas ou E est un tore et G un groupe de type fini (en caracteristique zero toujours) a ete traite par Laurent [Lal. Lorsque G est 1'enveloppe divisible d'un groupe de type fini et X est une courbe, Liardet [Lil a demontre que X (-k) n G est soit un ensemble fini soit un ensemble forme de racines de l'unite et que cette derniere possibilite ne se produit que dans certains cas bien precis. Le travail recent de Ruppert [Rul etudie les solutions en racines de 1'unite de systemes d'equations algebriques. Lorsque k est un corps fini tout element de E(k) est de torsion et it est done impossible d'etendte le theoreme A au cas ou G = E(k). La situation, lorsque k est un corps de fonctions de caracteristique positive, a ete etudiee par Voloch et Abramovich [Vol, [Ab-Vol.

Remarque 1

:

si en plus on suppose que 1'ensemble des Tp Bi du

theoreme soit choisi de facon minimale, alors les Tp Bi seront necessairement les composantes irreductibles de X (k) n Es. En particulier, ils sont uniquement determines par X et S.

Remarque 2 : notre demonstration montre que les sous-varietes semiabeliennes Bi du theoreme A peuvent etre realisees comme des stabilisateurs par translation de composantes irreductibles de fermes de la forme TQ1X n TQ2X n ... n TQ,,X. On peut se demander si les sous-varietes abeliennes apparaissant dans les travaux de Raynaud, Hindry et Faltings puissent etre construites de maniere analogue.

72

J. BOXALL

Le §3 est consacre a une etude de 1'effectivite dans la demonstration du theoreme A.

Je remercie vivement L. Moret-Bailly pour la lecture approfondie d'une version preliminaire de ce travail.

2. - Varietes semi-abeliennes Rappelons que par definition une variete semi-abelienne E est une extension d'une variete abelienne par un tore; elle est alors definie par une suite exacte

1-+T-4 E-->A->0, ou T est un tore et A une variete abelienne definis sur k. A 1'aide de la theorie de la structure des groupes algebriques, on voit aisement que tout sous-groupe algebrique connexe d'une variete semiabelienne est une variete semi-abelienne. Il s'ensuit que si V est une sousvariete (fermee) de E, alors le stabilisateur By de V par l'operation des elements de E(k) par translation est le produit d'une sous-variete semiabelienne et d'un groupe fini. On designe alors par By la composante neutre de By et le groupe fini par Hv. Pour toute sous-variete V de E, on designe par TQV le translate de V

par -Q E E(k). On peut alors ecrire : By = nQEV(k)TQV. On en tire que si V est irreductible, alors dim(Bv) < dim V et si dim(Bv)= dim V alors By = By et V est le translate de By par un element de E(-k). Soit a nouveau S un ensemble fini de nombre premiers, ft(S) le monoide

multiplicatif engendre par S. Pour tout n E f2(S), on designe par E[n] le groupe des points d'ordre n de E(k) et l'on pose Es = UfEQ(S)E[n]. Soit r s le groupe de Galois de k(Es) sur k. Pour tout 1 E S on designe par r(l) le rang du module de Tate T1(E) (on a alors r(l) = 2dimA+dimT n

si l est different de la caracteristique de k et 0 < r(l) < dim A si 1 est egal a la caracteristique de k). L'operation de r s sur ES induit une representation

de I'S dans le groupe des automorphismes continus Aut(Es) de Es. On fixe un choix des bases des TI(E), ce qui induit un isomorphisme continu entre Aut(Es) et DIES GLr(l) M); on obtient ainsi une representation p de rs dans ce dernier groupe. Posons L = 11IES l', oft l'on a ecrit 1' = 1 si l est impair et 1' = 4 si 1 = 2. Pour tout n E fl(S), on designe par k,, le corps k(E[n]) et par r,, le groupe de Galois de k(Es) sur kn. Soit N le plus grand element de Il(S) tel que kN = kL. On a alors, pour tout n divisible par L :

P(rn) S fJ

(I+l°rdi(fl)Mr(I)(ZI))

lES

(oft I designe la matrice identite et Mr(I) (Z1) designe 1'algebre des matrices carrees d'ordre r(l) a coefficients dans Z1). Soit 0 un generateur topologique

SOUS-VARI9'ItALGEBRIQUES DE VARIETE`S SEMI-ABE`LIENNES SUR UN CORPS FINI 73

de FN : k etant un corps fini tout element de rN s'ecrit de maniere unique dans la forme eb avec b E 7L = 1 im Z/nZ. Pour tout 1 E S designons par OI n

la l-composante de p(O). D'apres la definition de N, on a OI = I mod lord, (N) mais Ol o I (mod lord,(N)+1) pour tout l E S. Pour tout 1 E S, definissons la matrice 4)1 par OI = I+lord, (N) (DI. Comme 0 ne laisse stable qu'un nombre fini d'elements de ES, 4DI est inversible (i.e. un element de GLr(I) (QI)). Soit el (P) 1'exposant de l dans l'ordre de P E ES.

Il s'ensuit alors que pour tout l E S on a :

e` (P) - ordi (det 'I) - ordi (N) < el (9P - P) < el (P) - ordi (N). LEMME 1. - Soit b un entier strictement positif. Alors pour tout P E Es

on a: el(P) - ordl(det 4)1) - ordi(N) - ordi(b) < el (95P - P) < el (P) - ordi (N) - ordi (b) . Par consequent, si l'on pose p = PIES lord, (det'Dt) alors E[n] C Es(kn) C_ E[µn] pour tout n E Q(S) divisble par N, Es (kn) designant les elements de ES rationnels sur kn.

Demonstration : le cas b = 1 est deja acquis. Si b > 1, on utilise 1'equation b

OiP-P=(Oi-I)P=b(OIP-P)+rCb(OI-I)rP r=2 b

= b1ordi(N)4,1P+ 1:Cblrord,(N) P. r=2

Puisque lIN pour tout I E S et 41N si 2 E S on verifie que la puissance exacte de 1 divisant Cb lr ord, (N) pour 2 < r < b est strictement superieur a ordi (b) + ordi (N) . On en tire que el ((Oi p - P)) = ordi (b) + ordi (N) + el ((BP - P)) d'ou l'encadrement de el (ObP - P). Pour montrer la derniere assertion on pose b = N . Soit alors P E Es (k)

un point d'ordre ,una avec a E cl(S) et a > 1 et soft 1 un element de S divisant a. Alors el (P) - ordi (det I) - ordi (N) - ordi (b) = ordi (a) > 1 et donc eb ne laisse pas stable P, c'est-a-dire P 0 ES(kn). On conclut que Es(kn) C E[pn] et l'inclusion E[n] C Es(kn) est triviale.

Remarque i : soit k;n l'unique extension de kN de degre m. Pour tout n E fl(S) divisible par N, 0* est un generateur topologique de Gal(k/kn).

74

J. BOY-ALL

On tire de la demonstration du lemme 1 que OP I (mod lord,(n)) mais OR I (mod lordi(n)+1) 11 s'ensuit que 0* est egalement un generateur topologique de kn et donc que kn = k'' pour tout n E Q(S) divisible par N. Ce lemme nous permet d'aborder la demonstration du theoreme A. Soit

n E I(S) divisible par N et soit V une sous-variete de E define sur

_n

k, que l'on supposera kn-irreductible.Supposons d'abord que By = (0).

Si P E V(k) Es, alors o P E V (T) n Es pour tout or E I, et donc P E V n Tp_PV(k). Si P E[pn], le lemme 1 montre que P West pas definie sur kn et donc nest pas stable par 0* : on peut donc choisir une puissance a de 0* de telle facon que 0 j4 aP - P E E[,an]. On trouve ainsi que

V(k)nEs c E[un]uU(vnTQV(k)nEs), Q

ou Q parcourt E[,an] \ (0). Puisque By = (0) et V est kn-irreductible, on conclut que toutes les composantes irreductibles des V n TQV sont de dimension strictement inferieure a celle de V. En prenant l'adherence de Zariski, on peut ecrire : (4)

v (T) n Es = { partie finie de E[pn] } U w (T) n Es, w

ou {W } parcourt un ensemble de varietes definies sur k,,,n que l'on peut supposer k,,,,-irreductibles.

On peut etendre cet argument au cas ou By

(0) en passant a la

variete semi-abelienne quotient E' = E/Bv. Soient /Gy et NV les entiers u et N associes a E' et soit Hv un sous-groupe fini de E tel que By = By x Hv. On a alors Bv,s = Bv,s ® Hv,s Si e(Hv,s) designe 1'exposant de Hv,s on obtient, pour tout n E I(S) divisible par Nve(Hv,s) : (5)

V(k) n Es = U P

uU

n Es,

w

ou P parcourt une partie de ELavn] et W un ensemble fini de varietes definies et irreductibles sur kµvn. On peut alors appliquer le meme raisonnement a chacune des varietes W (en replacant k par ki,). Le theoreme A peut alors etre demontre par recurrence en la dimension de X.

En outre, comme les W sont des kn-composantes des V n TQV, on conclut que x (T) n Es est une reunion finie de translates de sousvarietes semi-abeliennes dont chacune est le stabilisateur d'une composante irreductible d'une intersection de translates de X (remarque 2 de l'introduction).

SOUS-VARIETE° ALGEBRIQUES DE VARIEIES SEMI-ABELIENNES SUR UN CORPS FINI 75

Remarque 2: lorsque E est une variete abelienne simple, on conclut que v (T) fl Es est une ensemble fini. Meme lorsque E est une variete abelienne quelconque on peut parfois demontrer que V (T) fl G est necessairement un ensemble fini pour certain sous-groupes G de E(T). Ceci est le cas par

exemple lorsque G est le groupe des points tues par une puissance d'un ideal premier p de degre un d'une sous-algebre commutative de rang 2 dim A de End A ® Q (voir [Bo2]). Rappelons que d'apres un theoreme bien connu de Tate [T], End A ® Q contient toujours une telle sous-algebre lorsque k est un corps fini.

Remarque 3 : 1'hypothese que k soit un corps fini West pas essentielle dans notre demonstration du theoreme 1. Si par exemple k est un corps de nombre (ou plus generalement un corps de type fini sur Q ou sur un corps fini) on peut verifier une version modifiee du lemme 1 qui permet de conclure

dans la meme maniere. On obtient ainsi une nouvelle demonstration des resultats originaux de Bogomolov ([Bgl], [Bg2]). Par contre, la methode de Bogomolov ne s'applique apparemment pas sur un corps fini : en effet it utilise le fait que sur un corps de nombres p(F) contient un sous-groupe d'indice fini du groupe des homotheties et cette propriete est fausse en general sur un corps fini.

3. - Un peu d'effectivite D'apres un resultat fondemental de Chow [Ch], toute variete semiabelienne E est quasi-projective. Fixons une fois pour toutes un plongement de E dans un espace projectif P". Le degre d'une sous-variete equidimensionnelle V de E relatif a ce plongement peut alors etre defini comme le cardinal de l'intersection de l'adherence de 1'image de V avec un sous-espace projectif generique de codimension egale a la dimension de V. Si V est un ferme de E, on definit son degre comme etant la Somme des degres de ses composantes irreductibles. Le degre jouit alors des proprietes suivantes : (a) Le degre d'une reunion finie de varietes est inferieure ou egale a la Somme de leurs degres ; en particulier, si W est une reunion de composantes irreductibles de V, alors deg W < deg V ; (b) Le degre d'une intersection finie de varietes est inferieure ou egale au produit de leurs degres; _ (c) Le degre est invariant par translation par un element de E(k) (car le morphisme defini par translation par un element de E(k) est plat).

(Pour plus de details sur la notion de degre, le lecteur pourra consulter [Fu], notamment le §8). Le but de ce paragraphe est de demontrer le theoreme suivant : THEOREMS 2. - Soit k un corps fini, soft g > 1 un entier soit E une variete semi-abelienne de dimension g et soit S un ensemble fini de premiers. Alors

76

J. BOXALL

it existe une constante effective K, dependant uniquement de k, S et de E, telle que pour toute sous-variete X de E definie sur k, on ait

deg(X(k) fl Es)
1, la representation de Q a valeurs dans

Mn(C) donnee par 4 : a H aln, a E Q. Le domaine D est donne par 1'espace de Siegel

= ' H , , z={zEMn,,(C)I tz=z,

1 (z-z)>0}

sur lequel agit le groupe modulaire F = Sp(2n, Z) ou

Sp(2n,Z)={M=I C

)EM2(z)ltM( _0°

0) M=(-o°

Ici A, B, C, D sont dans Mn(Z). On a r = rE pour une famille E de representants des classes d'isomorphisme de varietes abeliennes de dimension n principalement polarisees. L'action de r sur D est donnee par z H (Az + B)(Cz + D)-1. Soit A une variete abelienne principalement polarisee isomorphe a Cn/A on A est un reseau dans Cn. On peut ecrire

A=Z(J w)®...®Z(J 71

Ou

w)

72n

f_t(fwfw)Ecn

pour une base W1, ... ,Wn de H°(A,1) sur C et une base yl, ... , yen de H1(A, 7L) sur Z. On peut choisir ces bases de H°(A, 52) et de H1(A, Z) de telle facon a ce que pour les matrices des periodes

Q2 = (J

Q1= ( 'yn

7n+1

0'...,J 72n CO)

on alt z = SZj 1522 dans Hn. On appelle z un module de A. Clairement, A est isomorphe a Az = Cn/(z.Zn + Zn) et Az est munie d'une polarisation principale C. On a une bijection

F\ln ^' {[(AzjC.)],z E -ln} L'hypothese que la polarisation soit principale nest pas cruciale : pour 1'enlever, it faudrait modifier le groupe r et a chaque variete abelienne polarisee de dimension n on peut associer un module (modulo F) daps 'Hn. Le Resultat principal donne donc :

PROPRIETES TRANSCENDANTES DES FONCTIONS AUFOMORPHES

85

PROPOSITION 1. - Soient A une variete abelienne polarisee define sur 0 et z E fl un module de A. Alors Z E 9{n, (0) si et settlement si A est de type CM.

Le Theoreme permet de deduire des resultats de transcendance sur les fonctions automorphes de Siegel definie sur 0. Soit K le corps de ces fonctions dont 1'existence est une consequence de celle de V, ou si l'on veut K est le corps des fonctions automorphes de Siegel qui sont des quotients de formes automorphes de Siegel a coefficients de Fourier algebriques. Alors le Corollaire donne : PROPOSITION 2. - Si z E l,,, (0) alors toute fonction automorphe de Siegel darts K et definie a z prend une valeur algebrique a z si et seulement si z est un point CM.

Un exemple de Freitag montre qu'en general les resultats donnes par le Corollaire, comme celui de la Proposition 2, sur la transcendance des valeurs des fonctions automorphes aux points algebriques non CM ne peuvent pas etre ameliores sans faire des hypotheses supplementaires sur le point non CM.

Exemple de Freitag [Fr] : prenons n = 2 dans 1'exemple des espaces de Siegel. Alors Freitag a montre que le corps K des fonctions sur 9-12 automorphe par rapport a r = Sp(4, Z) et definies sur Q est engendre par trois fonctions fl, ,((f2, f3 telles que sur

D'={z= l

0

0 T2

I

E912IT1,T2EN1

onait: fl (Z)

0,.f2(z)

9(71)+9(72),f3(z)

9(Tl)-1+9(T2)

avec G4 = G4(7-) et G6 = G6 (T), T E 9-l les series oil 9( T) = d'Eisenstein (normalisees) de poids 4 et de poids 6. On peut choisir z E V avec Ti un point CM et T2 un point algebrique mais non CM (et g(Tl), 2 T) GG4(T)

g(T2) 0 0). Le point z sera alors un point algebrique non CM auquel deux fonctions dans K algebriquement independantes prennent une valeur

algebrique. Seulement une troisieme fonction dans K algebriquement independante de ces deux fonctions et definie en z prendra une valeur transcendante en z.

Les fonctions modulaires de Hilbert : la demonstration du Resultat principal peut donner un corollaire plus fort. En effet, on n'a pas toujours

besoin de savoir que tous les coefficients du point non CM z sont des nombres algebriques, pour avoir un f E K tel que f (z) soft transcendant. Dans le cas des fonctions modulaires de Hilbert on a par exemple :

86

P. B. COHEN

PROPOSITION 3. - Soit F un corps totatement reel de degre g = [F : Q] et E) l'anneau des entiers de F. Le group r = PSL(2, O) agit sur )-l9 et on design par K le corps des fonctions I -automorphes definies sur Q. Alors, si z = (zl, ... , z9) E x9 n'est pas un point CM et si zi E 7-l(J) pour un seul i = 1, ... , g, it existe un f E K avec f (z) transcendant.

On va donner une demonstration (differente de celle qui figure dans ICSWI) de la Proposition 3 a la fin de cet article. La condition qu'il s'agit d'un seul i = 1, ... , g dans la Proposition 3 peut donner lieu a des points dans 1'espace de Siegel 7-ig dans l'image d'un plongement de 1-l9 dans 7-lg dont tous les coefficients sont transcendants mais auxquels les fonctions

modulaires de Siegel definies sur 0 ne prennent pas toutes des valeurs algebriques.

We de la demonstration du Resultat principal

:

les details de la

demonstration du Resultat principal sont donnes dans notre manuscrit [CSW]. Le fait que (ii) implique (i) est une consequence de la normalisation choisie pour D (d'ailleurs si B est une variete abelienne de type CM, alors par la theorie de la multiplication complexe, B est isomorphe sur C a une variete abelienne definie sur (0). Comme dans le cas elliptique, c'est la demonstration que (i) implique (ii), c'est-a-dire que si A,, nest pas de type CM alors z 0 D(Q) qui utilise la theorie des nombres transcendants. La demonstration de Schneider utilisait une fonction auxiliaire, polynome en certaines fonctions elliptiques de Weierstrass. Notre demonstration en dimension superieure utilise le Theoreme du sous-groupe analytique de G. Wiistholz [Wii] (donc une fonction auxiliaire donnee par des fonctions abeliennes est implicite) et des renseignements tres explicites tires des travaux de Shimura, notamment de [Shi]. Prenons le cas ou A est simple, A E [Az], Az E E, z c D = D(L, 4))

avec L = Endo(A). On suppose que Az n'est pas de type CM et donc que dim(D) > 0. On demontre ensuite que l'hypothese que z E D(Q) entraine une contradiction. Soit w1, ... , wi,, une base du Q-espace vectoriel HI (A, S2(O).. Alors it existe -ii, ... , rym E Hl (A, Z), m = i' avec, pour f.Yi w

t (f7, wl ... , f 'j

E (C-,

AQ=A®zQ=1: 4)(L)J w. 7=1

'Yi

Comme A est definie sur 0 on a 4)(L) C MA(O). Donc les elements de L induisent des relations de dependance lineaire sur 0 entre les periodes

fy w, ou w E H° (A,1

), y E Hl (A, Z).

D'autre part, dans la construction de Shimura le domaine D se decom-

pose en un produit de g = [F

:

Q] (avec les notations deja donnes)

PROPRIETES TRANSCENDANTES DES FONCTIONS AU7OMORPHES

87

domaines irreductibles D = D1 x

.

.

. x V9,

et donc le point z s'ecrit : z= v=1' Les z, sont des matrices a coefficients algebriques (par hypothese) qui sont des quotients de matrices

Les matrices Q', k = 1, 2 ont leurs coefficients des combinaisons Qlineaires des f wi, i = 1,.. , n, j = 1,... , m : c'est implicite dans la .

construction de Shimura. Si z est une matrice a coefficients algebriques it y a donc une relation de dependance (0-lineaire entre les periodes f3 wi, i = 1 ... , n, j = 1, ... , m et on peut montrer que cette relation est non triviale.

C'est ici qu'intervient l'argument de transcendance qui dit que les relations non triviales de dependance lineaire sur 0 entre les fj wi, i = 1,. .. , n, j = 1,. .. , m proviennent toutes de relations non triviales de dependance lineaire sur 4 ) (L) entre les fi w" , j = 1, ... , m. En effet, dans [CSW] nous demontrons un resultat que G. Wustholz a annonce sans demonstration dans [Wu,ICM] : LEMME. - Soit A une variete abelienne define sur 0 et isogene auproduit

direct Al' x ... x AN de uarietes abeliennes simples Aµ, dim(A,) = nµ, definies sur Q et deux-d-deux non isogenes. Alors le Q-espace vectoriel VA

engendre par toutes les periodes des d ferentielles dans H° (A, 1l) est de dimension : dimo(VA) =

Comme les If-,,. w, j = 1, ... , m sont independants sur 4)(L) l'hypothese

z E D(Q) entraine la contradiction voulue (en fait meme 1'hypothese z E D (Q) pour un seul v aurait ete suffisante : voir la Proposition 3 pour une application de cette remarque). Pourtant, lorsque L est strictement contenu dans Endo(A) on utilise l'hypothese plus forte sur z. Si L est strictement contenu dans Endo(A) et A est simple alors A est isomorphe a A., ou A,, appartient a une famille E' C E d'elements d'un S(L', 4', p') ou L' = Endo(Az-). Soit D' = D(L', I'). Si dim(D') = 0 alors A est de type CM. Si dim(D') > 0 alors la discussion precedante montre que le point z' ne peut pas etre algebrique. Pour deduire la transcendance de z de celle de z', it faut utiliser des proprietes de rationnalite de certains plongements modulaires, definis sur (0, de D' dans D. Un argument facile

88

P. B. COHEN

de plongement modulaire permet aussi de traiter le cas on A nest pas simple.

Afin d'illustrer l'idee de la demonstration du Resultat principal demontrons la Proposition 3.

Demonstration de la Proposition 3 : le quotient PSL(2, O)\'Hg est 1'espace des modules d'une famille E de varietes abeliennes polarisees de M9 (C) donnee par : dimension g dans S(F, p) avec D : F 4) : 0 H diag(aj(0),...,ag(6)),

ou al, ... , ag sont les plongements galoisiens de F dans T18. On est donc dans le cas du TYPE a) avec m = 2. Soit A une variete abelienne define sur Q et isomorphe a AZ E E on z E D. L'espace H° (A,1 ) se decompose en g sous-

espaces propres pour 1'action induite de F. Tous ces sous-espaces sont de dimension 1 sur F. Soient cal, ... , wg des generateurs correspondants et soient ryi, 72 des elements de Hl (A, Z) qui engendrent Hl (A, Q) sur F. On peut choisir cette base de telle sorte que : Wi

- J7i f72 Wi

zi =

soit dans 7-l et que z = (zi)i-i.....g soit le point qui correspond a A. Si zi E 0 alors on sait que zi n'est pas dans ai (F) et donc par le Lemme it doit y avoir un element M de Endo(A) qui n'est pas dans fi(F). Mais les elements

de 4)(F) commutent aux elements de Endo(A). Donc L4, = 4)(F)(M) est un sous-corps de Endo(A) totalement imaginaire de degre 2dim(A) et une extension CM de F, d'ou it vient que A est a multiplication complexe. Manuscrit recu le 7 fevrier 1994

PROPRIE7ES TRANSCENDANTES DES FONCTIONS AUIOMORPHES

89

REFERENCES

[CSW] P. B. COHEN, H. SHIGA, J. WOLFART. - Criteria for complex multiplica-

tion and transcendence properties of automorphic functions, preprint J. W. Goethe-Universitat, Frankfurt am Main (1993). [Mil J. S. MILNE. - Canonical models of (mixed) Shimura varieties and automorphic vector bundles, "Automorphic forms, Shimura varieties and L-functions", Vol. 1, ed. by L. Clozel, J. S. Milne, Ann Arbor 1988, Academic Press (1990), 283-414. [Sch] Th. SCHNEIDER. - Arithmetische Untersuchungen elliptischer Integrate, Math. Ann. 113 (1937), 1-13. [Shi] G. SHIMURA. - On analytic families of polarized abelian varieties and automorphicfunctions, Ann. Math. 78 (1963), 149-192. [Will G. WosTHOLZ. - Algebraische Punkte auf analytischen Untergruppen algebraische Gruppen, Ann. of Math. 129 (1989), 501-517. [Wii,ICM] G. WUsTHoLz. - Algebraic Groups, Hodge Theory and Transcendence, Proc. ICM Berkeley 1986, Vol. 1, AMS, (1987), 476-483.

Paula Beazley COHEN UA 747 CNRS

College de France 3 rue d'Ulm F-75005 Paris France

Number Theory Paris 1992-93

Supersingular primes common to two elliptic curves E. Fouvry and M. Ram Murty

1. - Introduction Let E be an elliptic curve over Q. Denote by 7ro(x, E) the cardinality of the set of the supersingular primes of E less than x. It has been conjectured by Lang and Trotter [L-T] that, when E has no complex multiplication, the following holds, when x -i 00 x (1.1) fro (x, E) ' CE tog x

where CE is a positive constant depending only on E, precisely defined in terms of Ga1(Q(Eto,.s), Q). The first significant step towards (1.1) is due to Elkies ([Ell]), who proved

that each elliptic curve over Q, has infinitely many supersingular primes. This result was improved by the authors, who proved THEOREM A ([F-M] Theoreme 1). - Let E be an elliptic curve over Q. Then, for every positive S, there exists xo(6, E) such that, for x > xo(6, E), the following holds : 1093 x . 70 (x, E) > (log4 x)'+6

Here logk is the k-fold iterated logarithm function. Note that the best upperbound for 7ro(x, E) is due to Elkies and Murty (1E121, [E131) and has the shape 7ro (x, E) = OE (x 4) , for any non CM-curve E, with the convention that CM means that the curve has complex multiplication.

In the direction of (1.1), we must also quote another result which asserts, very vaguely speaking, that the Lang-Trotter Conjecture is true on average. More precisely, let a, b be two integers with 4a3 + 27b2 let Ea,,b be the elliptic curve defined by the equation y2 = x3 + ax + b,

then we have :

0 and

92

E. FOUVRY and M. R. MURTY

THEOREM B ((F-M] Theoreme 6). - For every positive e, we have, for x - oo, the asymptotic relation 'ro(x,Ea.,b)

\3

IaI x2+E, AB > xz+E A familiar way to write (1.1) is to say that the probability for a prime p to be supersingular for E is (1.2)

CE 2

f' 1

and we are led to the problem of the primes supersingular for two given elliptic curves E and E'. We say that two elliptic curves over Q are in general position when none of them has complex multiplication and when they are not isogenous over Q. Let us recall that, when E has complex multiplication,

we have 7ro (x, E) - 2 iog , and if E and E' are isogenous over (0, they have the same supersingular primes apart from the prime divisors of the conductors of these curves ; in other words, we have (1.3)

iro(x,E,E') =7ro(x,E) -OE(1) =iro(x,E') -OE,(1),

where 7ro(x, E, E') is the cardinality of the set of primes, less than x, supersingular for E and E'. Then, following (1.2), it is natural to think that if E and E' are in general

position, the probability for a prime p to be supersingular for E and E' should be CE E' P

where CE,E, is a positive constant depending only on E and E'. Such a probabilistic assumption of independence appears in IL-T] page 37, and leads to the conjecture (1.4)

7ro(x, E, E') - CE,E' loge x (x -- 00)-

when E and E' are supposed to be in general position. This conjecture seems extremely hard to prove - for the moment, nobody knows how to prove that fro (x, E, E') - oo when x -> oo - since the set of primes in question is very, very sparse (heuristically as sparse as the following set connected with the Mersenne conjecture : {p < x; 2" - 1 is a prime}). The bulk of this paper is to prove that (1.4) is true on average, in the same philosophy as Theorem B proves (1.1) on average. We will prove

SUPERSINGULAR PRIMES COMMON It) TWO ELLIPTIC CURVES

93

THEOREM 1. - For every positive E, we have for x -> oc, the asymptotic relation 0(xi Ea,b> r"'a',b') ..'

(1.5) IaI

x2+E, AB, A'B' > x-2+E

Let S(A, A', B, B') be the sum studied in (1.5). To be allowed to say that Theorem 1 proves (1.4) on average, we must check that the contribution to S(A, A', B, B') of the pairs of curves (Ea,b, Ea',b,) which are not in general position is negligible. So we denote respectively by SCM(A, A', B, B') and S1--(A, A', B, B') the contribution of those pairs with Ea,b having complex multiplication and with Ea,b and Ea',b' isogenous over (Q. The first contribution satisfies : SCM(A, A', B, B') < {(a, b); Ial < A, JbI < B, Ea,b is CM}1

E Y, 70(x,Ea',b') Ia'IFjkX=

ai /l(gi)2k_3J(gi,

Z)_kF(gi

< Z >),

(7)

aiI',,.g

(X =

If F is a Siegel modular form of weight k with respect to r and X E 7-1(I) C 7-1(I ,,,,) we obtain the representation of the ring 7-1(r) on the finite dimensional space of Siegel modular forms (Hecke operators). It is known that eigenfunctions of all Hecke operators form a basis of the space of all Siegel modular forms. We have identified the Hecke ring of the symplectic group with its image (see (6)) in the ring 7-1(F ), which also contains two subrings isomorphic to the Hecke ring 7{(SL2(Z)) : (8)

x(r)

7{(x(,)

"

7-l(SL2)

.

It is enough to define the embeddings j± for the generators

T(p) = SL2(Z)diag(1, p)SL2(Z) and T(p, p) = SL2(Z)diag(p, p)SL2(Z) of the ring 7-l(SL2(Z)). By definition we have

j_(T(p)) = T_(p) = I 00diag(l, p, p, 1)I 00,

j+(T (p)) = T+ (p) = r0diag(1,1, p, p)r,,

j-(T(p,p)) = A-(p) = r.diag(p,p2,p,1)r0, j+(T(p,p)) = A+ (p) = r.diag(p,1,p,p2)r00 The statement that the mapping j_ is a homomorphic embedding is clear, because there is a one-to-one correspondence between the left cosets in the decomposition of the double cosets T(p), T(p,p) and T_(p), A_(p) (see [G1] and [G5] where more general embeddings have been constructed). The mapping j+ is dual to the embedding j_ with respect to the involution * of

the Hecke ring l(r

)

*:

I -gr- -> r-µ(g)g-11F00

The next lemma is a special case of a general result proved in [G I].

ARJ7HMETICAL LIFTING AND ITS APPLICA77ONS

109

LEMMA 1. - The polynomial QP(X) splits over the ring -l(F,,,) :

QP(X) = j_ (QSL(X)) (1 + pzP(VP - p)X 2) j+(QsL (X )), where thefirst and the thirdfactors are the j f -images of the Heckepolynomial

QSL (X) = 1 - T(p)X + pT(p, p)X2 for the group SL2 (Z) and

VP rEP-1Z/Z

rEP 1Z/Z

1

0 1

0 0

0

0 0 0

0

1

0

0

1

0

r

Proof : using the elementary divisor theorem for the symplectic group we can calculate the images (6) of the generators of 'H (IF) in the Hecke ring

of the parabolic subgroup l (F

)

:

T(p)=T_(p)+T+(p), T1,=A_ (p)+A+(p)+r diag(1,p,p2,p)r,,,+LP(VP 1).

The coefficients T_(p) and A_(p) of the polynomial j_(QPL(t)) = 1 T_ (p)t + pA_ (p)t2 have the same decompositions as sums of left cosets as the elements T (p) and T (p, p) in the Hecke ring of SL2 (Z) and it is easy to verify that the following identities hold : A_(p)T+(p) = p20PT_ (p), T_ (p) (V P - p) = 0,

A_ (p) (VP

- p) = 0.

Using the antiautomorphism *, we have that T_ (p)A+(p) = p20PT+(p),

(OP - p)T+(p) = 0,

(OP - p)A+(p) = 0.

Taking into consideration the identity T_ (p)T+ (p) = pr. diag (1, p, p2 , p) r. + (P3 + p2) OP,

which one can easily check, we obtain the factorization of the lemma.

There are two representations of the Hecke ring f(F,,,) on the space of the Fourier-Jacobi coefficients of Siegel modular forms. The first one is the representation "Ik " on the space of all Jacobi forms of weight k (homogeneous modular forms with respect to Fj, defined in (7), and the second is the representation on the space of Fourier coefficients of r,,,invariant functions F(Z) f.. I I k X := the

mtrt Fourier-Jacobi coefficient of the function Fl k X X.

110

V. GRITSENKO

The following formulae are clear from the definitions (see [G1] for more general statements) fm(T,z)IIkT+(n)=fmnik T+(n)(Z)exp(-27rimw), (9)

fm(T,z)IIkT-(n)= llif-/nIk T_(n)(Z)exp(-27rimw),

otherwise, where by T± (n) we denote the j±-images of the standard Hecke element ,

SL2(Z)diag(a,b)SL2(Z).

TSL(n) a6=n alb

To make our notation shorter we set (fmnik T+(n))(Z)exp(-27rimw), fmnik T+(n) (10) (fmlk T_(n))(Z)exp(-27rimnw). milk T_(n) These are Jacobi forms of index m and mn respectively. COROLLARY 1. - Let

F(T, z, w) _

fm (T, z) exp(27ri mw) m>1

be a Siegel cusp form of weight k which is an eigenfunction of all Hecke operators. Then for any natural n and prime p the following identity holds in the ring of formal power series X6 = Qp,F(X) E fnp6l k T+ (P) 6>0

(fn+fnIkT-(p)X+pfP IkA-(p)X2)Ik(1+p(Vp-p)A X2), where

f

fmIk(1+p(V -p)A X2) = 1 (1 - p2k-4X2) fm,

if m - 0 mod p, otherwise.

Proof : taking the j+-image of the formal power Hecke series for SL2 (Z)

we have >6>0T+(p6)X6 = j+(QSL(X))-1. The function F(Z) is an eigenfunction, thus Qp,F (X) fn = fn I l k Qp(X) and we obtain with help of Lemma (11) 1 the following identities in the ring of the formal power series

Qp,F(X)Y,fnp6lkT+(p6)X6 =Qp,F(X)EfnllkT+(p6)X6 = 6>0

fnllkQp(X)(1-T+(p)X

6>0 +pA+(p)X2)-1 =

M l k (1 - T_ (p) X + pA- (p) X 2) (1 + p(Op - p) APX 2) .

To finish the proof we can use the formulae (9).

ARI7HMETICAL LIFTING AND 175 APPLICATIONS

111

COROLLARY 2. - Let F(Z) be the same as in the previous corollary. Let t be a natural number such that ft 0 0 and all Fourier-Jacobi coefficients ft1d of the modular form F, where d > 1 is a divisor of t, are identically equal to 0. Then the following identity holds for sufficiently large Re(s)

L(2s - 2k + 4, Xt) L ft,, (T, z)IkT+(n) n-s = ft (T, Z) ZF (S), n>1

whereL(2s-2k+4, Xt) is the Dirichlet LfunctionwiththeprincipalDirichlet character modulo t.

Proof : one has to apply the identities of previous corollary successively

for all primes p with X = p-' and to take into account the standard estimation

(T, z) I = O((v/m) _ 2 e2"'"`y2/v) of Fourier-Jacobi coefficients

(see [KSD.

A generalization of these results to the case of Spn can be found in [G1].

3. - Jacobi lifting. In full analogues with (1) one can define the space fmk(r[t]) of all modular forms of weight k with respect to the paramodular group r[t] (see (4)).

In this section we construct an injective map from the space of Jacobi forms of index t > 1 and weight k (i.e., from the space of modular forms on

the parabolic subgroup I'j into the space of modular forms with respect to the paramodular group of level t. THEOREM 3. - Let q5(T, z) be a Jacobi form of weight k and index t > 1 with the following Fourier expansion

O(T, z) = ) ,

f (n, 1) exp (2iri (nT + lz)).

n,LEZ t> 12

If the zeroth Fourier coefficient f (0, 0) of the Jacobi form 0 is not 0, we also suppose that the weight k > 4. Then the following function (see (10)) CO

m2-k

G,6 (7-, z, w) = f (0, 0)Ek(T) +

Ik T_ (m)) (T, z) exp (27ri tmw)

nt=1

is a modular form of weight k with respect to the paramodular group T[t], where Ek(T) sk + n>1 Qk-1(n)exp (27ri nr) is the Eisenstein series of weight k on SL2 (7G).

112

V. CRITSENKO

Let us make some remarks about this theorem. If index t = 1, the map -* G,6 coincides with well-known the Maass or the Saito-Kurokava lifting (see [EZ]). The theorem shows that the Maass lifting is only the first member in the infinite series of liftings connected with Jacobi forms. Thus for any Siegel modular form fm(T, z) exp (27rimw)

F(T, Z, W) = m>O

we can construct a infinite series of lifted functions

that defined a

"section" of the following infinite product

F -+ l `mk(r[m]). mEN

We may rewrite at least formally the definition of the form G,5 using multiplicative notations. Let f (0, 0) = 0 and 1(Z) = q(T, z)exp (27ri tw). Then Em2-kT

(11) Gm(Z)= I k

(1-T_(p)p2-k+T

(m) _q' Ik 11

m=1

(p,p)p3-2k)-1

P

where the p-factor in the infinite product is the j--image the Hecke polynomial for SL2(Z) (see Lemma 1).

j_(QPL(p2-k)) of

It is interesting, that we can rewrite the classical theta-function in the same terms. To this end let us define the Hecke rings l(SL2) = ,1-l(SL2(Z), SL2(Q)) and 7i(Fo) = 1-l((Fo, I'o(Q)) of the special linear group

and its parabolic subgroup r0 = {

(

1 si)' b E Z }. Like in the

case of Sp4(Z) (see (6)) we can define an embedding'1-l(SL2) -* R(IO)We may continue the comparison with (8) and define an embedding of the multiplicative semigroup N-1 or, more generaly, the polynomial ring Q[x-1] into l(r'0). (Q[x-1] is isomorphic to the Hecke ring?({ 1}, N-1) of the trivial group, consisting only of the unity!) By definition n-1

7-

--

[n-1]

ro

0

no 1

ro = ro

n

I E l(ro) .

0

We can interpret Z-periodic functions of the complex variable r as automorphic functions with respect to the parabolic subgroup r0 C SL2(Z) (compare with the definition of the Jacobi forms). If we take the representation of the Hecke ring 9-l(F0) on the space of Z-periodic functions (automorphic with respect to F o) we obtain, for instance, that exp (27ri r) I [n-1]

ARITHMETICAL LIFTING AND ITS APPLICA77ONS

113

= exp (27ri n2T). As a consequence, we can represent the classical thetafunction as a sum over a semigroup of the Hecke operators {[n-1], n E N} instead of as a sum over the lattice Z. Namely, 0(T) =

exp (27ri n2T) = 1 + 2 nEZ

exp (27ri T) I [n-1], [n 1]E7{({1}, N-1)

or using some formal notation 0(T) = 1 + 2 exp (27ri r) I fl(1 -

[p-1])-1

= 1 + 2 exp (27ri r)I j_(((1)).

P

From this point of view the lifting (11) is a generalization of the last formal identity.

Proof of Theorem 3 : the function G4,(Z) is the sum of Jacobi forms of indices mt for m > 0 (the Eisenstein series is a Jacobi form of index 0). Thus G4, is invariant with respect to the action of the subgroup I'". and, moreover, with respect to I [t] = F (Q) f r[t]. Let us calculate the Fourier expansion of Go :

1

Go (Z) =f(0,0)Ek(7)+

mk-1

rn> 1 ad=m

f (n, 1) exp (27ri (n

dk

b mod d

4tn>l2

aTd b

+ laz + tmw))

(f(o, 0)0'k-1 (m) exp (27ri mtw)

= f (0, 0)Ek(T) + m >1

ak-1 E f(dnl,l)exp(27ri(niaT+alz+adtw)) I

+ a d =m

/

4tdn1>l 2

"1; 0

= P0,0 ) (`- B2k + 1: Qk_ 1( m)(ex P( 27ri mT) + ex P(7ri mtw k

+

m>1

E

ak-1

f (a2 , 1) exp (21ri (n7- + lz + mtw)).

4tmn>l2 al(n,l,m)

m>l,n>1

This expansion shows us that GO (T, z, w) is invariant with respect to exchanging of the variables (T - tw, w -k t-1T). The element

Wt=I tot

Ut

),

where Ut=I

00

114

V. GRITSENKO

realizes this transformation. Hence Golk Wt = (-1)k Go.

(12)

Moreover we have Gm I k Jt = Gk, where Jt is the element from the definition of the parasymplectic group (see § 1), since

where I =

WtIWtI = Jt,

0

0

1

0

O1

0

0

0

0

0

0

1

E 1700.

It is easy to see that the element Jt and the group r,,,, [t] generate the paramodular group r[t]. The theorem is proved. From the definition of the function Go follows the following COROLLARY. - The lifting

J:

k,t - `mk(r[t]),

J(O):=Gk

is injective and satisfies the following commutative relation

J(O)IkT_(m) = J(OIkT (m)) We would like to compare the lifting of Theorem 3 with the analytical theta-lifting connected with dual reductive pairs (see, for example, [Ku]). The space of Jacobi forms fitk,,, is isomorphic to a subspace of modular forms of a half-integral weight with respect to an appropriate congruence subgroup of SL2 (Z) (see [EZI). There is an isogeny between the paramodular group F[t] and a special orthogonal group of type SO(2, 3) (see (G41). Let

us consider the theta-lifting for the pair (SL2, SO(2, 3)), i.e., the integral operator with a theta-function of an even quadratic form of signature (2, 3) as a kernel (see (01], [RS], [Ko]). It will give us a map from the modular forms of a half-integral weight into the space of modular forms with respect to a congruence subgroup of r[t]. We would get the full paramodular group r[t] only for an unimodular even quadratic form of signature (2, 3), which does not exist ! The next defect of the theta-lifting is non-existence of the theta-integral

for modular forms of small weights. We shall see below, that in order to prove Theorem 1 about the moduli spaces we need modular forms of weight 3. Moreover, it is not easy to construct the theta-lifting of Eisenstein series, but in context of the arithmetical lifting we can take not only the Eisenstein

115

ARITHMETICAL LIFTING AND ITS APPUCA77ONS

series, but we can also lift a constant function that gives us so-called singular modular forms (see [G4]).

To finish this short discussion we would like to add, that Theorem 3 is a particular example of a general lifting from the space of Jacobi forms defined on 1H11 x C' (see [G31 and [G41).

Now we shall prove Theorem 1.

Basis to the geometric theory of automorphic forms is the fact that automorphic forms of special weights correspond to sections of canonical line bundles on algebraic varieties. Let F E 03(r[t]) be a modular form of weight 3. The holomorphic differential form on the Siegel upper-half plane WF = F(Z) A dZ = F(T, z, w)d-r A dz A dw is F[t]-invariant and defines an element of the zeroth cohomology group H°(At, Q3 (At)), where Q3 (At) is the sheaf of canonical differential forms on At. The complex variety At is not compact and has a lot of singularities. Due to Freitag we have the following simple criterion about continuation of canonical differential froms on a singular variety to its non-singular model. LEMMA (Freitag). - The elementw c- H° (At, SZ3 (At)) could be extended to

a canonical dferentialform on a non-singular model it of a compactification of the variety At if and only if the differential form w is square integrable. See (F], Hilfsatz 3.2.1.

It is known, that WF is square-integrable for the cusp modular form F. Thus we have the following identity for the geometrical genus of the variety At

p9(At) = h3'°(At) = dims 63(rab[t]) (see §1). If F E 9JTk(F[t]), then FIk J1 E'!Jtk(rab[t]), since J1gJ1 = tg-1 for any g c r[t]. Theorem 3 gives us examples of modular forms with respect to r[t]. It is easy to show that the Satake compactification of F[p] \ 1112 has two one-dimensional components, which are isomorphic to SL2(7L) \ H1

(see [HKW]). Thus the restrictions of the lifting Go (0 E 93Z" P) to the boundary of r[p] \H2 are modular forms of weight k with respect to SL2(Z).

They are identically equal to 0 for odd k and we automatically get cusp forms. Consequently, using the lifting of Theorem 3 we have the following estimation for the geometrical genus of the moduli variety

2]

m-

p9 (At) ? dimc with

9Jt3,n

{2j+2}12- [

=

2

{2j + 10}12 = lI k j 12

-

1

if k if k

0(m),

2 mod 12,

2mod 12.

116

V.. GRITSENKO

The formula for the dimension of the space of Jacobi forms has been obtained in [EZ) (see also [SZ)). For a prime number p > 11 we have pg (AP) > 0, that proves Theorem 3. Corollary from Theorem 3 gives us even more. THEOREM 3bs. - The variety it is not unirational if t has a prime divisor greater than 11. Proof : let us take a Jacobi form 0 of weight 3 and index p. In accordance with (9) 0 Ik T_ (m) E fii3 p,,,, and the lifting J(0I k T_ (m)) = J(q) I k T_ (m) is a cusp form of weight 3 with respect to F[pm].

It is possible to prove, that the variety At could be unirational only for finite numbers of t. The maximal such t is 36. This subject will be developed in more detail in the separate publication [G6) (see also [G41).

4. -Analytical continuation of ZF (S) In this section we shall construct an analytical continuation of Lfunction ZF(s) using a variant of Rankin-Selberg convolution of two Siegel

modular forms, proposed in [KS]. The first function in this convolution will be the eigenfunction F(Z) and the second one will be a lifting of some Fourier-Jacobi coefficient ft. The lifting J(ft) has been defined (see (11)) as action of the "infinite product" on the Jacobi form ft, hence it is not a surprise that the Rankin-Selberg convolution of this form with an eigenfunction has an Euler product. In order to get the functional equation of ZF (s) we consider in §5 more "advanced" variant of this integral, in which we take a convolution with respect to the parabolic subgroup of the maximal normal extension of the paramodular group F[t]. LEMMA 2. - Let F (Z) be a cusp form which is an eigenfunction of all Hecke

operators and t be the same as in Corollary I of Lemma 1. Let Gt = J (ft) be the lifting of the Fourier-Jacobi coefficients ft of F. For Re s > 3 the following identity holds Irk-2t-(s+k-2)

< ft, ft > ZF(s + k - 2) = L*(2s, Xt) f

F(Z)G(Z)Eo(Z, s)I YIk-3dXdY, 00(t)\H12

where

< ft, ft >=

f

xC

ft (r, z)ft(T, z)v-3exp(-47rt y2 v-1)dudvdxdy k

117

ARITHMETICAL LIFTING AND ITS APPLICATIONS

is the scalar product of Jacobi forms, Eot (Z, s) is the Eisenstein series of the congruence subgroup roo (t) = Sp2 (Z) n r [t]

I Y(ry < Z >)Isv(ry < Z >)-s, (Z=X+iY=

Eot(Z, s)=

(ZT

Z

7E r, \ro0 (t)

T = u+iv, z = x+ iy, I YJ = detY and L*(2s, Xt) = 7r-8r(s)L(2s, Xt) (see Corollary 2).

Proof : the integral on the right hand side is the Rankin-Selberg convolution of two modular forms on roo(t). As usual in this method, we may pass to the integral over a fundamental domain of the parabolic subgroup r"' of roo (t)

r,,,\IH[2={01

The operators T_ (m) and T+(m) are connected by the duality * (see §1), thus using the standard Hermitian consideration it is not difficult to prove

that < ftm, m2-kftlkT-(m) >_ < ft.IkT+(m), ft > (see [F], Chapter N, and [KS]). We can finish the proof using Corollary 2.

Proof of analytic continuation of ZF(s)

:

the Eisenstein series

Eot (Z, s) is reduced to a sum of so-called Epstein zeta-functions (see [K], [Kr]). Let us introduce the following positive definite quadratic form corresponding to the variable Z = X + iY E 1H[2

Pz

)

(Y0

=

Y-1

)]

[\ X E

(M[N] = tNMN).

Then

Pry=Pz[try] andY(ry )-1=16z[I tD I] for-y

=(A D )ESP4(R).

The quotient v(7 < Z >)/I Y(ry < Z >)I is equal to the (2,2)-entry of the matrix Y(ry < Z >) -1 and one can rewrite the series L(2s, Xt)Eot(Z, s) as a sum of Epstein zeta-functions of the quadratic form Pz : L(2s, Xt)Eot(Z, s) =

Pz[N]-s = 1` t-'%(s, g, 0, Pz),

N=t(nl,'2,n3,n4)EZ4 nl,n2,n3=Omodt, (n4,t)=1

9=(0,0,0,94)

g4Et-1Z\Z (t9q,t)=1

V.. GR175ENKO

118

where for g, h E ]R4 (see [Ep], [T))

exp(27ritNh)Pz[N+gj-s.

g,h,Pz)= NEZ4 N+gj4O

It is known (see (Ep)) that the function (* (s, g, h, Pz) = 7r`F(s)((s, g, h, Pz) has the meromorphic continuation to the s-plane, satisfies the functional equation (13)

(*(s,g,h,Pz) =exp(-27ri )(*(2-s,h,-g,Pz1)

where < g, h >= g1h1 + . - + g4h4, and has simple pole with residue 1 at -

s = 2, if h is integral, and simple pole with residue -1 at s = 0, if g is integral. As a corollary of the integral representation of Lemma 2, we have the meromorphic continuation of the function ZF(s) of Theorem 2. Moreover, if t > 1 then the vectors g are not integral and ZF(s) could have a pole only at s = k.

If t = 1 (that could be possible only for even weight k), then the Eisenstein series 7r-sF(s)((2s)Eol(Z,s) is equal to the Epstein zetafunction c* (s, 0, 0, Pz), satisfies the functional equation (13) and has two poles at s = 0, 2 with residues :1 respectively. It gives us the functional equation of Theorem 2 in the case t = 1. Moreover the residue Ress-_kLF.(s) is proportional to the scalar product < F, G1 > of the Siegel modular form F and the form G1 = J(fl) containing

in the Maass subspace, which is invariant with respect to the action of Hecke operators. This scalar product is zero if F orthogonal to this subspace, thus ZF(s) is entire for such F. Otherwise F = G1 and the residue of ZF(s) at s = k is equal to 7r2-k< F, F >/< fl, fi >. We consider the case t > 1 below.

5. -Functional equation of the Spin L-functions for Siegel modular forms with the first Fourier-Jacobi coefficient fl - 0 The integral representation of the Spin L-function obtained above gives

us its meromorphic continuation, but the Eisenstein series, which is the kernel of the integral, has no good functional equation. As was shown in the proof of Theorem 3, the lifting J(ft) is "nearly" invariant with respect to a normal extension F [t] of the paramodular group r[t] generated by the group I'[t] and the element Wt (see (12)). To get the functional equation for ZF(s) one has to construct the second variant of the Rankin-Selberg convolution for this new group, since in that case the Eisenstein series satisfies a good functional equation. Without loss of generality we may restrict ourselves to the case of a prime number t.

119

ARITHMETICAL LIFTING AND ITS APPLICATIONS

LEMMA 3. - Let F be a cusp form which is an eigenfunction of all Hecke

operators. Let us assume that its first Fourier-Jacobi coefficients f, (T, z)

vanishes. Then there exists a prime number p such that fP (7-, z) is not identically equal to zero.

Proof : let us consider the Fourier expansion F(Z) _ E a(N) N E'B

exp (27ri tr(NZ)), where the sum is taken over the set SB2 of all positive

definite semi-integral symmetric matrices N = (l72 1/2) . The Fourier coefficient a(N) depends only on the class of the quadratic form N : a(N) _ a(tXNX) for any X E SL2(Z). If there is a primitive N ((m,1, n) = 1) such that a(N) # 0, then we may take any prime p represented by the quadratic form N. For such prime numbers fp(T, z) 0 0. Let us suppose that a(N) = 0 for all primitive matrices N. Consider the Fourier-Jacobi expansion of F

F(T, z, w) = L fm (T, z)exp (2iri mw). nx>r> 1

The form F is an eigenfunction, therefore we have the following relation between the Fourier-Jacobi coefficients of F and FlkT(e) for any divisor e of the index r (ed = r) :

fd{FjkT(e)}(T,z) = (fr{F} 1kT+(e))(T,z) = AF(T(e))fd{F}(r,z) - 0,

where T(e) is the Sp4(Z)-Hecke operator with index e, )'F(T(e)) is its eigenvalue and T+(e) is the j+-image of the SL2(Z)-Hecke operator (see (7), (9), (10) and the proof of Lemma 1). In the Fourier expansion of the Jacobi form fr a(N)exp (27ri (tr(NZ))),

f, (r, z)exp (27ri rw) = N=

*

r

E'B2

0, where there are no a(N) with a primitive N. If a(l e1/2 eed2)) ed = r, e > 1 and (m,1, d) = 1, then it is easy to see, that the function (frlkT+(e))(T,z), that is identically equal to 0 in accordance with the previous considerations, has at least one non-zero Fourier coefficient? ! The lemma is proved. In the next lemma we consider two "trace" operators for Siegel modular forms on Sp4 (Z) which send them to modular forms on the paramodular group and on the group I * [t] respectively.

120

V. GRITSENKO

LEMMA 4. - Let

F(Z) = 1: f. (-r, z)exp (2iri mw) E Mk (SP4 (Z)), m> 1

then the functions

Fp = FIkVp + Flk Jp

Fp =Fr+FPIkWP,

and

(see (1), (12) and Lemma 1) are modular forms of weight k with respect to the paramodular group T[p] and its (maximal) normal extension r- [pi = r[p] U F[p]W9,

respectively. Moreover the function F; (Z) has the Fourier Jacobi expansion FP (Z) =

z)exp (2iri pmw)

where =Pfpm+p-(2k_6) fr

fpm

+p-(k-3)fmlkT_(p)

IkA-(p)

Proof : the first part of the lemma is evident. To calculate the FourierJacobi expansion we can rewrite the trace operator as follows

Fp = 1: FIko(x)+FlktJlJp xEp-17/Z 0

+xEp E 'Z/Z FI k

0 0

1

0

0

0

0(x)Wp + FIk

0

0

Ol

0

1

0

0 1

0

0

0

-1

01

0

0

0

1

0

JPWP.

The first sum gives us only coefficients with indices divisible byp, the second sum is equivalent to the action of the operators A_ (p)AP 1 and the last two summands coincide with the action of the Hecke operator (0,/P-)-1T_(p). LEMMA 5. - Let p be prime. Then the Eisenstein series

EP(Z, s) = 7r-'F(s)(1 +p s)((2s)

>2

ryEro

I Y(1 < Z >)I sv(ry < Z >)-s, [p1\r* [p]

ARrITIMETICAL LIFTING AND ITS APPLICA77ONS

121

has a meromorphic continuation to the whole s-plane and is invariant with respect to the transformation s -> 2 - s. Proof : as in the proof of Lemma 2, Et (Z, s) can be represented as a sum of Epstein zeta-functions. The last rows of representatives of r,, [p] \ r* [p]

form the set of all Z[p-1]-primitive (primitive outside p) vectors of the following types : (pa, pb, pc, d) with (p, d) = 1;

p(a, b, c, d) with (b, p) = 1; f (a, pb, c, d) without a - c - 0 mod p.

Therefore there is a representation of Ep* as a sum of Epstein zeta-functions. A simple computation shows that

EE(Z,s)=i 'r(s)p24(

Pz[t(pa,pb,pc,d)]-s+p-$Pz[t(a,pb,c,d)])

(a,b,c,d)EZ4\{0} C(s,2,0,Pz)+p_ 1 E

g=(0,92,0,0) 92 mod p

p

h=(0 0 O h4) h4 mod p

(*(s,0,

h p

,Pz)

Using the functional equation(13) and the identity PZ 1 = Pz [J1] we get the functional equation for E*.

In accordance with Lemma 2 and with the identity (12), the product

(Fp(Z) + (-1)kFpIk Wp(Z)) Gp(Z) (det y)k

(where Gp = J(fp)) is invariant with respect to the action of r* [p] and we can construct an integral analogue to the integral of Lemma 2 for the group

r*[p] LEMMA 6. - Let F(Z) be a cusp form of weight k on Sp4 (Z). Let F be an eigenfunction of all Hecke operators with the first FourierJacobi coefficient f 1(T, z) = 0 and let p be a prime number for which f p (rr, z) 0 identically. For Res > k + 1 the following identity holds p3-2k7rk-2 (p2 -k+1 + (_1)kp 2 )) < fp, fp > ZF(s) _

(FF(Z) + (-1)kFpIkWp (Z)) Gp(Z) E, (Z, s - k + 2)1 yak-3dXdy, r' [p] \H2

where < fp, fp ># 0 is the scalar square and Gp = J(fp) is the Ufting of the Jacobi form fp.

122

V. GRITSENKO

We note that the functional equation for L-function ZF(s) of Theorem 2 follows from the above representation and the functional equation for the Eisenstein series EP (Z, s) from Lemma 5. Proof of Lemma 6: applying the same unfolding arguments with EE as in Lemma 2 we find that the integral equals the following Dirichlet series

+2r(s)r(s - k + 2)(1 - P -(s-k+2))-1 L(2s - 2k + 4, Xp)

C

fmpIkT+(m), fr >

m>1

where

MS

are the Fourier-Jacobi coefficients of the function Fr+(-1)kFpl k

W. These coefficients contain three summands, as has been shown in Lemma 5. Thus we have

1: m>1 (14)

E

fmplkT+(m) = ms

[pfPr+p (2k-6)fTIkA-(p)+p (k-3)fTIkT-(p)] IkT+(m) MS

m> 1

which is reduced to three Dirichlet series. The first series one can calculate using Corollary 2. For the third sum one gets

(15) L(2s-2k+4, Xp) E fmI kT (p)T+(m) =p-s(fpI kT_(p)T+(p)) ZF(s) m> 1

To prove this we may use the identity (16)

)pfmp = f m p l l k T (p) = f m I k 7- (p) + f m p 2 I k T+ (p),

where AP is the F-eigenvalue of the Hecke operator T(p). Thus the series (15) is equal to L(2s - 2k + 4, Xp)

(Apfmp - fmp2 Ik T+(p)) I kT+(m) m-s. m> 1

The operators T+(p) and T+(m) commute, thus using Corollaries 1 and 2 of Lemma 1 we see that the last sum is equal to

(Af - (ff2 - fplkT-(p)ps)IkT+(p))ZF(s)

123

ARrIHMETICAL LIFTING AND ITS APPLICATIONS

Applying (16) again for mp = p2 and taking into account the assumption that f, - 0 we get (15). The calculation of the second sum in (14) can be done as follows. The standard identity between elements in the SL2 (Z)-Hecke ring T(p)T(M) P_ T (m) + pT (p, p)T (P) gives us after the "plus" j+-embedding

T+(p)T+(P) = T+ (m) + pA+(p)T+(). In Lemma 1 we have seen that A_(p)T+(p) = p20PT_(p) and A_(p)A+(p) = p40P. Therefore fmlk (A-(p)L 'T+(mp))

(mp)-3

m> 1

=p2-s E fmlkT-(P)T+(in)m-s _p5-2sY fvlIkT+(l)zP1-s m> 1

1>1

= (p2-sfpl k T-(p)T+(p) -

p2k-1-23)ZF(s)L(2s - 2k + 4, Xn)-1'

where we have used (15) and Corollary 2. In the second and third sums there is a summand of type f,IkT_(p)T+(p), which we shall calculate in the next lemma. LEMMA 7. - Let fP(T, z) be a Jacobi form of index p and weight k such that fP I k T+ (p) - 0. Then

fplkT-(p)T+(p) =p2k-6(p3 - (_l)kp2)f

.

Proof : one can prove the lemma almost entirely "inside" the formal Hecke ring 7-l(I ,,.). As in the proof of Lemma 1

T-(p)T+(p)=(pTj(p)+p3+p2)OP, where Tj(p)=I diag(p-1,1,p,1)I',. We note that the operator IkTj(p) coincides up to a constant with the Hecke-Jacobi operator Tj(p) defined in §4 of [EZ]. On the other side it is easy to check that (17)

T'+(p)T-(p) =

(T.I(p)VP

where

_ P

x,y,rEP-'Z/Z

I

+ pVP + 'P)AP, 0

0

y

1

y

r

0

0

1

x

0

0

0

1

1

-x

V GRITSENKO

124

Let us take the standard expansion of fP(T, z) with respect to the basic Jacobi functions 2

Op,,, (-r, z) = E exp (27rip(1 + 2

2p)

IEZ

T + 27ri(2pl +,u)z

(see IEZ], §5). If

ff(T, z) =

0li(7)BP,µ(T, z), µ mod 2p

then after obvious calculations with Gauss sums one gets ff(T, z)I k ..y = p2

E

Oµ(T)eP,-µ(T, z)

Ii mod 2p

(this is the only place in the proof of the lemma in which we have to use Jacobi forms themselves). The invariance of fP(T, z) with respect to the mi-

nus identity matrix -E4 is equivalent to the identity µ = (-1)_,(T, z), thus ffIk P = (-1)kp2 fP. Moreover, from (17) we get fP I k T(p) =

2p 2 fP,

0

if k is even, if k is odd,

since by our assumption fpl k T+ (p) = 0. This prove the lemma.

To finish the prove of Lemma 6 and to get the functional equation of Theorem 2 one has to collect the three sums in (14) together and to take into account the result of Lemma 7. We have proved in §4, that the function ZF(s) could have only one pole for t > 1. Together with the functional equation stated above it gives us that the Spin L-function is entire function on the whole s-plane if the first Fourier-Jacobi coefficients of the cusp form F vanishes. Thus Theorem 2 is proved completely.

Manuscrit recu le 26 octobre 1993

ARI7TIMETICAL LIFTING AND 175 APPLICA77ONS

125

REFERENCES

[A] A. N. ANDRIANOV. - Eulerproducts corresponding to Siegel modularforms of

genus 2, Russian Math. Survey 29 (1974), 45-116. [EZ] M. EICHLER, D. ZAGIER. - The theory of Jaccobi forms, Progress in Math. 55,

Birkhauser, Boston, Basel, Stuttgart, 1985. [Ep] P. EPSTEIN. - Zur Theorie allgemeiner Zetafu ctionen Math. Ann. 56, (1903), 614-644. [Ev] S. A. EvDOKIMOV. - A characterization of the Maass space of Siegel cusp forms of degree 2, Matem. Sbornik 112 (1980), 133-142 (Russian) ; English

transl. in Math. USSR Sbornik 40 (1981), 125-133. [F] E. FREITAG. - Siegelsche Modulfunktionen, Grundlehren der math. Wissensch., 254, Springer, Berlin, Heidelberg, New York, 1983. [G 1 ] V. A. GRITSENKO. - The action of modular operators on the Fourier-Jacobi co-

efficients of modular forms, Matem. Sbornik 119 1982, 248-277 (Russian) ; English transl. in Math. USSR Sbornik 47 (1984), 237-268. [G2] V. A. GRITSENKO. - Jacobi functions and Euler products for Hermitian modularforms, Zap. Nauk. Sem. LOMI 183 (1990), 77-123 (Russian) ; English transl. in J. Soviet Math. 62 (1992), 2883-2914. [G3] V. A. GRITSENKO. - Jacobi functions of n-variables,Zap. Nauk. Sem. LOMI 168 (1988), 32-45 (Russian); English transl. in J. Soviet Math. 53 (1991), 243-252. [G4] V. A. GRITSENKO. - Modular forms and moduli spaces of abelian and K3 surfaces, Mathematica Gottingensis Schrift. des SFB "Geometrie und Analysis", Helt 26, 1993, p. 32; appears in St.Petersburg Math. Jour. 5 (1994). [G5] V. A. GRITSENKO. - Induction in the theory of zeta-functions, Preprint 91097 University Bielefeld, 1991 p. 76; appears in St.Petersburg Math. Jour. 5 (1994). [G6] V. A. GRITSENKO. - Moduli spaces of abelian surfaces (in preparation). IKW] K. HULEK, C. KAHN, S. H. WEINTRAUB. - Theta functions and compactification

of moduli spaces of polarized abelian surfaces, 1993.

[I] J. IGVSA. - Theta function, Grundlehren der math. Wissensch., 254, Springer Verlag, Berlin, Heidelberg, New York, 1972.

126

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[K] W. KOHNEN. - On character twists of certain Dirichlet series, Mem. of the Fac.

of Science Kyushu University, series A, Mathematics 47 (1993), 103-119. [KS] W. KOHNEN, N.-P. SKORUPPA. - A certain Dirichlet series attached to Siegel

modular forms of degree two, Invent. Math. 95 (1989), 449-476. [Ko] H. Ko,nvA. - On construction ofSiegel modularforms of degree two, J. Math. Soc. Japan 34 (1982), 393-411. [Kr] A. KRIEG. - A Dirichlet series for modular forms of degree n, Acta Arith. 59 (1991), 243-259. [Ku] S. KUDLA. - Seesaw dual reductive pairs, Automorphic forms of several variables, Progress in Math. 46, Birkhauser, Boston, Basel, Stuttgart, 1983, 244-268. [L] R.P. LANGLANDS. - Euler products, Yale Univ. Press, 1971.

[01] T. ODA. - On modular forms associated with indefinite quadratic forms of signature (2, n - 2), Math. Ann. 231 (1977), 97-144. [02] T. ODA. - On the poles of Andrianov L-functions, Math. Ann. 256 (1981), 323-340. [P-S] I. I. PYATETSKII-SHAPIRO. - Automorphic functions and the geometry of clas-

sical domains, Gordon and Breach, New York, 1969. [RS] S. RALLIS, G. SCHIFFMANN. - On a relation between SL2 cusp forms and cusp

forms on tube domain associated to orthogonal groups, Trans. Amer. Math. Soc 263 (1981), 1-58. ISZI N-P. SKORUPPA, D. ZAGIER. - A trace form for Jacobi forms, J. reine and angew. Math. 393 (1989), 168-198. [T] A. TERRAS. - Harmonic Analysis on Symmetric Spaces ans Applications, I, Springer Verlag, Berlin, Heidelberg, New York, 1983.

Valeri GRITSENKO

Department Steklov Mathematical Institute St. Petersburg FONTANKA 27

191011 ST. PETERSBURG RUSSIA

Number Theory Paris 1992-93

Towards an arithmetical analysis of the continuum Glyn Harman

1. - Introduction Since the set of real numbers is uncountable, almost all (in a variety of senses) real numbers are effectively indescribable. Our curiousity forces us, however, to attempt to describe the irrationals in terms of their relation to the "known" set of rationals. An elementary theorem given by Dirichlet (1842) provides the simplest such relation. For every real a, and any given N > 1, there exist coprime integers m, n

such that (1)

a

n 0.

I

Manuscrit recu le 2 septembre 1993

TOWARDS ANARI7HME77CAL ANALYSIS OF77IE CONTINUUM

137

References [11 R.C. BAKER. - Diophantine Inequalities, Clarendon Press, Oxford 1986. [2] R.C. BAKER and G. HARMAN. - On the distribution of app` modulo one,

Mathematika, 38 (1991), 170-184. [3] H. BEHNKE. - Uber die Verteilung von Irrationalitaten mod 1, Abh. Math. Sem. Hamburg, 1 (1922), 252-267. [4] E. BOREL. - Une contribution a I'analyse arithmetique du continu,

Journal de Mathematiques Pures et Appliquees, (5eme serie), 9 (1903), 329-375. [5] J.W.S. CASSELS. - Some metrical theorems in Diophantine approximation I, Proc. Cambridge Phil. Soc., 46 (1950), 209-218.

[6] J.-R. CHEN. - On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica, 16 (1973), 157-176. [7] I. DANicic. - Contributions to Number Theory, Ph. D. Thesis, London 1957. [8] R.J. DUFFIN and A.C. SCHAEFFER. - Khintchine's problem in Metric Diophantine approximation, Duke Math. J., 8 (1941), 243-255. [9] P. ERDOs. - On the distribution of the convergents of almost all real numbers, J. Number Theory, 2 (1970), 425-441. [10] P.X. GALLAGHER. - Approximation by reduced fractions, J. Math. Soc.

of Japan, 13 (1961), 342-345. [11] G.H. HARDY and J.E. LrrrLEwoOD. - The fractional part of nJB, Acta

Math., 37 (1914), 155-191. [12] G. HARMAN. - On the distribution of ap modulo one, J. London Math. Soc., (2) 27 (1983), 9-18. [13] G. HARMAN. - Diophantine approximation with a prime and an almost

prime, J. London Math. Soc., (2) 29 (1984), 13-22. [14] G. HARMAN. - Diophantine approximation with almost primes and two

squares, Mathematika, 32 (1985), 301-310. [15] G. HARMAN. - Metric diophantine approximation with two restricted variables I, Math. Proc. Cambridge Phil. Soc., 103 (1988), 197-206. [16] G. HARMAN. - Metric diophantine approximation with two restricted variables III, J. Number Theory, 29 (1988), 364-375. [17] G. HARMAN. - Some cases of the Dufn and Schaeffer conjecture, Quart. J. Math. Oxford, (2) 41 (1990), 395-404. [18] G. HARMAN. - Numbers badly approximable by fractions with prime denominator, preprint Cardiff, 1993.

138

G. HARMAN

[19] D.R. HEATH-BROWN. - Diophantine approximation with square-free integers, Math. Zeit., 187 (1984), 335-344. [20] H. HEILBRONN. - On the distribution of the sequence n29 (mod 1), Quart. J. Math. Oxford, (1) 19 (1948), 249-256. [21] A. HuRwITZ. - Uber die angenaherte Darstellung der Irrationalzahlen durch rationaleBriiche, Math. Ann., 39 (1891), 279-284. [22] C. HUYGENS. - Projet de 1680-81, partiellement execute d Paris, d'un

planetaire tenant compte de la variation des vitesses des planetes dans leurs orbites supposees elliptiques ou circulaires, et consideration de diverses hypotheses sur cette variation, in CEuvres Completes de Christian Huygens, 21, 109-163, Martinus Nijhoff, La Haye (1944). [23] H. IwANIEC. - On indefinite quadratic forms in four variables, Acta Arithmetica, 33 (1977), 209-229. [24] A. KHINTCHINE. - Zdr metrischen Theorie der diophantischen Approximationen, Math. Zeit., 24 (1926), 706-714. [25] A.D. POLLINGTON and R.C. VAUGHAN. - The k-dimensional Duffin and

Schaeffer conjecture, Mathematika, 37 (1990), 190-200. [26] A.M. ROCKETF and P. SzUsz. - Continued Fractions, World Scientific, Singapore-New Jersey-London-Hong Kong 1992. [27] J.D. VAALER. - On the metric theory of Diophantine approximations,

Pacific J. Math., 76 (1978), 527-539. [28] R.C. VAUGHAN. - Diophantine approximation by prime numbers III, Proc. London Math. Soc., (3) 33 (1976), 177-192. [29] R.C. VAUGHAN. - On the distribution of ap modulo 1, Mathematika, 24 (1977), 135-141. [30] I.M. VINOGRADOV. - The method of trigonometric sums in the theory of numbers (translated from the Russian by K.F. Roth and A. Davenport), Wiley-Interscience, London 1954. Glyn HARMAN

School of Mathematics University of Wales College of Cardiff, 23 Senghenydd Road, P.O. Box 926, CARDIFF CF2 4YH

United Kingdom

Number Theory Paris 1992-93

On A-adic forms of half integral weight for SL(2)/Q Haruzo HIDA

1. - Let S be the two-fold metaplectic cover of S = SL(2)/z and fix a prime p > 5. In this short note", we want to describe a technique of lifting a family of complex automorphic representations of S(A) to a "Aadic automorphic" representation II of S(A(P°°)), where A is a one variable power series ring over an appropriate p-adically complete discrete valua-

tion ring, and A(P°°) is the adele ring A of Q the p and oo-components removed. Then we will have a A-adic version of a result of Waldspurger [Wa2]. We begin with the study of p-adic cusp forms of half integral weight

and prove in Section 3 that the classical cusp forms of weight k + . is dense in the space of p-adic cusp forms of half integral weight if k > 2 (Theorem 1). Then we study A-adic forms of half integral weight in Section 4 by combining the techniques of Wiles [Wi] (introduced for integral weights) and the representation theoretic technique of Waldspurger [Wal,2]. Taking

the limit shrinking the congruence subgroup, we get the desired A-adic representation of S(A(P°°)) (Proposition 1). Then we prove the weak multiplicity one theorem for p-ordinary A-adic automorphic representations (Theorem 2 in Section 4). Although our construction is just the combination of these two existing techniques, we get a fairly strong result on p-adic standard L-functions of G = GL(2)/Q. That is, a certain ratio of the restriction of 2-variable p-adic standard L-functions [K] to the line interpolating The author is partially supported by an NSF grant. The final touch to the paper was given while the author was visiting the Isaac Newton Institute for Mathematical

Sciences, Cambridge, England. The author acknowledges the support from the Institute for the month of April in 1993. Some part of the work presented in this note was actually done in 1988 in order to construct a p-adic standard L-functions for GL(2) restricted at the center critical line (Theorem 4). The construction of two variable p-adic standard L-functions was later done by K. Kitagawa [K] using a different method.

140

H. HIDA

the central critical values is shown to be square in the field of fractions of the Iwasawa algebra A (Theorems 3 and 4), which is the A-adic version of a result of Waldspurger ([Wa2] Corollary 2) we alluded to. A further scrutinizing of the representation we constructed might bring us a sharpening of this result giving a A-adic version of the result in M. However to make our presentation short, we will not touch this subject in the present account. Another interesting point which awaits further study is the behavior of the specialization 7rv,t=2 of irreducible factors it of 11 at weight 2. In [GS],

Greenberg and Stevens gave an interesting limit formula of the derivative

of the p-adic standard L-function at the center critical point, when the L-function has an exceptional zero at this point. This is the unique case where the specialized automorphic representation It t=2 of S(ZP°°)) (supplemented with the p-component) becomes super cuspidal at p although the integral image of 7rwt=2 under the Shimura correspondence is special and p-ordinary. Thus the study of the behavior of the other local components of Trwt=2 might cast some new insight upon the p-adic analog of the conjecture of Birch-Swinnerton Dyer formulated in [MTT]. Although I have only worked out here the result for SL(2) defined over Q, our idea works fine for SL(2) over general number fields. However, in the general case, the many variable standard p-adic L-functions defined on the spectrum of the p-adic Hecke algebra are not yet constructed.

2. - Let A be a congruence subgroup of level prime to p. When we consider modular forms of half integral weight, we assume that A is contained in ro(4). We write O1(pa) = A n r1(pa) and A(pa) = A1(p) n ro(pes). We use the same notation introduced in [H1] Sections 1 and 2 for classical modular forms. In particular, for each integer k and an algebra A, Pk+(1/2)(AI(pa); A)) stands for the space of A-integral cusp forms of half integral weight k + 2 with respect to Ai(pa), while for each integer SK,(AI (pa); A)) stands for the space of A-integral cusp forms of integral

weight ,c. Here the A-integrality is given by the q-expansion at the cusp oo. For each Dirichlet character X modulo Npa, Pk+(1/2)(ro(Npa); X; A) consists of cusp forms g in Pk+(1/2)(r1(Npa);A) with 9Ik+(1/2)a = X(d)9 for each or = I a d

)

E ro (N), where glk+(1/2)0'is the action of or defined in

[H 1J (2.2a) which is a little different from the normalization of [Sh 1] p. 447. Our normalization is :

9Ik+(1/2)0'(z) = g(a(z))j(a, z)-1J(a,

z)-k

for a = (c

b

d

J

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)IQ

141

00

where J(o, z) = (cz + d) and j (a, z) = 0(a (z))/9(z) for 9(z) = >exp(27rin2z).

n--oo

By [HI] Theorem 2.2 or its proof, Pk+(1/2)(r1(Np°); A) is stable under this action of a E 1'o (Npa). We now give an interpretation in adelic language following [Wa2] III. We write S for the algebraic group SL(2)lz. We write S for the two fold metaplectic cover of SL(2) defined in [Wa2] II.4. Thus S(Qp), S(A) and S(R) have meaning, where A is the adele ring of Q. In other words, we have a non-splitting exact sequence of groups :

1-->{±1}--S(A)->S(A)--* 1, where A is either A, Qp or R. Now let us describe the 2-cocycle /3 giving the ), we put x (a) = d

extension S(Q,,) for a place v of Q. For each a = (a or c according as c = 0 or c # 0. We also put

sv(a) _

(c,d)v if cd

d

0, v is finite, vp(c) is odd, otherwise,

1

where (c, d)v is the Hilbert symbol at v (that is, Artin symbol (d, Qv) of d). Then we have

VC-(d'w)

= (c, d)v f for the

Qv(a> a') = (x(a) , x(aTv(-x(a)x(a'), x(aa'))vsv(a)sv(a')sv(ao') .

For a e S(Q), the product s(a) = IIvsv(av) is well defined. Similarly we may define /(a, a') = nv3(av, a',) for a, a' E S(A). Then we identify S(A) with S(A) x {±1} under the multiplication law given by (g, e)(h, e') = (gh, )0(g, h)ee'). By the product formula of the Hilbert symbol, ,Q(a, a') =

s(a)s(a')s(aa') for or, a' E S(Q). Thus a F--> (a, s(a)) gives a section : S(Q) -> S(A). We identify S(Q) with its image in S(A). We also identify the standard maximal compact subgroup S02(]R) with ][8/Z by 0 1--k cos 2x9 sin 27x9

sin 27r9 cos 27r9

Then the pull back image of S02(R) in S(R) can

be identified with ][8/2Z. We write the corresponding element r(9) in S(R) for an integer k and C, = {r(9) 10 E R/2Z}. Then r(9) - e((k + 2)9) and e(9) = exp(27ri9) is a character of C. Via (g, e) H g(i) E H, we have

S(R)/C,, = H for the upper half complex plane H. Let e : A/Q -> C be the standard additive character such that e(xo,,) = exp(27rix,,,,). We write ev for the restriction of e to Qv for each place v, and we define 'yv(t) to be the Weil's constant with respect to ev and the quadratic form tx2 on Qv [W] p. 161. We put, following [Wa2],'(t) = (t, t),,-y,, (t)-y (1)-1. Then we

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(t,t')v5'v(t)5'v(t'), 1 for arbitrary v, and ye(t) = 1 if t E Ze and s 54 2, ry52(1) = '2(5) = 1 and 5'2(3) = 5'2(7) = -i. Let Uo (N)

(a d)

E S(2) I c E Nz 5

(2 = IIC

and write Uo (N)Q for the f -component of Uo (N). Defining for or = ( a c

d)

E

S(Q2)

J 'Y2(d)(c,d)2s2(a) ry`2 (d)

if C 74 0,

if c = 0,

we can check that E extends to a character of Uo(4)2 x {±1} in S(Q2) non-

trivial on {+1}. Let x be a character of (Z/NZ)" with X(-1) = 1. For (a (u,E) E Uo(N) x {±1}, we define X(u) = X(d) if u = d). We then consider the space of functions f satisfying :

(m 1)

f(ax(u,E)r(B)) = e2(u2)EX(u)f(x)e((k + 2)6)

for a E S(Q), (u,E) E Uo(N) x {±1} and r(O) E C. We impose another condition at oo :

(m2) D f = (k(k 2- 2) ) f for k' = k+ 2 for the Casimir operator D at oo . We write Pk+(1/2) (N, X; C) for the space of functions satisfying (ml - 2)

which are cusp forms. Writing J(g, z) = cz + d for g = I a b ) E SL2 (R) and z E H, we can identify S(18) = { (g, t(g, z)) Ig E SL2 (R) , t(g, z) : holomorphic on H

with t(g, z)2 = J(g, z) }.

The product is then given by (g, t(g, z))(h, t(h, z)) = (gh, t(g, h(z))t(h, z)). We have a natural inclusion map S(l) , S(A) and S(Q) - S(A). We have

the theta series : 0(z) : E00 exp(27rin2z) defined on H. As is well known, n--oo

ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)/Q

143

putting j(ry, z) = B(y(z))/O(z), j(y, z)2 = (dl) J(ry, z) if y E ro(4). Thus ry H (y, e2(y)j(y, z)) defines an inclusion of ro(\4) into S(R). It is known that the extension splits over U1 (4) = { I a b

i

E S(7G)

I

c E 42 and

a - d - 1 mod 47L}. Thus we have by the\strong g approximation theorem that S(A) = S(Q)U1(4)S(R). We can identify these two realizations by S(A)E) (g, t(g, z)) 1-- (g, t(g., z)J(g", z)-1/2) where the square root is taken so that -ir/2 < arg(cz + d)1/2 < 7r/2. For each cusp form f E Pk+(1/2)(N, X; Q, we define F : H - C by F(z) = f((g,1))J(g,i)k+(112) for z = g(i) (g E S(R)). Then as shown in [Wa2[ Proposition 3, f H F induces an isomorphism :

(2.1)

T'k+(1/2)(ro(N), X; C)

Pk+(1/2) (N, X, (C)

When f is cuspidal, the holomorphy of F follows from (ml - 2). Let us prove the above isomorphism. We have put F(z) = f ((g, 1))J(g, i)k+(1/2) for g E S(IR). Then

F(y(z)) = f (('Y.g,1))J(yg, i)k+(1/2) Suppose y c Fo(N). Then note that

(y.g, l) _ (ygyf 1,1) _ (y, S(-Y))(g'yf 1, 1)(1, S('Y-'),3(-Y,g'yf 1))

_ (y, S(y))(g, 1)(yf 1, 1)(1, S(y-1)0(g, yf 1)Q(y,gyf 1))

Since,3 is a 2-cocycle, /3(h, k)/3(g, hk) =,3(gh, k),3(g, h). This shows

(y, s(y))(g, 1)(yf

1

1)(l, S(y-1))3(7g,yf 1)0(y,g))

Thus :

F(y(g)) =f((y,s(y))(g,l)('Yf1,1)(1,s(_ 1))3(yg,yf1)/3(y,g)))J(yg,i)k+(1/2) =f((g, l)(yf 1, 1)(1,S(7-1)0(yg,yf 1)13(y,g)))J(yg, i)k+(1/2) =S(y-1)0(yg, yf 1)Q(y, g)E2(yf 1)X(y f 1) f ((g, 1))J(yg, i)k+(1/2)

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Since J(y9, i)1/2 = Q('Y, 9)J(y, z)1/2J(9, i)1/2 and

y

_

a

c

x(") = X(d)

if

b

d) ' we see

F(y(z)) = s(y-1)/3(79,yf 1)E2(yf 1)X(d)J(7,z)1+(1/2)f(9,

1)J(9,i)k+(1/2)

= s(y-1)0(79,yf 1)E2(yf 1)X(d)F(z)J(y,z)k+(1/2)

Thus we need to prove s(ry-1)0(ryg,yf 1)E2(yf 1) = (d)%y2(d). If c = 0, the

both sides are trivial. Thus we may assume that c # 0. The case c # 0 is treated in [Wa2] p. 388.

For any open subgroup U of Uo(4), we write Tu = S(Q) n US(IR). We write Pk+(1/2) (U; C) for the space of holomorphic cusp forms on S(A) satisfying (m2) and (m' 1)

f (ax(u, e)r(O)) = E2((u2i E)) f (x)e((k + 2)9) for u E U and a E S(Q) .

Then Pk+(1/2) (Fu; C) - Pk+(1/2) (U; C). Thus we can transfer the rational

structure from the classical side to the adelic side to have the spaces Pk+(112)(U; A) for any subalgebra A of C.

3. - In this section, we first prove the density theorem of low weight classical cusp forms in the space of p-adic cusp forms of half integral weight. Using this fact, we describe another way, much closer to Weil's original definition in [W] and due to Shimura [Sh2], to define S(A). By the strong approximation theorem, we have a bijection :

{congruence subgroups of S(Z) of level prime to p} Z = {open subgroups of S(Z(P))}

A=

fl s(z) -,& : the closure of A in S(Z(P)),

where Z(P) = fl Z e. We put t =/P

S.(0; A) =U SK.(AI(pa); A) and Pk+(1/2) (S; A) =U Pk+(1/2) (A, (pa); A) a

a

Let 0 be the ring of Witt vectors with coefficients in an algebraic closure FP of FP and K be the field of fractions of 0. Let 1l be the completion

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2) /Q

145

of an algebraic closure Q of Qp under its standard p-adic norm I Ip. We take an embedding : K - 1l. = QP and fix two embeddings Q -> C and Q , SlP for an algebraic closure Q of Q. Put A = 0 fl Q and S,, (A; 0) _ SK.(0; A)_®A 0, Pk+(1/2) (A; O) = Pk+(1/2) (,a;_A) ®A O. We write S(0; O)

(resp. P(0; 0)) for the p-adic completion of S, (0; 0) (resp. Pk+(1/2) (0; 0)), which is independent of ic (resp. k) if c > 2 (resp. k > 2). This fact is proven in [H2] and [H6] for integral weight and is conjectured in [H 1 ] for half integral weight. Now we can give a proof of this fact for half integral weight.

THEOREM 1. - If k > 2 and p > 3, we have an isomorphism preserving q-expansions : Pk+(1/2) (A; 0)

Pk+(3/2) (O; O) .

Proof : let U be an open subgroup of G(2) (G = GL(2) /z) and Y(U) be the corresponding open modular curve. Suppose that U D G(Zp) and we put

U(pa)=

sESIsP=(0

)modP}.

For each positive integer N, we put (N = exp( jet). Then Y., = Y(U(p-)) has a model over A = Z[1/6N, (N] for the level N of U which is the moduli space parametrizing an elliptic curve E with U-structure and a Drinfeld

style level structure at p; that is, a morphism ¢ : Z/p'Z -+ E of group schemes such that

E [O(P)] is of degree pa as a relative Cartier divisor PEZ/p'Z

(see [KM] Chapter 1 or [H71). Suppose that U C Uo(4). We can compactify Ys, adding cusps to get the proper curve X, which is regular proper over Zp [KM]. Let w/Y be the invertible sheaf corresponding to weight 1 modular

forms studied in [KM]. Let Ia be the Igusa curve containing the cusp 00 which is the irreducible component of Xq mod p'. If we consider the pordinary moduli problem 0 : pp. C E of generalized semi-stable elliptic curves, it gives an open subscheme Ua of X,, whose fiber at p is Ia-{super singular points}. Then there exists a unique invertible sheaf w1/2 on Ua such that wl/2 = Wand O E r(Uc/c, wl/2). By the q-expansion principle and p > 3, 9 is a section defined over Z P, We first suppose that U is contained in the principal congruence subgroup of level 24. Then the Dedekind ri function is a section of Ho(Uc.,w1/2). Writing w(2 + (k/2)) for w1®2 ®° IU., we

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consider the following commutative diagram : 0 1

H°(U.,w(k+

2))

H°(U.,w(k+ 2) 0 Z/p°Z)

0Z/p`xZ

177

l77

H°(U,,wl(k+ 1)) ®Z/paZ

H°(Ua,wl(k + 1) ® Z/paZ)

lp

1

H°(Ua,O(D)) 0 Z/p`Z

H°(U., O(D) 0 Z/paZ)

1 0,

E

where D is a cuspidal divisor given by div(a) =

(ords(ij))s

sEU.,orde (,j)>0

and the first horizontal maps are given by the multiplication by 77. Here we

regard D as a closed subscheme of Ua in a natural way, and O(D) is its structure sheaf. The first row is exact. When k > 2, deg (w (k + 2) ®A Q) > deg(1 x,/A ®A Q). Thus the Riemann-Roch theorem tells us the vanishing of H1(X,y, w (k+ 2)) ®AQ = H1(U.y, w (k+ 2)) ®AQ. Since w (k+ 2) is Aflat, this shows the vanishing of H1(Uy, w (k + 2 and the exactness of the second row. Since the vertical maps are injective, we have a commutative diagram whose rows are exact if k > 2 0

H°(Uy,w(k + 2)) 0 Z/ppZ

H°(U.y, w(k + 2)) 0 Z/ppZ

H°(U.y, O(D)) 0 Z/ppZ

0

E Ea

H°(U.y,w(k+2)) ®Z/ppZ

H°(Uy,w(k+ 1))O Z/p'Z

H°(Uy, O(D)) ® Z/p8Z

ONA ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2),,

147

where 0 < /3 < a < y and Ea is the modular form on U. of weight 1 with E,, - 1 modp«. Taking injective limit with respect to y, we write

H°(U,w(t)) = lim H°(Uy, w(t)) . 'Y

Then we have by the p-adic density theorem of integral weight modular forms, if k > 2 0

0

HO (U., w(k + )) ® Z/pOZ

E.

H°(U.,w(k+

®Z/p'Z 2))

2

H°(U.,w(k + 2)) 0 Z/pa7G

H°(U,,.,w(k + 1)) 0 Z/p'Z

H°(UU, O(D)) ®Z/p'7G

H°(Uc, O(D)) ®Z/p Z

.

This shows the p-adic density theorem for half integral weight if 24 1 N. If not, we just use restriction and transfer maps and recover the result in general if p > 3. Put

S,(A) = _U S,,(0; A) and P,,,(A) = _U PK,(0; A) DEZ

S(O)

= U S(0; O)

,

DEZ

and P(O) = U P(0; 0).

DEZ

AEZ

If f E S,,(A), one can find I' such that f E SK,(I;A). Then for each x E S(A(P°°)) (A(P°°) = {x E A xP = x,). = 0}), one can find I

u E f c S(A(°")) and y E S(Q) such that x = wy, where f is the closure of T in S(A(°°)) (A(°°) = {x E A I x = 0}). Some time ago, Shimura defined the action of x E S(A(P°°)) on f by f' = f I y [Sh2]. Then he showed that the action is a smooth action of S(A(P°°)) on Sk (Q( 6)), where Q(b) = Q[(N I (p, N) = 1] is the maximal abelian extension of Q unramified at p. Using Katz's theory of p-adic modular forms (see [H7] Chapter 2), it is easy to check that the action of S(A(P°°)) preserves S,, (A) and extends

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to S(O) by p-adic continuity. Note that the representation of S(A(P°°)) we obtained is smooth, but not of finite type. I like to call this representation the p-adic automorphic representation of S(A(P°°)). According to Shimura [Sh2l, we can give a definition of S(A(P°°)) as follows :

S(A(PO°))={(x,v)ES(A(P°°))xGL(P(O)) I (f")2 = (f2)x for all fEP(O)}. Then we have an exact sequence : 1-* {±1 } -> S(A(P°°)) - S(A(P°°)) -1.

It is basically shown in [Sh2l that any x E S(A(P°°)) is liftable to an automorphism v of Pk+(1/2) (Q( 6)). Since x preserves A-integrality, v keeps

A-integrality and hence gives an automorphism of P(O). This shows the surjectivity of 7r. There is an alternative way of showing the surjectivity of it. One can check that the action of S(Z(P)) is liftable to half integral weight by multiplying half integral weight cusp forms by 71 (or 0), because the action

of S(Z(P)) preserves A-integral structure of integral weight cusp forms. It is easy to check the liftability of the action of upper triangular matrices. Thus by the Iwasawa decomposition, every x E S(A(P°°)) is liftable. By definition, we have a smooth p-adic "automorphic" representation of S(A(P°°)) on P(O).

Although we do not have a good action of S(Q) on P(O), we can at least define an action of the maximal split torus T(7LP) = ZP < in S(7LP). Take a subgroup A corresponding to Z E Z. Thus its level N is prime

to p. We assume that ro(4) D A. When A is a Zr algebra, we can show multiplying by 0 as done in [H 1l §3 that Pk+(1/2) (O1 (pr); A) is stable under

the action ofZN= 7LP" x (Z/NZ)" for the level N of A, which is given for (A 1 (pr); A) by f E Pk+(1/2) (3.1)

fz= f kzP

o,,, for arz E SL2 (7G) with vz

= I\

z-1 0 ) 0

Npr.

z 11 mod

This action of ZP" extends by continuity to P(O).

4. - We put W = 1 + pZP in ZP" Z. Then W = ZP as topological groups, and ZP" = W x a for the subgroup p of (p-1)-th roots of unity. Simplifying the notation, we write Pk+(1/2)(Npa; A) for Pk+(1/2)(L'1(Npa); A). We put,

for O- D E Z and a character E of W modulo pa, Pk+(1/2)(A(pa);E; A)={fEPk+(1/2)(AI(pa);A)

If I z=E(z)4f for z E W},

where A(pa) = Ai(p) n Fo(pa) and A is a ring either in SZP or in C containing all the values of e on W. We now consider the action of the

ON A-ADIC FORMS OF HALF INFEGRAL WEIGHT FOR SL(2)/Q

149

Hecke operator T(q2) for each prime q on Pk+(1/2)(A1(pa); (C). As shown in [Shl] Theorem 1.7, we know

a(n,fIT(q2))=a(p2n,f)+q-1(2)a(n,flq)+q-'a(n/g2,flg2) ifg{Np°, (4.1)

a(n,flT(g2)) = a(p2n, f) if

glNp«,

where N is the level of A and q E ZN (= ZP < x (Z/NZ)") acts on f as in (3.1). This combined with [H 11 Theorem 2.2 shows that Pk+(1/2) (O 1(pa); 0)

is stable under T(q2). In particular, we can define the idempotent e in Endo(Pk+(1/2)(A1(p"); 0)) by taking the limit :

e = n-,oo lim T(p2)n!

(4.2)

We write M°'`1 for eM for any module M with an action of e. Hereafter we allow as a base ring a finite extension of the ring of Witt

vectors with coefficients in ]FP and write the ring as 0 and its field of fractions as K. All the definitions we have given for the ring of Witt vectors carry over to this slightly general situation by extending scalar to 0 from the ring of Witt vectors. Write A = 0[[W]] for the completed group algebra of W. Then A is isomorphic to the one variable power series ring O[[X]] via u 1--f 1 + X if we fix a generator u E W. We fix an algebraic closure 1L of the quotient field L of A and consider the algebraic closure of K in QP as a subfield of E. For each normal integral domain II in 1L finite over

A, let X(II) = Homo_alg(ll, S2p) be the space of all Qp valued points of Spec(II) and A(ll) be the subset of arithmetic points, that is, those 0-algebra homomorphisms P : II -+ QP such that P(ry) = -1k(P) for an integer k(P) > 0 on a neighborhood of the identity of W. Thus sp(ry) = P(-y)ry-k(P) defines a finite order character of W, whose order will be denoted by Pr(P)-1. We write A(II; 0) = {P E A(II) 10 D P(1)}. For each congruence subgroup A (with level N) associated with ,& E Z, let ]F(O;1) be the space of II-adic cusp forms. Thus f E IF(A; II) is a formal q-expansion : (n/N, f)qn/N E ] [[q1/N]] n=o

whose specialization f(P) = E P(a(n/N, f))gn/N E P(II)[[g'/N]] at PEA(II) n=o

is a classical cusp form in Pk(p)+(1/2) (A(pr(P) ), sp; Stp) for all P E A(ll) with

sufficiently large k(P) > 0. When A = r1(N) (4 I N), we write 1F(N;11) for lP(O; II). Since A is a regular local ring of dimension 2, 1 is A-free. Fixing a

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base {ij} of l over A, we can write formally that f = Ejfjij. Then it is easy to see that fj is a A-adic form. Thus P(0; II) = IP(A; A) OA II. There is another interpretation of the above space of A-adic forms. We first identify A with the measure algebra on W having values in O. Then to each f E P(0; A), we associate a p-adic measure 0 fW qdf on W having values in O[[g11N]] by (4.3)

fW Odf =

J

Oda(n/N, f)q N E O[[ql/N]].

n=1 W

Writing Xp(w) = Ep(w)wk(p) for each arithmetic point P (that is, the character of W corresponding to P), we have fW xpdf = f(P) E Pk(p)+(1/2) k(P) >> 0} (0(pr(p)), p; Q p) for sufficiently large k(P). Since {xp I

spans a dense subspace of continuous functions on W having values in K, as a measure, df has values in P(O). In particular, the new measure 0 H fW qdf s for s E S(A(P°°)) again comes from a A-adic form f s E IP(O3; A) for a suitable congruence subgroup O3 corresponding to I

Os E Z. Thus, we have a natural action of S(A(P°°)) on P(1) _ U IP(A; II). ZEZ

Similarly, we have an action of Hecke operators T(q2) and the group Z on IP(N; II). Writing c : w 1--> [w] for the tautological character of W into A,

we know that w E W acts on F(N; II) via t, that is, f I w= [w]f. Since the projector e naturally acts on Pk+(1/2) (O) and hence on P(O), e again acts on P(0; II) and P(1[). We note this fact as

PROPOSITION 1. - As long as q is prime to the level N of 0, we have Hecke operators T(q2) given by (4.1) and the ordinary projector e on IP(A; II), and the metaplectic group S(A(P°°)) naturally acts on P(II) through a smooth representation. Here the smoothness means that the stabilizer of each vector

in the representation space is open in S(A(P°°)).

We can think of the corresponding notion of II-adic cusp forms for integral weight modular forms (cf. [H5] Chapter 7). We briefly recall the definition. For o E Z, a formal q-expansion f c II[[g1/N]] is called an II-adic cusp form of integral weight if f(P) E Sk(p) (A(pr(p)), Ep; Q p) whenever P is arithmetic and k(P) is sufficiently large. We write S(A; II) for the space of II-adic cusp forms (of integral weight). Then similar to Proposition 1, we have Hecke operators T(n) (cf. [H5] Chapter 7) and the ordinary projector e on S(o; II). In this case, e is given on the space of p-adic cusp forms by

e = lim T(p)". The group S(A(P°°)) naturally acts on U S(0; II). We n-*oo DEZ

actually need to have G(A(P°°))-action (recall G = GL(2)lz). Note that

ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)/Q

151

G(A) = G(Q)G(7G)G+(R) for the identity connected component G+(R) of G(R). For each open subgroup U of G(2), we consider cusp forms f : G(A) ---> C satisfying : (M 1)

f (axu) = f (x) det(u,,,) J(u,,., i)-k for u E UC,,.R" ;

(M2)

Df= rk(k2 -2)l f; u ) xf Idu=0forallx E G(A).

(M3) J(Q\A

1

We write Sk(U; (C) for the space of functions f satisfying (M 1-3). Choosing

a complete representative set R = R(U) for G(Q)\G(A)/UG+(R) in G(2), (Ftut-1 = S(Q) f1 tUt-1S(R)) for each we can define Ft E Sk(FtUt-1; cC)

t c R by Ft(z) = f(tg)det(g)-1J(g,i)k, where g E G+(TR) such that g(i) = z. Then it is easy to see Sk(U;C) = ®tERSk(FtUt-1;(C). We then define Sk(U; A) by the image of ®tERSk(FtUt-1; A). We can take R inside

R = { (0

0

/

1 a E 7L(P) }. We always choose R in this way. Then we have

e and T(p) we/ll defined on Sk(U; Slr). Let U = {U : open subgroup of G(Z(P))}.

Write Uo = U x GL2(ZP) for U E U. Taking R(Uo) in R so that R(Uo) D R(Vo) if V D U for all U, V E U, we define S(U; II) = ®t ER(U(,)S(rtUot-1; II)

and S(l[) _ U S(U; II). Using the stability of U S(0; ) 2 for P E A(A; 0), then (*)

Sord(A; A)/PSord(A;

A) -

Skrd(A(pr(P)), EP; 0)

.

This implies that there are only finitely many, bounded independently of weights, of complex irreducible automorphic representations of G(A) which is p-ordinary and of conductor dividing Np. On the other hand, one has the Shimura correspondence : Sh : {irreducible holomorphic automorphic representations of S(A) of weight k + 1 } --> {irreducible holomorphic automorphic

representation of G(A) of weight 2k}.

By a result of Waldspurger, there exists a bound M > 0 such that (i)

#Sh-1(7r) < M for all k, if C(7r) I Np,

where C(7r) is the conductor of 7r. If if is p-ordinary (that is, the eigenvalue of T(p2) in is a p-adic unit), Shff) is p-ordinary. Moreover, if we write V for the space of i, we have a positive bound M' independently of weights

(but depending on 0) such that (ii)

Then (i) + (ii)

dime H°(0(p), V) < M. ranko Pk+(1/2) (A(p); 0) < M"(0) independently of k

for a positive bound M"(o). Take a subset

in pord (A; A)

which is linearly independent over A. Then we can find m rational numbers nl,... , n,,,,, such that D = det(a(n1, Off)) # 0. Therefore for arithmetic P with k(P) sufficiently large and ep = id, gti(P) is and element of Pord (p); 0) and D(P) 0. In other words, {oi(P)}ti is linearly independent over 0. Therefore m < M". This implies that rankA pord (A; A) < M". As we will see later, Ford (A; A) is actually free of finite rank over A. Then all the assertion follows from the weak multiplicity one theorem of Waldspurger by reducing the A-adic reprensentation modulo P.

ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)

/Q

153

Thus we have the A-adic Shimura correspondence :

Sh : {irreducible A-adic ordinary automorphic representations of S(A)} -- {irreducible A-adic ordinary automorphic representations of G(A)}. Suppose II = Sh(II). We write 9rp = II mod P and 7rp = Sh(arp). Then 7rp for an arithmetic P is a scalar extension of classical representation if k(P) >

2. This means that one can supplement a (unique) local representation at p with irp to get a complex automorphic representation if k(P) > 2, which we again write 7rp. Similarly 9rp is associated with a complex automorphic representation of the metaplectic group if k(P) is sufficiently large, because we can only prove the metaplectic version of (*) under the assumption that k(P) is sufficiently large. Here note that 7rp # II mod P but 7rp = Sh(arp) = II mod p2, because representations of weight k + .

correspond to those of weight 2k. Here we used the group structure of Spec(A)(O) = Homgr(W,Ox) to define p2. The above fact characterizes the A-adic Shimura correspondence. By (*), the prime to p-part C(II) of the conductor of 7rp is independent of P. Moreover the central character of II can be written as L02 for a finite order even character V) modulo 4pC(II), where t is the tautological character of W into A" composed with the "norm" character : (A(P00))x 9 x --* IxI-lw-1(x) E ZPx for the Teichmuller character w. We put Op = cp2/)w k( ) for each arithmetic P. As a striking consequence of his theory, Waldspurger expressed the square of a certain ratio of two Fourier coefficients of a cusp form of half integral weight by a ratio of L-values attached to the image under the Shimura correspondence. Applying this result, we get a A-adic version of his result : I

THEOREM 3. - For each pair (m, n) of positive square free integers with m/n E HII4NPQ , we find two elements 4 and T in II such that if k(P) > 1 or 1/i2p # 1, we have : 4,(P)2 q,(P)2

L(2,7rp L(2,7rp

P1Xm)

as long as

L(2,EP ®0P'Xm) 0, where Xt is the quadratic character associated with Q(f ). 6. - We now start filling the details with the argument in Section 5. Fix a character V of (Z/NpZ) x . For each arithmetic point P E A(A), we define

154

H. HIDA

a character by of ZN by Op(z) = /i(z)Xp()z-k(P) _,0ePW-k(P)(z), where z 1--> < z > is the projection to W and w is the Teichmiiller character. We now prove

PROPOSITION 3. - The dimension of P%+1/2)(A0(pr(P))>V)P;1 )) is

bounded independent of P E A(A) if k(P) > 1 (the dimension depends on ,& E Z).

To prove the proposition, we prepare several lemmas. Let £ be a prime

and put

Ur=Ur,e={( a

d)esL2(z)Icomodr},

For each character x of Ze modulo £r and a Ur-module M, we write M(x)

for the x-eigenspace. That is, M(x) _ {m E M I (a Ca c

b d/

E

d

) m = x(d)m for

Ur}. When the reference to the level fr is necessary, we write

M(i'.r, x) in place of M(x).

LEMMA 1. - Let 7r be an irreducible admissible representation of the metaplectic covering group S(Qe) of SL2 (Q) and V denote its representation

space. Let x be a character of Qe modulo fr. Suppose that 7r appears as a local factor of a holomorphic automorphic representation of weight k + (1 /2) (k > 2). Then the dimension of V (.fir, x) is bounded independent of V and x (but it depends on r).

Proof : when 7r is special or principal, then we can realize it as a subquotient of the induced representation space 13,. = 13µ,eQ of a character A of the standard Borel subgroup, as in [Wal, 11.2] and [Wa2, II], for the .part ee of the standard additive character e of A/Q and a quasi character

u of the standard Borel subgroup of §(Q t). Since the left translation by the upper triangular matrices of S(Qe) is already prescribed on Bµ, any function in C3µ is determined by its restriction to SL2(Zt) x {±1}. Then for each given open compact subgroup U of SL2(Ze) x {±1}, the dimension of H°(U, Bµ) is bounded by the index 2(SL2(Z() : U). A more effective bound can be obtained using the explicit calculation of the space x)

done in [Wa2) Proposition 9 (p. 417) (see also Lemma 3 in the text). We then have dim(B,(Qr, x)) < 2(r + 1). This settles the problem in the case of non-super cuspidal representations. Let II be a holomorphic automorphic representation of S(A) of weight k + a (k > 2) having it as its factor

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ON A ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2),Q

at e. Let W be the space of the e-component of the automorphic representation of GL2(A) corresponding to II by the Shimura correspondence. Using the notation of [Wall V.4 (p. 99), we mean by W the f-component of V'(e, V) ®x. We know from [Cl that dim W (er, x2) < r + 1. By [Wa2) V, Proposition 5 (p. 404), V (er, x) is a subspace of the space spanned by, with the notation in (Wa2l, i,,,e o j,,,e(w)(fr,,,) for r sufficiently large (if

r > max(2ve(2) + 1, ve(C(x))) for the conductor C(x) of x), where w E W(er,x) and v E (Qell"(V)/(Qell")2). Here fr,,, is a SchwartzBruhat function on He = {x E M2(Qt) I Tr(x) = Of determined by (r, v) as specified in [Wa2) Chapter V. The choice of v E Qell" is bounded by #(Qellx/(Qell>)2) which is 4 if e > 2 and 8 if e = 2. Thus we have, for general V,

dim(V (er, x)) < 8(r + 1) for r sufficiently large . This finishes the proof. LEMMA 2. - Let it be an irreducible admissible representation of S(Q1)

with representation space V. Suppose that it is super cuspidal. Then, for sufficiently large m, T(t) annihilates V (er, x) if r > 0. Proof : note that

Uo(er)(( 0 e0 and

((e0

1)Uo(er) =

U uE7Z. /1 "Ze

uf-M ((e0 f-M ),1) _

((e0

e--M.

)

eo )'i) ((0

1

,

i)Uo(er)

u) 1). )

Thus for v E H°(Uo(er), V), we define an operator T(f-) by

vIT(em)= UEZe /1"Z

((e0

ue-'n ) e-,n

,

1) v .

The operator T(fm) coincides with the Hecke operator (TT)m acting on V (f', x) defined in [Wa2) III.3, pp. 388-389. Then we have

vit(((0

eo )'1)) I_'_Zt it( ( 10

1

,1)vdu=0

for sufficiently large m by the definition of super cuspidality. As shown in (Wa2] Lemma 4, p. 389, we know that TI = e(3-2k)/2ye(ex(e)-1T(e2) for T(e2) defined in [Shl], where ye(t) = (t, t)e-Ye(t)rye(1) 1. Thus we know the lemma from the above result. Here we should note that the definition of our space of modular forms of half integral weight is different by the character k

(=1) from that of [Wa2l, and thus we do not replace x by Xo as was done in [Wa2l for these formulas.

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H. HIDA

LEMMA 3 ([V)). - Suppose that f > 2. Let V = 13,. and let X be a continuous character of Q into Cx. We consider the Hecke operator Tyr) = Suppose that r > Sup(vt(C(X)), X(2)ry'e(2)-1f(2k-3)/2Te.

vt(C(fLX)C(ItX-1))), where C(X) is the conductor of X. Then we have the following assertions :

(i) If both µX and pX-1 are non-trivial on Zellx, then T(P) is nilpotent on V (fr; X) for r > 0; (ii) Suppose that pX-1 is unram flied but pX is ramified. Then we can decompose V(Fr; X) = N ® V(C(X); X) so that T(e2) is nilpotent on N and V (C(X); X) is one dimensional on which T(e2) acts by scalar multiplication of XV)Q

(iii) Suppose that uX is unramified but uX-1 is ramified. Then we can decompose V (Fr; X) = N ® V (C(X); X) so that T (f2) is nilpotent on N and V (C(X); X) is one dimensional on which T(2) acts by scalar multiplication

of

x()Q(2k-1)/2/l(e)

;

(iv) Suppose that both µX and

µX-1

are wuamified. Then we can de-

compose V (.fr; X) = N (D V(; X) so that (I) T(t2) is nilpotent on N, (ii) V (i?; X)

is two dimensional, and (iii) we have a base {vl, v2 } of V (2; X) such that vlIT(t2)=X(&)f(2k-l)/2µ(2-1)v1

with some constant c.

and v2 IT(j2)=x (t)e(2k-1)/2A(f)v2+cv1 l

Proof : write v(e) for the exponent of 2 in C(e) for any character of e of Z.11'. As shown in [Wa2) Proposition 9, p. 417, under the assumption v(µX-1) of r > Sup(v(X), 1), V(2'; X) 0 if and only if r > v(pX) +

As long as r > v(X) and r > 1, T(22) sends V(2r; X) to V(; X) (cf. [Wa2] Lemma 7 or [H2) (8.6)). This shows that for sufficiently large M. V (2r; X) IT(f2m) is contained in V (C(X); X) or V (f; X) if v(X) < 1.

Unless X is quadratic, v(X) = v(X2) since 2 > 2. Thus if X2

id,

then v(pX) + v(µX-1) > Max(v(µX), v(,X-1)) > v(X). If moreover both v(pX) and v(/4X-1) are positive, then v(pX) + v(pX-1) > v(X) and thus V (C(X); X) = 0. Therefore T(22) is nilpotent on V (jr; X) if X2 id and if both v(jX) and v(pX-1) are positive. Now suppose that X2 = id and both v(µX) and v(pX-1) are positive. Then if X id, then C(X) = 2

and V (C(X); X) = 0 because v(pX) + v(pX-1) > 1. If X = id, then again V(2; X) = 0 because v(µ) + v(µ) > 1. Thus T(22) is nilpotent if both v(pX) and v(µX-1) are positive. Now suppose that v(pX-1) = 0 but v(µX) > 0. Then X2 54 id because v(µX) = v(X2) > 0 (and hence v(X2) = v(X)), and V(C(X); X) is one dimensional by [Wa2] Proposition 9. Moreover

by [;a2] Proposition 10, (ii), we know that T(f2) acts on V(C(X); X) by X(f)2(k/2)-lµ(2-1) Thus we can decompose the scalar multiplication of V(fr;X) = N (3 V(C(X);X) such that on N, T(f2) is nilpotent, and on the one dimensional space V(C(X);X), T(f2) acts via the multiplication

ON A ADIc FORMS OF HALF INTEGRAL WEIGHT FOR SL(2)/Q

of

X(f)f(k/2)-1µ(f-1). Suppose v(µX-1)

157

= v(µX) = 0 and x # id. Then

x2 = id because v(ltX) = v(X2) = 0. By [Wa2] Proposition 2, V(f; x) is 2-dimensional, and there is a base {v1, v2} of V(f; x) such that v1 V2

I T(f2) =

x(f)f(k/2)-1µ(f-1)v1

I T(f2) =

and

X(f)&/2) -11L(f)V2 + f(k/2)-21'e(f)-1x(f)(f - 1)v1

.

Thus we can decompose V (P'; x) = N ® V (t; x) such that on N, T (j2) is nilpotent and on the 2-dimensional space V (f; x), it acts by the above formula. Next suppose that v(µx-1) = v(ax) = 0 and x = id. Then V (f; x) is 2-dimensional, and we can find a base {v1, V21 by [Wa2] Proposition 10 such that (v1 + v2) E V(1; x) and

vl IT (f2) =

x(f)f(k/2)-1µ(f-1)v1

and v2 I T(f2) =

X(f)f(k/2)-1µ(f)v2

+ cv1

with some constant c. The value of c is given by [Wa2] p. 420. This shows

that V (f'; x) = N ® V (f; x) such that on N, T(f2) is nilpotent, and on the 2-dimensional space V(f;x), T(f2) is an automorphism described

as above. Finally we assume that v(µx) = 0 but v(µx-1) > 0. Then v(µX-1) = v(X-2) > 0 and hence v(x2) = v(x). Thus again by [Wa2] Propositions 9 and 10, V(C(x); x) is one dimensional and T(f2) acts on X(f)f(k/2)-1µ(f). Therefore, we can decompose it by the multiplication of V (rr; x) into V (C(x)) X) ® N, where on N, T(f2) is nilpotent and on the one-dimensional space V (C(x); x), it acts by the scalar X(f)f(k/2)-1µ(f). LEMMA 4 ([Wa l ] Proposition 18, p. 68). - Let p* be an irreducible admissible representation of PGL2 (Qe) and let p be the corresponding

irreducible admissible representation of S(Q1) via Well representation with respect to the additive character ee (1; E Qell"). Then we have

Equivalence class of p* 7r(µ> µ-1) (µ2 a) 11 2 a, jI 7 a 1/2 ) a(µ, A -1 ) (11

Equivalence class of p r{lx{

u(a1/2 a-1/2)

Supercuspidal Supercuspidal

Supercuspidal

aµxe

where et is the standard additive character of Qt and ee (x) = et(t;x) and we have used the notation of [Wa 1 ] Propositions 1 and 2.

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H. HIDA

Here note that irf,xe (resp. &µx4) with respect to ep is isomorphic to µxev (resp. vµx{,,) with respect to e,v, and hence the right-hand side is well defined independent of the additive character. A cusp form f E S,c(A,(pr);C) is called ordinary at p if f I T(p) = Af and IAIp = 1. An automorphic representation it of GL2(A) spanned by a holomorphic primitive form f is called ordinary at p if f is ordinary at p. LEMMA 5 (e.g. [H3) § 2). - Let 7r be a unitary holomorphic automorphic representation of GL2 (A). Suppose that 7r is irreducible and ordinary at p. Then the local component 1rp of 7r is either a principal series representation it (a, )3) with unramfied a or a special representation a(a,,0) with unramified a. Let f be the primitive form of weight k on GL2 (A) belonging to it and

write µ for the central character of it and A(T(p)) for the eigenvalue of T(p) on f. Then if 7r = 7r(a, p) and a and,3 are both unramified, then a(p) + /3(p) = p(1-k)I2A(T(p)), a(p)/3(p) = µ(p) and I A(T(p))I p = 1. If7rp = 7r(a, /)) and,3 is ramified, thena(p) = p(l-k)/2A(T(p)), a(p)O(p) = µ(p) and I A(T (p)) I p = 1. If itp = a(a,,3), then 7r,,. is of weight 2 and A(T (p)) = a(p).

LEMMA 6. - Let F be a number field of finite degree. Let p be a cuspidal automorphic representation of PGL2 (FA) and let R be the set of all cuspidal automorphic representations of S(FA). Define for each integral ideal N of F,

R(p; N) = {7r c R 17rv = pv for all v outside N}

,

where 7rv denotes the corresponding representation of PGL2(Fr) via Weil representations defined in [Wa 1, V.41 (where it is written as : T see Lemma 4). Then we have

V'(e, T) ;

#R(p; N) C #{H INF, cj(f)fz and Dc;,(f) E L Thus D-1(IIf1 +

+ If,) D ll ord(N;1[) and

£=1

hence prd (N; II) is of finite type over II as II-module, because II is noetherian. Now we see by definition that ]EDOrd(N; II) = npJpord(N; IIp) where P runs

over all prime ideals of height 1, lip is the localizaton at prime P and lord (N; 1p) = pord (N; II) ®Q IIp. This shows that p rd (N; II) is II-reflexive and

hence if II = A, then prd (N; A) is A-free of finite rank. Since we already know that Ford (N; 1[) = pord (N; A) ®A II, we conclude that Ford (N; 11) is II-free

of finite rank. PROPOSITION 5. - Let PEA(II). Then each f EPk(P)+(1/2) (A(pr(P)), "FP; 0)

can be lifted to an ordinary A-adic form f E p

rd (A; II)

such that f(P) = f .

Proof : it is sufficient to prove the assertion for II = A. Let E(X) E A[[q]] be the A-adic Eisenstein series (cf. 1H51 § 7.1) such that for the generator

w=1+pofW

l E(Q)=(Q(w)-1) {LP(1-k(Q),EQw-k(Q))/2+y( Q()d-1)gn1 J 00

n=1 0 0 so that (i) fi(P) E Pk(P)+(1/2) Ep; O) for all i and all P with

k(P) > a, and (ii) there exist integersni such that det(a(ni/N,fj))(P)#0 ifk(P)>a. Then fi (P) are linearly independent over O. Thus pord (0; II)/ppord (0 1) injects into Pk(P)+(1/2) (A(pr(p)), EP; 0). Surjectivity of the morphism follows from Proposition 5. COROLLARY 2. - Let fl,... , fr be a base of pord(A; II). Then we can find

integers nl,... , nr so that det (a(ni/N, fj)) E F. Proof : let fl, ... , fr be a base of Pk(P)+(1/2) (A(pr(p)), Ep; O). let = be a prime element of O. If det(a(ni/N, f3) - 0 mod wO for all choice of integers fl, ... , nr, then { fi mod zo} are linearly dependent and hence we can find A, E 0 not all divisible by w such that EiAi fi = 0 mod ruO. Then E Pk(P)+(1/2)(A(Pr(P)) Ep;O)

but w-1 Ai are not all in O. This contradicts to the fact that {f} forms a base. Thus we can find the ni's so that det(a(ni, fj)) E Ox. Now applying this argument to a base {fi(P)} by choosing P with sufficiently large k(P), we find that det(a(ni/N, fj))(P) E Ox which implies that det(a(ni/N, fj)) E Ix

Analogs of all the assertion so far we proved in this paragraph holds for Sord (U; II) in an obvious sense (see [H5] Chapter 7). In particular, the statement corresponding to Corollary 1 for Sord(A; II) holds if k(P) > 2.

7. - We now restate Theorem 3 in the language of p-adic Hecke algebras. Let hord(N; 0) be the p-adic ordinary Hecke algebra defined in [H5] § 7.3. Let us recall the definition. The algebra hors (N; 0) is the Asubalgebra of EndA(Sord(N; A)) generated by T(n) for all n. There is another description of the algebra. Writing h,rd(Np'; 0) for the 0-subalgebra of Endo(Skrd(Npr; 0)) generated by T(n) for all n, we have a natural isomorphism : hord(N; 0) = l4im hk''d(Npa; 0) if k > 2, which takes T(n) to a

T(n) [H2]. Under the natural pairing < h, f >= a(1, f I h), we know HomA (hors (N; 0), A) = §ord (N; A) and (7.1)

HomA (Sord (N; A), A) - hord (N; O)

.

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H. HIDA

We have a smooth representation of G(A(P°°)) on Sord (II) = U Sord (U, II) _ UEU

eS(l 2). Pick an arithmetic point P with k(P) > 2

and consider the localization Ap at P. Then by the control theorem, Sord(Ap) ®A K(P) for K(P) = Ap/P is a semi-simple GL2(A(P°°))module. Since there are Zariski dense arithmetic points in Spec(A) at which the control theorem holds, we see that Sord (L) is semi-simple as a G(A(P°°))-module. Thus Sord(L) is a sum of irreducible subspaces. The multiplicity is one by the control theorem combined with the multiplicity one theorem in classical situation. Since the proof of the factorization theorem

in [JL] § 9 is purely algebraic, it carries over to our situation, and each irreducible factor it of §ord(L) is factored into the tensor product of local representations : it = ®e P're. Let A; hord(C; 0) -> II be a primitive Aalgebra homomorphism. Then by the control theorem, we have a unique automorphic representation ir(P) = ®e7re(P) corresponding to A mod P for

P E A(II) with k(P) > 2. Thus A corresponds a unique factor it = ir(A) of S°rd (L) and 7re(P) = ire mod P. We write V (7r) for the subspace of §ord (j[) on which S(A(P°°)) acts via it. Thus for each arithmetic point P with k(P) > 2, Ap(T(n)) = A(T(n))(P) is an algebraic number. Then for each Dirichlet character cp, we can define the complex L-function : 00

L(s, Ap

E n=1

(p(n)Ap(T(n))n_s

.

Note that L(s, ir(P)) = L(s + k (p)-', Ap) is the standard L-function of ir(P). As is well known, the L-function L(s, Ap ®cp) has a motivic interpretation. Since II is an integral domain, we see that ZC = ZP" x (Z/CZ)" E) z F--, A() E II is a character, where is the operator induced by the central action of z E (Z(P)) X C G(A(P°°)) on S(II) (see (2.1)). In particular, it restriction to µp_1 x (Z/CZ)" gives a character Oo : µP_1 x (Z/CZ)" --> QP . We regard this character as a character of ZC composing the projection : ZN -> µP_1 X (Z/CZ)" = (Z/CpZ)" and call it the character of A. We now consider the following conditions on A :

(He). Writing xe = 7r (a, 0) when Ire is principal (f a(-1) = Q(-1) = 1;

p), we have

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ON A-ADIC FORMS OF HALF INTEGRAL WEIGHT FOR SL(2) /Q

(Hp). Oo = Vi2 for an even character 0 modulo N for N divisible by C and 4.

Under this condition, the automorphic representation associated to A is in the image of the Shimura correspondence (see [Wa2] Proposition 2). Now we consider the automorphism of A which takes w to IDm (w E W) for m prime to p. This ring automorphism extends to an automorphism a,,, of II if II is sufficiently large. For each P E A(ll), we denote p2 for

P 0 0'2. Then k(P2) = 2k(P) and Ep2 = Ep. As constructed in [K] and [GS], for each character cp of (Z/NpZ) 1, there is a two variable p-adic Lfunction Gp(P, Q; A ®cp) defined on X(1) x X (A) interpolating the value L(k(Q), Ap ®EQlwk(Q)cp) for (P, Q) E A(II) x A(A) with 0 < k(Q) < k(P). Here is a result slightly stronger than Theorem 3 : THEOREM 4. - Let A : h°rd (C; 0) -> II be a primitive A-algebra homomor-

phism Suppose (He) and (Hr). Then for any pair (m, n) of two square free positive integers with m/n E fl Q2, there exists an element 1 in 1K such LINp

that for all P E A(1[) with k(P) > 1, ifCp(P2, P; A ® /-IX L)

4,(P)2 _ Gp(P2, P; A

0, we have

®V)-1Xn.)Op(n/m)(n/m)k(p)-(1/2)

1Cp(P2, P; A (D V)-1X'.,,)

where Xt is the quadratic character corresponding to Q(f ). Here note that under our assumption on (m, n), (m/n) is prime to Np. Proof : we take P E A(ll) with k(P) sufficiently large. Let cp be a Dirichlet

character. Then for the least common multiple N' of C and the conductor of cp, we find A 0 cp : h(N'; 0) --* 1[ such that A 0 o(T(n)) = cp(n)A(T(n)). Then the character of A ®cp is given by 02V2 . Taking even cp with sufficiently

large 2-power conductor, we may assume that the conductor C' of A (D V is divisible by 16. If we replace A by A ®cp, the role of V) will be replaced by the L-value appearing in the Vicp. Since A ®'-1X,,, = (A ®cp) ®(p assertion of the theorem is unchanged even if we replace A by A 0 . Thus we may assume that 16 1 C (hence Tr satisfies the condition (H2) in [Wa2] p. 378). Let f be the cusp form in Pi(p)+1/2(I o(N2pr(p)) gyp; flu) which is a linear combination of the base defined in [Wa2] Theorem 1 for 7r (p2). Let

us take f E p"rd(N2;1) such that f I T(q2) = a2 0 A(T(q))f for all prime q outside Np and f(P) = c f with 0 7 c E 0. Such f exists by Corollary 1. Then by [Wa2] Corollary 2, for any Q E A(ll) such that f(Q) is classical, we have : a(m, f)2(Q)L(k(Q), AQ2 0

1GQ'X.)OQ(n/rn)(n/m)k(Q)-(1/2)

= a(n, f)2 (Q)L(k(Q), AQ2 0 OQ1X,,,,)

164

H. HIDA

To get the p-adic interpolation, we need to remove certain Euler factor at p and divide the special value by a certain period. However the Euler factor and the period are the same for n and m under the condition of the theorem. Thus using two variable p-adic L-functions, the above identity can be stated as : a(m, f)(Q)2GP(Q2, Q; A ®V)-1xn.) I

Q(n/m)(n/m)k(Q)-(1/2)

= a(nf) (Q)2GP(Q2, Q; A ® W-'X.). If £P(Q2, Q;

A®V)-1xn) = 0 for all Q

as above, the p-adieG-function GP(A®

'-1xn) vanishes. Hence there is nothing to prove. If L ,(A ®/-1xn) #0, by the assumption of the theorem, GP(A ®O-1x,,,,) # 0. Then we may assume

that GP(P2, P; ® -1xm)Gp(P2 P; A ® 0 by moving around P. Then we may assume by Theorem 1 of [Wa2] that the m-th and n-th Fourier coefficients of f are both non-zero. Therefore -IX")

0. Thus we can take 4) = a(n, f)/a(m, f). Now we have the a(m; f)a(n; f) evaluation property of 4) described in the theorem for almost all P. Note that .CP(P, Q; A ®0-1xn) for a fixed n is a p-adic analytic function of (P, Q) (see

[K] and [GS]). Thus as long as the removed Euler factor does not vanish,

we get the result. The only case where the Euler factor vanishes is the case where k(P) = 1 and the character of ir(P2) is trivial. However this case is excluded because of the vanishing of the p-adic L-function in the denominator at (P2, P). = 0 4=* L(k(P), Apt ®1/Ip1xn) = 0 if either k(P) > 1 or 02 # 1, Theorem 3 follows from Theorem 4. Since GP(P2, P;

A®z0A00-1)(n)

Manuscrit recu le 20 juin 1993

ONA-ADIC FORMS OF HALFWIEGRAL WEIGHT FOR SL(2)/Q

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References [C] W. CASSELMAN. - On some results of Atkin and Lehner, Math. Ann.

201 (1973), 301-314. [GS] R. GREENBERG and G. STEVENS. - p-adic L-functions and p-adic periods of modular forms, Inventiones Math. 111 (1993), 407-447. [H1] H. HIDA. - p-adic L functions for base change lifts of GL2 to GL3, Perspective in Math. 11 (1990), 93-142. [H2] H. HIDA. - On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math. 128 (1988), 295-384. [H3] H. HIDA. - Nearly ordinary Hecke algebras and Galois representations of several variables, JAMI inaugural conference proceedings, 1988 May, Supplement to Amer. J. Math. (1990), 115-134. [H4] H. HIDA. - A p-adic measure attached to the zeta functions associated with two elliptic modular forms H, Ann. 1'Institut Fourier 38 No 3 (1988), 1-83. [H5] H. HIDA. - Elementary theory of L -functions and Eisenstein series, LMS Student Texts tenbfbk 26, Cambridge University Press, 1993. [H6] H. HIDA. - On nearly ordinary Hecke algebras for GL(2) over totally real fields, Adv. Studies in Pure Math. 17 (1989), 139-169. [H7] H. HIDA. - Geometric modular forms, Proc. CIMPA Summer School at Nice, 1992. [JL] H. JAcQuET and R.P. LANGLANDS. - Automorphic forms on GL(2), Lecture notes in Math. 114, 1970. [KM] N.M. KATZ and B. MAZUR. - Arithmetic moduli of elliptic curves, Ann.

of Math. Studies 108, Princeton University Press, 1985. (K] K. KITAGAWA. - On standard p-adic L functions of families of elliptic

cusp forms, preprint. [MTT] B. MAZUR, J. TATE and J. TEITELBAUM. - On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones Math. 81 (1986), 1-48. [Sh 1 ] G. SHIMURA. - On modularforms of half integral weight, Ann. of Math. 97 (1973), 440-481. [Sh2] G. SHIMURA. - On certain reciprocity laws for theta functions and modular forms, Acta Math. 141 (1978), 35-71.

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[V] M.-F. VIGNERAS. - Valeurs au centre de symetrie des fonctions L associees awcformes modulaires, Seminaire de Theorie des Nombres, Paris 1979-80, Progress in Math. 12, Birkhauser (1981), 331-356. [Wall J.-L. WALDSPURGER. - Correspondance de Shimura, J. Math. pures et appl. 59 (1980), 1-133. [Wa2] J.-L. WALDSPURGER. - Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. pures et appl. 60 (1981), 375-484. [W] A. WEIL. - Sur certain groupes d'operateurs unitaires, Acta Math. 111, 143-211. [Wi] A. WILES. - On ordinary A-adic representations associated to modular forms, Inventiones Math. 94 (1988), 529-573. Haruzo HIDA

Department of Mathematics UCLA

Los Angeles, Ca 90024 U.S.A.

Number Theory Paris 1992-93

Structures

sur les reseaux

Jacques MartinetN

PREMIERE PARTIE : rappels sur les reseaux

1. - On note E un espace euclidien de dimension n, souvent identifie par le choix d'une base orthonormee de E. La norme d'un vecteur x E E est N(x) = x.x, le carre de la norme euclidienne 11x1j. Par reseau, on entend un sous-groupe discret A de E de rang n. La norme de A est N(A) = minxEA,x#o N(x). On pose S(A) = {x E A I N(x) = N(A)} et s(A) =

Le determinant de A est le determinant de la matrice de Gram d'une base de A (matrice des produits scalaires deux a deux des vecteurs !'IS(A)I.

de la base). L'inuariant d'Hermite d e A est y (A) = N(A). det(A)et la constante d'Hermite pour la dimension it est rye,, = SUPA -y (A).

On dit qu'un reseau A est entier si le product scalaire de E est a valeurs entieres sur A, et qu'il est pair si ses vecteurs sont de norme paire. Le reseau dual de A est A* = {x E E I Vy E A, x. y E Z}. Les reseaux entiers sont les reseaux qui sont contenus dans leur dual. Ceux qui sont egaux a leur dual sont dits unimodulaires; ce sont les reseaux entiers de determinant 1.

Les reseaux que nous rencontrerons seront tous proportionnels a des reseaux entiers. Dans ce cas, it existe une plus petite norme qui les rend

entiers. On definit l'invariant de Smith d'un reseau A en considerant le reseau entier A' qui lui est ainsi associe; le couple (A'*, A') de Zmodules libres de rang n possede lui-meme un invariant de Smith (suite des "facteurs invariants" ou "diviseurs elementaires"), qui est l'invariant

L

de Smith Smith(A) de A. Si Smith(A) = (al, ... , a,,), on a a,z = 1 et Smith(A*) = (a , _an-1 , ... , a). a, 2. - Soit E' un sous-espace de E de dimension r coupant A suivant un reseau A' de E'. Alors, E'1 coupe A* suivant un reseau A'1 de E'1, et *

Recherche effectuee au sein de I'unite mixte C.N.R.S.- Enseignement Superieur U.R.M. 9936

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J. MARTINET

l'on a entre determinants la relation

det(A') = det(A). det(A") . Considerons le cas particulier dans lequel it existe une similitude u de A sur A*, que nous prenons egale a 1'identite dans le cas unimodulaire. On associe alors a tout reseau A' comme ci-dessus le reseau relatif A' = u(E')1 n A C A. En designant par le rapport de similitude, on obtient la formule

det(A' ) = A' det(A). det(A') .

3. - Soit A un reseau. Nous appellerons defaut de perfection de A la difference entre la dimension ('2 1) de 1'espace Ends(E) des endomorphismes symetriques de E et celle du sous-espace de Ends (E) engendre par les projections sur les directions des vecteurs minimaux de A; on appelle relation d'eutaxie toute expression de 1'identite de E comme combinaison lineaire de ces projections. On dit que A est parfait s'il est de defaut nul et qu'il est eutactique s'il possede une relation d'eutaxie a coefficients positifs. On sait (theoreme de Voronoi) que A est extreme (c'est-a-dire qu'il realise un

maximum local de son invariant d'Hermite) si et seulement s'il est parfait et eutactique. (En exprimant les endomorphismes de E dans un couple de bases (13,13*) ou 13 est une base de A, on transforme ces definitions geometriques issues de [Be-M1] en les definitions classiques de la theorle des formes quadratiques.)

Une condition suffisante de perfection, due a Barnes, est 1'existence d'une section hyperplane parfaite de meme norme et de n vecteurs minimaux independant en-dehors de cette section (perfection relative). Soit Ao un reseau de dimension no. Cette condition de perfection relative est verifiee par les reseaux de dimension no+ 1 dont le determinant est minimum parmi ceux possedant A0 comme section hyperplane de meme norme. Les reseaux

faiblement lamines au-dessus de Ao sont ceux que l'on obtient en iterant le procede ci-dessus, et l'on parle de reseauxfortement lamines dans le cas de ceux qui sont de determinant minimum dans chacune des dimensions no, no + 1, no + 2.... (ce vocabulaire est emprunte a Plesken et Pohst qui ont etudie les variantes des procedes de lamination dans lesquelle on considere des reseaux entiers de norme donnee). Les reseaux lamines sans autre precision sont ceux qui ont ete obtenus par Conway et Sloane par "laminations fortes" au-dessus de Ao = {0} auquel est attribue la norme 4 ([C-S], ch. 6); pour n < 8, ces reseaux, notes A,,,, sont les renormalisations a la norme 4 des reseaux {0}, Z, A2, A3, ID4,1Th5, E6, E7, ]E8, puisque la cons-

tante d'Hermite est atteinte dans ces dimensions sur les reseaux qui leurs sont semblables ("theoreme de Blichfeldt-Vetchinkin"; Korkine et Zolotareff

pour n < 5, Barnes pour n = 6).

STRUCTURES ALGEBRIQUES SUR LES RESEAUX

169

Certains resultats de perfection et d'eutaxie que nous presentons dans cette note ont ete obtenus en utilisant deux programmes de Batut, l'un calculant le rang des projections sur les vecteurs minimaux d'un reseau defini par une matrice de Gram et indiquant s'il existe une relation a coefficients d'eutaxie egaux, et l'autre dormant une base de 1'espace des relations qui existent entre ces projections et l'identite, ainsi que divers programmes disponibles dans le systeme PART.

L'inuariant dHermite dual de A, introduit dans [Be-M1], est ^1' (A) _ (N(A)N(A*))1/2 ; sa borne superieure sur les reseaux ("constante de BergeMartinet" de [C-S31) est notee y,,,. On dit que A est dual-extreme si son invariant y,,, est un maximum local. Pour qu'il en soit ainsi, it suffit ([BeM1], 3.20) que A soit extreme et que A* soit eutactique. 4. - Rappelons les definitions de quelques reseaux classiques (cf [C-S], ch. 4). Soit (Ei), 0 < i < n (resp. 1 < i < n) la base canonique de Zn+1 (resp. de 7Ln). On pose

An = {xEZn+1I

xi=0} et IIDn={xE7LnI>,xi-Omod 2}.

Ce sont des reseaux pairs de norme 2. Le dual de D. est le reseau cubique centre, de norme 1 lorsque n est > 4. 11 est isometrique au sous-reseau de Z muni de la forme 4 E xiyi defini par les n-1 congruences x1 = X2 xn mod 2. Pour n pair > 8, soit 1D ,+ = IIDn U (El + E2 +

+ En)Dn. On obtient un reseau isometrique en considerant le2double systeme de congruences

XI -x2-...-xnmod2 et sur Zn muni de la forme 4 E Xi yi. Sous cette forme, on voit que IIDn est isometrique a son dual, et meme qu'il est unimodulaire pour n - 0 mod 4, pair pour n = 0 mod 8. On pose B8 = D8 , et l'on definit E7 (resp. E6) comme l'orthogonal dans ]E8 d'un vecteur minimal (resp. d'un sous-reseau isometrique a A2), cf. No 2 (a isometrie pres, les choix faits ci-dessus sont sans importance). Les reseaux de racines sont les sommes orthogonales de reseaux de racines irreductibles, isometriques a Z, An (n > 1), IIDn (n > 4) ou En (n = 6,7,8). Ces derniers sont extremes, ont des duals eutactiques, et sont donc aussi dual-extremes. DEUXIEME PARTIE : autour du reseau de Coxeter-Todd

5. - Soit A 1'anneau des entiers d'un corps de nombres K totalement reel ou de type C.M., de degre q. On note x H x l'involution de K (1'identite

170

J. MARTINET

dans le premier cas), et l'on munit K de la forme bilineaire TrK/Q(\µ) (ou parfois d'une forme qui lui est proportionnelle), ce qui fait de A un reseau entier de I[8 0 K.

Soit a un ideal de A stable par l'involution de K. On considere sur A' les congruences suivantes : moda (CO Al-A2-..._A,,,,, (C2) (C'2)

=0 mod a =0 mod a2,

qui definissent des reseaux de dimension n = qm. La congruence C'2 n'interviendra qu'en meme temps que la congruence C' I et seulement lorsque m est un multiple de la norme de a. En notant d le discriminant du corps K (i.e. le determinant du reseau A), on trouve pour les determinants des reseaux definis par les congruences C l (resp. C2, resp. Cl et C'2) les valeurs IdImNK/Q(a)2(m-I) (resp. IdImNK/Q(a)2 resp. jdImNK/Q(a)2m).

On observe que, pour m E a, le reseau defini par Cl ou par Cl et C'2 est encore entier lorsqu'on munit A de la forme -LTr(A7) : on a en effet A77

=AlDµt=mA1µ1moda.

i

Les determinants donnes ci-dessus sont alors a diviser par mq"`.

6. - Dans les numeros 6 a 9, sauf dans la remarque 8.3, A est l'anneau Z[w] (w2 + w + 1 = 0) des entiers d'Eisenstein. Les congruences C 1, C2, C'2 ont ete considerees dans les annees cinquante par Coxeter, Coxeter et Todd, et Barnes ([Cox), [Cox-T], [Bar]).

Soit n = 2r + t. Le reseau Lr de Barnes est forme des elements de Ar+t qui verifient la congruence C2 et dont les t dernieres coordonnees

sont reelles. Les reseaux Ln sont parfaits pour n > 5 et r > 2 ([Bar] ; cela se voit par reduction a la dimension 5 en utilisant des arguments de perfection relative). Pour n = 2r > 6, ces reseaux sont extremes et dual extremes ([Bar], [Be-M1]). Dans le cas n = 6, r = 3, considere initialement par Coxeter ([Cox]), on trouve un reseau semblable a E6*, et l'on obtient donc E6 par la congruence C 1 avec la forme Tr. s est defini par les congruences C 1 Le reseau de Coxeter-Todd, note K12, et C3, avec la forme Tr. Cette definition par congruences, jointe au fait que

s a son dual, montre que x H lwx est un isomorA -- A2 est semblable phisme de K12 sur son dual, un resultat note par Conway et Sloane, qui 11

1'interpretent en faisant remarquer que K12 est Z[w]-unimodulaire ([C-S], ch. 4, § 9). Une variante de cette construction, analogue a la definition de 1, 1, 1 1, 1). Dn (cf. n° 3) consiste en 1'adjonction a L62 du vecteur 1

i (l

Sous cette forme, on voit immediatement que K12 est extreme et dualextreme (on a des resultats analogues dans toutes les dimensions multiples de 6 et

STRUCTURES ALGEBRIQUES SUR LES RRSEAUX

171

> 12).

Il est facile de verifier que le reseau A6 - IE6 se plonge dans K12. Comme les reseaux A,,, realisent la constante 7,n pour n < 6, ce sont les reseaux de plus petit determinant contenus dans K12 pour les dimensions comprises entre 0 et 6; c'est la serie K,. pour 0 < n < 6. La methode du no 2, appliquee

en prenant u = (x H 11 l.x), permet de construire une suite descendante D K7 D K6 de reseaux dont les determinants sont K12 D K11 . minimaux pour les dimensions comprises entre 12 et 6. On obtient une suite K, 0 < n < 12 en raccordant les deux suites en dimension 6. Cela est bien connu depuis Leech (et egalement entre les dimensions 12 et 24 que nous examinerons plus loin), cf. IC-S], ch. 6, § 1. Toutefois, comme on va le voir, ces plongements ne sont pas compatibles avec les Z[w]-structures qui existent naturellement sur D4 et sur 1E6 (on a rencontre une telle structure dans le cas de IE6, et l'on peut identifier ID4 a l'ordre de Hurwitz 931 des quaternions usuels sur Q, puts plonger A dans 931 par

w I. -1+i+9+k 2

7. - Nous nous interessons maintenant a des reseaux A pour lesquels le produit scalaire est de la forme Tr o h ou h : A -f A est une forme hermitienne (nous dirons simplement Z[w]-reseaux), et nous considerons les plongements qui sont des isometries pour les structures hermitiennes, ce qui est plus restrictif que Metre seulement une isometrie pour la structure euclidienne qui s'en deduit. Le theoreme suivant sera demontre au no 9. 7.1. THEOREME. - Soit A un Z[w]-reseau entier de norme 4. Si n = 4

et si det(A) est < 81 (resp. si n = 6 et si det(A) est < 243), alors A est Z[w]-semblable a D4 ou a L4 (resp. a E6 ou a E6) Ces reseaux ont des A-bases (el, e,2) (resp. (el, e2, e3)) formees de vecteurs minimaux,

et sont definis par les suites de produits scalaires (el.e2i e1.we2) (resp. (el.e2, el.we2 i el.e3, el.we3, e2.e3, e2.w63)); des choixpossibles pour ces quatre reseaux sont les suites (0, 2), (1, 1) (resp. (0, 2, 0, 2, 0, 0), (1,1,1, 1, 1, 1)). [Le th. 7.1 prouve en particulier l'unlcite a Z[w]-isometrie pres des reseaux ID4 et E6. Felt a demontre un resultat analogue par une formule de masse pour le reseau K12 dans son article [Fe] consacre aux reseaux Z[w]-unimodulaires. Des resultats d'unicite concernant en particulier IID4 sur Z[(8] et sur Z[(121 et ]E6 sur Z[(9] figurent dans [Be-M2], th. 4.3 et 4.6].

En examinant les produits scalaires entre vecteurs minimaux de K12, on s'apercoit qu'iI n'est pas possible de plonger A4 - IID4 dans K12 en tant que Z[w]-reseau, et donc non plus A6 - E6. En revanche, la definition de K12 montre que 1'on peut plonger L et L3 - E*. En utilisant la methode du no 2, on construit une suite croissante de Z[w]-reseaux Kn (n pair) plonges dans

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J. MARTINET

K12, que l'on complete pour n impair en prenant le reseau de determinant minimum parmi ceux qui sont contenus dans Kn+1 et contiennent Kn_1. Ces reseaux Kn, comme les K, , sont bien definis a un automorphisme de K12 pres.

On voit tout de suite que 1'on a Ki = Al - Z, K2 = A2 - A2, K111 = K11, K12 = K12, et que, pour 4 < n < 8, Kn est isometrique a Lr avec r = L J . Le reseau K3, connu des cristallographes (cf. [C-S31) est caracterise a2similitude pres comme le reseau d'invariant ry3 minimum parmi ceux qui satisfont l'inegalite s > 5. Une verification informatique a partir d'une matrice de Gram de K12 montre :

7.2. PRoPosrrioN. - Les reseaux Kn (resp. Kn) sont parfaits sauf pour n = 7 et n = 8 (resp. n = 3 et n = 4) oft le defaut de perfection est egal a 1. Le tableau suivant decrit les principaux invariants des reseaux Kn :

K3 K4 Ke Ks K Ks

reseau det(Kn)

36

81

s(Kn)

5

9

Ks

Kio

162 243 486

729

972

972

36

54

81

135

15

Smith(K') i 12.3 i 9.32 6.33

27

35

18.33 I' 9.3 4

36.3 62.33

Signalons que les reseaux Kio et Kio* (pour lequel on a s = 120) sont extremes et donc dual-extremes. [Voici une construction explicite de ces deux reseaux. Par division par 1 - w, on transforme le vecteur minimal (0, 0, 0, 0, 1 - w, -(1 - w)) de K12 en le vecteur (0, 0, 0, 0, 1, -1) de K12, dont l'orthogonal permet de definir Kip par les congruences Al =- 1\2 =- A3 = 1\4 = A5 mod a et Al + A2 + A3 + 1\4 - A5 = 0 mod a2 sur Z(w)5 muni de la forme hermitienne AA + A2A2 + A3A3 + A4A4 + 2A5A5. On volt que les 135 couples de vecteurs minimaux de K10 sont representes par 34 = 81 vecteurs de composantes de la forme w' et 6.9 = 54 vecteurs obtenus par permutation des 4 premieres composantes de vecteurs de la forme (w'(1 w), -wj (1 - w), 0, 0, 0). On obtient le reseau dual en remplacant dans la forme hermitienne 2.A5A5 par 2A5A5 et en divisant par 1 - w, et les 120 couples de vecteurs minimaux proviennent de 81 vecteurs comme ci-dessus, de 4.9 = 36 vecteurs obtenus par permutation des 4 premieres composantes de vecteurs de la forme (w'(1 - w), 0, 0, 0, -wj (1 - w)) et des 3 vecteurs (0, 0, 0, 0, 3w')].

Grace a des programmes de Batut, on verifie qu'il existe dans les cas de K11 et de K9* une unique relation d'eutaxie. Pour une indexation convenable des directions de vecteurs minimaux, elles ont les formes respectives 41

12

Id

= d

i=2

pi

et

Id = p1 + d

- i=2

pi.

STRUCTURES ALGI;BRIQUES SUR LES RESEAUX

173

On montre que la section de K11 (resp. de KO) par 1'hyperplan orthogonal a la premiere direction minimale definit le reseau Kio (resp. K$), alors que

les autres directions minimales sont asociees a K1o (resp. a des reseaux K8 isometriques au reseau P8 de Barnes, note A(2) dans IC-S], ch. 8, § 6) [Ces proprietes d'eutaxie s'interpretent par 1'existence de deux orbites de plans hexagonaux engendres par des vecteurs minimaux dans K12 = K12 et dans Kio , signalons que K10 possede une section parfaite K9 de meme determinant (972) que

K9, mats avec s = 82 au-lieu de s = 81, les reseaux Kg et Kg ont ete trouves par Barnes ([Bar], II, p. 221)].

8. - En plongeant K12 dans le reseau de Leech A24 et en utilisant la methode du no 2, on complete la suite Kn jusqu'a la dimension 24. II est clair que Yon obtient une suite de sections de A24 qui sont de determinant minimum parmi les reseaux contenant ou contenus dans K12, et que l'on a la relation de symetrie det(Kn) = det(K24_n). On peut proceder de ]a meme facon avec la serie Kn. On commence par munir A24 d'une Z[w]-structure compatible avec le plongement K12 -f A24; on indiquera dans la quatrieme partie comment realiser un tel plongement sur un ordre maximal du corps de quatemion de centre Q ramp en {3, oo}, ce qui est un resultat plus precis. La methode du no 2 permet de prolonger

la suite K,, jusqu'a la dimension 24, les reseaux obtenus etant des Z[w]reseaux pour n pair; on a les relations de symetrie det(K,',) = det(K24_n) et les egalites K13 = K13 et K,, = An pour n = 22,23,24 (alors que la coincidence de Kn et de An a lieu des la dimension 18) ; on definit de meme un reseau K16 a partir de K$ .

Pour etudier ces reseaux K, au-dela de la dimension 12, on utilise la determination par Plesken et Pohst ([PI-PI) des reseaux faiblement lamines

pour la norme 4 au-dessus de K12. Ces auteurs ont trouve un reseau en dimension 13, qui est K13 = K13, deux en dimension 14 qui sont K14 et K14, puis, au-dessus de l'un d'eux, qui ne peut etre que K14, une suite de reseaux de determinants det(K'' ), uniques a isometrie pres, sauf en dimension 16 oit it y a deux reseaux, que l'on distingue par leurs invariants s, qui prennent les valeurs 1218 et 1224. 8.1. TrICOREME. - Le reseau K16 est le reseau d'invariant s = 1224.

Demonstration (1) ( H. NAPIAS). On repere le reseau K22 dans A24 a l'aide de matrices de Gram, on construit la suite descendante des K. jusqu'a la dimension 17, et l'on distingue les sections Kl6 et K6 par leurs orthogonaux. [Elle a egalement montre qu'un seulement parmt les 37 vecteurs minimaux de K17 a pour orthogonal Klg dans K17, resultat analogue a ceux que l'on a observes pour Kli et K9*1.

(2) Par adjonction a L2 du vecteur vi = 1 1 (1, 1, 1, 1, 1, 1, 0, 0, ...)

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J. MARTINET

pour 2m > 12 puis du vecteur v2 -1, 0, 1, -1, 0, 1, -1, 0, 0, ...) pour 2m > 16, on obtient des reseaux de determinant 3'n puis 3ri-2, obtenant K12 pour m = 6, puis un reseau A de determinant 36 pour m = 8, qui contient visiblement K12 ainsi qu'une suite de sections en dimensions n = 15,14, 13 de determinants det(K,,). On a ainsi construit celui des deux reseaux de Plesken-Pohst qui est K16, et l'on verifie facilement que 1'on a s(A) = 1224. [On construct K18 par adjonetion a L98 de vi, v2 et v3 = ll

(0, 0, 0,1,1,1,1,1,1),

et l'on en deduct des constructions explicites de K17 et de Kist.

8.2. Remarque. - Les reseaux Kn et K;,, n > 12 et K16 sont parfaits. Cela se voit en controlant la perfection relative a partir de la dimension 12. Il est probable, mais non demontre, que les constructions par lamination pour

une norme donnee donnent dans ce cas particulier les reseaux faiblement

lamines au-dessus de K12, un resultat qui entrainerait directement la perfection, comme dans le cas des reseaux An consideres par Conway et Sloane.

8.3. Remarque. - Prenons A = Z[(9] et a = ((1 - (9)2), et soit R6m le reseau defini par la congruence E Ai - 0 mod a4 sur ((a2)-,9 T). On a R6 - E6 ([Cral), d'ou R12 ^' K12 et (R18, 1 1C9 (1, 1, 1)) -- K1'8; on trouvera dans [Bay-M] une construction de K12 comme module de rang 1 sur Q((21).

9. - Nous demontrons maintenant le th.7. 1. 9.1. LEMME. - Soit A un reseau pair de dimension n et de norme m. mny,-n. Alors, A possede n Supposons uerifiee l'inegalite det(A) < 1,+2 vecteurs minimaux independants.

Demonstration. L'inegalite de Minkowski sur les minima successifs d'un reseau montre qu'il existe des vecteurs el, C 2 ,- .. , en de A verifiant

l'inegalite N(el)N(e2) ... N(en) < y7 det(A), qui entraine que l'on a N(el)N(e2) ... N(en) < + n. On peut supposer ces vecteurs ranges par normes croissantes. On montre que ce sont des vecteurs minimaux en raisonnant par recurrence sur leur indice. On a en effet

N(el) ... < '"+2m''; si l'on suppose que el, ... , ei_1 sont de norme m, on trouve pour la norme de ei les inegalites N(ei) < N(ei_l)N(ei)n-i+l

m(

)1/(n-i+l) < m+2, donc N(ei) < m puisque A est pair, d'oU 1'egalite

N(ei) = m,

U

[Pour n = 2,3,. .. , 8, la borne du lemme est egale a 18,48,96, 192, 288, 324, 3241.

9.2. TiiEOREME. - SoitA un Z [w] -reseau entier de norme4, de dimension 4 (resp.6), et de determinant < 96 (resp. < 288). Alors, A est Z [w]-semblable

STRUCTURES ALGEBRIQUES SUR LES RI;SEAUX

175

d D4 ou a L2 (resp. a E6 ou a ]EE). En particulier, les Z[w]-structures sur les reseauxD4, L2 et E6 sont uniques a Z[w]-isometrie pres.

Demonstration. Le lemme montre que A contient n = 4 (resp. n = 6) vecteurs minimaux independants et donc, compte tenu de 1'action de Z[w], qu'il contient un sous-reseau A' possedant une base de la forme (x, wx, y, wy) (resp. (x, wx, y, wy, z, wz)). Le reseau A' est determine par la

donnee des produits scalaires a1 = x. y, bi = x.wy (resp. a1 = x.y, b1 = x.wy, a2 = x.z, b2 = x.wz, a3 = y.z, b3 = y.wz) qui sont majores par 2 en valeur absolue. On a det(A') < 2n det(A2)n/2 = 12n/2 et det(A') > 24 det(D4) (resp. det(A') > 26 det(E6), d'ou la majoration [A A'] < 3 avec

egalite seulement pour n = 6, A' _- A2 I A2 I A2 et [A A'] = 3 (car 2 est inerte dans Q[w]), cas dans lequel A est de determinant 26.3, donc semblable a E6, et oii 1'existence d'autres vecteurs minimaux que ceux des orbites de x, y, z permet encore de supposer que l'on a A' = A.

On peut supposer que x.y est > 0 et minimum parmi les valeurs absolues des produits scalaires des vecteurs minimaux de A' appartenant a deux orbites distinctes, et utiliser 1'automorphisme w'-4 w2 de Z[w] pour echanger x.wy et x.w2y. On voit tout de suite que, en dimension 4, A = A' est obtenu en prenant pour (al, b1) l'un des 4 couples (0, 0), (0, 1), (0, 2), (1, 1), conduisant a des reseaux de determinants respectifs 144, 121, 64, 81, d'ou le theoreme dans cc cas. Dans le cas de la dimension 6, nous avons d'abord prouve "a la main"

1'assertion d'unicite de la Z[w]-structure de Es, qui entraine le resultat analogue pour E6. On observe pour cela que le produit scalaire de deux vecteurs minimaux de EE nest jamais nul, cc qui permet de definir A' en prenant pour (al, bl, (12, b2i a3, b3) l'une des suites (1, 1, 1, 1,1, 1) ou (1, 1, 1, 1, 1, -2), et la seconde se ramene a la premiere en remplacant y par x + w2y. On acheve la demonstration en controlant sur ordinateur qu'il n'y a pas de determinant dans l'intervalle ] 192, 243[, et que la valeur 243 du determinant, lorsqu'elle ne provient pas d'une suite sans produit scalaire nul, correspond a un reseau de norme 2. [Variante pour l'assertion d'unicite concernant 1E6 : on considere un vecteur minimal x de E6 ; on verifie que le reseau 1E6 fl (Z [w]x) L, qui est de norme 2 et de determinant

9, est isometrique a A2 I A2 ; on en deduit que E6 s'identifie a un reseau de la forme (A2 I A2 I A2, 11. (x, y, z)), et l'on observe que x, y, z doivent etre des unites de Z[w] pour que le nombre de vecteurs minimaux soft superieur a celui de A2 I A2 I A2. Par multiplications a droite par des unites, on se ramene au cas

x = y = z = 11.

10. - Nous terminons cette premiere partie par quelques remarques sur la constante ryn. Sloane, dans une lettre a 1'auteur ([S!)), a donne pour

176

J. MARTINET

certaines dimensions < 24 des exemples de reseaux sur lesquels l'invariant

yn prend des valeurs relativement grandes. Pour n < 9, ce sont ceux de [Be-M1], 4.6 et 4.7. Pour n = 10, it indique la valeur 4 pour ryn2, atteinte sur deux reseaux semblables a leur dual (dont D+), cf. [C-S3]; le couple (K'0, Kio*) fournit la meme valeur. Le resultat propose est le meme pour "ii, atteint en particulier sur les reseaux Ail et K11, qui sont tous deux dual-extremes ([Be-M 11, § 4, (a) pour le premier, n° 7 ci-dessus pour le second).

H. Napias a montre que Yon -yn2(K18) = 8 et yn2(K21) = 9. Nous avons rencontre pour la premiere fois le reseau K18 dans un travail de Souvignier ([Soul) consacre aux sous-groupes maximaux de Gln,(Z), dont nous avons extrait le premier exemple d'un reseau L de dimension 21 avec ry21(L) > y'21(A21) (L et son dual sont extremes, et l'on a y21(A21) = 8
0 un entier et soit n = 4m. On on munit 971 de la forme Tr(Aji) et l'on pose J. = { (A,, , A.) E 9311T I Al + ... Am = 0 mod a}. On verifie

que J,, est un reseau de norme 4, primitif sauf pour n = 4 ou n = 8 ou l'on trouve une renormalisation de IlD4 et de E8, dont le dual s'identifie a .L.(9J1)m. En identifiant D4 a la derniere composante de J,J, et en coupant par les orthogonaux des sections de ID* semblables a {0}, A1, A2, A3, llD4. on

obtient des reseaux Jn, Jn_1, J,t_2i J,i,_3 et un reseau qui s'identifie a J,_4, ce qui definit Jn pour tout n. On a ainsi construit les analogues pour 992 des reseaux L Lnj2J construits par Barnes sur 1'anneau des entiers d'Eisenstein. 11.1. PROPOSITION. - Pour tout n > 1, J,, est un reseau entier de norme 4, qui est une section hyperplane de Jam,+1. It possede les invariants suivants :

STRUCTURES ALGEBRIQUES SUR LES RESEAUX

n = 4h n = 4h + 1 n = 4h + 2 n = 4h + 3

det(Jn) = 22h+4 det(Jn) = 22h+5 det(Jn) = 3.2(2h+4) det(Jn) = 22h+6

177

s = 12h(4h - 3) s = 4h(12h - 7) s = 12h(4h - 1) s = 3(16h2 + 4h + 1)

It est parfait quelque soit n, extreme pour n - 0 ou 1 mod 4, et est dualextreme lorsque n est divisible par 4. En outre, pour n < 12, Jn est un reseau [amine An, et l'on a plus precisement J12 ^-, A 2 et J11 - Aii". Enfin, pour n = 4h + 2 > 9 et 2 E {1, 2, 3}, Jn est de norme e+1 et la configuration S(J,n) est semblable a S(AQ).

Demonstration. Le calcul du determinant et du nombre de vecteurs minimaux ne presente pas de difficult. On verifie aussi facilement que Jn est relativement parfait par rapport a Jn_1, d'ofi 1'assertion de perfection.

On montre que Jn est extreme pour n - 0, 1 mod 4 en montrant qu'il contient un reseau de norme 4 semblable a D, ce qui assure qu'il est dualextreme pour n = 4m vu que (M)m est eutactique. La suite des valeurs des determinants pour 0 < n < 12 montre tout de suite qu'il s'agit de reseaux lamines, que l'invariant s permet d'identifier en dimensions 11 et 12. Enfin, on determine S(J,ry) par recurrence descendante en identifiant J,, a une projection de Jn+1

12. - L'analogue pour l'ordre de Hurwitz des reseaux D8 = E8 et K12 est le reseau A16 de Barnes-Wall (la notation A16 des reseaux lamines est justifiee ci-dessous), que l'on peut definir au choix par le double systeme de congruences Al = A2 = A3 = A4 mod a et Al + A2 + A3 + A4 = 0 mod a2

sur 9314 muni de la forme 2 >

1

Trd(AJi) ou par adjonction a J16 de

[Plus generalement, on definit de facon analogue un reseau J;, pour tout n > 16 divisible par 8, ayant le meme determinant (4'') que 931''1.

On demontre comme dans les cas de DI et de K12 le resultat suivant : 12.1. PROPOSITION. - L'application q'-4 q(1 + i) A16 stir son dual; en particulier A16 est de norme 2.

est une isometric de

Il resulte de cette proposition que, pour tout vecteur minimal x de Ai6, 931x est un reseau de dimension 4 isometrique a ID4. Les orthogonaux de ses sections minimales {0}, A1, A2, A3i ) 0 le reseau de )R ® H'' muni du produit scalaire (a,µ) 2Trp/Q(aTrdH/F(Aµ)), de dimension n = 4[F: Q]m, defini sur Al

=A2=...=A,,,,mod T et

=0 mod T'.

Al

[S'il y a de la decomposition au-dessus de 2 dans F/Q, on peut choisir un couple T T' pour chaque Ideal de F au-dessus de 2].

15.1. MiEOREME. - Sous les hypotheses ci-dessus, le reseau defini par la condition (*) est entier et pair et de norme > 4 lorsque m est > 3.

Demonstration. La demonstration de l'integralite se fait par completion en p (notee par le symbole -), ce qui permet de ramener les calculs de norme

reduite a des calculs de determinants d'ordre 2. Par une identification convenable de l'algebre locale a une algebre de matrices, on peut faire en sorte que l'on ait K

K ZK/ \ZK

\2K p/

et

, - \p 2 K

182

J. MARTINET

Identifiant alors Ai E 931 a une matrice de la forme

que les congruences de la condition (*) deviennent

(xi \ zi

Yi

ti), on constate m

yi - y1 mod p et ti - t1 mod p (1 < i < m),

et

xi i=1

i=1

zi

0 mod

Comme la norme reduite dans une algebre de matrices n'est autre que le determinant, on a m

m

A Ai = T riti - yizi = (Exi)t] - (Ezi)J1 = Omodp, i=1

i=1

i=1

i=1

ce qui prouve qu'il s'agit d'un reseau entier pair. Pour minorer la norme de A _ (A1, A2, ... , A,n) suppose non nul, on distingue trois cas :

si les Ai ne sont pas dans T, on a N(A) > m min Nrd(A) > 3minNrd(Ai), donc N(A) > 4;

Si les A, sont dans T et si deux d'entre eux sont non nuls, on a N(A) > 4 puisque les produits A Ai sont dansT n ZF = 2ZF; si un seul des Ai est non nul, c'est un element de 'a3 l', 43' = p931, d'ol

encore le resultat dans ce cas.

16. - Nous prenons maintenant pour H 1'algebre H3 de centre Q ramifiee en 3 et a l'infini, munie de sa base (1, i, j, k) verifiant les relations

i2 = -1, j2 = -3,ij = -ji = k, et donc les relations supplementaires

k2 = -3,jk = -kj = 3i,ki = -ik = j, et pour ordre maximal l'ordre 9713 = 931 de base 1, i, w, iw sur Z ou w = - 2 jest une racine de

]'unite d'ordre 3. (Le choix de 931 importe peu, les ordres maximaux de H etant conjugues, comme dans le cas des quaternions de Hurwitz.) Les unites de 93Z sont {±1, ±w, ±w2, ±i, ±iw, ±iw2 } ; elles forment un groupe isomorphe au

groupe quaternionien d'ordre 12. Le theoreme ci-dessous donne une construction explicite d'une structure de 9313-reseau sur K12, dont ]'existence a ete prouvee it y a peu par Gross (IGro]) :

16.1. THEOREME. - Le reseau construit a l'aide de la condition (*) avec m = 3 sur L'ordre 9313 est isometrique au reseau de Coxeter-Todd.

Demonstration. Le theoreme 15.1 montre qu'il s'agit d'un reseau de norme au moins 4, dont on voit tout de suite qu'il est entier en tant que 9313-reseau et de determinant 36. 11 est donc unimodulaire en tant que 9313-reseau, et donc en particulier en taut que reseau sur 1'anneau des

STRUCTURES ALGEBRIQUES SUR LES RSEAUX

183

entiers d'Eisenstein. Le theoreme de Felt ([Fe)) cite au no 7 entraine qu'il est isometrique a K12. [Le theoreme de Felt montre qu'il s'agit meme d'une isometrie en tant que 7G[tWIreseau; Ch. Bachoc vient de demontrer que K12 est meme unique a 9723-isometrle presj

17. - Soit F un corps quadratique reel, de discriminant d impair. Nous considerons maintenant le corps de quaternions H ramifie exactement aux deux places infinies de F, et nous supposons que l'unite fondamentale a de F est de norme -1, ce qui equivaut au fait que la differente de F possede

un generateur totalement positif, en l'occurence a = ev, si bien qu'un ordre maximal 971 de H, muni de la forme TrK/Q(a-1Trd(Aµ)), definit un reseau Z-isometrique a E8. Un corps de quaternions Ho de centre Q peut etre plonge (d'une infinite de facon) dans un corps gauche H du type ci-dessus : it suffit de choisir un corps F dans lequel les nombres premiers ramifies dans Ho sont inertes ou ramifies dans F/Q et de prendre H = F ®Q Ho, les invariants locaux aux places finies de F etant alors tous nuls. Un ordre arbitraire 0 de Ho etant contenu dans l'ordre 7GF4.7 de H, lequel est a son tour contenu dans un ordre maximal 931 de H. on voit que E8 peut etre muni d'une structure de i:7-reseau sur n'importe quel ordre de quaternions totalement defini sur Z. On verra plus loin d'autres exemples du meme type; signalons simplement ici qu'un resultat analogue s'applique a A16. En appliquant a l'ordre 971 la construction par le double systeme de congruences (*), on obtient un reseau unimodulaire pair A de dimension n = 8m que nous notons simplement Un, sans mettre en evidence dans la notation sa dependance a priori des choix de H, 971,'3, T'. Il est fort possible que la classe d'isometrie de Un (en tant que Z-reseau) ne depende pas de ces choix. C'est ce qu'on constate en dimension 8 (resp. 24), puisque E8 (resp. A24) est alors l'unique reseau unimodulaire pair (resp. et de norme 4, theoreme de Conway). Le cas de la dimension 32 a ete resolu par Coulangeon ([Coul), qui a caracterise U32 comme le reseau unimodulaire pair d'invariant

de Venkov maximum qui est associe au code de Reed-Muller. Quant a la dimension 16, on trouve E8 I E8, comme on le volt en considerant 1'application (A, p) - (A +;t, A - µ). [Pour n > 40, les repartitions des normes redultes dans les suites (Al, A2,. . . , Am ) definissant des vecteurs minimaux sont des permutations de (1, 1, 0, . . . , 0) ; on en

deduct 1'egalite s(Un) = 15n(n - 7) pour n > 40. Les resultats pour n = 24 et n = 32 (et aussi pour n = 40) decoulent de la theorie des fonctions O ; on a s(U24) = 98280 et s(U32) = 73440. Pour n = 32, on dolt ajouter aux 15n(n - 7) = 12000 vecteurs ci-dessus 61440 vecteurs assoctes a la repartition

(1,1,1,1); pour n = 24, on ajoute a 15n(n - 7) = 6120 la contribution des

184

J. MARTINET

permutations de la repartition (2,1,1), soft 92160 vecteurs.

On connaitlsl en dimension 40 quelques autres reseaux unimodulaires pairs de norme 4. Pour celui de McKay (cf. [C-S], ch. 8, § 5), d'apres McKay, le groupe d'automorphismes ne serait pas transitif sur ]'ensemble de ses vecteurs minimaux, ce qui entrainerait que U40 ne lul est pas isometrique. Nous Ignorons st notre reseau U4o coincide avec 1'un des reseaux construits par Eva Bayer dans [Bay] ou par Ozeki dans [Oz]].

Revenons au double systeme de congruences (*) defini par deux ideaux

a gauche maximaux T et T' au-dessus de 2 d'un ordre maximal i3 d'un corps de quaternions Ho de centre Q. Si on choisit un corps F dans lequel 2 et les nombres premiers ramifies dans Ho sont inertes, on plonge comme cidessus Ho dans H et i7 dans un ordre maximal fit de H, et ces plongements transforment le reseau A0 associe au double systeme de congruences en un reseau defini de facon analogue sur 931 a ]'aide d'ideaux maximaux audessus de 2 contenant respectivement ¶ 3 et T'. Ceci s'applique en particulier au cas on 971o est l'ordre note'9J13 au no 16

en prenant F = Q(/5) ou p est n'importe quel nombre premier congru a 5 ou 11 modulo 24, par exemple p = 5. En prenant m = 3, on en deduit une construction du reseau de Leech A24 sur 9R3, utilisee par Tits ([Ti]) dans le

cas du corps F = Q(v), et un plongement de K12 dans A24 compatible avec la 9713-structure dont nous avons muni K12 au no 16, qui justifie la construction de la serie K i au-dela de la dimension 12 que nous avons faite au no 8.

On peut faire une remarque analogue avec ]'ordre 9312 de Hurwitz. On verifie que la construction par double congruence des reseaux J4,,n faite au debut du no 12 conduit au reseau A12 " lorsque l'on prend m = 3 (lorsque m est impair, le determinant calcule au no 12 doit etre multiplie par 24), et l'on en deduit le plongement connu de A12 " dans A24 en tant que reseaux sur l'ordre de Hurwitz.

Manuscrit recu le 8 mars 1993

Pl

Je remercie Eva Bayer pour les references concernant les reseaux de dimension 40

SMUCTTJRES ALGEBRIQUES SUR LES R$SEAUX

185

BIBLIOGRAPHIE

[Bar] E.S. BARNES. - The construction of perfect and extreme forms I, II, Acta

Arith. 5 (1959), 57-79, 461-506. [Bay] E. BAYER-FLUCKIGER. - Definite unimodular lattices having an automorphism of given characteristic polynomial, Comm. Math. Helvet. 59 (1984), 509-538. [Bay--M] E. BAYER-FLUCKIGER et J. MARTINET. - Formes quadratiques liees aux

algebres semi-simples, J. refine angew. Math. (1994), a paraitre. [Be-M 1 ] A.-M. BERGS et J. MARTINET. - Sur un probleme de dualite lie aux spheres

en geometrie des nombres, J. Number Theory 32 (1989), 14-42. [Be-M2] A.-M. BERGS et J. MARTINET. - Reseaux extremes pour un groupe d'automorphismes, Asterisque 198-200 (1992), 41-66. [Be-M-S] A.-M. BERGS, J. MARTINET et F. SIGRIST. - Une generalisation de l'algorithme de Voronoi pour les formes quadratiques, Asterisque 209 (1992), 137-158. [C-S] J.H. CONWAY et N.J.A. SLOANE. - Sphere Packings, Lattices and Groups,

Springer-Verlag, Grundlehren no 290, Heidelberg, 1988. [C-S11 J.H. CONWAY et N.J.A. SLOANE. - Complex and integral laminated lattices,

Trans. Amer. Math. Soc. 280 (1983), 463-490. [C-S2] J.H. CONWAY et N.J.A. SLOANE. - Low-dimensional lattices. III. Perfect forms, Proc. Royal Soc. London A, 418 (1988), 43-80. [C-S3] J.H. CONWAY et N.J.A. SLOANE. - On Lattices Equivalent to Their Duals,

a paraitre. [Cou] R. COULANGEON. - Expose au Sem. Th. Nombres de Paris, (]anvier 1993).

[Cox] H.S.M. COXETER. - Extreme forms, Canad. J. Math. 3 (1951), 391-441. [Cox-T] H.S.M. COXE'I'ER and J.A. TODD. - An extreme duodenary form, Canad.

J. Math. 5 (1953), 384-392. [Cra] M. CRAIG. - Extreme forms and cyclotomy, Mathematika 25 (1967), 4456.

[Fe] W. FELT. - Some Lattices over Q(/), J. Algebra 52 (1978), 248-263.

186

J. MARTINET

[Gro] B. GROSS. - Group representation and lattices, J. Amer. Math. Soc. 3 (1990), 929-960. [La] M. LAIHEM. - Communication privee. [Oz] M. OZEKI. - Examples of even unimodular extremal lattices of rank 40,

J. Number Theory 28 (1989), 119-131. [P1-P2] W. PLESKEN and M. POHSr. - Constructing Integral Lattices With Prescribed Minimum. 11, Math. Comp. 60 (1993), 817-825. [Q] H.-G. QUEBBEMANN. - An application of Siegel's formula over quaternion

orders, Mathematika 31 (1984), 12-16. [Si] F. SIGRIST. - Lettre electronique du 11 septembre 1990 a l'auteur. [S1] N.J.A. SLOANE. - Lettre a l'auteur du 11 mai 1992. [Soul B. SOUVIGNIER. - Diplomarbeit, Aachen,1991.

[Til J. TITS. - Quaternions overQ(f), Leech's lattice and the sporadic group of Hall-Janko, J. Algebra 63 (1980), 56-75.

Jacques Martinet Mathematiques, Universite Bordeaux I 351, cours de la Liberation 33405 TALENCE cedex

Number Theory Paris 1992-93

Construction of Elliptic Units in Function Fields Hassan Oukhaba

i. - Introduction Let k be a global function field and Fq be its field of constants. Fix a place o0 of k. Let Ok be the Dedekind ring of elements of k regular outside of oo, and k,, be the completion of k at oo. For each finite abelian extension F of k we let OF be the integral closure of Ok in F. We know that OF is a Dedekind ring with a finite ideal class

number h(OF). As usual we denote by OF the group of units of OF, and p(F) C OF the finite multiplicative group of non zero constants of F. We have p(k) = Ox = F9 , and in general the quotient group OF/µ(F) is a free abelian group of rank rF - 1, where rF is the exact number of places of F sitting over oo. Now suppose that F C k, which means that the place oo splits completely in F/k. Suppose in addition that one of the following two conditions holds.

1) The extension F/k is unramified. 2) One, and only one, prime divisor of k ramifies in F/k and deg (oo) = 1.

Then one knows that there exists a subgroup EF of OF called the group of elliptic units of F. It is a Galois module generated by the torsion points of certain Drinfeld Ok-modules. It's elements are also obtained as finite products of special values of elliptic functions. The group EF has finite index in OF, cf. ] 10]. Actually when only one prime does ramify in F/k we had succeed to construct subgroups of finite index in OF even if deg (00) > 1,

cf. 191. Unfortunately, the index formula obtained then contains a factor depending on deg (oo) and which is hard to control. When deg (00) = 1 this factor is equal to 1 also and the index formula is just what one can expect. But in general this factor increases proportionaly to deg (00). This means that the subgroups so constructed are not sufficiently large when deg (00) > 1. Hence, one could suppose that there is possibility to obtain larger subgroups of OF, in other words to obtain more units of OF, using

188

H. OUKHABA

new techniques of constructions. This is what we propose to do in the present paper. Our aim here is to define £F, the group of elliptic units of F. We shall expose some of its interesting properties, precise the nature of its elements and calculate its index in O. As we shall see the description of £F is rather easy and almost canonical. Moreover, the "exponential function", which we are going to redefine in the next section, is the only basic material of its construction. Finally we would like to draw the attention to the work of D. Kersey, cf. [7] chap. 12 and 13, which was one of our source of inspiration.

Some supplementary notations Let F C k,,, be a finite abelian extension of k such that the place 00 splits

completely in F/k. let b C Ok be an ideal of Ok prime to the conductor of F/k. Then we will write (b, F/k) for the automorphism of F/k associated to b by the Artin map. Moreover if q is a prime ideal of Ok then qF will denote

the product of the prime ideals of OF sitting over q. Finally, if m C Ok is an ideal of Ok then we know that there exists a maximal finite abelian extension of k whose conductor divides m and which is contained in k,,,,. It will be denoted by H,,,.

2. - Some preliminaries In this section we recall some definitions and results, necessary in the sequel. The reader is invited to consult [1], [4], [9] or [11], where are proved

all the results stated here. Let S2 be the completion at o0 of the algebraic closure of k, Then we call a lattice of l every finitely generated projective Ok-module, contained into Q. To such a lattice r C 0 one can associate its exponential function defined on 1 by :

er(z)`ifnzJJ(1- z). 7Er

ry

7#0

We know that er is defined everywhere and is entire and IFq-linear. It is also an epimorphism and we have er(z) = 0 if, and only if, z E r. Moreover the equation eyr(xz) = x er(z) holds for every x E S2" and z E Q.

When F is contained into a lattice r of SI such that r and r have the same rank as Ok-modules then er and er are related by the formula : (1)

er(z) = P(r/r; er(z))

,

where P(r/r; t) is a linear polynomial whose roots are all simples and

CONSTRUCTION OF ELLIP77C UNITS IN FUNCTION FIELDS

189

constitute the finite group er(r). Its leading coefficient is

6(r

-) arll

)-i

( pEn/r R er(p) p#0

where p describe a complete system of non zero representatives of r modulo F. Let K(oo) be the constant field of k,,.. It is a finite extension of ]Fq. We have [K(oo) 1Fq] = deg (oo). Let us choose s (once of all) a sign-function of :

k,,, i.e., a co-section of the inclusion map K(oo)" -4 k' such that s(z) = 1 if Iz - 11,, < 1. Then one can associate to each lattice r of S2 of rank 1 its s-discriminant OS(r) E SZX and ar an Fq-automorphism of K(oo) such

that X6 (r, x-1 r) = A, (r)

Nx-1 s(x)°r

for all x E Ok\{0}. In the above formula, w,,, is just the number of non zero elements of K(oo), i.e., w... = K(oo)". On the other hand Nx is by definition the exact number

of congruence classes of Ok modulo the ideal xOk. One can show that A3(zr) = z-w°°OS(r) for all z E V 1, we define

M : Imt -* Z/mZ, with m = wM/wk, to be the following surjective morphism XPM

nT1,T2...... e

((,l

- 1)(T2 - 1) ... (Te - 1)/

T1, 72 .....Te

dfn

E

' 'Tl ,T2 ,...,Te

T1,T2......

e

M(T1) - I" CXPM(T2) - 11... CWM(Te) - 11 wk

wk

Wk

J

These operators are well defined, as will be made clear in some further work; here we just need to have IF(') and 0M) to our disposition, and the following lemma relating them :

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

199

LEMMA 1. - For any element A of IM we have

M (wkA) _ ` (l (A)

mod.

wM 1

.

wk)

Proof : obvious. dfn

F fl H(1) and, for q" the conductor of F/k; dl [FH(1) F] = [H(1) F(1)] so that FH(1)] and d2

PROPOSITION 8. -Let F(1) dfn

: [Hqn : : put d1 Z/mZ and' F(), ) : IF(,) -> 7G/m7L, d1d2 = [Hqn : F]. Let WF(1) : IF(1) with m = wF(,) /wk, be the surjective morphisms defined as above. Also put W = wkw,,,h. Then the group SF is formed of all the products

(18)

LI

TT

1PF(1)

L

[bhwo]m(,wkdld2

=1

v

na,a' (Na' - 1) ; whence in terms of automorphisms

where Va = a' EY

11 Ta k

11 T6T2bd1 = 1.

aEY

bEZ

This equality is a necessary and sufficient condition for the sum

'a (1-Ta)+Cd1

M

aEY Wk

Tb)

bEZ

to be an element of IF(1) ; yet this sum itself (which belongs to wkIF(1)) is congruent modulo IF(1) to our n, (7) which here is

E na,a'(NQ -Ta')(1 - Ta)+wkdl a,a'EY

mbTgnl(1 -Tb) bEZ

hence condition 2) is proved. Condition 1) is trivial. On the other hand, by the Lemma 1 applied to the sum (*) we have 0(2

F(j 2 ))

Y' nT(T))l T

naa

(_) (O(T )-1/

a,a'EY

Mb )(Tgnl)

+ d1

bEZ

-d

b 4 (Tq^1)

bEZ

-dl

(1 - 4'(Tb) ) 1\

Y

(1 -

Wk

J

(Tb) )

(mod. wF(') wk

(mod. wF(1)

Wk

MT(T) I

Wk

(mod. wF(1) Wk

)

CONSTRUC77ON OF ELLIPTIC UNITS IN FTJNC77ON FIELDS

203

where as above we have put MT

E

df"

mb;

o6EGul (F/k)

o61 F.(1)=-qn

hence condition 3) is also satisfied. This concludes the proof of Proposition 8.

6. - The index formula Take F C koo to be, as in section 5, a finite abelian extension of k such that the conductor of F/k is equal to q", where q is a prime ideal of Ok. We want to calculate the index of the group £F in O. The technique we will use is well known, cf 191 or [121. Let a E Gal (F/k). Then for each rational integer a > 0 we put dfn

to F(a) =

OF(1) (a)aWF(a)

where Q E Gal(F(l)/k) is such that & = aIF(1). It is obvious that ta,F(a1)/ta,F(a2) E OF, for all al, a2 E Gal (F/k). Moreover we have the action

ta'F(a1) ° Cta,F(a2)

ta,F(aia) ta,F(a2a)

for all

Cr E

Gal (F/k).

Let us denote Ta,F the subgroup of OF generated by the quotients ta,F(a)/ta,F(a'), or, a' E Gal (F/k). We know that the group Ta,F has finite index in OF, cf. [91 or [ 121. We have (19)

[OF

:

Ta,F] = wkea(F) h(hF) (Wc)[F:k]-1

where ea(F) is a positive integer, equal to 1 if F n H(l) = k ; otherwise we have (20)

ea(F)

ST

11 (1 - X((q, F n H(1)/k)) + awkh[F : F n H(1)]) X541

where x runs through the set of all non trivial characters of Gal (FnH(l) /k).

The fact that ea(F) # 0 implies that the quotients ta,F(a)/ta,F(1), a E Gal (F/k) and a 1, constitute a maximal system of independant elements of O. In particular we have NF/F(1) (T4,F)= Ta,Fn H(1).

204

H. OUKHABA

In other words the group Ta,F n H(1) is generated by the quotients to F(T)/ta F(1),T E Gal (F(1) /k) where we have put F(1) = F n H(1) and

fj

to F(T) df"1

ta,F(T),

for all T E Gal (F(1)/k).

rE Gal (F/K) 4I F.(1) =r

This leads to the identity

(Ta,F n H(1))n= Ta F n H(1), for all n > 0.

(21)

Moreover the group Ta,FnH(1) has finite index in OF(1) given by the formula [OF(1) : Ta,F n H(1)] = wkea(F)

(22)

h(OF(u (w")[F(1):k]-1 h

Nota Bene. Let us recall also that the subgroup OF(1) of OF(1) formed of all the products 11 OF(1)(r)

j

such that

rE Gal (F(1)/k)

1

nr(T) E IF(1) has a finite index in OF(1), cf. [91 or

rE Gal (F(1)/k)

[ 121, given by the following formula (23)

[OX F(

: 0 F(1)= U)k(wkw

h)[F(1):kl-1

h(OF(1)) [H(1)

:

F(1)]

PROPOSITION 9. - Let a be a positive integer. Then the group dfn

Za,F = TaWkh F F(1) has finite index in OF, given by the formula (24)

[oF :

ZaF]=Wk(Wkw.h)[F:k]-1

h(OF) [H(1)

:

F'(1)

Proof : on one hand we have the isomorphism Za,F/T"F OF(1)/Ta F n OF(1). On the other hand one can check that Ta F n OF(1) Ta F n H(1). This leads to the following identity [OF : Za,F] [OF(1)

:

(Ta,F n H(1))wkhl = [OF : Ta,F ]

[OF(1)

:

_

OF(1)1

205

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

which allows us to conclude the proof using Formulas (19), (21) and (23).o One can notice the inclusion EFkw°°h C Za,F, for all a > 0. In fact when EFkw°°h

I a then the group group of the products w,,,,

fi

(25)

can be characterized as follows. It is the

fi

ta,F(a)ma

of Cal (F/k)

OF(,) (-r)"

TE Gal (F(1)/k)

such that of Gal (F/k)

ma E IF

E

2')

nT E IF (1)

TE Gal (F(1) /k)

3') We have the congruence

Y

d1 PF(,) (

nr(T))

TEGal (F(1)/k)

where

0(mod. ww()'\

,

TEGal (F(1) /k)

M,(7-) is the element of IF(1) such that MT =

>'

mo,

T E Gal (F(1) /k).

o EGal (F/k)

LEMMA 2. - We have a well defined morphism

Za,F -, (Z/mZ), M = WF(,) /wk,

which associate to the element (25) of Za,F the lefthand side of the above congruence 3'). This morphism is onto and its kernel isjust the group EF kwoo h

so that we have

[ZaF

:

wkwooh

EF

wF'(1) Wk

Proof : all we have to prove is that the congruence d1 ' F(j)

(

nr(T))

MI(T))+WF(i) ( TE Gal (F(1) /k)

( \

lWkh

fJ ta,F aE Gal (F/k)

Wk

TE Gal (F(,) /k)

occurs whenever the element

f, TEGal (F(1)/k)

0(mod . wF(,)

OF(1)

206

H. OUKHABA

of Za,F is equal to 1. But in this case one can see easily that the product

H uEGal (F/k)

E Ta,F n H(1),

ta,F(Q)m

which means that ma = ma', if 0 F(1) = 11'F(1); and then one can write using the definition of ta,F(a) for a E Gal (F/k) ta,F(a)'n'

OF(1)(T)'n')a[F:F(1)1

= (

aEGal (F/k)

[J

TEGal (F(1)/k)

fi

x

TEGal (F(1)/k)

(f

PF(a))"*

,

o EGal (F/k)

°I FM=

where we have put m' = ma if a E Gal (F/k) and T E Gal (F(1) /k) are such that a,F(,) = T. Now the formula OF(1)(T

VF(a)m')wkix=

)1

,TE

Gal(F(i)/k),

already proved in 191, chap. IV, leads to the equality :

11

awkh[F : F(1)]-"x-4]

8FU)

=1

rEGal (F(1)/k)

which is equivalent to the condition (26)

nT + m'T + mrawkh[F : F(1)] - m;.Tq = 0

for all T E Gal (F(1) /k). Now we have d14

M.(T)) =

[F

)/F(1) (

TEGal (F(1)/k)

F(1)J mTTgn (T))

TEGal (F(1)/k)

= [H q" : H(1)] 0(1 ( F(j) )

E

m'T(TTq^1))

TEGal (F(1) /k)

Nq

"-1(Nq-1) Wk

= -g1F(j) (

m, T

TEGal (F(1)/k)

E TEGal (F(1)/k)

,

1

)-1)

Wk

m7(1 -T)(1 - Tq1)).

CONSTRUCTION OF ELLIPTIC UNITS IN FUNCTION FIELDS

207

On the other hand the above condition (26) allows us to write ,j(F(1)

nT(T)

1

m'.)(r))

F21'

/TEGal (F(l)/k)

TEGal (F(1) /k) (2) l

mT(1 - T)(1 - Tq 1) I .

1

TEGal (Fhl /k)

The Lemma 2 is proved. PROPOSITION 10. - The quotient group OF 1,6F is finite. The exact number

of its elements is given by the index formula

[o : EF]J = L

h(OF) [H(1)

:

F(1)]

Manuscrit recu le 4 decembre 1993

208

H. OUKFIABA

References [ 1 ] V.G. DRINFELD. - Elliptic modules, Math. USSR-Sbornik, 23 (1974), 561-

592. [2] S. GALOVITCH, M. ROSEN.-Theclass number ofcyclotomicfunctionfields,

Journal of number theory, 13 (1981), 363-375. [3] S. GALOVITCH, M. RosEN. - Units and class groups in cyclotomic function

fields, Journal of number theory, 14 (1982), 156-184. [4] D.R. HAVES. - Explicit class field theory for global function fields, Studies

in algebra and number theory (Rota G.C (ed)), New-York, Academic Press, (1979), 173-217. [5] D.R. HAVES. - Elliptic units in function fields, In proc. of a conference on

modem developments related to Fermat's last theorem, D. Goldfeld ed. Birkhailser, Boston, 1982. [6] D.R. HAVES. - Stickelberger elements in function fields, Compositio. Math, 55 (1985), 209-239. [7] D. KUBERT, S. LANG. - Modular Units, Grundleh der Math. Wiss., 244 (1981), ed. Springer. [8] H. OuKHABA, G. ROBERT. - Etude d'un ideal particulier associe a un caractere de Dirichlet d'ungroupefini, Seminaire de Theorie des Nombres de Bordeaux 3 (1991), 117-127. [9] H. OUKHABA. - Fonctions discriminant, formules pour le nombre de classes et unites elliptiques; le cas des corps de fonctions (associes a des courbes sur des corps finis), These (Grenoble, Institut Fourier, Juin 1991).

[10] H. OUKHABA. - Groups of elliptic Units in Global function fields, in Proceedings of the Workshop at the Ohio State University, June 17-26, 1991. 1111 H. OUKHABA. - On discriminant functions associated to Drinfeld Modules

of rank 1, Journal of number theory, 47 (1994). [12] G. ROBERT. - Unites elliptiques, Bulletin Soc. Math. France, Supplement 36, Decembre 1973. [ 13] J. Yu. - Transcendence and Drinfeld modules, Invent. Math. 83 (1986), 507-517. Hassan OUKHABA Equipe de Mathematiques URA CNRS 741 16, Route de Gray France - 25030 Besancon Cedex

Number Theory Paris 1992-93

Arbres, ordres maximaux et formes quadratiques enti8res Isabelle Pays

On salt depuis Lagrange que tout entier naturel est une somme de quatre canes, et, d'apres Jacobi (1828), que le nombre de representations d'un entier en somme de quatre canes est

r4(m) = 8 Ed (m > 1), dim 4td

les representations obtenues en permutant 1'ordre ou en changeant le signe des composantes etant comptees separement. Les preuves connues de cette formule sont de nature analytique (analyse complexe, formes modulaires, fonctions elliptiques, ...). Parmi les nombreuses references, citons E. Landau [8, pp. 146-150] qui determine le nombre de representations d'un en-

tier en somme de quatre canes en utilisant les formules sur le nombre de decompositions d'entiers en somme de deux canes (qu'il a etablies auparavant de maniere tout a fait elementaire); G.H. Hardy et E.M. Wright [7, p. 314], J.V. Uspensky et M.A. Heaslet [15, pp. 450-458], ainsi que E. Grosswald [6, pp. 30-36] donnent des preuves basees sur des identites qui peuvent etre derivees des proprietes des fonctions elliptiques ou simplement verifiees "a la main"; dans [9, p. 3331, W. Scharlau exploite le fait que la somme de quatre canes est une forme quadratique avec un seul element dans son genre pour deduire la formule de Jacobi; A. Robert [ 111 et B. Gordon [4] etablissent la formule de Jacobi a partir de resultats sur les formes modulaires (ce qui necessite un peu d'analyse complexe). Toutes ces preuves utilisent ou bien des identites un peu "mysterieuses", ou alors du materiel assez sophistique. Pour des references concernant l'origine et les developpements historiques, nous renvoyons le lecteur au recueil de L.E. Dickson [3, Chap. VIII, p. 2851. Signalons aussi un article de G. Rousseau [ 121, oft l'auteur donne un moyen pour construire des representations d'un entier en somme de quatre canes a partir de fractions continues.

I. PAYS

210

Nous proposons ici une nouvelle preuve a caractere purement algebrique et geometrique de la formule de Jacobi. Cette preuve est tout a fait elementaire : les prerequis sont a peine un peu plus qu'un cours de premier cycle en algebre. La preuve que nous presentons decoule de resultats plus generaux sur le nombre de representations d'une puissance quelconque d'un nombre premier par certaines formes quadratiques a quatre variables, obtenus au moyen d'actions de "groupes de quaternions" sur "l'arbre de SL2 (Q p)". L'article se presente comme suit : Au § 1. on rappelle la definition d'une algebre de quaternions. Les ordres maximaux dans une algebre de quaternions rationnelle permettent de definir les formes quadratiques entieres que l'on examine plus loin. Au §2 on decrit la construction de 1'arbre a partir des ordres maximaux de M2(Qp). C'est au §3 que

l'on explique la relation entre l'action d'un certain groupe sur l'arbre et les representations d'une puissance d'un nombre premier par les formes quadratiques associees (au § 1) aux ordres maximaux. On montre au §4 que, lorsque l'ordre maximal est principal, on peut obtenir le nombre de representations d'un entier quelconque (et non plus uniquement d'une puissance d'un nombre premier). Cela conduit a une nouvelle preuve de la formule de Jacobi.

1. - Algebres de quaternions et ordres Nous renvoyons aux ouvrages 11 ], (101 et 1161 pour les preuves detaillees

des resultats mentionnes dans ce paragraphe.

Soit K un corps de caracteristique differente de 2 et soient a et b deux elements non nuls de K. L'algebre de quaternions (a, b)K est l'algebre

admettant une base de quatre elements sur K, notes 1, i, j, k, avec la multiplication definie par les relations i2 = a, j2 = b, k = i.j = -j.i. Le conjugue du quaternion q = Xi + x2i + x3j + x4k, note q, est defini par

q = xi - x2i - x3j - x4k. La norme reduite du quaternion q, notee n(q), est definie par n(q) = q.q = x1 - axe - bx3 + abx4. La trace reduite du quaternion q, notee t(q), est definie par t(q) = q + = 2x1. Il est bien connu qu'une algebre de quaternions est soit a division, soit isomorphe a 1'algebre de matrices M2(K).

Les corps consideres ici sont soit le corps (global) Q des nombres rationnels, soit un des corps (locaux) Qp des nombres p-adiques ou IR le corps des nombres reels. Sur un corps local (ici IR ou Qp), it y a une unique algebre de quaternions a division, a isomorphisme pres. Sur R, it s'agit de 1'algebre des quaternions de Hamilton, IEII = (-1, -1)a. Soit H = (a, b)q. Quitte a multiplier a et b par des carres convenables, on peut supposer que a et b sont dans Z. Pour reconnaitre si Hp = (a, b)Qp est a division, on utilise le symbole de Hilbert (a, b)p. L'algebre (a, b)Q est

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

211

a division si et settlement si (a, b) p = -1. Nous renvoyons le lecteur a 114, p.391 pour le calcul de ce symbole. Notons toutefois que (a, b)P = 1 pour presque tout p (c'est-a-dire pour tout p sauf un nombre fini d'entre eux).

Le discriminant de H est le produit des nombres premiers p pour lesquels 1'algebre de quaternions H ® Q, est a division

disc(H) =

11

p.

p premier (a,6)p=-1

Soit R un anneau principal de caracteristique differente de 2, K son corps de fractions et H une algebre de quaternions sur K (nous envisageons en particulier le cas on R = Z ou Z[P] = {ap' la, n E Z} avec K= Q ou alors R = Z P, l'anneau des entiers p-adiques, avec K = Q p). Nous designons par R" le groupe multiplicatif des elements inversibles de R.

Un ordre de H sur R est un sous-R-module de H de rang 4 qui est aussi un anneau. Les elements d'un ordre ont la propriete d'etre entiers sur R, c'est-a-dire que leur trace et leur norme appartiennent a R. Un ordre maximal est un ordre qui n'est contenu proprement dans aucun autre ordre. Voici deux exemples qui nous seront utiles.

Exemple IL. Dans H = (-1, -1)q le Z-module 0' de base (1, i, j, k) est un ordre de H sur Z. De meme, le Z-module 0 engendre par 1, i, j,

a= (1+i+j+k)/2 est un ordre de H. On note que t(a) = 1 etn(a) = 1. L'ordre 0' nest pas maximal car it est contenu dans l'ordre 0. Exemple 2. Soit R un anneau principal et K son corps de fractions. Alors M2 (R) est un ordre de M2 (K).

Les formes quadratiques que nous allons examiner sont les formes normes d'ordres sur Z d'une algebre de quaternions H sur Q. Soit 0 un tel ordre et soit (el, e2, e3, e4) une base de 0 sur Z. La forme norme de 0 par rapport d la base e est

n(x) = n(> Xei) _

XiXjt(eiej)

Xi n(ei) + i Zp U oc normalisee par la condition v(p) = 1, et la fonction It : H --> Z U oo definie par :

µ(x)=max{nEZIxEp"O}

pourx

0

et

µ(0) = 00. Cette fonction satisfait les proprietes suivantes : PRopweTes. - Pour x, p E H et a E Q, , on a : 1. µ(x) = oo si et seulement six = 0. 2. µ(x + y) > min{µ(x), µ(J)}.

3. µ(xy) ? i(x) + µ(J)

4. µ(xa) = it(x) + v(a). 5. ;L(x) =,u(x). De plus, en designant respectivement par OX et par H" les groupes multiplicatifs des elements inversibles de 0 et de H,

a) Pourx E Hx, on ax E 0 " sietseulementsiµ(x) = µ(x-1) =0.

I. PAYS

214

b) Pour x E H', les conditions suivantes sont equivalentes : i) x E pa0" pour un certain a E Z. ii) it(x) + µ(x-1) = 0.

iii) x0x-1 = 0. c) Pour x, y E H, si x satisfait les conditions equivatentes de la propriete precedente, on a : p(xy) = µ(x) + µ(y) = µ(yx) et µ(xyx-1) = µ(y).

d) Pourx E H", µ(x-1) = µ(x) - v(n(x)) (oCl n design la norme de H).

Demonstration : les proprietes 1 a 4 sont toutes evidentes. La propriete 5 decoule immediatement du fait que tout ordre d'une algebre de quaternions est stable par la conjugaison quaternionienne (car y = t(x).1-

x). Si x E 0", alors µ(x) = 0 car les elements de p0 ne sont pas inversibles dans 0; on a alors de meme µ(x-1) = 0. Reciproquement, si /1(x) = µ(x-1) = 0, alors x et x-1 sont tous deux dans 0, done x E 0". Cela prouve la propriete (a). Si x E pa0" pour un certain a E Z, alors p(x) = a et /1(x-1) = -a, done µ(x) +EL(x-1) = 0. Inversement, six E H"

est tel que u(x) + u(x-1) = 0, soit x = pay pour a = µ(x) et pour un certain y E 0 N p0. On a alors a(y) = 0 et µ(x-1) _ -a + µ(y-1). La relation µ(x) + li(x-1) = 0 entraine alors : µ(y-1) = 0, done y E Ox par la propriete (a). Cela demontre 1'equivalence des conditions (i) et (ii) de (b). Par ailleurs, la condition (i) entraine evidemment (iii). Reciproquement, si

x satisfait la condition (iii), on ecrit encore x = pay pour a = µ(x) et pour un certain y E 0 N pO; comme 0 - M2 (Zp), it nest pas difficile de verifier qu'alors OyO = 0. Or, de la relation xOx-1 = 0, on deduit que yO = Oy; on a done

yO=Oy=OyO=O, ce qui montre que y E O" et x E paQx et acheve la demonstration de la propriete (b). Pour etablir la propriete (c), on observe que, d'apres la propriete 3, µ(xy) ? l2(x) + t1(y)

14) = µ(x-Ixy) a(x-1) + A(xy) Lorsque µ(x-1) = -µ(x), on en deduit immediatement que µ(xy) _ µ(x) + µ(y). La relation lz(yx) = µ(x) + µ(y) se demontre de maniere analogue, et la relation a(xyx-1) = µ(y) se deduit des deux precedentes. Enfin, la propriete (d) resulte des proprietes 4 et 5, car x-1 = x.n(x)-1. Soient maintenant 01 et 02 des ordres maximaux de H, et soient x1 et x2 E H" tels que :

01 =x10xi1

et

02

=x20x21.

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

215

On pose d(01, 02) = -µ(xi 1x2) - µ(x2 1x1) E 7L.

Pour voir que la fonction d est bien definie, it faut verifier que le second

membre ne depend pas du choix de xl et x2. Si xi E H" est tel que Oi = xi0x'i 1 pour i = 1, 2, alors x'i 1xi0x%lx' = 0 pour i = 1, 2, donc, par la propriete (c) ci-dessus, on a

µ(x,l ix2) = µ(xti

I 1x2) = µ(x'1 XI) + µ(x 1x2) + µ(x2 1x'2)

1

et, de meme, 1L(x'2 1x1) = p(x'2 1x2) + p(x21xi) + ji(xi 1x'1). Des lors, µ(x1 1x2) + p(x'2 1x'1) = p(xi 1x2) + p(x21xi), ce qui prouve que d est bien definie. On a en fait d(01, 02) > 0, car, d'apres la propriete 3, µ(x1 1x2) + µ(x2 1x1) < 11(x1 1x2.x21xi) = 0.

PROPOSITION 2.1. - La fonction d est une distance sur l'ensemble des ordres maximaux de H. Cette distance est invariante par conjugaison, c'est d-dire que pour x E H" et pour 01, 02 des ordres maximaux de H,

d(x0lx-1, x02x-1) = d(01, 02)

Demonstration

:

it

est clair par definition que la fonction d est

symetrique. De la propriete (b), on deduit que d(01, 02) = 0 si et seulement Si 01 = 02. L'inegalite triangulaire decoule de la propriete 3 et l'invariance par conjugaison est evidente puisque (xx1)-1(xx2) = x1 1x2. On obtient alors l'arbre des ordres maximaux de H. TI-IEOREME 2.2. - Le graphe X dont les sommets sont les ordres maximaux de H et dont les aretes sont les couples (01i 02) d'ordres maximaux tels que d(01, 02) = 1 est un arbre, c'est-d-dire un graphe connexe et sans circuit. De plus, cet arbre est (p+ 1)-regulier, c'est d-dire que chaque sommet est l'origine de p + 1 aretes.

Demonstration : montrons d'abord que le graphe X est connexe. II suffit de montrer que tout ordre maximal 0' est lie a 0 par un chemin du graphe. On raisonne par induction sur la distance de 0 a 0. L'enonce est evident si cette distance est 1, puisqu'alors 0 et 0' sont lies par une arete. Il suffit donc de prouver que si la distance de 0 a 0' est n > 1, alors i1 existe un ordre 0" a distance n - 1 de 0 et a distance 1 de 0'. Soit 0' = xOx-1. Quitte a multiplier x par une puissance convenable

de p, on peut supposer p(x) = 0, c'est-a-dire que x E 0 N p0. Comme O/pO ^ M2 (1FP), la trace induit une forme bilineaire non degeneree sur

I. PAYS

216

O/pO; on peut donc trouver u E 0 N p0 tel que t(xu) ¢ pp, ce qui entrain bien sur que xu E 0 , pO. Soit alors y = xu + p- in (x). Comme d(O, 0') = n > 1, on a, par la propriete (d),

-IL(x-1) = v(n(x)) = n > 1, d'ou y E 0 N pO, c'est-a-dire, µ(y) = 0. Par ailleurs, x-1 y = u + p-lx,

donc µ(x-ly) _ -1, et de la relation n(x-1 y) = n(u) + p-lt(xu) + p-2n(x)

on tire : v(n(x-ly)) = -1. D'apres la propriete (d), on en deduit

-tc(x-ly) - IL(y-lx) = 1. Par ailleurs, comme v(n(x)) = it, on dolt avoir v(n(y)) = n - 1, donc

-µ(y) - 12(y-1) = n - 1. Des lors, l'ordre 0" = yOy-1 possede les proprietes requises.

Montrons ensuite que le graphe X ne contient pas de circuit. Soit 01i ... , On un chemin sans aller-retour, c'est-a-dire, (1)

f d(Oi,Oi+1) = 1 pouri = 1,...,n- 1 d(Oi, Oi+2) > 0 pour i = 1, ... , n - 2.

Pour prouver que ce chemin n'est pas un circuit, it suffit de montrer que d(Oi, On) = n - 1.

Ecrivons Oi = xiOx-1 pour i = 1 , ... , n et, pour i = 1, ... , n - 1 xi lxi+1 = pa`yi pour un certain yi E 0 N pO et ai = µ(xi lxi+1) E Z. D'apres la propriete (d), on a, pour i = 1, ... , n - 1,

p(xi+1xi) = ai - v(n(p'' yi)) = -ai - v(n(yi)) Des lors, d(Oi,Oi+1) = -N(x lxi+1) - µ(xi+lxi) = v(n(yi)), et les conditions (1) ci-dessus s'ecrivent :

v(n(yi)) = 1 pour i = 1, ... , n - 1 v(n(yiyi+l)) - 2,a(yiyi+l) > 0 pour i = 1, ... , n - 2.

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Vu la multiplicativite de la norme, la premiere condition entraine : pour v(n(yiyi+l)) = 2 pour i = 1,... , n-2; par ailleurs, comme yti E i = 1, ... , n - 1, on a yjy2+1 E 0 pour i = 1, ... , n - 2, done µ(yiyi,+1) > 0. Ces observations conduisent a reecrire les conditions ci-dessus sous la forme :

v(n(yti)) = 1 pour i = 1, ... , n - 1

(2)

I µ(ytiyti+l)

= 0 pour i = 1,...,n - 2.

Montrons alors, par induction sur m, que µ(y1... ym) = 0. C'est clair pour m = 2. Supposons donc µ(y1... yri.-1) = 0 et µ(y1 ... ym) > 0, c'est-a-dire,

yl...ym EPO. On a alors (3)

(yl

... y.-1 + pO).(ym + pO) = 0

dans O/p0.

Comme O/pO est isomorphe a une algebre de matrices carrees d'ordre 2 sur IF7,, on peut considerer yj + pO,... , ym + pO comme des operateurs lineaires sur un espace vectoriel de dimension 2 sur IFr. Ces operateurs sont non nuls puisque yj ¢ pO, et non inversibles puisque v(n(yi)) = 1; ils sont donc tous de rang 1. De meme, yl ... ym-1 + pO est de rang 1 puisque µ(y1 ... ym-1) = 0. ;equation (3) indique alors que : Im(ym + pO) = Ker(yl ... ym-1 + pO).

Par ailleurs, on a aussi Ker(yl

... y.-1 + p0) = Ker(ym-1 + pO),

donc

(ym-1 + PO) (Y. + PO) = 0

et par consequent µ(ym-lym) > 0, contrairement a l'hypothese. On a donc bien µ(y1 ... ym) = 0 pour tout m = 1, ... , n - 1. Un calcul direct donne alors d(Ol, On) = v(n(yl ... yn-1)) - 2µ(y1 ... yn-1) = n - 1, ce qui acheve de demontrer que le graphe X ne contient pas de circuit.

Pour prouver que X est (p + 1)-regulier, comme la conjugaison par tout element de H" induit un automorphisme de X et que tout ordre maximal est conjugue a 0, it suffit de montrer qu'il y a p + 1 ordres a

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distance 1 de 0. Or, it y a une correspondance bijective entre 1'ensemble des ordres a distance 1 de 0 et l'ensemble des ideaux a droite I de 0 tels

que 0 Q I Q p0, qui associe a tout ideal I son ordre de stabilisateurs a gauche :

O9(I)={xeHIxICI}

et a tout ordre 0' a distance 1 de 0 l'ideal pO'0 (pour etablir la bijectivite de cette correspondance, it est utile de remarquer que si x E 0 '. p0 et v(n(x)) = 1, alors OTTO est un ideal bilatere de 0 contenant proprement p0, donc 0770 = 0 puisque 01p0 f-- M2(]Fp) est simple. Si a present 0' est un ordre a distance 1 de 0, on peut ecrire :

0' = xOx-1 pour un certain x comme ci-dessus, d'oCi

p0'O = xOTO = xO. Reciproquement, 09(xO) = x0x-1).

Comme par ailleurs les ideaux a droite I tels que 0 12 p0 sont en bijection avec les ideaux a droite non triviaux de 0/p0 ^ M2 (IFp), qui sont au nombre de p + 1, it y a bien p + 1 ordres a distance 1 de 0.

3. - Actions de sous-groupes et representations d'entiers Soit H une algebre de quaternions sur Q et_ 0 un ordre maximal de H sur Z. La forme quadratique quaternaire a coefficients entiers que nous allons etudier est la forme norme sur 0, c'est-a-dire (voir aussi § 1) que si e = (el, e2i e3, e4) designe une base de 0, la forme s'ecrit 4

Xiei) _ i=1

Xi n(ei) + i

i 1.

Demonstration : soit 0' un ordre maximal de Hp. On sait que tous les ordres maximaux de Hp sont conjugues, donc 0' = xOpx-1 pour un certain x E Hp >'. Quitte a multiplier x par une puissance convenable de p, on peut choisir x E Op. Considerons alors l'ideal a droite I de 0 define par

I= n(OgnH)

n(x0p n H),

qi4P

c'est-a-dire l'ideal dont les localises sont Iq = °q pour q p et Ip = xOp. D'apres l'hypothese, cet ideal est principal, donc I = yO pour un certain

y E 0. Comme Iq = Oq pour q

p, on a y E Coq pour q 54 p, donc

y E CA[P]". Par ailleurs, de la relation Ip = XOP = Y°p,

on deduit que l'ordre des stabilisateurs a gauche de Ip est

'g(Ip) = X0pX-1 = JOPJ-1+

c'est-a-dire que 0' = yOpy-1. Cela prouve que 0[-!]' agit transitivement sur les sommets de 1'arbre Xp. Il en resulte en particulier que On est 1'ensemble de tous les sommets a distance n de Op. Cet ensemble contient

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pn-1(p + 1) elements, puisque I'arbre Xp est (p + 1)-regulier, d'apres le theoreme 2.2. Remarque : plus generalement, Vigneras [ 16, p.147, Prop.3.3] a montre que le nombre d'orbites de sommets de Xp sous 1'action de O[P] est egal au nombre de classes de O. La demonstration n'est pas aussi elementaire que celle du theoreme precedent, car elle utilise un theoreme puissant d'Eichler.

On a choisi au debut de ce paragraphe un nombre premier p qui ne divisait pas le discriminant de H. Voici maintenant ce qui se produit lorsqu'au contraire p divise le discriminant de H, c'est-a-dire lorsque Hp = H 0 Qp est une algebre a division. Comme precedemment, on dit que deux elements x, y E 0 sont associes (a droite) s'il existe un element inversible u E Ox tel que x = yu; on note alors x - y. THEOREME 3.3. - Lorsquep divise le discriminant de H, alors les elements

de R(pn) sont tous associes, pour tout n > 0, de sorte que le quotient R(pn)/ - contient un seul element, si R(pn) nest pas vide. Si l'ordre 0 estprincipal, alors R(pn) est non vide, pour tout n > 1.

Demonstration : supposons que R(pn) est non vide. Soient x, y E R(pn) ; alors x et y sont inversibles dans Oq = 0 0 Z. pour q # p, et donc, xOq = YOq = Oq.

En p, l'algebre de quaternions Hp = H®Qp est isomorphe a l'unique algebre de quaternions a division sur Qp, et Op est l'anneau devaluation de Hp U10, § 12]). De plus, tout ideal a droite de Op est bilatere et principal et est donc de la forme irPOp, ou 7rp est une uniformisante de Or,. Comme la valuation p-adique de n(7rp) est 1, on a en particulier x0p = 7rnop = yOp.

Ainsi, xOp = yOp pour tout p, donc xO = yO et x - y. Cela prouve que tous les elements de R(pn) sont associes. Si 0 est principal, alors l'ideal I dont les localises sont Oq pour q 54 p et 7rpOp en p est principal; soit I = 7r0 pour un certain it E O. On a alors

0

7rO 2 pO,

donc p = 7rir' pour un certain 7r' E 0, et n(7r)n(ir') = p2.

Si n(-7r) = ±1, alors 7rO = 0; si n(7r) = ±p2, alors 7rO = pO. Comme ces deux egalites sont exclues, on doit avoir n(7r) = fp,

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donc R(p) est non vide. De plus, pour tout n > 1, on a n(7rn) = ±p', donc R(pn) est non vide pour tout n > 1. Dans le cas particulier ou la forme norme est definie positive, ce qui

revient a dire que l'algebre de quaternions H est telle que H 0 IR est isomorphe a 1'algebre (-1, -1)R des quaternions d'Hamilton, it est clair que les equations n(x) = -pn n'ont pas de solution et que les equations n(x) =

pn n'en ont qu'un nombre fini. Les resultats precedents permettent de denombrer ces solutions. L'ordre 0 de l'algebre de quaternions rationnelle H etant fixe, notons r(pn) (resp. rp(pn)) le nombre de solutions (resp. de solutions primitives) x E 0 de 1'equation n(x) = pn, c'est a dire le nombre d'elements de R(pn) (resp. Rp(pn)). COROLLAIRE 3.4. - Supposons que laforme norme soit definie positive. Si p est un nombre premier qui ne divise pas le discriminant de L'algebre H, alors pour tout n > 1, rp(a)n) = 10' I Ionl oil An est L'ensemble des sommets de I'arbre Xp qui sont dans la meme orbite que O. sous l'action de 0[-1]" et a distance n de CAP, et [n/21

r(pn) = IOX I

E k=0

.

oil [n/2] est le plus grand entier inferieur ou egal a n/2. Si p est un nombre premier qui divise le discriminant de H. alors pour tout n > 1,

r(pn) = 0 ouI0xI. Si de plus L'ordre 0 est principal, alors pour tout nombre premier p qui ne divise pas le discriminant de H,

rp(pn) =

I0XI. pn-i(p+ 1)

et

np+1

lox

p

-1

-1 r(pn) = et sip divise le discriminant de H,

pourtoutn> 1 pour tout n > 1,

I

r (pn) = I C X I

pour tout n > 1.

Demonstration : Si p ne divise pas le discriminant de H, les formules pour rp(pn) resultent directement des theoremes 3.1 et 3.2 ci-dessus, puisque rp(pn) = 0 < I ' I Rp(pn)/

I.

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Les formules pour r(p') s'en deduisent, car les solutions non primitives de n(x) = p' sont de la forme x = pkxk ou Xk est solution primitive de n(x) = p"-2k. De meme, si p divise le discriminant de H, les formules pour r(p') decoulent du theoreme 3.3. Pour completer l'information donnee dans ce corollaire, remarquons que la structure du groupe O" est connue pour les ordres maximaux des algebres de quaternions definies positives : PROPOSITION 3.5. - Si 0 est un ordre maximal d'une algebre de quater-

nions H sur Q telle que H 0 R est isomorphe a (-1, -1)R, alors O" /{f1} est cyclique d'ordre 1, 2 ou 3 sauf dans deux cas :

si H = (-1, -1)Q, alors 0'/ f ± 11 est isomorphe au groupe alterne A4; si H = (-1, -3)Q, alors O" /{±1} est isomorphe au groupe symetrique S3. Demonstration : voir [ 17, th. 5, p. 269]. Exemple : revenons a 1'exemple 1 avec l'ordre Ode base (1, i, j, a) dans

1'algebre H = (-1, -1)Q. Le discriminant de 0 vaut -4 et le discriminant de H est egal a 2. Des lors, 0 est maximal. Pour voir que c'est un anneau principal, on montre que les elements de 0 satisfont un algorithme de division euclidienne ([ 13, p. 98, lemme 3]). L'ordre du groupe O" des elements inversibles de 0 est 24, d'apres la proposition 3.5. Des lors, pour tout p 2, le nombre de representations primitives de p"` par la forme : q0 (Xl, X2, X3, X4) = X1 + X2 + X3 + X4 + X1X4 + X2X4 + X3X4.

est egal a rp(p') = 24(p +

1)p"-1

(p

2),

et le nombre de representations de 2 est donne par :

r(27z)=24

n>0.

4. - Ordres principaux Dans cette section, 0 designe un ordre maximal principal dans une algebre de quaternions rationnelle H dont la forme norme est definie positive. On se propose de montrer que, grace au fait que 0 est un anneau principal, it est possible de donner le nombre de representations d'un entier

positif quelconque (et non pas seulement des puissances d'un nombre premier) par la forme norme de O.

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LEMME 4.1. - Soient a et b des entiers positifs premiers entre eux. Si

x E 0 est tel que n(x) = ab, ators it existe y, z E 0 tels que x = yz et n(y) = a, n(z) = b.

Si y et z sont des elements de 0 tels que n(y) = a et n(z) = b, alors yz0 + a0 = yO et Oyz + 0b = Oz. Demonstration : soit x E 0 tel que n(x) = ab. On considere l'ideal x0 + a0 de 0. Comme 0 est principal, on a :

x0 + a0 = yO pour un certain y de 0. Soient x = yz et a = yy'. Alors n(x) = ab = n(y)n(z) et a2 = n(y)n(y'), donc n(y) est un commun diviseur de abet de a2. Comme

a et b sont premiers entre eux, n(y) divise a. Par ailleurs, de la relation xO + aO = yO, on tire aussi

xx'+aa' = y pour certains x', a' de 0. En prenant la norme des deux cotes, on deduit :

n(y) = n(xx') + n(aa') + t(xx'aa') = n(x) n(x') + a2n(a') + at(xx'a') = a(bn(x') + an(a') + t(xx'a')). Comme x, x' et a' sont dans 0, on a t(xx'a') E Z, donc le facteur de a dans le membre de droite est un entier. Il en resulte que a divise n(y). Comme a et n(y) sont tous les deux positifs, on a que n(y) = a et donc aussi n(z) = b. Par ailleurs, si y et z sont des elements de 0 tels que n(y) = a et n(z) = b alors, comme 0 est principal, on a :

yzO+aO=dO pour un certain d dans 0. Les arguments du debut montrent que n(d) = a. Par ailleurs, de yj = a, on deduit que aO C yO. Des lors, yzO + aO = dO C yO.

Ainsi i1 existe u E 0 tel que d = yu. Comme n(d) = n(y) = a, on a n E 0" et donc d0 = yO. Avec ce lemme, on peut donner le nombre de representations d'un entier quelconque par la forme norme de 0; comme precedemment, on note

R(m) =Ix E0I n(x)=m} et

r(m) = JR(m)I, ou m est un entier (positii) quelconque.

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T1ii orEME 4.2. - 1. Lafonction r(m)/ IOx I est multiplicatioe, c'est-d-dire que si a et b sont des entiers positifs premiers entre eux, alors :

- lox I

r(ab)

r(a)

lox I

r(b)

lox I

2. Soit m un entier positif. On a

r(m) = I

I

(

d) dim pgcd(d,disc H)=1

Demonstration : 1. Soient a et b des entiers positifs premiers entre eux. La multiplication

dans 0 definit une application : R(a) x R(b) -i R(ab), qui est surjective d'apres le lemme 4.1. Pour demontrer la premiere partie de 1'enonce, it suffit de prouver que tout element de R(ab) est l'image de lox I elements de R(a) x R(b), puisqu'alors

r(ab) =

IR(a) x R(b)I

r(a)r(b)

IoxI

IOxI

.

Fixons x E R(ab). Si (y, z) et (y', z') sont des elements de R(a) x R(b) tels que

yz=x=y'z',

alors d'apres la seconde partie du lemme 4. 1 on a : xO + aO = yO = y'O.

Des lors, it existe un element inversible u E Ox tel que y' = yu (et donc z' = u-1z). Cela prouve que les elements de R(a) x R(b) qui ont x pour image sont les couples (yu, u-Iz), oli u E O. Le nombre de ces couples est bien egal au nombre d'elements de Ox. 2. On a deja calcule, dans le corollaire 3.4, le nombre de representations des puissances d'un nombre premier p : si p ne divise pas disc H, alors :

et si p divise disc H, r(Pn)

IoxI

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On voit ainsi que les fonctions

(r

et (T, d) prennent la mil-me valeur dIm

pgcd(d,disc H)=1

lorsque m est une puissance d'un nombre premier. Comme ces deux fonctions sont multiplicatives, elles doivent prendre la meme valeur pour tout M. Le theoreme 4.2 s'applique aux ordres principaux dans les algebres

de quaternions rationnelles definies positives, qui sont au nombre de cinq [16, p.1551. L'ordre principal est alors unique (a conjugaison pres) [16, p.26, Cor. 4.111. Voici la liste des cinq formes quadratiques (a Zequivalence pres) auxquelles le resultat s'applique ainsi que, pour chacune, la formule pour le nombre de representations d'un entier quelconque. - Le nombre de representations d'un entier positif n par la forme

Xi +X2+x3+X4 + X1X4 + X2X4 + X3X4 est

241: d, din 2{d

- Le nombre de representations d'un entier positif n par la forme

Xi

+X2+X3 +X4 +X1X4+X2X3

est

121: d, din 3$d

- Le nombre de representations d'un entier positif n par la forme

x2 + 2X2 + 5X2 + X4 + X1X2 + X1X4 + X2X4 + 5X2X3 est

6Ed, dIn 5{d

- Le nombre de representations d'un entier positif n par la forme X1 + X2 + 2X3 + 2X4 + X1X4 + X2X3 est

41: d, din 7{d

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- Le nombre de representations d'un entier positif n par la forme X1 + 4X2 + 13X3 + 2X4 + X1X2 + X1X4 + X2X4 + 13X2X3 est

2> d. dln 13{d

Nous allons maintenant deduire la formule de Jacobi pour la somme de quatre carres a partir de la premiere de ces formules. La forme quadratique Xi +X2 +X3 +X4 est la forme norme de l'ordre 0' de base (1, i, j, k) dans l'algebre de quaternions H = (-1, -1)Q. Cet ordre n'est pas maximal : it est strictement contenu dans l'ordre maximal 0 de base (1, i, j, (1+i+j+k)/2). On ne peut donc pas lui appliquer la technique developpee ci-dessus. Cependant, les relations entre 0 et 0' sont telles qu'il est quand meme possible de deduire le nombre de representations d'un entier en Somme de quatre carres a partir du nombre de representations d'un entier par la forme norme de 0. Pour eviter la confusion, la notation r(m) designe, dans la fin de ce paragraphe, le nombre de representations de m par la forme norme de 0 tandis que r4(m) designe le nombre de representations de m en somme de quatre carres (qui est la forme norme de 0'). Commencons par indiquer ]a relation entre 0 et 0'. PROPOSITION 4.3. - Soit x un element non nul de 0. Si n(x) est paire, alors x E 0'; si n(x) est impaire, alors x est associe (a droite) a 8 elements de 0' (et a 16 elements de 0 N 0').

Demonstration : par rapport a la base (1, i, j, a) de 0, la forme norme s'exprime de la maniere suivante : n(xi + 2x2 + 353 + ax4) = xi + x2 + x3 + x4 + 51x4 + 52x4 + 53x4.

Des lors, pour x = x, + i52 + jX3 + ax4, n(x) -= (xl + x2 + x3)2 + (xl + x2 + 53)x4 + x4 mod 2.

Comme la forme quadratique X2 + XY + Y2 est anisotrope sur le corps a deux elements, on a donc n(x) E 2Z si et seulement si xl + X2 + 53 - 54 0 mod 2. En particulier, si n(x) est paire, alors 54 est pair, ce qui entraine :

xe0'.

Si n(x) est impaire, alors (x + 20)(5 + 20) = 1 + 20, donc T + 20 est l'inverse de x + 20 dans l'anneau quotient 0/20. Par ailleurs, on montre

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aisement (voir par exemple [13, p.1001) que x est associe a droite a un element de 0'; pour terminer la demonstration, on peut donc supposer x E 0'. Si U E 0' est tel que xu E 0', alors dans 0/20, on a u + 20 = a; (xu) + 20,

ce qui prouve que u E 0', puisque Y(xu) E 0' et 20 C 0'. On a donc xu E 0' si et seulement si u E 0", ce qui prouve que les associes a droite de x qui sont dans 0' sont en bijection avec 0", d'ou la proposition, car

10'x1=8. La formule de Jacobi se deduit alors aisement de la formule pour le nombre de representations par la forme norme de 0 : THEOREME 4.4. (Jacobi). - Le nombre de representations d'un entier positif m en somme de quatre carres est donne par

7'4(m) =8() 'd). dim 4{d

Demonstration : soit, comme precedemment :

R(m)={xE01n(x)=m}. Si m est impair, on deduit de la proposition precedente que chacune des classes d'elements associes a droite qui constituent R(m) contient 8 elements de 0' et 16 elements de 0 . 0'; donc r4(m) = 3r(m) =

d.

d1m

Si m est pair, alors d'apres ]a proposition precedente R(m) C 0', donc

r4 (m) = r(m) = 24 Ed. djm 2{d

On peut encore exprimer ce resultat comme suit :

r4(m) = 8(E d +

2d) =

dim

dim

dim

2{d

2{d

4{d

Dans les deux cas, on a donc bien le resultat annonce. manuscrit recu le 22 fevrier 1994

ARBRES, ORDRES MAXIMAUX ET FORMES QUADRATIQUES

229

Bibliographie

[1] A. BLANCHARD. - Les corps non commutatifs, Presses Universitaires de France, Paris, 1970. [2] J.W.S. CASSELS. - Rational Quadratic Forms, Academic Press, London, 1978. [3] L.E. DICKSON. - History of the Theory of Numbers, vol. II, Chelsea Publishing Co., New York, 1952. [4] B. GORDON. - An Application of Modular Forms to Quadratic Forms, BA-

thesis, 1975. [5] H. GROSS. - Darstellungsanzahlen von quaternaren quadratischen Stammformen mit quadratischer Diskriminante, Comment. Math. Helv 34, (1960) 198-221. [6] E. GROSSWALD. - Representations of Intergers as Sums of Squares, SpringerVerlag, New York Berlin Heidelberg Tokyo, 1985. [7] G.H. HARDY and E.M. WRIGHT. - An Introduction to the Theory of Numbers,

5th ed., Oxford University Press, 1979. [8] E. LANDAU. - Elementary Number Theory, Chelsea Publishing Co., New York, 1966.

[9] G. ORZECH ed. - Conference on Quadratic Forms 1976, Queen's Paper in Pure and Applied Math. 46, Kingston, Ontario, Canada 1977. [10] I. REINER. - Maximal Orders, Academic Press, London 1975. [11] A. ROBERT. - Introduction to Modular Forms, Queen's Paper in Pure and Applied Math. 45, Kingston, Ontario, Canada 1976. [12] G. ROUSSEAU. - On a construction for the representation of a positive integer

as the sum offour squares, L'Enseignement Math 33, (1987) 301-306. [13] P. SAMUEL. - Theorie algebrique des nombres, Hermann, Paris 1967. [14] J.-P. SERRE. - Cours d'Arithmetique, Coll. SUP, Presses Univ. France, Paris, 1970. [15] J.V. USPENSKY and M.A. HEASLET, Elementary Number Theory, McGraw-Hill,

New York and London, 1939.

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[16] M.-F. VIGNERAS. - Arithmetique des algebres de quaternions, Lecture Notes in Math 800, Springer-Verlag, Berlin Heidelberg New York, 1980. [17] M.-F. VIGNERAS. - Simplification pour les ordres des corps de quaternions

totalement definis, J. Reine Angew. Math. 286/287, (1976) 257-287. [18] A. WEIL. - Sur les sommes de trois et quatre carres, L'Enseignement Math 20 (1974) 215-222.

Isabelle PAYS

Universite de Mons-Hainaut Avenue Maistriau, 15 B-7000 MONS BELGIQUE

Number Theory Paris 1992-93

On a conjecture that a product of k consecutive positive integers is never equal to a product of mk consecutive positive integers except for 8.9.10 = 6! and related questions T.N. SHOREY

For an integer m > 2, we consider the equation

(1) (x+1) .

(x+k) _ (y+1) . . . (y+mk) in integers x > 0, y > 0, k > 2.

We replace x + 1 by x and y + 1 by y in (1) for observing that it is identical to considering equation

x(x+1) . . (x+k-1) = y(y+l) . . . (y+mk-1) in integers x> 0, y>0, k>2. If m = 2, equation (1) has a solution given by 8.9.10 = 6! (2)

x=7,y=0,k=3.

MacLeod and Barrodale [8) observed that this is the only solution of (1) with

m = 2 and k < 5. I give their proof for k = 2. We write equation (1) with m = 2 and k = 2 (3)

(x + 1)(x + 2) _ (y + 1)(y + 2)(y + 3)(y + 4).

By putting u = y2 + 5y, we have (x + 1) (x + 2) _ (u + 4) (u + 6).

Notice that

(x+32 _ 1)2 2, it is proved in [13] that equation (1) implies that max (x, y, k) < C

where C is an effectively computable(i) number depending only on m. We have not been able to replace C by an absolute constant. It is likely that equation (1) with m > 2 has no solution. (5)

Now, we give a sketch of the proof that equation (1) with m > 2 implies that max(x, y, k) is bounded by a number depending only on m.

As pointed out earlier, we secure that x and y are large as compared with k. We re-write equation (1) as

(y + 1) ... (y + mk) (mk)! k! (1)

(mk)!

k!

All the constants appearing in this paper are effectively computable.

233

ONA CONJECTURE CONCERNING 8.9.10=6!

We count the powers of 2 on both the sides to obtain

k < ord2 ( k! ') < ord2

(x + 1

k! x + k )

< max ord2 (x + i) < - 1 y > yo.

For extracting k-th roots on both the sides of equation (1), we need to introduce some notation. We write ink

(9)

(z + 1) . . . (z + mk) =

Aj (m, k)zmk-j. j=0

Further, we determine rational numbers

Bj=Bj(m,k) withl yo, we conclude that £(Y) = 0.

(17)

Now, I give two proofs to exclude the possibility (17).

We assume (17). Then (18)

L(X,Y) = (X - O(Y))

(Xk-1 + R1(Y)Xk-2 +

... + Rk-1(Y))

where Rj(Y) E Q[Y] for 1 < i < k. By equating the terms independent of X in the factorisation (18), we observe that the polynomial

(Y+1)...(Y+mk)-k!

236

T.N. SHOREY

is reducible over the field of rational numbers. Now, we apply a result of Brauer and Ehrlich 151 to conclude that k!

>k-lmk - 1)! 2(

[(mk - 2)/2]!

The right hand side is an increasing function of m and the inequality is not valid for m = 4. Therefore, we derive that m = 3 which implies that k = 2. Now, by looking at the constant term of £(Y), we observe that B3 (3, 2)

- 4 = W.

This is not possible, since B3 (3, 2) is a rational number. Next, we turn to the second proof to exclude the possibility (17). We as, i,,,,, jl, sume (17). Then, there exist pairwise distinct integers ii, , jm.

such that O(Y)+1 = and b(Y)+2-(Y+ji)...(Y+j,,,.)

Thus (19)

By putting Y = -ii in (19), we have

(ji - ii) ... U. - ii) = 1 which implies that m = 2. Then, we observe from (19) that

jl + j2 = it + i2 , 3132 = i1i2 + 1. Consequently (jl - j2)2 = (

2-4

which is not possible.

This completes our sketch of the proof of (5). Now, I mention some extensions of (5). Let f (X) be a monic polynomial of positive degree with rational coefficients. For an integer m > 2, we consider the equation

(20) f(x+1). . f(x+k) = f(y+1). . f(y+mk) in integers x, y, k > 2.

237

ONA CONJECTURE CONCERNING 8.9.10=6!

If f is a power of an irreducible polynomial, Balasubramanian and Shorey [3] proved that equation (20) with

f(x+j)#0 for 1 0, y > 0, k > 2 are integers satisfying (22)

(y+(mk- 1)d),

x(x + d) ... (x + (k - 1)d)

then max (x, y, k) is bounded by a number depending only on d and m. In fact, this is an immediate consequence of the following result ([ 14, Theorems 1,21): For e > 0, there exists a number Cl depending only on m and e such that equation (22) with max (x, y, k) > C1 implies that

d>

(23)

y(1-E)/(+n+1)

log d > (

M m(m + 1)

- e)K

where K and M are positive numbers given by K2 = k log k

(24)

,

M2 = m(m - 1)/2.

We observe from (23), (24), (25) and (22) that max (x, y, k) is bounded by a number depending only on d and m. Further, for positive integers d1, d2 and m > 2, Saradha and Shorey [ 151 considered a more general equation than (22) : x(x +

(x + (k - 1)d1) =y(y +

(mk - 1)d2)

(25)

in integers x > 0, y > 0, k > 2.

It was shown in [15] that equation (25) with m = 2 implies that either max(x, y, k) is bounded by a number depending only on d1, d2 or k = 2, d1 = 2d2, x = y2 + 3d2y. On the other hand, equation (25) with m = 2 is

238

T.N. SHOREY

satisfied whenever the latter possiblities hold. If m > 2, it was proved in [ 151

that there exist numbers C2 and C3 depending only on dl, d2, m such that equation (25) implies that k < C2 and moreover max (x, y) < C3 unless (*) d1 /c 2" is a product of m distinct positive integers composed of primes not exceeding m and m > a(k) where

a(k) =

14 for 2 < k < 7 50 for k = 8 I exp(klogk - (1.25475)k - logk + 1.56577) for k > 9.

This includes a result on the case dl = d2 = d mentioned above, since (*) is never satisfied. Further, we derive that equation (25) with 3 < m < 14 or 3 < m < 2568, k > 9 implies that max (x, y, k) is bounded by a number depending only on d1, d2, m. Finally, Saradha, Shorey and Tijdeman (17] showed that condition (*) is not necessary. Consequently, we conclude that equation (25) with m > 2 implies that max (x, y, k) is bounded by a number depending only on d1, d2, m. This is a consequence of a more general result which we describe now. For distinct positive integers £ and

m with gcd(f, m) = 1 and £ < m, Saradha, Shorey and Tijdeman [ 17] proved that there exists a number C4 depending only on d1, d2, m such that if x > 0, y > 0 and k > 2 are integers satisfying

x(x + di)

.

(x + (fk - 1)dl) = y(y + d2) ... (y + (mk - 1)d2),

then max(x, y, k) < C4

unless f = 1, m = 2 which corresponds to equation (25) with m = 2 and we refer to the result already stated in this case. By applying the theory of linear forms in logarithms, it is shown in [ 17] that the preceding assertion is also valid for k = 1 provided that f E {2, 4} and (f, m) = (3, 4). Now, we turn to equation (25) with m = 1. In this case, there is no loss of generality

in assuming that x > y and gcd(x, y, dl, d2) = 1. Then Saradha, Shorey and Tijdeman [ 161 proved that equation (25) with m = 1 implies that there exists a number C5 depending only on d2 such that either

x=k+l,y=2,d1=1,d2=4 or

max(x, y, k) < C5.

We observe that

(k+ 1). (2k)=2.6...(4k-1) fork= 2,3....

239

ON A CONJEC'TJRE CONCERNING 8.9.10=6!

since the right hand side is equal to

2k(2k)!/(2.4... (2k)) = (2k)!/k!.

Therefore, the above possibilities for the case d1 = 1, d2 = 4 cannot be excluded. Further Saradha, Shorey and Tijdeman [18] showed that equation (25) with m = dl = 1 implies that y < k2d2/12 and furthermore k < d2 - 2 unless y 2(mod 4) and d2 = 21 for some integer 2 > 2. In the case m = 2, dl = 1, Saradha, Shorey and Tijdeman [ 18] proved that equation (25) implies that y(y + (2k - 1)d2) < (0.44)k4d2 and furthermore

k < d2 - 2 unless k < 35 and d2 = 2e for some integer 2 _> 2. These results are applied in [181 to determine all the solutions of equation (25) with m = d1 = 1, d2 E 12,3,5,6,7,9, 10} and m = 2, dl = 1, d2 E {5, 6}. Let a and b be positive integers. We consider (26)

a(x + 1) ... (x + k) =b(y+1)...(y+k+2) in integers x > 0,y > 0,k > 2,2 > 0.

Equation (1) is a particular case of (26), namely, a = b = 1 and k + 2 is an integral multiple of k. ErdOs [71 conjectured that there are only finitely many integers x > 0, y > 0, k > 2, 2 > 0 with k + 2 > 3 and x > y + k + 2 satisfying

(26). This is a difficult problem. The assumption k + 2 > 3 is to exclude Pell's equations and the assumption x > y + k + 2 is to guarantee that the two blocks of consecutive integers in equation (26) are non-overlapping. Mordell 191 proved that equation (26) with a = b = 1, k = 2, 2 = 1 implies

that x = 1, y = 0 and x = 13, y = 4. Mordell's result initiated much of research in this direction. Avanesov [ 1] confirmed a conjecture of Sierpinski

by proving that x = 0, y = 0; x = 3, y = 2; x = 14, y = 7; x = 54, y = 19 and x = 118, y = 33 are the only solutions of equation (26) with a = 3, b = 1, k = 2, 2 = 1. Tzanakis and de Weger [20] determined all the solutions of equation (26) with a = 1, b = 2, k = 2, 2 = 1. Boyd and Kisilevsky [4] showed that x = 1, y = 0; x = 3, y = 1 and x = 54, y = 18 are the only solutions of equation (26) with a = b = 1, k = 3, 2 = 1. Cohn 161 proved that equation (26) with a = 1, b = 2, k = 4, 2 = 0 is satisfied only if x = 4, y = 3. Further, Ponnudurai [ 10] showed that x = 2, y = 1 and x = 6, y = 4 are the only solutions of equation (26) with a = 1, b = 3, k = 4, 2 = 0. Let us consider equation (26) with 2 = 0. We re-write equation (26) with

2=0 as (27) (axk-byk)+Al(axk-1-byk-1)+ +Ak-1(ax-by)+Ak(a-b) = 0 where A1,. .. , Ak are given by (28)

F(z)=(z+1)...(z+k)=zk+Alzk-1+

+Ak.

240

T.N. SHOREY

If a = b, we observe from (27) that (29)

(xk

- yk) + Al

(xk-1 _ yk-1) +

... + Ak-1(x - y) =0

which implies that x = y, since all the summmands in (29) are of the same sign. Now, we assume that a # b. Shorey [191 showed that there exists a number C6 depending only on a and b such that equation (26) with £ = 0, x > y+k and k > C6 implies that the first summand in (27) is positive and the second summand in (27) is negative (then all the summands in (27) following the second one will be negative). This is equivalent to saying that

equation (26) with e = 0 and x > y + k implies that either k < C6 or k = [a + 1] where

a =log

(a)/ log (y

We have not been able to exclude the latter possibility. This is the case if we allow C6 to depend also on P(x) and P(y).(2) In fact, Shorey [191 showed

that equation (26) with .£ = 0 and x > y + k implies that max(x, y, k) is bounded by a number depending only on a, b, P(x) and P(y). Saradha and Shorey [11) extended these results to equation (26) where $ is not necessarily equal to zero. Now, we give a sketch of the proof that equation (26) with x > y + .£ + k

implies that max(x, y, k, e) is bounded by a number depending only on a, b, P(x) and P(y). The proof depends on Gel'fond - Baker theory of linear forms in logarithms. We assume (26). By (26) and (28), we observe that

0 < aF(x) - bF(y)yl = Uk + AlUk_1 + ... + AkUO

(30)

where

Uj=ax'-by'+e for0 0.

(32)

We write Cl, c2, ... , c12 for positive numbers depending only on a, b, P(x)

and P(y). We may assume that x > cl with cl sufficiently large. In view (2)

For an integer v > 1, we write P(v) for the greatest prime factor of v and we

put P(0) = P(1) = 1.

ONA CONJECTURE CONCERNING 8.9.10=6!

241

of the result of Shorey [ 191 mentioned above, we may suppose that f > 0 which implies that k + $ > 3. Then, it is proved in [ 11, Corollary 21 that

x - y< c2tx(log x)/k.

(33)

As in the proof of (5), we count the power of 2 on both the sides of (26) for deriving that

fgeG 9 annihilates the class group, so we can study C as a module over ZI [G] IN. If (1 denotes a primitive l-th root of unity, we have an isomorphism Z1[G]/N -2- Zl[(l] Ors

> (l

showing that A = Z1 [G] IN is a discrete valuation ring whose maximal ideal is generated by a - 1, with or a generator of G. The residue class field of A is the finite field of l elements IF1. Every finite A-module M is isomorphic to a module of the form $

11 A/A(°-1)"i i=1

with n;, E Z>1 for i = 1, 2, ... , s. Thus, we can specify the isomorphism

REDEI-MATRICES AND APPLICATIONS

247

class of the A-module M by giving the sequence of integers dimF,(M(o-1)k-1/M(°-1)k).

rk = #{i: ni > k} =

Note that r1(M) = s and that {rk(M)}k is a decreasing sequence with rk(M) = 0 for k sufficiently large. The evaluation of r1 (C) amounts to doing genus theory for the field K. More precisely, class field theory associates to C an unramified extension H of K, called the 1-class field of K, for which the Galois group Gal(H/K) corresponds to an is canonically isomorphic to C. The quotient unramified extension H1 of K that is known as the genus field of K. It is the maximal unramified extension of K that is abelian over Q, and Gal(H1/Q) is isomorphic to the elementary abelian 1-group G x C/C°-1. If x denotes a Dirichlet character generating the character group X of G that corresponds to K, we can write x as a product x = flit= 1 xi, where t is the number of primes that ramifies in the extension K/Q and xi is a character of conductor a power of some ramifying prime pi and of order 1. The conductor of xi is equal to pi if and only if pi # 1. The field H1 corresponds to the group of Dirichlet characters C/Ca-1

Xi =

ftx.

i=1

It follows that Gal(H1/K) has order 1t-1, i.e. the (a-1)-rank r1 (C) is equal to t - 1, where t is the number of ramifying primes in K/Q. The subgroup Co = C[o, -1] of G-invariant ideal classes in C is known as the subgroup of _ambiguous ideal classes. As G is cyclic and C is finite, the order of CG = H° (G, C) equals the order of H1(G, C) = C/C°-1. It is not difficult to check that Cc is generated by the t classes [pi] of the ramified primes pi of K. The order of CG is 1t-1, so there is exactly one additional relation between these classes that is independent of the obvious relations [pti]=0. The Redei-Frohlich theorem gives a description of r2(C) by combining the two descriptions of ri (C). Note that as abelian groups, we have : C/C(o-1)2

C/C4 { C/Co-1 X C

if l = 2; 1/C(°-1)

2

if l> 2,

and that the 1-rank of C/C1 is equal to the sum EI-I rk. The theorem is based on the observation that r2 (C) can be obtained from an explicit description of the natural map

0 : C[v - 1]

C/Ca-1.

P. STEVENHAGEN

248

This is a homomorphism between elementary abelian 1-groups, so it can be viewed as a linear map between vector spaces over IF1. With this terminology, the (v - 1)2-rank of C is nothing but the Fl-dimension of the kernel of 0. This dimension can be given in terms of the rank of a certain matrix over F1, called the Redei matrix of K, as follows. 1. THEOREM (Redei-Friihlich). - Let K/Q be a cyclic extension of prime

degree l with group (v) and conductor f , and let X : (Z/ f Z) * -> IF1 be a generator of its character group, with values taken in the additive group IF1. Let pl, p2, ... , pt be the primefactors off , andX = Ei-1 Xi the corresponding decomposition of X. Then the 1-primary part C of the narrow class group of K has (Q - 1)2-rank r2 (C) = t - 1 - rankF, R, where the entries ai3 E IF1 of the Redei matrix R = (ai,j )i. j=1 are defined by : ai3 = Xi(pj)

if i

j;

t

Y. aid = 0. i=1

Proof : as C[v - 1] is generated by the classes of the ramified primes pi, we have a natural surjection p : Ft --f C[ai - 1] that maps the j-th basis vector ej to the class of p;. The group C/C°-1 is canonically isomorphic to the subgroup Gal(Hi/K) of Gal(Hi/Q) under the Artin map. We know H1 explicitly from genus theory : it is the compositum of the cyclic fields Q(Xi) of conductor a power of pi corresponding to the characters Xi. Each character Xi furnishes an isomorphism Gal(Q(Xi)/Q) -' F1, and they can be combined into an isomorphism EB 1Xi : Gal(Hi/Q) - lFi. The Redei map R : Fl -> Ff is defined as the composed map R : Fit

P . C[a - 1] - C/Ca-1 => Gal(Hi/K) C Gal(H1/Q) _

E)xi ------

Fit

of vector spaces over Fl. As the kernel of p is of dimension 1, one has (2)

r2 (C) = dime [ker 0] = dimF, [ker R] - 1 = t - 1 - rankF, R,

as desired. The image of a basis vector ej is the Artin symbol of p; in Gal(Hi/K). If i j, the restriction of this symbol to Gal(Q(Xi)/Q) is the Artin symbol of pj, and this is mapped to ai3 = Xi (pj) by Xi. For the diagonal entry aii the Artin symbol and Xi (pi) are not defined, but we can use the fact

that ®Xi maps Gal(Hi/K) to the hyperplane {(ai)i E IF' : Ei_1 ai = 0}. The desired identity F_i_1 ai3 = 0 follows immediately.

REDEI-MATRICES AND APPLICATIONS

249

The Redei matrix R is by definition a singular (t x t) -matrix since the sum of its rows is zero. It is said to have maximal rank if the rank equals t - 1.

Obviously, the rank is maximal if and only if r2 (C) = 0. We will meet this condition in the next section when investigating the solvability of the negative Pell equation.

The field H2 corresponding to the quotient C/C(°-1)2 is the central 1-class field of K, i.e. the largest unramified extension E of H1 that is normal over Q and for which the group extension

0 -p Gal(E/Hl) ---+ Gal(E/Q) -) Gal(Hi/Q) -40 is a central extension. In Frdhlich's terminology [[711, the central class field

H2 is a field of class two : its Galois group 11 = Gal(H2/Q) is not in general abelian but its lower central series has length at most two. This is equivalent to saying that the commutator subgroup [S2, S2] is contained in the center of S2, or that [ci, [SZ, Ii]] = 0. The Redei-FrOhlich method enables us to obtain Gal(H2/K) from very simple rational data. More precisely, we

can determine ri = ri(C) for i = 1, 2 in terms of the prime factors of the discriminant, and this leads to (A/A°-i)rl-r2 X (A/A(U-1)2).2 Gal(H2/K) =A f (Z/2Z)rl-r2 x (Z/4Z)r2 if l = 2; if 1 > 2. l (Z/JZ)rl+r2

The first isomorphism is an isomorphism of modules over the ring A = Z1 [G] IN, the second is an isomorphism of abelian groups.

3. - Applications As a first application, we will obtain divisibility results for the real cyclotomic class numbers hn of the type discussed in the introduction. Recall that h,+ is the class number of the maximal real subfield Fn = Q(Sn + (n-1) of the cyclotomic field of conductor n.

3. LEMMA. - Suppose that l > 2 and that the l -class group C of the field K in theorem 1 has (v - 1)2-rank r2(C) = r. Then jr divides h+, with n the conductor of K.

Proof : as K is real of conductor n, it is contained in Fn. The genus field H1 of K is equal to H n F, so H2Fn/Fn is an unramified abelian extension of Fn of degree [H2 : H1] = jr. This degree divides hn by class field theory, so we are done. This lemma provides us with an easy method of constructing infinitely many n for which hn is divisible by an arbitrarily high power of a prime number

250

P. STEVENHAGEN

1. One simply takes those n for which Fn contains a subfield K of degree l over Q for which the Redei-matrix is of rank much smaller than t - 1. By taking it equal to the zero matrix, the following result is obtained. 4. THEOREM. - Let n be divisible by t distinct primes congruent to 1 mod l

such that each of these primes is an l-th power modulo all others. Then divides h, .

It-1

For fixed t, there are infinitely many pairwise coprime n satisfying the hypothesis of the theorem. This follows easily from Dirichlet's theorem on primes in arithmetic progressions. If t -1 primes congruent to 1 mod l have

been chosen such that each is an l-th power modulo the others, the t-th prime that makes the hypothesis of the theorem hold true can be chosen 1)12(t-1)]-1 This follows from an infinite collection of Dirichlet density [(1from the Gebotarev density theorem, as the condition on the t-th prime is that for each of the t -1 previous primes p, it splits completely in the field Ep that is obtained by adjoining an l-th root of unity (l and to the subfield of degree l in the p-th cyclotomic field. The fields Ep are all of degree (l -1)12 over Q, and they are linearly disjoint over Q((l). As a very special case, we obtain the claim made in the introduction that h132 is divisible by 3 for a set of primes p of Dirichlet density 1/18. For odd 1, results similar to those in the preceding theorem have been proved by Cornell and Rosen [[3], 19841 using cohomological methods that

go back to Furuta [[9]]. For t = 2 and t = 3 their results are identical to those following from lemma 3, for large t their method is better. Neither method gives any result for the prime conductor case t = 1. For l = 2, the lemma and the arguments given above have to be adapted for several reasons. First of all one has to take care of the ramification of real primes. This leads one to consider only those quadratic fields K that have real genus fields, i.e. real quadratic fields for which all odd prime divisors of the discriminant are congruent to 1 modulo 4. One only obtains a divisibility result 2r-1 h ,which is weaker than lemma 3. In order to find 2T hn one needs to show that H2 is real. This can sometimes be done using a method of Scholz [[25]]. Secondly, one has to adapt the density computation given above as the fields Ey have smaller degree, a statement equivalent to the quadratic reciprocity law. Rather than working out all details here, we give a characteristic example. It is stronger than the cohomological result in [[3]].

5. THEOREM. - Let p and q be primes congruent to 1 modulo 4 that are mutual quadratic residues, and suppose that the fourth power residue symbols (q) 4 and (P) are equal Then hr9 is even. 4 It is not difficult to see that we obtain the claim in the introduction for q = 13. For n = pq = 5 29 = 145 it follows that hi45 is even. This is one

REDEI-MATRICES AND APPLICATIONS

251

of the two values n < 200 with n not a prime power for which h+ > 1. In fact, one can use Odlyzko's discriminant minorations to show [13111 that h145 = 2.

There are no results of a similar algebraic nature in the prime conductor case t = 1, and we cannot produce infinite families of primes p for which hP is even. See [12811 for a more complete discussion.

A second application of the technique of Redei matrices arises in the study of the solvability in integers of the negative Pell equation x2 - Dy2 = -1, where D > 1 is a squarefree integer. With ED a fundamental unit in Q(om) and N the norm to Q one has x2

- Dye = -1 is solvable in integers e= NED = -1.

Indeed, if the equation is solvable there are units of norm -1, so the fundamental unit cannot have norm + 1. Conversely, if NED = -1 it may be that ED is not in Z[/], but as its cube ED always is we still get an integral solution to the equation. As it is more natural to work with discriminants

than radicands, we will further take D to be a quadratic discriminant and say that the negative Pell equation is solvable for D if the equation x2 - Dy2 = -4 has integral solutions. If the equation is solvable for D, then D is positive and -1 is a quadratic residue modulo every prime divisor

of D, so D must be in the set D of real quadratic discriminants that are not divisible by any prime congruent to 3 mod 4. A question that has been studied by many people but that is still completely open is the following. 6. PROBLEM. - Let D(-1) C D be the set of real quadratic discriminants for which the negative Pell equation is solvable. Decide whether the limit lim X-

#{D E D(-1) : D < X} 00

#{DED:D 1. The condition that the genus field H1 is real is equivalent to the requirement that D is in D, since H1 is obtained from K by adjoining a square root of (-1)(P-X)/2 for each odd prime divisor p of D. If H = H1 this condition is also sufficient for solvability of the negative Pell equation. 8. LEMMA. - The negative Pell equation is solvable for D E D if the Redei matrix of Q (v/D) has maximal rank

Proof : the condition implies an equality H1 = H2, so H = H1 is real and we are done by the previous lemma. If D E D has t distinct prime divisors, the corresponding Redei matrix R is by the quadratic reciprocity law a symmetric (t x t)-matrix over F2 whose

rows and columns add up to zero. Let R' be the (t - 1) x (t - 1)-minor obtained by leaving out the last row and column from R. If D ranges over the subset Dt of D consisting of those discriminants that have exactly t distinct prime divisors, it is intuitively clear that the corresponding Redei minor R'D behaves like a random symmetric (t - 1) x (t - 1)-matrix over ]F2, i.e. that 1imX-00

#{DEDt:D 1 be an integer and q a prime power. Then there are An(q) = q(n21)

fl

11Dt and D(-1) = Ut>1Dt(-1) make it very plausible that the lower density of D(-1) in V is not smaller than a.. However, I do not know how to prove this. The problem is that each Vt is a subset of zero density in V, and it seems non-trivial to prove a density result for D from a density result for each of the subsets Dt. The preceding argument can be further refined in order to obtain a still higher value of the lower density of Dt (-1) in Dt. In particular, we will push the limit value for t -* oo over the value 1/2 that has been suggested as a possible value [[ 17]]. However, we need to pass to the next higher level, i.e. the field H3, in order to do this.

4. - Higher levels In principle, the Redei-Frohlich method for determining r2 (C) can be extended to determine inductively all values rk (C). Having defined the first Redei map

R = R1 : Ft --> C[v - 1] -- C/Co-1 -4 Fit, one can repeat the procedure and consider the higher Redei maps Rk :

kerRk_1 -> C[Q -1] n

C(o_1)k

Just as in the case k = 1, we obtain the (o -

1

can

C(0-1)k-1

/C(o-1)k.

1)k+1-rank from this map by

rk+1(C) = dimF, [ker Rk] - 1 = rk (C) - rankF, Rk,

which is the analogue of (2). Despite the close analogy, a serious complication arises for these higher levels. For k = 1, we were able to embed c(a-1)k-1 /C(a-1)k in a canonical way in a vector space of dimension t over ]Fl. This was due to the fact that we could describe the genus field H1 very explicitly in terms of Dirichlet characters. The fields Hk for k > 2 are no longer abelian over Q, and no general method is known to describe them explicitly. This is a serious drawback that accounts for the fact that there is no generalisation of theorem 1 to higher levels that is of a comparable simplicity. For the same reason, we do not have general density results for these levels that resemble those in the preceding section.

Only in the special case where k = 1 = 2, there is a more explicit version of the theory that goes back to Redei [[22]] and was further developed by Frohlich [[7]]. We can formulate it in modern terms as follows.

Let D be a quadratic discriminant, and D = jlt=i dg its factorization into prime power discriminants. The set V of discriminantal divisors of D is defined as the set of divisors d of D of the form d = llt=1 d7 with ei E 10, 1}. This is in a natural way a vector space of dimension t over F2 with a canonical basis consisting of the divisors d1. The natural surjection

REDEI-MATRICES AND APPLICATIONS

255

V -> C[2] maps dti to the class of the ramified prime ai of K that divides dti. As the genus field H1 of K = Q(v/D) is generated over Q by the square roots dti, Kummer theory tells us that the Galois group Gal(Hi/Q) can be seen as the dual space V* = Hom(V, IF2) of V. The kernel of the Redei map R1 : V -- V * consists of divisors d E V for which the associated Artin symbol oa E Gal(H/K) is the identity on H1. The kernel of the dual map Ri : V = V** -> V* consists of those d E V for which Vd- is left invariant by the Artin symbols of all ideals that have order 2 in the class group. Note that D itself is always in this kernel. A decomposition D = Dl D2 with D1 E ker Ri is called a decomposition of the second kind. These decompositions are characterized by the fact D1 is a square modulo all prime divisors of D2 and vice versa. The prime 2

needs special attention here. Given a decomposition D = Dl D2 that is of the second kind, Redei explicitly constructs a quadratic extension of Q( Dl D2) that is cyclic of degree 4 and unramified over K. This is possible since the equation x2 - D1y2 - D2 Z2 = 0 has non-zero rational solutions by Legendre's theorem and the assumption on the decomposition. For a primitive integral solution (x, y, z) with well-chosen 2-adic behavior, the extension E that is generated over K by a square root yD, of x + y D has the desired properties. The extension E/K depends on the choice of the solution, but the quadratic extension EH1 = H1(ryD1) of H1 does not. Every element v E Gal(H2/Hi) is determined by its action on the elements 'D, for D1 E kerRi, so we can view this Galois group as a subspace of Hom(kerRi,IF2) _ (kerRi)*. With these identifications, we can describe the second Redei map : R2 :

kerRi -> C[2] fl C2 -> C2/C4

Gal(H2/Hi) C (kerRi)*

explicitly as an F2-linear map between vector spaces of dimension 1+r2(C)The 8-rank r3(C) of the narrow class group of K is given by the formula r3 (C) = r2 (C) - rankF, R2,

which is non-negative as R2 is always singular. As soon as one chooses a basis for ker R1 and for the space ker R*1

of decompositions of the second kind, R2 is given by a matrix whose entries describe the action of the Artin symbols va coming from d E ker R1

on explicit elements 7d' with d' E ker Ri . Note the equality R1 = Ri in case D is in the set D of discriminants that are of interest for the negative Pell equation. The entries of R2 are quadratic symbols of quadratic

irrationals and can be computed rather easily. Redet's paper [[2211 has numerous identities that express these `new number theoretic symbols', as he calls them, in rational terms, and FrOhlich 11811 does the same in a

P. STEVENHAGEN

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more systematic way. However, these expressions are usually given in terms of the chosen solution of x2 - D1y2 - D2 Z2 = 0, and this makes it difficult to obtain density results in terms of the prime divisors of D. Special cases have been dealt with by Morton [[141-[1611, and density statements for the

behavior of C/C8 have been proved by the author [[26]] in the case that t - 1 prime divisors of the discriminant are fixed and the last one varies. For t = 2 this yields results that had been known for some time. In the previous section, we showed that the negative Pell equation is solvable

for all D in a subset of density at of Dt since these D have r2 (C) = 0. Following an idea that goes back to Redei [[21]] and Scholz [[25]], we can use the 8-rank theory to enlarge this set even further by looking at those D that have r2 (C) = 1. We will indicate briefly how this is done.

The density Qt of the set of D E Dt having r2 (C) = 1 follows easily from proposition 9. One has ,Qt = at if t is even and pt = (1 - 21-t)at if t is odd. Note that limt_,00)3t = limt_,00 at = a... For D as above, there is exactly one non-trivial decomposition D = D1D2, and one can show that the higher Redei matrix R2 equals R2

-

D2 4 \ (D2) 4

(D I /4

which has to be interpreted in the obvious way as a matrix over ]F2. As the biquadratic residue symbols have value f1, there are 4 possible values for this matrix, and they each occur for a set of D that has density .114,Qt in Dt. If (D )4 and (D )4 are both equal to 1, the matrix R2 is the zero matrix and H2 is strictly smaller than the 2-Hilbert class field H. In all other cases,

its rank is one and H = H2. We can determine whether H2 is real by a generalization of the argument used in proving theorem 5. One has :

H2isreal

\Da)4=1. (:;-)

It follows that H = H2 is totally complex and the Pell equation is not solvable if the biquadratic residue symbols are not equal, which happens for a collection of discriminants of density,Qt/2 in Dt. If both symbols equal

-1, the Pell equation is solvable, and this happens for a set of density Qt/4. Taking together the two collections of D for which the Pell equation is solvable, we conclude that for each fixed t, the set Dt (-1) has lower density at + 1)3t and upper density 1 - 2 Qt inside V. For increasing t these values rapidly converge to : s 4

a,, = .52428.. .

1 - 2 a = .79029.. .

REDEI-MATRICES AND APPLICATIONS

257

It remains a challenging problem to deduce any non-trivial density result for D(-1) in D. Hardly anything is known about the distribution of the ranks rk(C) when

k > 4. Some numerical data are available in the quadratic case 1121, [27]], mainly for cyclic C. The best known example is probably that of the quadratic field Q(v/=p), with p a prime congruent to 1 mod 4. In this case

the discriminant D = -4p has two prime divisors and C is a non-trivial cyclic 2-group. It follows from theorem 1 that the order of C is divisible by 4 exactly when p - mod8, and the 8-rank results quoted above imply that the order is divisible by 8 if and only if p splits completely in Q((s, V/-1-+-i),

which is a non-abelian field of degree 8 over Q. All numerical evidence suggests strongly that the order of C is divisible by 16 for a set of primes of density 1/16, but the existing techniques do not even suffice to show that this happens infinitely often. The question is closely related to the 2-adic behavior of the fundamental unit e, in the field Q(J), see [[27]].

Note added in proof It is now conjectured that the limit value in problem 6 exists and equals 1 - a,, = .5805775582..., with a,,, as in section 3, see [[29]]. The heuristics

can be extended to the case of quadratic orders [[30]]. They have been confirmed by extensive computer calculations [[1]].

Manuscrit recu le 7 septembre 1994

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References [1] W. BosMA, P. STEVENHAGEN. - Density computations for real quadratic

units, preprint (1994). [2] H. COHN, J.C. LAGARIAS. - On the existence of fields governing the 2-

invariants of the class group of Q(Vl p) as p varies, Math. Comp. 41, 711-730 (1983). [3] G. CORNELL, M.I. RosEN. - The $-rank of the real class group of cyclotomic

fields, Compositio Math. 53, 133-141 (1984). [5] J. E. CREMONA, R.W.K. ODONI. - Some density results for negative Pell

equations; an application of graph theory, J. London Math. Soc. (2) 39, 16-28 (1989). [6] A. FROHLICH. - The generalization of a theorem of L. Redei's, Quart. J. Math. Oxford (2) 5, 130-140 (1954). [7] A. FROHLICH. - On fields of class two, Proc. Lond. Math. Soc. (3) 4, 235-256 (1954). [8] A. FROHLICH. - A prime decomposition symbol for certain non Abelian numberfields, Acta Sci. Math. 21, 229-246 (1960). [9] Y. FURUTA. - On class field towers and the rank of ideal class groups, Nagoya Math. J. 48, 147-157 (1972).

[10] G. GRAs. - Sur les 1-classes d'ideaux dans les extensions cycliques relatives de degre premier 1, Ann. Inst. Fourier, Grenoble 23,3, 1-48 (1973).

[111 J. HURRELBRINK. - On the norm of the fundamental unit, preprint, Louisiana State University (1990). [12] E. INABA. - Uber die Struktur der i-Klassengruppe zyklischer Zahlkorper vom Primzahigrad 1, J. Fac. Sci. Imp. Univ. Tokyo, section I, vol. W 2, 61-115 (1940). [13] J. MACWILLIAMS. - Orthogonal matrices over finite fields, Amer. Math. Monthly 76, 152-164 (1969). [14] P. MORTON. - Density results for the 2-classgroups of imaginary quadratic fields, J. reine angew. Math. 332, 156-187 (1982).

[15] P. MORTON. - Density results for the 2-classgroups and fundamental units of real quadratic fields, Studia Scientiarum Math. Hungarica 17, 21-43 (1982). [16] P. MORTON. - The quadratic number fields with cyclic 2-class groups, Pac. J. Math. 108, 165-175 (1983).

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[17] R. V. PERLis. - On the density of fields with N(e) _ -1, preprint, Louisiana State University (1990). [18] L. REDEI, H. REICHARDT. - Die Anzahl der durch 4 teilbaren Invarianten

der Klassengruppe eines beliebigen quadratischen Zahlkorpers, J. reine angew. Math. 170, 69-74 (1934). 1191 L. REDE!. - Arithmetischer Beweis des Satzes fiber die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengn.ippe im quadratischen Zahlkorper, J. refine angew. Math. 171, 55-60 (1935).

[20] L. REDEI. - Uber die Grundeinheit and die durch 8 teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkorper, J. reine angew. Math. 171, 131-148 (1935). [211 L. REDEI. - Uber einige Mittelwertfragen im quadratischen Z iilkorper, J. reine angew. Math. 174, 131-148 (1936). [22] L. REDE!. - Ein neues zahlentheoretisches Symbol mitAnwendungen auf die Theorie der quadratischen Zahikorper, J. reine angew. Math. 180, 143 (1939).

[231 L. REDEI. - Bedingtes Artinsches Symbol mit Anwendungen in der Klassenkorpertheorie, Acta Math. Acad. Sci. Hung. 4, 1-29 (1953). [241 L. REDEI. - Die 2-Ringklassengruppe des quadratischen Zahlkorpers and die Theorie derPellschen Gleichung, Acta Math. Acad. Sci. Hung. 4, 3187 (1953). [251 A. SCHOLZ. - Uber die Losbarkeit der Gleichung t2 - Due = -4, Math. Zeitschrift 39 [261 P. STEVENHAGEN. - Class groups and governing fields, Publ. Math. Fac.

Sci. Besancon, annee 1989/90, 1-94 (1990). [27] P. STEVENHAGEN. - On the 2-power divisibility of certain quadratic class

numbers, J. Number Theory 43 (1), 1-19 (1993). 1281 P. STEVENHAGEN. - Class number parity for the p-th cyclotomic field, Math. Comp. 63 no. 208 (to appear, 1994). [291 P. STEVENHAGEN. - The number of real quadratic fields having units of negative norm, Exp. Math. 2 (2), 121-136 (1993). [301 P. STEVENHAGEN. - Frobenius distributions for real quadratic orders, J. Theorie des Nombres Bordeaux (to appear, 1995). [31] F.VAN DER LINDEN. - Class number computations of real abelian number fields, Math. Comp. 39, 693-707 (1982). Peter Stevenhagen Faculteit Wiskunde en Informatica Plantage Muidergracht 24 1018 TV Amsterdam, Netherlands e-mail : psh@fwi . uva. n1

Number Theory Paris 1992-93

Decomposition of the integers as a direct sum of two subsets R. Tijdeman

1. - Introduction Two subsets A and B of a set C induce a decomposition of C if every

element of C has a unique representation a + b with a E A, b E B. Notation : C = A T B. We call A and B complementing C-pairs. A first study of such pairs arose in the forties from Hajos' proof of Minkowski's conjecture on systems of linear inequalities. Hajos reduced this conjecture to an equivalent statement on decompositions of finite abelian groups, which he was able to prove. A survey of the work on decompositions of finite abelian groups is given in Section 2. The question of characterising all complementing Z-pairs seems first to have been stated by de Bruijn in 1950. De Bruijn came to the problem while he studied bases for the integers. Let A be a finite set of integers including 0. A set of integers {bl, b2,. ..I is called an A-base whenever any integer x can be expressed uniquely in the form 00

x=

00

Eibi

i=1

(Ei E A,

IEiI < oo). i=1

if it can be rearranged in the form h2d3.... } where h denotes the cardinality of A and dl, d2, d3, .. .

An A-base is called simple {dl, hd2,

are integers. De Bruijn [21 considered the special case where the elements of A have no common factor and where h is a prime. He conjectured that under these assumptions A ® B = Z implies that B is the set of multiples of h. He remarked that a proof of his conjecture would imply that every A-base is simple. For later work on A-bases we refer to de Bruijn [61, Long and Woo [161, Swenson and Long [281. In 1974, Swenson 1271 showed that there is no effective characterisation of all complementing i-pairs. More precisely, he showed that any two finite

sets of integers A, B with the property that all sums a + b (a E A, b E B)

R. TIJDEMAN

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are distinct, can be extended to two infinite complementing Z-pairs. For a similar construction, see Post [20]. In contrast, there is a particularly nice characterisation of all complementing Z>o-pairs. The result, which was implicit in the work of de Bruijn [5], was rediscovered by Vaidya [30]. It is obvious that A fl B = {0} and 1 E A U B. Suppose 1 E A. Then A and B are infinite complementing Z>opairs if and only if there exists an infinite sequence of integers {mi}i>1 with mi > 2 for all i, such that A and B are the sets of all finite sums of the form 00

CK)

a = E x2iM2i, b = E x2i+111'I2i+1 i-o

i-o

i

where 0 1.If j=1

A or B is a finite set, a similar characterisation holds with the change that the sequence {mi} will be of finite length r and the only restriction on xr is that it be nonnegative. C.T. Long [ 151 gave a corresponding characterisation of all complementing C-pairs in case C = {0, 1, ... , n - 1}. He also showed

that in this case the number C(n) of complementing C-pairs is the same as the number of ordered nontrivial factorisations of n. The number C(n) is determined by

2C(n) = E C(d)

(n E Z>1).

din

Hansen [ 14] and Niven [ 181 generalised these results to a characterisation of the complementing pairs of the set Z>o x Z>o. Long [ 151 made the interesting

observation that it follows from the above characterisation that if A and B are infinite sets such that A ®B = Z>o then A ®(-B) = Z (see also Brown [1]). In particular, we can take for A the set of finite sums of odd powers of 2 and for B the set of finite sums of even powers of 2. Eigen and Hajian [31] showed that if A and B are infinite sets such that A ®B = Z>o, then there exists a continuum number of sets b such that A ® B = Z.

In Section 3 we formulate a conjecture which, if true, provides an inductive characterisation of all complementing sets A, B for which the cardinality of A is fixed integer n. It was already observed by Hajos [12] and de Bruijn [5] p. 240 that B is periodic if A is finite. In this way the problem is reduced to a finite problem which can be stated in terms of finite cyclic groups. If n is a prime number, then our conjecture coincides with de Bruijn's one stated above. This conjecture was proved by Sands [24] in 1957. We shall show how a proof of our conjecture can be derived from Sands' results if n is a prime power. The general case remains open.

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We shall further show by a combinatorial argument that if m is coprime to

n and A ®B = Z, then mA ®B = Z (we define mA = {mala E A}). On using this result we give an alternative proof of de Bruijn's conjecture.

The problem of characterising all sets B such that A $ B = Z in the special case where A consists of the finite sums of odd powers of 2 was posed to me by Yu. Ito. He was interested in the problem because of his joint research with S. Eigen and A. Hajian on exhaustive weakly wandering sequences for ergodic measure preserving transformations [71, [91, [8]. Such a characterisation will be presented in Section 4.

2. - Decomposition of finite abelian groups About one century ago Minkowski [ 171 proved the following fundamental

result on the geometry of numbers : Let h1i ... , n be homogeneous linear forms in the variables x1, ... , xn with real coefficients and determinant 1. Then there exist integers x1,. .. , xn, not all zero, such that Ill

(1)

1,...,IU

1.

Since L;1i ... , n may have integer coefficients, the equality signs cannot be deleted. Minkowski conjectured that (1) can be replaced by IS1I < 1,...,ISnI < 1

(2)

unless at least one of the linear forms has integer coefficients. Minkowski proved the statement for n < 3. Several mathematicians worked on it and in 1940 it was known to be true for n < 9. In 1941 Hajos [11] established Minkowski's conjecture in the affirmative. His proof consists of three parts : (1) reduction to some equivalent geometric statement on k-multiple lattice tiling of the unit cube, (ii) further reduction to the equivalent group theoretic statement given below.

(iii) proof of this group theoretic statement.

Hajos' theorem is very fundamental and has various aspects. Fary [101 reformulated it as a result on the structure of commutative compact topological groups. Now we state Hajos' result in terms of group theory. Let G be a finite abelian group with unit element 1. A subset of G is called a simplex if it is of the form {1, a, a2, ... , ae-1 } where a E G has order > e. Notation [a]e, or briefly [a]. It is clear that [a] is a subgroup of G if and only if a has order e. We say that G is the free product of the sets Al, ... , A. if every element

of G has a unique representation a1 Hajos' theorem reads as follows.

an with aj E A; for j = 1, .

.

.

, n.

R. TIJDEMAN

264

If G is the free product of n simplices, then at least one of the simplices is

a subgroup of G. Hajos' proof has been simplified by Redei [22] and Szele [29]. Szele [29] p. 57 conjectured that Hajos' theorem would hold true for any decomposition of the finite abelian group G. A simple example (cf. [ 131 p. 185) shows

that this is false. Let G be the cyclic group defined by a8 = 1 and let A = {1, a2}, B = {1, a, a4, a5}. Then none of A and B are subgroups of G whereas G is the free product of A and B. Note, however, that a4B = B. A subset A of G is said to be periodic whenever there exists an element g E G, g 1, such that gA = A. De Bruijn [2] conjectured that if G is a finite abelian group of order > 1 and G is the free product of the sets A and B, then A or B is periodic. He observed that the assertion is not true if G is the infinite cyclic group generated by g. Szele (cf. [ 131 p. 185) made

the same observation. He took for A the product of the subsets {1,g2}, { 1, g32}.... and for B the product of the subsets 11, g-1 }, 11, g-4 }, {1, g-16}, .... Here G is the free product of A and B and none of A and B is periodic. { 1, g8 } ,

Some years earlier, however, Redei [21] had published two examples of Hajos which show that de Bruijn's conjecture is false. The simplest example refers to the abelian group generated by the elements a, b, c of orders 4, 4, 2 respectively. This group is the free product of the nonperiodic sets {1, a}, {1, b} and {1, a2, ab2, a3b2, c, a2bc, a2b3c, b2c}.

Later, Hajos [ 131 showed that any finite cyclic group the order of which is the product of three pairwise relatively prime numbers > 1, two of which are composite, can be represented as the free product of two nonperiodic subsets. He gave an explicit example of order 180 = 9 x 4 x 5, which is the smallest number satisfying the conditions. Let us follow de Bruijn in calling a group good if any factorisation of G as a free product of A and B implies that A or B is periodic and otherwise bad. De Bruijn [3] extended Hajos' result by showing that if n = dld2d3 with (d1, d2) = 1, d3 > 1 and both d1 and d2 are composite numbers, then the cyclic group of order n is bad. The smallest order of this type is 72. De Bruijn [4] gave the explicit example (g72 = 1) g18 926 g34

90 98, 916 B : g12, 917 918 921 924 941 945 948 954 960 965, 969 A :

,

On the other hand, cyclic groups of the following orders have been proved to be good (p, q, r, s are distinct primes) : p' ' (A > 1) (Hajos [12]), pq, pqr (Redei [231), p"q (A > 1) (De Bruijn [41), p2g2, p2qr and pqrs (Sands [241).

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

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This covers all cyclic groups. Already in 1947 Redei [211 had shown that the non-cyclic group of order p2 is good. The problem was completely solved by Sands [25, 261 who determined all good finite abelian groups. Sands [241

further proved that if the finite cyclic group G is the free product of the subsets A and B and the cardinality of A is a prime power, then either A or B is periodic. This had been conjectured by de Bruijn ([3] p. 371) for the case that the number of elements of A is prime. Hajos 1 121 proposed the question whether every decomposition of a finite abelian group G is quasiperiodic. A factorisation of G as free product of A and B is called quasiperiodic if either A or B, B say, can be split into a number of parts B1, B2, ... , B,,,, (m > 1) such that ABi = giAB1 (i = 1,. .. , m) where the elements gl,... , gn form a subgroup of G. De Bruijn's example is quasiperiodic as we can take B1 = {g12 917 918 924 941 965} and gi = 1, 92 = g36. Hajos' example is quasiperiodic as we can take A = {1, a, b, ab}, B1 = {1, a2, ab2, a3b2}, B2 = {c, a2bc, a2b3c, b2c} and gi = 1, g2 = c.

De Bruijn [4] obtained some partial result on Hajos' question.

3. - A is finite Suppose A E B = Z where A is finite. Let A = {ao, a1,. .. , an_i }. Since,

for any integer x, we have (A - x) E B = Z, we may assume without loss of generality that 0 = ao < al < ... < an_i. If x = a + b with a E A, b E B, put (x)A = a, (x)B = b. The following result of Hajos [121 and de Bruijn 121 p. 240 reduces the problem of characterising all complementing Z-pairs to a problem on finite sets which can be stated in terms of finite cyclic groups. LEMMA 1. - The sequence {(x)A}XEZ is periodic. If the period length is L

then n divides L and B is periodic mod L.

Proof : put M = an_i. Consider the nM + 1 vectors ((i)A, (i + 1)A, ... , (i + M - 1)A) for i = 0,1, ... nM.

By the box principle at least two vectors are equal, for i = s and i = t with

s < t, say. Hence (x)A = (x+t-S)A forx = s,s+1,...,s+M- 1. Suppose k is the smallest integer with k > s + M and (k)A

(k + t - $)A

-

If (k)A # 0, then put k = a + b with a E A, b E B. We infer that (b)A = 0 and s < k - M < b < k. Hence (b + t - s)A = (b)A = 0 which implies b + t - s E B. Since k + t - s = a+ (b + t - s), we obtain (k + t - S)A = a = (k)A, a contradiction. If (k + t - s)A 0, a similar argument yields a contradiction. Thus (x)A = (x + t - s)A for all x > s. By

R. TIJDEMAN

266

symmetry we also have (x)A = (x+t-s)A for all x < s. Let the period length of {(x)A}xEZ be L. Since Z = U o 'jai + B}, all ai have the same density in the sequence {(x)A}xEZ. Therefore, they occur with the same frequency in one period. This implies n1 L. Since the elements of B are precisely the integers x with (x)A = 0, we have that B is periodic mod L. 0 By simple transformations each complementing Z-pair A, B with A

finite can be reduced to the standard situation that A is represented by < an_1 < L with gcd(ao, al, ... , an-1) = 1 and B is ao = 0 < a1 < a2 < < bn,._ 1 < L. Here L = nm. represented by U'- 1(bi +ZL) with 0 < b1 < Namely, 0 = a + b for some a E A, b E B. By taking A - a in place of A and B + a in place of B we have 0 E A n B. If gcd(ao, a1, ... , an_1) = d > 1, then 7G=

(3)

A

®Bjd

j forj=0,1,...,d-1

where Bj = {b E Bib - j(modd)}. The elements of A/d are coprime and we have obtained d complementing Z-pairs with coprime a's. It is obvious that B can be represented as indicated. Without any trouble we can add or subtract multiples of L from the elements of A to obtain the required structure. The problem is now reduced to the decomposition problem for the cyclic group of residue classes mod L (where we only know that L is a multiple of the cardinality of A).

It will be clear from the previous sections that a characterisation of all complementing i-pairs is not a simple matter, even if we assume one of the subsets to be finite. However, in the latter case, a kind of inductive characterisation would be possible, if we could prove the following

statement. CONJECTURE. - If A®B = Z, 0 E An B, gcdaEA a = 1 and A has exactly

n elements, then there exists a prime factor p of n such that all elements of B are divisible by p.

Suppose the statement is true. Then the elements of A are equally distributed among the residue classes mod p. We can make a splitting in

p complementing i-pairs as indicated in (3), with d = p and A and B interchanged. So the problem is reduced to the decomposition problem for

the cyclic group of residue classes mod(L/p) and the procedure can be repeated. The following examples show that p is not determined by L and n.

L=12, n=6, A=10,1,4,5,8,91, B = {0, 2(mod 12)}, p=2; L=12, n=6, A=10,1,2,6,7,81, B = {0, 3(mod l2)}, p=3.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

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If the conjecture is true, then every such decomposition is quasiperiodic

in accordance with Hajbs' conjecture stated at the end of the previous section. For we can split A into residue classes mod p, Ao, &... , A,-1, say, and Ai +B = pZ+i for i = 0, 1, ... , p-1. If the number n of elements of A is a prime, then the conjecture implies that A represents a complete residue system mod p and every element of B is divisible by p. This is precisely the conjecture of de Bruijn stated in the introduction. A proof of this conjecture can be obtained by combining results of de Bruijn and Sands. De Bruijn ([2)), p. 241) provided an argument which implies that his conjecture is true if the following statement is true : if the finite cyclic group G is the

free product of the subsets A and B and the cardinality of A is prime, then either A or B is periodic. As remarked in the previous section, the latter statement was proved by Sands. I shall present a completely different proof of de Bruijn's conjecture (Theorem 2). Subsequently I shall extend

de Bruijn's argument to the case where the cardinality of A is a prime power. By combining this with Sands' general result we obtain a proof of my conjecture, stated above, in case n is a prime power (Theorem 3). We start with a result without any restriction on n. THEOREM 1. - Let A ®B = Z with 0 E A fl B and cardinality n of A finite.

Then, for any integer h with gcd(h, n) = 1, we have hA ® B = Z.

We need some lemmas. Let A= {ao, a1, ...,an-, } with ao = 0. LEMMA 2. - For any integer x

{(x+ao)A,(x+al)A,...,(x+an_1)A}=A, {(x-ao)A,(x-a1)A,...,(x-an_1)A} =A. Proof : suppose (x + ai) A = (x + aj) A. Then x + ai - Q1 = x + aj - /32 for some 01, 32 E B. Hence ai + 32 = aj + ,31. Since such a representation is unique, we have i = j. The proof of the second statement is similar. 0 LEMMA 3. - Let q be a prime power with gcd(n, q) = 1. Then, for any integer x, {(x + gao)A, (x + ga1)A, ... , (x + qan-1)A} = A.

Proof : let q = pk, p prime. Put D = {(a, a, ... , a) E Agla E A}. Define ByLemma2 we have

f : A9 ->Aby(a1ia2i...,aq)'--1 n-1

U f (a1, a2, ... , aq-1, aj) = A. j=0

R. TIJDEMAN

268

Hence f (a) (a E A9) assumes each element of A exactly nq-1 times. Note that f (al, a2, ... , aq) does not change value if we permute al, a2, ... , aq. If (al, a2.... , aq) contains entry aj exactly h times (j = 0, 1, ... , n - 1), then it has precisely q!

lo! ii! .. In-I! permutations in Aq. This multinomial coefficient is divisible by p, unless all but one lo, equal zero, that is (al, a 2 ,--. , aq) E D. It follows that

f assumes on Aq\D each value of A a number of times which is divisible by p. Since p-n9-1, we infer that f assumes each value of A on D. Thus f (D) = A. 0

LEMMA 4. - Let q = -1 or a prime power with gcd(n, q) = 1. Then

qA®B=Z. Proof : we first show that all numbers {qa + b}aEA, bEB are distinct.

Suppose al, a2 E A, 31, /32 E B are such that qal +,31 = qa2 + 02. If q = -1, then the assertion follows from a2 + 01 = al + /j2. Otherwise qai - /j2 = qa2 - /3i. This number has a unique representation a +,3 with a c A,,3 E B. It follows that -,0 + qa1 = a + /32, -0 + qa2 = a +,31. Hence (-,3+gal)A = (-/3+ga2)A. By Lemma 3 we obtain al = a2 and therefore 13i =/32 By Lemma 1, B is periodic mod mn for some positive integer m. Hence B consists of m residue classes mod mn. Let {bo, b1, ... , bm_1 } be a set of representatives. Since by the first paragraph all numbers qal + bj (i =

0, 1,...,n- 1, j = 0,1,...,M- 1) are in distinct residue classes modmn, these mn numbers represent a complete residue system mod mn. Hence qAED B=Z. 0

Proof of Theorem i : since h can be written as the product of prime powers and factors -1, each coprime to n, we reach the conclusion by repeated application of Lemma 4.

0

COROLLARY. - Let h be an integer with gcd(h, n) = 1. Then h(aI - a2) _ 01 -,32 for some al, a2 E A, ,Ol, /j2 E B implies al = a2, 01 = Q2.

Proof : we have hal = (hal + /32)hA = (hat + /31)hA = hat.

0

Subsequently we show how a proof of de Bruijn's conjecture can be derived from Theorem 1.

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269

ThEOREM 2. - Let A®B = Z with 0 E AnB, the elements of A are coprime

and the cardinality n of A is prime. Then every element of B is divisible by n. Proof : since B is periodic mod mn for some m, we assume without loss of generality that A consists of the nonnegative integers ao = 0, a1, ... , an_1 and that bo, . . . , b,,,,_1 are integers with 0 < bo < . . . < b.,,,,_1 < mn such that B = U oi(b; + mnZ). Put Bo(z) = 1 + zb1 + zb2 + + zb^-1 and (4)

B(z) _ E zb = BO(z) (1 + zmn + z2mn + ...) =

Bo(Z) nen

bEB b>O

Note that every pole of B is an mn-th root of unity. We shall show that it is an n-th root of unity. Set Ah(z) = 1 + zhal + Zha2 + + zhan 1 . Then for every h > 0 with gcd(h, n) = 1 we have, by Theorem 1, 00

Ah(z)B(z) = > zk - Ph(z) =

(5)

1 1

k=0

z

- Ph(z)

where Ph(z) E Z[z]. Hence every pole # 1 of B is a zero of Ah. Let ( be a pole of B with (# 1. Put Sk = E o (k-; for k E Z. Since Ah(() = 0 whenever gcd(h, n) = 1, we have sh = 0 whenever gcd(h, n) = 1. By the theorem on elementary symmetric functions (formulae of Newton-Girard) we obtain, since n is prime,

IIn-1 j-o (z-(a') =zn+cn where cn is some constant. Since ao = 0, we have cn = -1. Therefore, is the complete set of n-th roots of unity. Since (ao = 1 (a1, ... , the a's are coprime, there exist integers to, ti, ... , tn_1 such that 1 = toao + tiai + + tn_lan_1. Hence, putting (i = e2nti/n ( = ((ao)to ((a1)tl ... ((an-1)tn-1 = (t for some t E Z. (an-1

Thus (is an nth root of unity. From (4) we see that every pole of B is simple and that (zmn -1) /(zn -1) divides Bo(z). This implies, by the choice of the b's, Bo(z) = (1 + zn + z2n

+... + z(,,,.-i)n)

(1 + fiZ+... + fn_izn-1)

for some coefficients fi, ... , fn-1. Since Bo has only m nonzero coefficients, = fn-1 = 0. Thus Bo(z) = (1 - zmn)/(1 Zn ) and we see that fi =

-

B(z) = (1 - zn)-1 = 1 + zn + Z2n + multiples of n.

, in other words, B consists of the 0

We use Sands' result on finite cyclic groups [241 to obtain the following generalisation of Theorem 2.

R. TIJDEMAN

270

THEOREM 3. - Let A ® B = Z With 0 E A fl B, gcdaEA a = 1 and A has exactly pt elements with p prime, t E Z>1. Then all elements of B are divisible by p.

Proof : let n = pt. By Lemma 1, B is periodic. Let L be its minimal period. If G denotes the group of residue classes mod L, then Z = A ® B furnishes a decomposition G = A* ® B* where A* and B* consist of the residue classes mod L determined by the elements of A and B. respectively. Note that L is divisible by pt.

For t = 1 the statement is true by Theorem 2. So suppose t > 1. We apply induction on t. It follows from Theorem 2 of Sands [24] that A* or B* is periodic. B* cannot be periodic because of the minimality of L, so it has to be A*. Note that the elements g with g + A* = A* form a subgroup Go of G. We shall show that Go contains the residue class Ll p (mod L). If not, Go contains L/q for some prime q # p. Then a E A* if and only if a+ vL/q E A* for all v E Z. Hence A* splits into subsets of size q. Since A* has pt elements, this is impossible. Thus A* is periodic mod L/p. Let A** and B** consist of the residue classes mod L/p determined by the elements of A and B, respectively. By the previous paragraph A** has pt-i elements and A** ® B** = Z/(L/p)Z. Put r = pt-1. Let A = {ao = 0, al, ... , ar_1} be a set of integers representing A**. Since gcdaEA a = 1

and every element of A is of the formal + wL/p, there exists integers + Vr_lar_1 + vrL/p = 1. vo, Vl, ... , Vr_1i yr such that voao + v1a1 + W e infer that the greatest common divisor d of do, a1, ... , aris coprime to L/p. Let h be the inverse of d mod(L/p). Then 1_

1_

1_

L

hao, hat, ... , har_1 = dao, dal, ... dar_1 (mod -) . P

Therefore, by Theorem 1 , d A = G do ao = 0, dal , ... , ar_ 1 } is a set of pt-I relatively prime numbers with d dA ® B** = hA ® B** = Z. Hence, by the induction hypothesis, all elements of B** are divisible by p. Since B** = B + (L/p)Z and p is a divisor of L/p, all elements of B are divisible

by p

0

4. - A is the set of finite sums of distinct odd powers of 2 We use the following notation. If n = f Ek b32 with b; E {0, 1} is the binary notation of n, then we write n = ±bkbk_1 ... bo. We say that bk is the first bit and bo the last. The bit b; is said to be at place j, for j = 0, 1, ... , k. If j is even, then b; is at an even place, otherwise at an odd place. If b; is the last bit 1 in the binary expansion of n, then ord2(n) = j.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

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Let A be the set of finite sums of distinct odd powers of 2 and A the set of the finite sums of distinct even powers of 2. Yu. Ito asked me to characterise all sets B such that A ® B = Z. Obviously A ®A = Z>o, whence A ® (-A) = Z. THEOREM 4. - The above A satisfies A ® B = Z if and only if B is such

that (i) if b, b' E B with b b', then ord2 (b - b') is even, (ii) the set B is maximal with respect to (i),

(iii) -A c A+ B. The first condition says that there is an even number k such that 2k I b - b', but 2k+1 { b - b'. The second means that B cannot be enlarged without affecting (i). The third condition is equivalent to saying that for

every element a E A there is an a E A such that a+ a E -B. Still another interpretation is that any finite collection of bits at even places can be completed to some nonpositive number in B by inserting suitable bits at odd places, zeros at even places and putting a minus sign in front of the number. Recently it was proved by Eigen, Hajian and Kakutani (32) that if F is a finite set of integers, then F can be extended to a complementary set B of A if and only if (i) holds for F.

Proof : (this proof was shown to me by Yu. Ito. A simpler proof of (i) can be found in S. Eigen, A. Hajian and S. Kakutani (32), Lemma 1). (ii) Suppose b V B and ord2(b-b) is even for every b E B. Since b = a+ b for some a E A, b E B, we have b - b = a (z- A, whence ord2 (b - b) is odd. NO Obvious.

(i) Put An = 22nA and Bn = An ®B. Then the 2n sets n-1 Bn +

ej2 2i+1,

eo, E1,

,Cn_1 E {0,1},

j=0

are disjoint and their union is Z. We claim that Bn + k 22n = Bn for k E Z. For n = 0 it is clear. Suppose the claim is valid for n = m. Then Bm+1 + k 22m C B,n + k

22m

= Bm = Bm.+1 U (Bm+1 +

22m+1).

Since B,n+1 and Bm,+1 + 22m+1 are disjoint sets of the same cardinality, we conclude that adding or subtracting 22m+1 to an element from one set

yields an element from the other set. It follows by induction on III that Bm+i + 1. 22m+1 = Bm+i + 22m+1 for odd 1 and B,,,,+1 + 1. 22m+1 = Bm+1 for even 1. This proves the claim.

R. TIJDEMAN

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Suppose B - B contains an element z with ord2 (z) is odd. Then

b - b' = z = (2k+1)2 21+1 = k . 221+2 +2 21+1 for some b, b' E B and k, 1 E Z. Since B C Bn for all n, we have

b=b'+k 221+2+221+1 E B1+1+221+1 Thus b E B1+, fl (B1+1 + 221+1) but these sets are disjoint.

The proof of the sufficiency part of Theorem 4 requires some lemmas. LEMMA 5. - If(i) holds, then every integer is represented at most once as

a+bwithaEA, bEB. Proof : suppose al + bi = a2 + b2 for some al, a2 E A and bl, b2 E B. Then al - a2 = b2 - bi. However, ord2(al - a2) is odd and ord2(b2 - bl) is even, unless al = a2, bl = b2. 0

LEMMA 6. - Let (i) hold. If b and b' are elements of B with bb' > 0 such that b = ±b2k-lb2k-2 ... bo, b' = ±b2k-lb2k-2 ... bo and b2j = b2i

forj=0,1,...,m-1.Then b3=b.forj=0,1,...,2m-1. Proof : clear.

LEMMA 7. - If (i) and (iii) hold, then every non-positive integer has a representation a + b with a E A and b E B.

Proof : we have 0 E A, whence 0 E A ® B. Consider n E Z 0. Put bo = no. Since bo E A, there exists some nonpositive element of B ending with bo by (iii). Let bi be the bit at place 1 of this element. Define a E {0, 1} such that, in binary notation, bibo - n1no + aiO(mod 4). Next consider n2nino + a10. Let b2 be the bit at place 2 of this sum. Then, by (iii), there is a nonpositive element in B such that the last two bits at even places are b2 and bo. Let b3 be the bit at place 3 of this element. Define a3 E {0,1 } such that b3b2bibo - n3n2nlno+a3OalO(mod 24). By considering n4n3n2n1no+a3OaiO and continuing the procedure, we eventually construct bits b2k+1, b2k,... , bl, bo and a2k+1, a2k-1, ... , a3i al such that (6)

b2k+lb2k ... bibo - -n + a2k+lOa2k-10 ... Oa1O (mod 22k+2)

and b2k+1 is the bit at place 2k + 1 of some nonpositive element b of B with b2k, b2k-2, ... , bo at the last k + 1 even places and zeros at all other even places.

DECOMPOSITION OF THE INTEGERS AS A DIRECT SUM

273

Note that, by Lemma 6, the bit at place 1 of b is bi (apply it for m = 1), the bit at place 3 of 6 is b3 (apply Lemma 6 for m = 2), and so on. Thus b ends with the 2k + 2 bits b2k+lb2k bibo, whence -b can be written as bibo with t E A. Further, observe that on both sides of (6) the numbers are nonnegative and less than 22k+2, by -n < 22k, so that actually in (6) both sides are equal. Put a = n - b. Then, for some t E A, t 22k+2 +b2k+lb2k

a

b

with a E A and b E B.

0

LEMMA 8. - Let B E Z be a set satisfying (i) and (iii) and such that A ® B represents every nonpositive integer, but not 1. Then every element of B - 1 has its last nonzero bit at an even place.

Proof : suppose b is a negative element of B such that the last nonzero bit of b-1 is at an odd place, at place 2m- 1, say. By (iii) there is a nonpositive bo in B with bo = b2 = = b2,,,_2 = 1 and b2k = 0 element b* _ -bt b1_1

for k > m. Since the last bit 1 of b - 1 is at place 2m - 1 we have 2m-1

Hence, by Lemma 6, b* - b(mod

22m+1). Thus

2m-1

It follows that b* -1 has zeros at all even places. Hence a :_ - (b* -1) E A and 1 = a + b* E A ® B. a contradiction. Suppose b is a positive element of B such that the last nonzero bit of b - 1 is at an odd place, at place 2m - 1 say. By (iii) there exists a negative element b' = -b11 b11_1 ... b o in B such that bo = b2 = =b2 m_2 = 1 and b2k = 0 for k > m. Since we have proved in the previous paragraph that the last nonzero bit of b' - 1 is at an even place, we find that bo = bi = _ U b2,,,,_1 = 1. Thus ord2(b - b') = 2m - 1 which contradicts (i).

Proof of the sufficiency part of Theorem 4 : by Lemmas 5 and 7 it remains to prove that every positive integer has a representation a + b with

a E A and b E B. Let n be the smallest positive integer without such a representation. Then n V B, since 0 E A. Put B = B - (n - 1). Then A ® B represents every nonpositive integer, but not 1. By Lemma 8 every element of b - 1 = B - n has its last nonzero bit at an even place. Put B* = B U {n}. Then B* is larger than B and ord2(b - b') is even for every b, b' E B* with b # b'. Thus B is not maximal with respect to (I), in contradiction to (ii). Hence every positive integer is contained in A ® B. 0

274

R. TIJDEMAN

Theorem 4 induces a similar characterisation for A. COROLLARY. - A ® B = Z if and only if B satisfies (i') if b, b' E B with b # b', then ord2 (b - b') is odd, (ii') the set B is maximal with respect to (i'),

(iii') -ACA + B. Proof: note that A=2A andA=2A U(2A+1).

'='. By 2A®2B=2Z, we have A3(2BU(2B+1))=Z. Hence 2B U (2B + 1) satisfies the conditions (i), (ii), (iii) of Theorem 3. It follows immediately, that B satisfies (i'), (ii'), (iii'). / .'. By (i') all elements of B are even or all are odd. In the latter case

we replace B by B + 1. This involves no loss of generality. Let t be the set of numbers of B divided by 2. Then t satisfies conditions (i), (ii), (iii) of Theorem 1. Thus A + B = Z. Hence 2A + 2B = 2Z and

A®B = (2A U(2A+1))®2B = 2A®2B U (2A+1)®2B = 2ZU(2Z+1) = Z. 0

It is obvious that conditions (i) and (ii) of Theorem 3 are not enough to

guarantee A ® B = Z. The set q satisfies (i) and (ii), but A ®A = Z o. Yu. Ito asked for some set of type A for which the complementing sets are characterised by (i) and (ii) only. He wondered whether A' = { (- 1) a/2 ala E

Al is such a set. P. ten Pas [19) showed that there exist sets B' and B" which both satisfy (i) and (ii) such that A' ® B' = Z and A' (D B" =,/: Z. Acknowledgement. I am indebted to Yu. Ito and J. Urbanowicz for useful discussions and to S. Eigen and Yu. Ito for remarks on an earlier version of this paper.

Manuscrit recu le 3 decembre 1993

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References

[11 J.L. BROWN. - Generalized bases for the integers, Amer. Math. Monthly 71 (1964), 973-980. [2] N.G. de BRUIJN. - On bases for the set of integers, Publ. Math. Debrecen 1 (1950), 232-242. [3] N.G. de BRUIN. - On the factorisation offinite abelian groups, Indag. Math. 15 (1953), 258-264. [4] N.G. de BRUIJN. - On the factorisation of cyclic groups, Indag. Math. 17 (1955), 370-377. [5] N.G. de BRUIJN. - On number systems, Nieuw Arch. Wisk. (3) 4 (1956), 15-17. [6] N.G. de BRUIJN. - Some direct decomposition of the set of integers, Math. Comp. 18 (1964), 537-546. [7] S. EIGEN and A. HAJIAN. - A characterisation of exhaustive weakly wandering sequences for nonsingular transformations, Comment. Math. Univ. Sancti Pauli 36 (1987), 227-233. [8] S. EIGEN and A. HAJIAN. - Sequences of integers and ergodic transformations, Advances Math. 73 (1989), 256-262. [9] S.EIGEN, A. HAJIAN and Y. ITO. - Ergodic measure preserving transformations

of finite type, Tokyo J. Math. 11 (1988), 459-470. [10] I. FARY. - Die Aquivalente des Minkowski-Hajosschen Satzes in der Theorie der topologischen Gruppen, Comm. Math. Hely. 23 (1949), 283-287. [111 G. HAJ6s. - Uber einfache and mehrfache Bedeckung des n-dimensionalen Raumes mit einem Wurfelgitter, Math. Z. 47 (1941), 427-467.

[12] G. HAJ6s. - Sur la factorisation des groupes abelien, Casopis Pest. Mat. Fys. 74 (1950), 157-162. [13] G. HAJ6s. - Sur la probleme de factorisation des groupes cycliques, Acta. Math. Acad. Sci. Hungar. 1 (1950), 189-195. [14] R.T. HANSEN. - Complementing pairs of subsets in the plane, Duke Math. J. 36 (1969), 441-449.

[15] C.T. LONG. - Addition theorems for sets of integers, Pacific J. Math. 23 (1967), 107-112.

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[16] C.T. LONG and N. Woo. - On bases for the set of integers, Duke Math. J. 38 (1971), 583-590. [17] H. MINKOwsKI. - Geometrie der Zahlen, Leipzig, 1896. [18] I. NivEN. - A characterization of complementing sets of pairs of integers, Duke Math. J. 38 (1971), 193-203. [19] P. ten PAS. - Complementing sets for Z (in Dutch), Leiden, 1990. [20] K. Posr. - Problem 71, Nieuw Arch. Wisk. (3) 14 (1966), 274-275. [21] L. REDEI. - Zwei Liickensdtze caber Polynome in endlichen Primkorpern mit

Anwendung auf die endlichen Abelschen Gruppen and die Gaussischen Surnmen, Acta Math. 79 (1947), 273-290. [22] L. REDEI. - Kurzer Beweis des gruppentheoretischen Satzes von Hajbs, Comm. Math. Helv. 23 (1949), 272-282. [23] L. REDEI. - Ein Beitrag zum Problem der Faktorisation von endlichen Abelschen Gruppen, Acta Math. Acad. Sci. Hungar. 1 (1950), 197-207. [24] A.D. SANDS. - On thefactorisation offinite abelian groups, Acta Math. Acad. Sci. Hungar. 8 (1957), 65-86. [25] A.D. SANDS. - The factorisation of abelian groups, Quart. J. Math. Oxford (2) 10 (1959), 81-91. [26] A.D. SANDS. - On the factorisation of finite abelian groups II, Acta Math. Acad. Sci. Hungar. 13 (1962), 153-159. [27] C. SWENSON. - Direct sum subset decompositions of 7G, Pacific J. Math. 53 (1974), 629-633. [28] C. SWENSON and C. LONG. - Necessary and sufficient conditions for simple A-bases, Pacific J. Math. 126 (1987), 379-384.

[29] T. SzELE. - Neuer vereinfachter Beweis des gruppentheoretischen Satzes vonHajos, Publ. Math. Debrecen 1 (1949), 56-62. [30] A. M. VAIDYA. - On complementing sets of nonnegative integers, Math. Mag.

39 (1966), 43-44. [31] S. EIGEN and A. HAJIAN. - Sequences of integers and ergodic transformations,

Advances in Mathematics 73 (1989), 256-262. [32] S. EIGEN, A. HAIIAN and S. KAKUTANI. - Complementing sets of integers - A

result from ergodic theory, Japan J. Math. 18 (1992), 205-2 10.

R. TIJDEMAN

Mathematisch Instituut R.U. Postbus 9512 2300 RA Leiden

The Netherlands

Number Theory Paris 1992-93

CM Abelian varieties with almost ordinary reduction Yuri G. ZARHIN

In this note we discuss the Hodge group Hdg(X) of a simple Abelian variety X of CM-type. It is well-known that dimQ Hdg(X) _< dim(X). Assuming that X has somewhere good almost ordinary reduction, we prove

that dimQ Hdg(X) = dim(X) and give an explicit description of Hdg(X).

1. - Almost ordinary Abelian varieties Let A be an Abelian variety defined over a finite field k of characteristic p. We call A almost ordinary if dim(A) > 1 and it has the same Newton polygon as the product of (dim(A) - 1)-dimensional ordinary Abelian variety and a supersingular elliptic curve. This means that its set of slopes is {0, 1/2, 1}

and slope 1/2 has length 2. For example, an Abelian surface is almost ordinary if and only if it is neither ordinary nor supersingular. One may easily check that if g = dim(A) > 1 then A is almost ordinary if and only if its p-rank equals g - 1, i.e., the group of "physical" points of order p is isomorphic to (Z/pZ)9-1. Almost ordinary varieties were studied by Oort 113] in connection with the lifting problem of CM Abelian varieties to characteristic zero. In particular, he proved that each almost ordinary Abelian variety can be lifted to characteristic zero as CM Abelian variety (recall 126] that each Abelian variety over a finite field can be lifted to characteristic zero as CM Abelian variety up to an isogeny). Of course, if we start with an (absolutely) simple Abelian variety over a finite field, then its lifting will be also (absolutely) simple. It follows from ([5], Th.7; 112], Th.4. 1) that polarized almost ordinary Abelian varieties of given dimension constitute subvarieties of codimension 1 in the moduli spaces of Abelian varieties. See also [ 141.

A special case of a theorem of Lenstra and Oort [61 asserts that, for each positive integer g > 1 and for each prime number p there exists an absolutely simple almost ordinary g-dimensional Abelian variety defined Supported by C.N.R.S.

Y.G. ZARHIN

278

over a certain finite field field of characteristic p. It was proven by Oort [ 13] that the endomorphism algebra of simple almost ordinary Abelian variety

(over finite field) is a number field of degree 2 dim(A). Notice (see Sect. 6.6 below), that each simple almost ordinary Abelian variety is absolutely simple.

One may easily check that each non-simple almost ordinary Abelian variety is isogenous either to the product of ordinary Abelian variety and simple almost ordinary Abelian variety or to the product of ordinary Abelian variety and a supersingular elliptic curve. Let A be an Abelian variety over a finite field k of characteristic p. We write FA for the multiplicative subgroup of C* generated by the eigenvalues of the Frobenius endomorphism of A [29, 30, 31]. It is known (1301, Sect. 2.1; [33], Sect. 4.1), that the rank rk(I'A) of FA is a positive number which does not exceed dim(A) + 1. The non-negative integer rk(I'A) - 1 is called the rank of A and denoted by rk(A) [31]. One may easily check ([31], Sect.

2.0), that 0 < rk(A) < dim(A) and rk(A) = 0 if and only if A is supersingular. Now, assume that A is simple and almost ordinary. In that case it is known ([71, Th. 5.7) that either rk(A) = dim(A) or rk(A) = dim(A) - 1. In addition, if dim(A) is even then rk(A) = dim(A), i.e., rk(FA) = dim(A) + 1. If rk(TA) = dim(A) then the endomorphism algebra of A must contain an imaginary quadratic field; see [7], Th. 3.6. H.W. Lenstra (see [31], pp. 286288) has constructed an example of 3-dimensional simple almost ordinary Abelian variety A with rk(FA) = dim(A). His construction also gives an example of a 3-dimensional absolutely simple CM Abelian variety having an almost ordinary reduction. 2. - Q-adic Lie Algebras

Let X be an Abelian variety defined over a number field K. We assume that K is sufficiently large, i.e., all endomorphisms of X are defined over K. We will also fix an embedding of K into the field C of complex numbers and consider K as a certain subfield of C. We write K(s) for the algebraic closure of K in C. We write G(K) for the Galois group of K. We write g for the dimension of X. Let E be the endomorphism algebra of X ; it is a finite-dimensional semisimple Q-algebra. For a positive integer in, we denote by X,,,, the group

{x E X(K(s)) I mx = 0}. It is well known that X. is a free Z/mZ-module of rank 2g. Let us fix a prime number £. Then one may define the ZI Tate module T1 (X) as the projective

CM ABELIAN VARIETIES WITH ALMOST ORDINARY REDUCTION

279

limit of the groups X,,,, where m runs through the set of all powers .£i and the transition maps are multiplication by 2. It is well known that T1(X) is a free Zi-module of rank 2g . Clearly, all X,,, are finite Galois submodules of X(K(s)), and the Galois actions for m = £ glue together to give rise to a continuous homomorphism

pi = pi,x : G(K)

Autz, Ti(X).

The image

Ge = Gi,x = Im(pi,x) C Autz Ti(X) is a compact £-adic Lie subgroup in Autze Ti (X ). Let us put VV (X) := TT(X) ®Ze Q

Clearly, Vi(X) is a Qi vector space of dimension 2g and one may

with a certain Zi-lattice of rank 2dim(X) in naturally identify Vi (X). In particular, AutZe Ti (X) becomes an open compact subgroup in AutQ, Vi(X). This allows us to regard pi as an .£-adic representation ([ 191):

pi = pi,x : G(K) - Autz, Te(X) C AutQ, Ve(X). We have

Gi C Autz, Ti(X) C AutQ, Vi(X). Clearly, Gi is a compact (and therefore) closed subgroup of AutQ, VV(X) and therefore is a closed 2--adic Lie subgroup. Let gi = gi,x C EndQ, Vi(X) be the Lie algebra of Gi [ 19]. A theorem of Faltings [4] asserts that of is a reductive Qt-Lie algebra, its natural representation in Vi (X) is completely reducible and the centralizer of this representation is E ®Q Qi. A theorem

of Bogomolov [1] asserts that gi is an algebraic Lie algebra containing homotheties Qtid. It follows that

gi,x = Qiid (Dg°,x. Here

9°,x := sl(Vi(X)) n gi,x is an algebraic reductive Qi-Lie algebra. Its natural representation in VV(X) is completely reducible and the centralizer of this representation is E ®Q Qi.

It is known that the rank of g° is a non-negative number which does not

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exceed g. If the equality holds then the Lie algebra is "as large as possible" and one may give an "explicit" description of go in terms of E; see (32], Th. 3.2; [331.

Let v be a non-Archimedean place of K such that X has a good reduction X (v) at v. Then GQ contains a Frobenius element Frv E Ge C AutQe V1(X)

canonically defined up to conjugation in G1 1191. If we view Frv as a linear operator in Ve (X), then its eigenvalues are just eigenvalues of the Frobenius endomorphism of X(v). In particular, if r(Frv) is the multiplicative group generated by the eigenvalues of Frv, then

I (Frv) = rX(v)

Notice, that the rank of ge is greater or equal than rk(r(Frv)) (see 1331, Corollary 2.4.1). This implies that the rank of go is greater or equal than

rk(r(Frv)) - 1 = rk(FX(v)) - 1 = rk(X(v)). We have

0 < rk(X(v)) < rkge < g = dim(X(v)). In particular, if rk(X(v)) = dim(X(v)) then rk(ge) = 9.

For example, if X(v) is a simple almost ordinary Abelian variety and g is even then (see the end of Sect. 1) rk(g°) = g

(recall that g = dim(X) = dim(X(v))).

The aim of the present paper is to prove that if X (v) is an almost ordinary Abelian variety then rk(ge°)

= 9

under an additional assumption that X is an absolutely simple Abelian variety of CM-type. (Compare with the corresponding results for Abelian varieties having a reduction of K3 type ([32], Th. 3.0 and Sect. 7.1).

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3. - Abelian varieties of CM-type Let X be an absolutely simple Abelian variety of CM-type. Then its endomorphism algebra E is a CM-field of degree 2g. We write a -4 a' for the complex conjugation on E. We write TE for the Well restriction RE/QGm of the multiplicative group Gm. Clearly, TE is a 2g-dimensional algebraic torus. Let UE be the g-dimensional algebraic subtorus of TE defined by the condition

UE(Q) = {a E TE(Q) = E* I aa' = 1}.

3.1. - The Hodge group We write V (X) for the first rational homology group H1 (X (C), Q) of X (C)

:

it is a 2g-dimensional Q--vector space. It also carries a natural

structure of 1-dimensional E-vector space. The choice of a polarization on X gives rise to a certain non-degenerate skew-symmetric bilinear form cp : V (X) X V (X) -+ Q

such that cp(ax, y) = cp(x, a'y)

for all x, y E V(X) and a E E. Let us choose a non-zero e E E with

E =-e. Then there exists a non-degenerate E-Hermitian sesquilinear form

0, : V(X) x V(X) -* E such that co(x, y) =

n'E/Q(e-i0E(x y))

where TrE/Q : E - Q is the trace map (see 121], [2], Sect. 4; [ 171, p. 531). If we change e by el then the form is multiplied by a non-zero totally real element el /e of E. The unitary group U(V (X ), 0) viewed as a Q-algebraic group does not depend on the choice of a and can be naturally identified with UE. In particular,

U(V(X), 0,) (Q) = {a E E* I aa' = 1}. Here we identify E with its image in EndQ V (X ). Its Lie algebra

uE := Lie(U(V(X), VE)) = Lie(UE) _ {a E E I a + a' = 0}.

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Let Hdg(X) be the corresponding Hodge or as it sometimes called the special Mumford-Tate group of X (see (10, 15, 17, 18, 11]). It is a connected commutative reductive algebraic Q-group. It is well-known ([ 17], p. 531)

that

Hdg(X) C U. Let f1Dg = f1DgX be its Lie algebra. Clearly,

13DgCuE={aEEIa+a'=0}. It is known that it is a commutative Q-Lie algebra, i.e., its rank and dimension coincide, and rk(f1Dg) = dirngp 11Dg < dimQUE = 9;

the equality holds true if and only if Hdg(X) = UE.

For example, it is known that this equality holds true when g is a prime (a theorem of Tankeev-Ribet [ 17, 23]). For arbitrary dimensions there is a Ribet's inequality (118], p.87) loge (2g) < dimQ Hdg(X)

(see also [81). For further properties and examples of the Hodge groups of CM-Abelian varieties see 118, 3, 8, 281. There is a well-known natural isomorphism of Qe-vector spaces

Vt(X) =V(X) ®QQ . It is known that for Abelian varieties of CM-type the Qt-Lie algebra g° is a commutative Qt-Lie algebra, i.e., its rank and dimension coincide. A theorem of Pohlman [16] asserts that the isomorphism of the Qt-vector spaces mentioned above gives us an identification 17Dg®QQt =g°

of commutative Qe-Lie algebras.

Clearly, if rk(g°) = g, then it follows easily that dimQ 13Dg = g and, therefore,

Hdg(X) = UE.

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3.2. - Remark. One may define the Hodge group Hdg(X) for any (complex) Abelian variety X not necessarily of CM-type [10]. It is a connected reductive algebraic (12-group which is commutative if and only if X is of CM-type. The Mumford-Tate conjecture [201 asserts that the 2--adic Lie algebra ge,x can be obtained from the Lie algebra of Hdg(X) by extensions of scalars from Q to Qt. The theorem of Pohlman cited above proves the MumfordTate conjecture for Abelian varieties of CM-type.

4. - Main result. The main result of the present paper is the following statement. MAIN THEOREM. - Let X be an absolutely simple g-dimensional Abelian variety of CM-type defined over a number field K and all endomorphisms of X are also defined over K. Let E be the endomorphism algebra of X. Assume that there exists a non-Archimedean place v of K such that X has a good reduction X (v) at v and X (v) is an almost ordinary Abelian variety X (v). Then Hdg(X) = UE.

In other words,

dime Hdg(X) = dim(X) = g.

4.2. - Remark. Assume that g is even and X (v) is a simple almost ordinary Abelian variety. Then dim(X(v)) = dim(X) = g is also even and, as we have already seen, g = rk(X(v)) < rk(g°) = dimQ Clag < dimQ UE = g and, therefore, dimQ ljag = dimQ UE = g.

This proves the Theorem under our additional assumptions.

4.3. - Remark. Assume that g is odd and X (v) is a simple almost ordinary Abelian variety. Then dim(X(v)) = dim(X) = g and, as we have already seen, g - 1 < rk(X(v)) < rk(ge) = dimQ fjag < dimQUE = 9 and, therefore, g - 1 < dimQ [jag = dimQ Hdg(X) < dimQ UE = 9-

4.4. - Combining the last two Remarks, we obtain that the Theorem follows from the next two lemmas.

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4.5. LEMMA. - Let X be an absolutely simple g-dimensional Abelian variety of CM-type defined over a number field K and all endomorphisms of X are also defined over K. Let E be the endomorphism algebra of X. Assume that there exists a non-Archimedean place v of K such that X has a good reduction at v and this reduction is an almost ordinary Abelian variety X (v). Then X (v) is a simple Abelian variety.

4.6. LEMMA. - Let Y be an absolutely simple g-dimensional Abelian variety of CM-type. Let E be the endomorphism algebra of Y. Assume that

dimQ Hdg(Y) = g - 1 = dimQ UE - 1. Then g is even.

4.7. - Remark. It is well-known ([ 17], Th. 0, p. 524) that the equality

Hdg(X) = UE implies that all Hodge classes on all powers of X are linear combinations of the products of divisors classes. In particular, all these Hodge classes are algebraic, i.e., the Hodge conjecture holds true for all powers of X. Since the Mumford-Tate conjecture holds true for Abelian varieties of CM-type [ 161, we obtain that all Tate classes on all powers of X are linear combinations

of the products of divisors classes. Indeed, by a theorem of Faltings [4], each 2-dimensional Tate class on an Abelian variety over a number field is a linear combination of divisor classes. In particular, all these Tate classes are algebraic, i.e., the Tate conjecture [24, 25] holds true for all powers of X.

5. - Proof of the Lemma 4.6. We start this section with the explicit description of Q-algebraic subtori in UE of codimension 1. This description had tacitly appeared in [6] and, later, was explicitly formulated and proved in [9]. In our exposition we follow 191.

Suppose E contains an imaginary quadratic subfield k. Let us define the algebraic subtorus SUE/k of UE by the condition SUE/k(Q) = {a E UE(Q) I NormE/k(a) = 1}. One may easily check that SUE/k has codimension 1 in UE. Clearly, its Lie algebra SUE/k := Lie(SUE/k) = {a E UE I TrE/k(a) = 0}.

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Here TrE/k : E --> k is the trace map. Notice, that this trace map commutes with the complex conjugation (if someone is unhappy with the definition of SUE/k by its Q-rational points then there is another description of SUE/k.

Namely, it is a Q-algebraic (connected) subtorus of UE such that its Lie algebra coincides with SUE/k). The following statement was proven in [9[, Sect. 7.3.

5.1. KEY LEMMA. - Let H be an algebraic subtorus of codimension I in UE. Then there exists an imaginary quadratic subfield k of E such that :

H = SUE/k.

5.2. - Since H := Hdg(X) is an algebraic subtorus of codimension 1 in UE, we obtain, applying the Key Lemma, that there exists an imaginary quadratic subfield k of E such that Hdg(Y) = SUE/k.

This means that CJ-0g = SUE/k.

Now, let us choose a non-zero e c k c E such that

Now, if we consider V (X) as a g-dimensional k -vector space, then the EHermitian form 0E gives rise to the k-Hermitian form '

E/k4'e : V(X) X V(X) --4k,

0(x,y) =TrE/k(0,(x,y))It follows easily that P(x, Y) = TrE/k(E-14(x, y))

for all x, y E V (X) and 0 is non-degenerate. Clearly, UE C u(V(X), ) :_

{aEEndkV(X) I 0(ax,y)+b(x,a'y)=0bx,yEV(X)} and

SUE/k C SUk(V(X),0) := {a E U(V(X),

)

1 TrV(X)/k(a) = 0}.

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Here

TV(X)/k : Endk V(X) - k is the usual trace map on the algebra of k-linear operators of the k-vector space V(X) (notice, that the maps TrE/k and Trv(X)/k coincide on E). So, we obtained that 49 = SUE/k C SUk(V(X),V)) It turns out that the inclusion hag C SUk(V(X),')

can be rewritten in terms of the action of k on the tangent space of X (see Well [27]). Namely, if Lie(X(C)) is the tangent space of the complexAbelian variety X then the inclusion means that Lie(X(C)) is a free k ®Q C-module ([91, Lemma 2.8; see also [ 17], p. 525). Since Lie(X(C)) is a g-dimensional complex vector space and k ®Q C = C ® C, the dimension g must be even. This ends the proof.

6. - Proof of the Lemma 4.5. By functoriality of Neron models, there is a natural embedding

E = End(X) ® Q -> End(X(v)) ® Q

and 1 E E acts on X(v) as the identity map. Notice, that E is a number field and

[E : Q] = 2 dim(X) = 2 dim(X(v)). The following proposition will be proved at the end of this Section. 6.1. PROPOSmoN. - Let Y be an Abelian variety over an arbitrary field

K and assume that the semisimple Q-algebra End°(Y) = End(Y) ® Q contains a numberfield F of degree 2 dim(Y) such that 1 E E is the identity automorphism of Y. Then there exists a K-simple Abelian variety Z over k such that Z is )C-isogenous to the power Zr of Z with r = dim(Y)/ dim(Z).

6.2. - Applying the Proposition 6.1, we obtain that there exists a k(v)simple Abelian variety Z over k(v) such that X (v) is isogenous to Zr for a certain positive integer r. In order to prove the lemma 4.5, we have only to check that

r=1.

First, notice, that each slope of the Newton polygon of X (v) has length divisible by r. Since the slope 1/2 has length 2, either r = 1 or r = 2. If r = 2 then X(v) is isogenous to Z2 and, therefore, 1/2 is the slope of the Newton polygon of Z with length 1. But it cannot be true, since the length

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of the slope 1/2 must always be even [30, 7], due to the fact that all the break-points of the Newton polygon are integral. This rules out the case r = 2. So, r = 1 and we are done.

6.3. - Proof of the Proposition 6.1. Assume that Y is not )C-isogenous to a power of a )C-simple Abelian variety. Then, using the Poincare reducibility theorem, one may easily check that there exist Abelian 1C-subvarieties Y1, Y2 C Y of positive dimensions, enjoying the following properties : a) the natural homomorphism Y1 X Y2 -> Y, (yl, y2) --' yl + y2

is an isogeny; b) Hom(Yi, Y2) = {0}, Hom(Y2, Yi) = {0}. This implies that

0 < dim(Yi) < dim(Y); 0 < dim(Y2) < dim(Y); End°(Y) = End°(Yi) ® End°(Y2). Let pri : End°(Y) --4End°(Y) be the corresponding projection homo-

morphisms. Clearly, if idy E End°(Y) is the identity automorphism of Y then pri (idy) E End° (Y) is the identity automorphism idy, of Y . This implies that Fi := pri(F) c End°(Yi) is a number field isomorphic to F; in particular, its degree equals 2 dim(Y) > 2 dim(Y) (i = 1, 2.) Now, in order to get a contradiction let us recall the following well-known fact (see [22], Sect. 5.1, Proposition 2). 6.4. SUBLEMMA. - If the endomorphism algebra of an m-dimensional Abelian variety contains a number field which, in turn, contains the identity automorphism, then the degree of this field divides 2m. In particular, it does not exceed 2m.

6.5. - Now, in order to finish the proof by coming to the contradiction, one has only to apply the Sublemma to the Abelian variety Yi of dimension m = dim(Y) and the number field Fi of degree 2 dim(Y) > dim(Y).

6.6. - Remark. Similar arguments prove that if k is a finite field and A is a gdimensional k-simple almost ordinary Abelian variety over k then A is absolutely simple. Indeed, for each extension k' of k the Abelian variety A' := A x k' is an almost ordinary Abelian variety and End° A' contains a number field End° A of degree 2g = 2 dim(A'), which, in turn, contains the

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identity automorphism. By the Sublemma, A' must be k'-isogenous to the power Z' of k'-simple Abelian variety Z. Now, the same arguments with the Newton polygons as in Sect. 6.2, prove that r = 1, i.e., A' = Z is k/-simple.

7. - Acknowledgements I am deeply grateful to H.W. Lenstra and B. Moonen for helpful discussions. This paper is a result of my stay in Paris in June-July of 1993 and I am deeply grateful to the Groupe d'Etudes sur les Problemes Diophantiens (Universite de Paris VI) for the hospitality. The support of the Universite Paris Nord is also gratefully acknowledged. I am grateful to Frans Oort who had read the manuscript and made many valuable remarks. My special thanks go to Daniel Bertrand and Larry Breen, whose efforts made my trip to France possible.

Manuscrit recu le 21 janvier 1994

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Yuri G. ZARHIN

The Pennsylvania State University, Department of Mathematics 325 McAllister Building, University Park, PA 16802, USA

e-mail address : [email protected] and Institute for Mathematical Problems in Biology Russian Academy of Sciences Pushchino, Moscow Region, 142292 Russia