Séminaire de Théorie des Nombres, Paris,

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1 Progress in Mathematics Volume 102 Senes Editm J. Oesterl6 A. Weinstein Séminaire de Théorie des Nombres, Paris, Sinnou David Editor Birkhauser Boston. Base1. Berlin

2 Sinnou David Département de Mathkmatiques Université de Paris-Sud Centre d'orsay F Orsay Cedex France "ne Library of Congress has catalogued this serial pblication as follows:"... Sgninaye Delange-Pisot-Poitou. Saninaire de théorie des nombres/saninaire Delange-Pisot- Poitou Boston: Birkhauser v.; 24 cm. - (Progress in mathematics) Annuai. English and French. Continues: Séminaire Delange-Pisot-Poitou. Séminaire Delange-Pisot-Poitou: [expésl 1. Numbers, Theory of - Periodicals. 1. Titie. II. Series: Progress in mathematics (Boston. MUS.) QA 24.S37a 512'.7'05dc Library of Congress [8510] AACR 2 MARCS CLP-Kuntitielaufnahme der Deutschm Bibliothek Séminaire de théorie des nombres: Séminaire de théorie des nombres. - Boston ; Basel ; Beriin : Birkhauser Teilw. auf d. Haupttitels. auch : Séminaire Delange- Pisot-Poitou 19891'90. Paris (Progress in mathematics ; Vol. 102) ISBN (Basel...) ISBN (Boston...) NE: GT Printed on acid-free Birkhauser Boston AU rights reserved. No part of this pubiication may be reproduced, stored in a retrieval system. a transmitted. in any fom or by any means. electronic, mechanical, photocopying, recording. or otherwise, without prior permission of the copyright owner. Pedssion to phot&py for intemal or pers& use of specific clients is granted by Birkhiuser Boston forlibraries and otherusers registered with the Copyright ClearanceCenter (CCC),pmvided thatthe base feeof.~0.00perc~~,~lus$0.20per~a~eis~aid &tly toc~c.21 ~on~ress~tr~t,~alan.~~01970, U.S.A. Special requests should be addressed directly to Birkhauser Boston. 675 Massachusetts Avenue, Cambridge. MA 02139, U.S.A. ISBN ISBN Camera-ready copy prepared by the editor using Tex. Printed and bound by Edwards Brothers. Inc., Ann Ahr. Michigan. Printed in the U.S.A. Séminaire de Théorie des Nombres Paris Contents Une remarque à propos des cycles de Hodge de type CM... 1 Y. ANDRÉ Galois representations and cohomology of GL(n, Z)... 9 A. ASH Modular forms and abelian vaxieties D. BLASIUS Sommes de Kloosterman généralisées : 1 'équation fonctionnelle M. CARPENTIER On the hermitian structure of Galois modules in number fields and Adams operations B. EREZ Searching for Solutions of z3 + y + z3 = k R. HEATH-BROWN Une généralisation d'un théorème de Terjanian Y. HELLEGOUARCH Lnteger points in a domain with smooth boundary M.N. HUXLEY Estimates for coefficients of L-functions. III W. DUKE and H. IWANIEC Principe de Hasse cohomologique u. JANNSEN Classes des corps surcirculaires et des corps de fonctions J.-F. JA ULENT et A. MICHEL Ideal Class Groups and Gaiois Modules W.C. W. LI Propriétés arithmétiques des substitutions C. MAUDUIT

3 Galois theoretic local-global relations in nilpotent extensions of algebraic number fields K. MIXAKE La racine 12-ième canonique de A(L)E:~II~~ G. ROBERT Pz dans les petits intervalles J. WU Les textes qui suivent sont pour la plupart des versions écrites de conférences données pendant l'année au Séminaire de Théorie des Nombres de Paris. Ce séminaire est organisé par la S.D.I.6180 du C.N.R.S. qui regroupe des arithméticiens de plusieurs universités et est dotée d'un conseil scientifique et éditorial. Ont été aussi adjoints certains textes dont la mise à la disposition d'un large public nous a paru intéressante. Les articles présentés ici exposent soit des résultats nouveaux, soit des synthèses originales de questions récentes ; ils ont en particulier tous fait l'objet d'un rapport. Ce recueil doit bien sur beaucoup à tous les participants du séminaire et à ceux qui ont accepté d'en réviser les textes. Il doit surtout à Monique Le Bronnec qui s'est chargée du secrétariat et de la mise au point définitive du manuscrit: son efficacité et sa très agréable collaboration ont été cruciales dans l'élaboration de ce livre. Pour le Conseil éditorial et scientifique S. DAVID

4 Séminaire de Théorie des Nombres Paris Une remarque à propos des cycles de Hodge de type CM Yves AND* 1. Introduction Dans sa preuve 121 du fait que la notion de cycle de Hodge sur une variété abélienne complexe est absolue (invariante par automorphisme de ê), P. Deligne s'appuie sur deux principes : (A) les cycles invariants sous le groupe qui fixe les cycles de Hodge absolus sont tous de Hodge absolus, plate. (B) la notion de cycle de Hodge absolu est invariante par déformation En outre cette méthode met en évidence des compatibilités aux classes de cohomologie étale, et à cette liste de résultats sont venues s'ajouter des propriétés p-adiques (D. Blasius, A. Ogus, J.-P. Wintenberger). La stratégie est la même : on remplace la notion de cycle de Hodge absolu par la notion enrichie pour laquelle on prouve (A) et (B); d'ailleurs, lorsqu'il ne s'agit que de compatibilités cohomologiques, (A) s'avère souvent formel. Si l'on cherche à aborder par ce biais la conjecture de Hodge pour les variétés abéliennes (les cycles de Hodge seraient combinaison linéaire de classes fon- damentales de sous-variétés ; ce qui expliquerait naturellement leurs mirifiques propriétés), on dispose de résultats partiels en direction de (B): ainsi d'après S. Bloch (11, les cycles algébriques dits "semi-réguliers" se deforment. En revanche, on ne peut guère espérer prouver un principe (A) pour les cycles algébriques. Le but de cette note est de montrer comment l'on peut se passer de ce principe.

5 REMARQUE A PROPOS DES CYCLES DE HODGE DE TYPE CM 3 En effet celui-ci n'est utilisé dans (21, via un argument "tannakien", que pour déduire du résultat que certains cycles dits "de Weil" sont absolument de Hodge (résultat acquis au moyen de (B), le fait que tous les cycles de Hodge sur une uariété abélienne X de type CM le sont absolument. Nous montrons beaucoup plusdirectement que tout cycle de Hodge surx est combinaison linéaire d'images inverses (via -Y --+ YJ) 11.2 cycles de Weil sur diverses variétés abéliennes YJ de type CM (construites à partir de X). En particulier, l'algébricité des cy- cles de Weil entraînerait la conjecture de Hodge pour les variétés abéliennes de type CM. Ceci ramène donc cette conjecture à trouver un représentant algébri- que déformable pour les cycles de Weil sur des variétés abéliennes convenablement choisies, par exemple comme au 3 ex. 1 ci-dessous (mais une difficulté supplémentaire apparaît du fait que les cycles de Weil, orthogonaux à une puis- sance de la polarisation, n'admettent aucun représentant effectif. Quoique A. Weil présente plutôt s/ces cycles 191 comme des contrexemples potentiels, on pourra noter les résultats positifs de C. Schoen 171 en petite codimension). 2. Structures de Hodge de Type CM Soit V une Q-structure de Hodge : ê = $ V P ~ Q = VqJ', ~ ; le poids p+q=n n étant fixé, une telle décomposition équivaut à la donnée d'un morphisme 1% du groupe du cercle S1 dans GL(V@W). L'adhérence de Zariski de l'image de S1 est le groupe de Hodge de V (encore appelé groupe de Murnford-Tate spécial), noté G(V). Les invariants de G(V) dans V et dans les espaces de tenseurs mixtes sur V coïncident avec les éléments de type (p,p): on les appelle cycles de Hodge. Les conditions suivantes sont équivalentes ([SI 16.1) : i) G(V) est un tore compact, ii) G(V) est abélien, et V est polarisable. Lorsqu'elles sont satisfaites, V est dite de type CM. Toute structure de Hodge de type CM est somme directe de sous-structures irréductibles. Supposant maintenant V de type CM et irréductible, et soit E le centre de l'algèbre de ses endomorphismes. Parce que G(V) est abélien, il est naturellement contenu dans le tore rationnel TE associé à E. Par irréductibilité de V, on en déduit que EndHodge V = E et que V E E comme E-espace vectoriel. Parce que G(V) est en outre compact, il est trivial ou bien se déploie sur un certain corps CA1 F C ê, qu'on peut choisir égal à la clôture galoisienne de E si E # Q. On voit donc que E est ou bien Q, ou bien un corps CM. Le cas E = Q ne correspond d'ailleurs qu'à des structures de Tate Q(n). La décomposition de Hodge est donnée par : cp étant une fonction sur S telle que p(s) + cp(7) = (constante) 12. De façon équivalente, le complexifié de h : S1 --t (2cp(s) - n ) < s 1, TEIW est le caractère où (< s I),E~ désigne la base duale de la base naturelie SES S des caractères de TE. Ainsi cp et V (qu'on note aussi V,) se déterminent mutuellement. L'irréductibilité de V se traduit de la manière suivante : faisons agir Gal(F/Q) sur cp par (ucp)(s) = cp(su-l) pour tout s E S; alors "V, irréductible" équivaut à (*) "le sous-groupe de Gal(F/Q) laissant cp invariant fixe E". Grâce à la relation V,++ i, V+, on voit aisément qu'on obtient tous les V+ (par produit tensoriel et twist à la Tate ou dualité) à partir de structures "type", pour lesquels 9 prend ses valeurs dans {O, 1). c'est-à-dire est l'un des 21E:Q112 types CM de E. Un tel V, est le H1 rationnel d'une variété abélienne (simple, pour cp satisfaisant à (*)) à multiplications complexes par E, bien déterminée à isogénie près. On en déduit que toute structure de Hodge V de type CM de poids pair 2p,.. avec V2J = O pour i < O ou j < 0, est facteur direct du HZP = A2PH1 d'une variété abélienne à multiplications complexes (non nécessairement simple). Nous montrerons au paragraphe 4 comment se déduisent les cycles de Hodge dans H~~ des cycles spéciaux introduits par Weil. 3. Cycles de Weil (déployés) Soit E un corps CM formé d'endomorphismes d'une Q-structure de Hodge V de Spe (1, O) + (O, 1). On dit que V est de Weil (relativement à E) si la condition suivante est réalisée : (**) il existe une forme E-hermitienne 4 sur V, admettant un sous-espace totalement isotrope de dimension p = dim V, et il existe un élément purement imaginaire de E, soit (, tel que treiq(()d définisse une polarisation de V.

6 REMARQUE A PROPOS DES CYCLES DE HODGE DE TYPE CM 5 2P On vérifie alors aisément que les éléments de A V (dont le E-vectoriel sous- E jacent est une droite) sont des cycles de Hodge, appelés cyles de Weil. D'après un résultat de Landherr (cf. e.g. 161 voir aussi 121 prop. 4.2). la condi- tion (**) détermine l'espace hermitien associé à 4. De plus. on peut naturelle- ment paramétrer les structures de Weil relativement à E de dimension 2p par [E/Q]/2 copies du domaine hermitien symétrique standard {U E Mp(ê), t ~ > Ü I), cf Exemple 1 (Weil) Soit Ifo de type (1,O) + (0,1), de rang 2p, muni d'un polarisation $0 (cône de formes alternées), et soit Wo un sous-espace isotrope maximal. On pose V = E, avec bigraduation de Hodge induite de ê via V 8 ê Y ê)'. Q Q Par extension de Go à V par E-linéarité puis composition avec la trace, on définit une polarisation $ de V, qui s'écrit automatiquement sous la forme tr~~~(c)4 avec C, comme en (**) (Wo@ E est un sous-espace isotrope de dimension p pour 4) cf Ecrivons Vo = H1(Xo, Q) pour une variété abélienne Xo à isogénie près. On a V = E,Q). Lorsque G(Vo) est un ResKIQ sp~(k) on a un ResK/Q Ur,-(Vo) pour un K c End Vo convenable, un argument de théorie des invariants 141, joint au théorème de Lefschetz, montre que les cycles de Weil sur T l sont algébriques. Exemple 2 (Shimura-Deligne) Soient pl,..., pz,, des types CM attachés à E, tels que C y, = constante sur S := Hom(E, ê) (cette constante est nécessairement p). Alors V = evqi est de Weil (121 p. 69). Le calcul des périodes des cycles de Weil correspondants conduit aux fameuses relations de périodes monomiales découvertes par G. Shimura 181. Remarquons que d'après le résultat de Landherr mentionné ci-dessus, l'exemple 2 peut être "déformé" sur l'exemple Cycles de Hodge "de type CM" THÊORCME. - Soient 13 un entier positif et V une Q-structure de Hodge de type (1,O) + (0, l), de type CM. Alors il existe un corps CM E, des structures de Hodge Vj de type CM de Weil (relativement à E) en nombrefini, et des morphismes de structures de Hodge Vj -+ V, tels que tout cycle de Hodge < E A V soit somme d'images de cycles Q 2 y de Weil < j E h Vj. Q Remarques : a) Si le centre de EndHodge V est le produit de corps E; (nécessairement CM), on peut prendre pour E le compositum des clôtures galoisiennes des E; dans ê. Il ne faut toutefois pas s'attendre à ce que les morphismes Vj --+ V soient Ei-linéaires. b) En interprétant V := H1(X, Q), A V := H2p(X, Q), Vj := H1(YJ, Q), Q et les VJ -+ V comme provenant de morphismes de variétés abéliennes Ar -t Yj (pour X convenablement choisi dans sa classe d'isogénie), le théorème se traduit en l'énoncé sur les variétés abéliennes figurant dans l'introduction. Preuve du théorème : Ecrivons V = BIG avec EndHodge = Ei. Quitte à substituer $V, cg E Q au facteur direct V pour E galoisien comme dans la remarque a) ci-dessus, on se ramène immédiatement au cas où E, = E. Ecrivons alors V = $ V,, iei V,,,, de sorte que VVi,, ê soit la composante de s S=Hom(E,ê) Hodge de type (pi(s), pi(?)). On a 2~ Le tore Ti agit sur A V en commutant à G(V): cette action étendue par E- Q linéarité se diagonalise dans les DV:;~. On en déduit que tout cycle de Hodge 2P 2P 2P [ EA V s'écrit J = C X jq où J décrit l'ensemble des suites (di,s)(;,s)e~x~ avec Q di,se{0,1).~di,,=2p,et0ùx, te,qje D v~~:~'.g(v)~q~=q~. d;,. J Remarquons ensuite que pour tout automorphisme u de E, les structures de Hodge V,; sont canoniquement isomorphes; l'isomorphisme ne respecte pas

7 REMARQUE A PROPOS DES CYCLES DE HODGE DE TYPE CM 7 l'action de TE mais envoie V,,,, sur Vu,i,,,. Posons Vj =. I / > ~, ~ z v::;: (avec (di,s) J := s-lyi), de sorte que via ces isomorphismes, on obtient un morphisme de structures de Hodge de type CM Vj --+ V et que 8 provient d'un élément ZP J c ( A VJ) BQ E, invariant sous G(VJ),~ (pour alléger. on a écrit j.j E 4, au lieu de $i,, lorsque j est l'élément de J d'indice (2, s)). Notons d'autre part Ij, s > (resp. < j, S I ), (j, s) décrivant J x S. la base naturelle des caractères (resp. cocaractères) de TE^. Si Cj est non nul, il engendre le caractère : de TE3. En outre, d'après le 5 2. la structure de Hodge VJ est donné par le cocaractère : Alors le fait que Cj soit invariant sous G(Vj),,y se traduit par la nullité de J,j < ahlj, 1 > pour tout a E Gal(E/Q). CjE En développant cette expression, on trouve : et puisque j = 2p : jej j $. - constante p, C' 3 - j J ce qui entraîne (5 3, exemple 2) que Vj est une structure de Hodge de Weil. De plus (J := [E : QI-' Au(: E$ Vj est un cycle de Weil. et ( est ueg4eiq) somme des images des (J via Vj -+ V puisque ( = [E : QI-1 Au6:. uegakeiq) Remerciements : cet article a été rédigé lors d'un séjour a Bonn au Max- Planck-Institut für Mathematik avec le support de la fondation von Humboldt; l'auteur remercie vivement ces institutions. ZP BIBLIOGRAPHIE 111 S. BLOCH. - Serni-regularity and de Rham cohomology, Inv. Math. 17, (19721, P. DELIGNE. - Hodge cycles on abelian varieties, notes by J. Milne, Springer Lecture Notes 900, S. LANG. - Complex multiplication. Springer Verlag, [4] K. mem. - Hodge classes on certain types of abelian varieties, Proceedings of a conference in honor of A. Weil. [51 N. SCI-IAPPACHER. - Periods of Heclce characters, Springer Lecture Notes 1301, W. SCI-IARLAU. - Quadratic and hermitian forms, Springer Verlag C. SCHOEN. - Hodge classes on self-products of a variety with an automorphism, Comp. Math. 63, (1988) G. SHIMURA. - Automorphic forrns and the periods of abelian varieties, J. Math. soc. Japm 31, (1979), A. WEIL c. in auvres Scientifiques, Springer Verlag, (3 volumes), Yves AND& I.H.P. 11, rue Pierre et Marie Curie PARIS CEDEX 05 Manuscrit reçu le 7 novembre 1990

8 Séminaire cle Théorie des Nombres pa.ris Galois representations and cohomology of GL(n, Z) Amer ASH' Since Serre [Se31 first conjectured the possibility of attaching Galois repre- sentations to higher weight modular forms for GL(2), and Deligne [Dl proved it, this notion has been expanded to apply to a large class of automorphic forms on more general reductive groups. Clozel [Cll], foliowing Langlands [La], has given a precise conjecture for GL(n). recailed below. 1 shall refer to it as the "central conjecture". Further discussion of the history, which should be traced backwards at least to work of Eichler, Shimura, and Weil, may be found in the 1st section of [Se31 and the introduction to [Cll]. One should add the remark that Serre [Se41 seems to have been the first to propose that al1 L-functions of motives might be L-functions of automorphic forms. In this survey lecture 1 would like to substitute "cohomology class" for "automorphic form". 1 have two reasons for doing mis : 1) Cohomology sometimes is a more concretely cornputable object. Over ê, it provides a set of automorphic representations which satisfy the hypotheses of the central conjecture, so that with them one can attempt to test the Conjecture. 2) Cohomology has an integral structure, so that an analogous conjec- ture can be made integrally, or modulo a prime. We can then try to prove cases of this new conjecture, which is of interest in itself and may also shed light on the centrai one. In outlining what is known about the central conjecture, we will see the importance of the notion of "selfduality". Then 1 wili present an experimental result [AFT] which contains the only evidence (albeit tiny) known to me of the

9 10 A. ASH GALOIS REPRESENTATIONS AND COHOMOLOGY OF GL(n, Z) 11 validity of the conjecture in the non-selfduai case. In the second part of the lecture, 1 will present my version of the conjecture "inodp" together with a discussion of the cases in which it is known to hold true. In the remainder of this introduction 1 will review the definition of the Hecke algebra and its action on cohomology classes. An adelic approach to the Hecke algebra is not adequate for our purposes, for the restriction map from the cohomology of a congruence group with modp coefficients to that of a subgroup need not be injective. Fix an integer IL > 1. A "Hecke pair" is a pair (r, S) with l? a subgroup of GL(n, Z), S a sub-semigroup of Gl(n, Q), and r c S. Two Hecke pairs (r, S) and (rr, Sr) are said to be compatible if ~lthough one might think (1) should be enough for our purposes, counterexam- ples in [A21 show that (2) is also needed. For example, define So(N;n) = {y E SL(n,Q) fi M(n,Z)I top row of (*,O,...,O) mod N) and ro(n; n) = So(N; n) n SL(n, Z). Then (ro(n; n), y So(N; 12)) is a congruence Hecke pair of level N. So is (r(n), SN(N)), of course. Given G and Gr any two groups, with right modules M and Mr respectively and a morphism a from (G, M) to (Gr, Mr), we denote the pull-back on coho- mology a* : H*(Gr, Mr) -t H*(G, M). When cu is an injective map of groups with cofinite image, we denote the transfer a, : H*(G, Ml) -4 H*(Gr, M'), where Adr is given the structure of G-module via the map a. Now we define the action of H(r, S) on cohomology. Let A be a ring, and M a right AS-module. For any s E S, set r(s) = s-lrs n r. We have two morphisms i, j of the pair (r(s), V) into (r, V) given by i(x) = x, i(v) = v; and j(z) = sxs-l. j(v) = us. For each,b E H*(I', M), define : TdP) = i,j*(p). The Hecke algebra of integral linear combinations of double cosets rsr, s E S will be denoted H(r, S). We shall use the notation T, for the double coset rsr in H(r, S). Define its degree by deg(t,) = [I' : r n s-'rs] = number of single cosets rx contained in the double coset rsr. If (r, S) and (rr, S') are compatible, there is naturally defined an injective map of algebras from H(rl, Sr) to H(r, S). See [AS] for more details. For natural numbers N and M, we make the following definitions : r(n) = {g E SL(n,Z)lg I(mod N)); GSnf(N) = {g E Ad(n,Z)lg = diag(l,l,...,1, *)(mod N) Snf(N) = {g E GSn.r(N)l det(g) > O). and det(g) is prime to M); We cal1 (r, S) a "congruence Hecke pair of level N" if Set H(N) = H(GL(n, Z), Gs~(1)). For any congruence Hecke pair (r, S) of level N and AS-module M. we let H(N) act on H*(r, M) by the formula above, via the injective algebra morphism H(N) + H(r, S). Note that if we increase the level N without changing r, this has the effect of throwing away the Hecke operators involving the finite set of new primes entering into the level. 1) Complex cohomology of congmence subgroups of GL(n, Z) The central conjecture [Cl 1, conjecture 4.51 would attach a mouve to each cuspidal automorphic representation II of GL(n) over a number field, as long as the infinity type of II is "algebraic". If II arises from a cuspidal cohomology Hecke-eigenclass cu of a congruence subgroup r of GL(n, Z) with coefficients in a complex rational representation M of GL(n, ê), then it satisfies this algebraicity condition. It is a fact that II is deterrnined up to isomorphism by the system of Hecke eigenvalues of a. From the motive, we get n-adic representations of Gq (the absolute Galois group of Q). Thus conjecture A below is a speciaiization of the central conjecture. Recall that H(N) is a polynomial ring over Z generated by the elements T(e, k) = rdiag(1,..., 1, l,...,e)r, with k Ps, where e runs over al1 primes not dividing AT and k = 1,..., IL

10 12 A. ASH GALOIS REPRESENTATIONS AND COHOMOLOGY OF GL(n, Z) 13 Suppose a # O and T(e, k)a = ~ (e, k)a. Then the Hecke polynornial attached to a at e is defined to be Also, for each & unramified in an extension E of Q, we write Frobe for the geometric Frobenius element in the Galois group of E/Q (defined up to conjugacy). Thus robe' acts on the residue field of a prime above e by raising to the [-th power. CONJECTURE A. - Let (I', S) be a congruence Hecke pair of level N, p a prime. Let.4 be the ring of integers of a numberfield F c ê, and.rr a prime in A above 13. Let M be an AS-module such that a3 is a rational finite dimensional representation of CL(?%, ê). Suppose a E ~"(r, M) is a cuspidal eigenclass for the action of the Hecke algebra H(N), with eigenvalues a(e, k) E A. Cuspidality here means that a is not a torsion class and the automorphic representation II corresponding to a in H~(I', ê) is cuspidal Then : (1) There exists afinite extension E off with ring ofintegers B, such that Jor any prime x in B abovep there exists a semisimple, continuous representations p : Gq + GL(n, B,) unramijled outside pn such that for al1 E not dividing pn lion. (2) p(c) is conjugate to diag(1, -1,1, -1,...), where c is complex conjuga- (3) For evey ram~ed prime q of Q, the conductor of p at q equals the conductor of II,. Remarks : acl(2) : the last entry is $1 if n is odd because p(c) must be the (n - 1)st symmetric power of diay(1, -1). ad(3) : the conductor of II, is defined in [JPSS]. As a step towards proving this conjecture, we have the following (where 1 have altered the notation to conform to this lecture) : THEOREM (Clozel,[C121). - Let II be a cuspidal, algebraic, regular and selfdual representation of G L(12, Aq). Suppose there exists a prime b such that xi, is square-integrable. If n r 2(mod4), we suppose there is another prime c # b such that xc is also square-integrable. Let F be an imaginary quadraticfield, Split at b and at c ifn = 2(inod 4). Then there exist : i) a numberfield E ii) afinite set S of prime numbers iii) a positive integer a iv) a compatible system W = (W,,r,) of x-adic representations of G~~(F/F), unramfied outside the places dividing S,.rr running over al1 the places of E, such lhat $tu is not in S and T does not divide w, one has for evey place v of F dividing W. and for eue y 172 > O, trace (~,(Fsob,)~) = a x trace (t,)rnq$n-l). Here, q, is the number ofelements of the residuefield at v, and t, is the Hecke matru at v of the base-change of II to F. This theorem is a ver- important step in the right direction, but note that it fails to verify the conjecture, even given its hypotheses, in two ways. First, the result holds only after the base change from Q to F. Second, only the a-th power of the desired Galois representation is being constructed, and we don't know if we can take a equal to 1. But the most important limitation of this theorem is that it only applies in the case that II is selfdual. For when II is selfdual, Clozel can deduce the x-adic representations from some motive associated to a Shirnura variety. The other hypotheses may be viewed as "technical". However, if II is not selfdual, even after allowing base change and twisting by a character, 1 don't know even a conjectured method of finding the conjectured motive or Galois representations. Such problematical representations are known to exist for GL(3)lQ. They anse from the cuspidal cohomology of congruence subgroups of Gl(3, Z) as follows : In [AGGI, the cuspidal cohomology H&(ro (N; 3), 43) was computed, for Prime values of N < 113. The cuspidal cohomology vanished except it was 2- dimensionai for N = 53,61,79,89. Unpublished computations of P. Green show that the next prime level with cuspidal cohomology will be 223.

11 14 A. ASH GALOIS REPRESENTATIONS AND COHOMOLOGY OF GL(n, Z) 15 Let's concentrate on the exarnple N = 61. From pp of [AGG] we find that the field generated by the Hecke eigenvalues is Q(w), with w = (1 + 0 )/2, and for some choice of Hecke- eigenclass a in H~u,,(I'o(61; 3), ê), we have the following table, with a! = eigenvalue of T(P, 1) : e ae a! mod J-3 ap - e mod J % O b O O 5-2f4w O 1 7-6w O w - 1 O 29 3+& 1-1 / The reasons for adding the last two columns will become clear later. In [APT] we show that for each e, P(a, e) is determined by ap. In fact we have P(a, e) = 1 - aex + Gex2 - e3x3. We also show that because the ap's are not ali real, the automorphic representation Il corresponding to oc is not selfdual. In fact for any base-change II of T to a finite extension of Q, I1 is not selfdual, even up to a twist. We remark that if there really is a motive underlying these eigenvalues, then P(a, e) should have roots of absolute value l/e for all e prime to N. This is in fact the case. If conjecture A holds for a, we can reduce the representation p mod T and use the fact that any representation of a finite group over a finite field can be realized over the field generated by its traces. We obtain then as a logical consequence of Conjecture A the following conjecture. CONJECTURE a. - Let q be a rational prime. Let a E H3(I'o(q; 3), M) be a cuspidal eigenclass for the action of the Hecke algebra H(q), with eigenvalues ae = a(@, 1) E A, a ring of integers. Let p be a rational prime and T a prime in A aboue p. Then : (1) There exists a semisimple, continuous representation p : Gq -+ G L(3, illa) unrarnijled outside pq such that P(a, L') = 1 - aex + &ex2- e3x3 = det(i - p(~robe)-l~) (mod T) for al1 e not diuiding pq. p-group. (2) p(c) is conjugate to diag(l,l, -1). (3) # q. and Iq denotes the inertia subgroup at q, then p(i,) is a In [APT] we seek to test this conjecture for q = 61 and a = J-3. We obtain an extension of Q which, as far as Our data on the Hecke eigenvalues goes, appears to be attached to the given cuspform modulo T. TNs extension is uniquely, indeed, highly overdetermined by the data. In this way we obtain a rnicroscopic, but positive, confirmation of this part of Langlands' philosophy in the non-selfdual case. We find, a posteriori, that there should be a congruence mod T between our form of level 61 and an Eisenstein series build from a Maass form on GL(2). We see no way ol predicting its existence a priori, nor indeed to prove the congruence ; i.e. to show that al1 the Hecke eigenvalues (not just those computed in [AGG]) match properly with the traces of Frobenius elements. 1 shali sketch bnefly the method used in [m. Assume a representation p exists as in the conjecture a, then the characteristic polynomial of the Frobenius element at e wili be determined by ap. In this way, the splitting in the fixed field M of Ker(p) will be controlled by the value of at. Also, we have the ramification information as laid out in the conjecture. This behavior of the primes in Q determines the fked field of I<er(p) uniquely (assuming p exists and is semisimple). Using the given data on the a,'~, we first proved that p is isomorphic to its contragredient and then that p preserves an alternating form. It follows that for some choice of coordinates, Im(p) c SL(2, Fa) x GL(1, F3). Hence p = o x w, where a : Gq --+ SL(2,F3) and w = det p rnust be the c~clotomicharacter. So finding p is equivalent to finding the representation in the following conjecture : CONJEWRE. - Let a be a cuspidal eigenclass for ro(61; 3) with Hecke eigenvalues ae. Then :

12 1 G A. ASH GALOIS REPRESENTATIONS AND COHOMOLOGY OF GL(n, Z) 17 (1) There exists a semisimple, continuous representation a : Gq -+ SL(2, IF3) unram$ed outside 3.61 such that for al1 l # 3,61. (2) p(c) = 1. ti- a(frobe)-' = ae - C (mod a) (3) ) is a 3-group, hence has order 1 or 3. It's not hard to see that if a exists as in conjecture 3.4, it must be surjective. - There is a filtration on SL(2, IF3) with successive quotients SL(2,F3) 4 PSL(2, F3) N A4 C3. Correspondingly. assuming that a exists, and M is the fixed field of Iier(a), we obtain a diagram of fields : M 1 c2 ramified outside 3 L Il' 1 c2 X cz ramified outside 3 1 c3 ramified outside 3 and q Q Moreover, each of Il', L and M are Galois over Q, with Galois groups C3, A4 and SL(2, F3) respectively. The splitting of finite primes in the fields is conditioned by (1) in the conjecture. The fields must be totally real by (2) of the conjecture. The indicated ramification data is dictated by (3) of the conjecture. We find IC, L and M in succession. We begin by looking for a Galois cubic extension K of Q, unramified outside 3 and 61, such that the splitting of primes in I< is compatible with (1) of the conjecture. Let e = 11. Then from the table we see that tra(frobl1)-l = al1-11 = O (mod a). If we list al1 the conjugacy classes of SL(2, IF3). we find that those with trace O al1 have even order. Therefore, a(frobll) must have even order, and hence must be / trivial when restricted to Il'. In other words, 11 must split in K. This determines h' uniquely to be the Galois cubic extension of Q ramified only at 61. kom [Gr] we gathered useful information about K. We relied on the computer algebra system REDUCE to find L and M. In fact, there exists a unique Cz x C2 extension L of K, unrarnified outside 3, and L/Q Galois. Now we seek a totally real quadratic extension M = L(6) such that M/Q is Galois, with Galois group SL(2, F3). unrarnified outside 3. A theorem of Serre [Se21 gives a criterion for an A4-extension L of Q to be liftable to an SL(2, F3) extension, and Crespo Kr1 gives an explicit procedure for constructing al1 such SL(2, F3) extensions, should any exist. Applying this procedure, we do in fact obtain M. which is uniquely determined. At this point the integers involved in the computations, for instance the coefficients of the irreducible polynomial of S2 over Q, have dozens of digits. Finally we check that the splitting of primes < 29 in M/Q indeed behaves according to conjecture a. - II) Mod p cohomology of congruence subgroups of GL(n, Z) Let p be a prime number and F a finite field of characteristic p. Let (r, S) be a congruence Hecke pair of level N. An "admissible" FS-module of level N is a finite-dimensional right IFS-module on which the elements of S with positive determinant act through reduction mod N. We consider the H(N)-action on the cohomology of r with coefficients in an admissible ES-module and try to attach modular Galois representations to eigenclasses. Counterexamples in [A21 show that adrnissibility is necessary here. In this context, 1 have modified conjecture A as follows [A21 : CONJEC~URE B. - Let (r, S) be a congruence Hecke pair of level N, p a prime and let M be an admissible IFS-module. Suppose P E H'(I', M) is an eigenclass - for the action of the Hecke algebra H(N), with eigenvalues a([, k) E F. Then there exists a semisimple continuous representation p : GQ GL(n, F) unramfied outside pn such that for al1 C not dividing pn.

13 18 A. ASH GALOIS REPRESENTATIONS AND COHOMOLOGY OF GL(n, H) 19 If conjecture A holds for an a, then conjecture B will hold for /3 = reduction of a(mod T), by reducing the Galois representation mod T. The main extra content of conjecture B consists of what it says about p's which are the reductions mod T of torsion classes in the cohomology. Conjecture B alm rnakes no reference to cuspidality, which no longer rnakes sense modp. When n = 2, by reducing modp a theorem of Eichler and Shimura [Sh, in weight 2 and one of Deligne [Dl in higher weight, we see that this conjecture holds true. (Men n = 2, there are not many torsion classes and they can be dealt with directly.) Conversely, by a conjecture of Serre [Se]. ali odd irreducible representations of Gq into GL(2, F) are supposed to arise this way. In [A21 1 showed that conjecture B holds in the following cases : (1) i =Oor 1; (2),û is a topological Chern class or Euler class; (3) p is an étale Chern class and p is a regular prime. (4) 12 = p - 1, i > (p- 3)/2, and /3 restricts non-trivially to Nr(P) for some p-group P c r. In (1)-(3). p may be taken to be a sum of characters of Gal(Q(CP)/Q) twisted by a character of Gq. In (4). p is induced from a character 5 : Gal(E/I<) + IFX of the Galois group of the class field of I< = Q(&) corresponding to the ideal classes prime to N modulo principal ideals generated by elements congruent to 1 inod N. Comment on the scope of (4) : By the last theorem in [Br], if a E Ha(GL(p - 1, Z), M) and a restricts non-trivially to the Tate-Farrell co- homology of GL(p - l, Z), then a restricts non-trivially to mme Nr(P). In particular, this applies to every a if i > virtual cohomological dimension of GL(p - 1, Z) = p(p - 1)/2, and also to very many a's for smaller i, by [AMI. Moreover, using [Al], one sees in the case M = IF that every 5 does arise. We also have a version of the principle "modp, ail modular forms are weight 2." In our context this translates into "modp, ail systems of Hecke eigenvalues occur in cohomology with 1-dimensional coefficients." In fact. let (r, S) be a congruence Hecke pair of level N. For any character a : (Z/N) + IF ', we define the ES-module E(a) to be IF with S acting via a O det O (reduction mod N). We have the foliowing theorem which reduces to "weight 2" : / THEOREM. - Let (r, S) a congruence Hecke pair of level N. M an admissible FS-module. Let Q> be a system of H(N)-eigenvalues occuring in H~(I', M). Then for some character a : --+ IFX, and some j 5 i. Q> occurs in ~j(r(n,, IF(&)) with respect to the Hecke pair (I'(N), SN(N)). 1 conclude with some more details on (1)-(4) while interspersing remarks on the other p-torsion classes of which 1 know. Certain "trivial" Hecke actions, which remind one of Eisenstein series, occur fkequently : a cohomology class /3 is said to be punctual up to a twist if there is a character a : (Z/N)X --+ IFX such that for aii s E S. Ts/3 = a($) deg(ts)/3. Denote by w : Gq -+ IFX the cyclotomic character of conductor p. LEMMA. - Let (r, S) be a congruence Hecke pair of level N, M an admissible FS-module and /3 E H*(r, M). if@ is an eigenclass for H(N), punctual up to twisting by the character E : (ZIN) -+ EX, then conjecture B holds for p. The representation p may be taken to be (1 $ w $ w2 $... $un-') 8 E. Al1 eigenclasses in Ho are punctual up to a twist. The same is not true for Hl, although we end up with a similar result when n > 2, after some work : THEOREM. - Suppose n > 2. Let (r, S) be a congruence Hecke pair of level N. and M an admissible OF S-module. Then conjecture B holds for any H(N)- eigenclass in H1(r, M). The representation p may be taken to be (1 $ w $ w2 $.. S, for some character II, : Gq -+ FX of conductor diuiding N2. Of course, when n = 2, the Galois representations attached to eigenclasses in H1(r, M) are those stemrning from modular curves, and are much more complicated than those in this theorem. Let (F, S) be a congruence Hecke pair of level N. Suppose /?' is a topological Chern class or the Euler class of r. It is known that /3 is a torsion class in the cohomology with trivial Z-coefficients and its annihilator in Z was computed in [EM]. Any such /3 is punctual. So are the Chern classes of Grothendieck, as studied by Soulé ISo21, when p is a regular prime. 1 don't know how the Hecke algebra acts on the more exotic Chern classes that arise when p is irregular. The 3-torsion in the cohomology of SL(3, Z) computed in [Sol] consists of characteristic classes. and hence is ail punctual. 1 don't know about the 2- torsion.

14 20 A. ASH GALOIS REPRESENTATIONS AND COHOMOLOGY OF GL(n, Z) 21 The proof of (4). presented in [A2], is lengthy, and (to me at least) interesting. It involves the classification of subgroups of order p in GL(p - 1, Z) via the class group of Q((,) and an interpretation of Frobe acting on class groups via rational matrices of C-power determinant acting on the subgroups of order p by conjugation. Again, the case n = 3 allows some interesting explicit computations. Theo- rem of [AS] shows that H3(r, M) contains in general many p-torsion Hecke eigenclasses for certain r and M. The reduction modp of each of these eigen- classes satisfies conjecture B. The corresponding Galois representations are symmetric squares of those attached to cusp forms of weight g + 2 for the classical modular group SL(2, Z). Unpublished computations of Philiip Green in 1986 on an IBM PC, using the methods of [AGG], have shown the existence of nontrivial p-torsion in H3(ro(N; 3), Z) in the following cases : p = 3, N = 127,137,151,193,211; p = 5, N = 136,197,211; p = 7, N = 167. Unfortunately, he did not compute the action of the Hecke operators. This should be done, and then conjecture B for the reductions modp of these classes should be investigated. There is also an additional 11-torsion class whenever p divides N - 1. These latter classes come from the boundary of the Borel-Serre compactification. It should not be hard to verify Conjecture B for their reductions modp, but 1 haven't checked this yet. REFERENCES [AI] A. A ~H. - Farrell cohomology of GL(n, Z), brael J. 67, (1989) [A21 A. AsH. - Galois representations at tached to modp cohomology of GL(n, P), preprin t. [&G] A. ASH, D. GRAYSON and P. GREEN. - Computations of cuspidal cohomology of congruence subgroups of SL(3, Z), J. Number Th. 19, (1984), [AM] A. ASH and M. Mc CONNELL. - Mod p cohomology of SL(n, Z), to appear in Topology. [APT] A. ASH, R. PINCH and R. TAYLOR. - An Â4 extension of Q attached to a nonselfdud automo~phic form on GL(3), preprint. [AS] A. ASH and G. STEVENS. - Cohomology of arithrnetic groups and congruences between systems of Hecke eigenvalues, J.f.d. reine u. angew. Math. 365, (1986), [Br1 K. BROWN. - Cohomology of Groups, Springer, New York, [Cl11 L. CLOZEL. - Motifs et formes automorphes : applications du principe de foncto- fiaté, in Automorphic Forms, Shimura Varieties and L-functions, Proceedings Manuscrit reçu le 13 septembre 1990 of the Ann Arbor Conference, L. Clozel and J.S. Milne eds, Academic Press 1, (1990), [Cul L. CLOZEL. - Représentations gdoisiennes associées aux représentations automorphes autodua.les de GL(n), preprint. * p. 9 : Itesearcli partially supported by NSF Grant No DMS and a grant froin SERC (UK). [Cr] T. CRESPO. - Explicit construction of type fields, J. Alg. 127, (1989) ID] p. DELIGNE. - Formes modulaires et représentations 1-adiques, Séminaire Bourbaki , no 355, and Lecture Notes in Math., Springer-Verlag 179, (1971),

15 22 A. ASH [EM] B. ECKMANN and G. MISLIN. - Galois action on algebraic matrix groups, Chern classes, and the Euler class, Math. Ann. 271, (1985) [Gr] M. GRAS. - Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q, J.f.d. reine u. angew. Math. 277, (1975), [JPSS] H. JACQUFT, 1.1. PIATETSKY-SHAPIRO and J. SHALIKA. - Conducteur des représentations des groupes linéaires, Math. Ann (1981), [La] R. LANGLANDS. - Automorphic representations, Shimura varieties and motives, Ein MGchen, Proc. Symp. Pure Math. 33, part 2, (1979), [Sel] J.-P. SERRE. - Sur les représentations modulaires de degré 2 de Gd@/&), Duke J. 54, (1987), Ise21 J.-P. SERRE. - L'invariant de Witt de la forme Tr(x2), Comm. Math. Helv. 59, (1984), [Se31 J.-P. SERRE. - Une interprétation des congruences relatives a la fonction T de Ramanujan, Serninar Delange-Pisot-Poitou 1967/68, no 14 ; # 80 in Collected Works, Volume II, [Se41 J.-P. SERRF,. - Résumé des cours de ; # 78 in Collected Works, Volume II, [Shl G. SHIMURA. - Introchction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, [Sol] C. SOULB. - The cohomology of SL(3, Z), Topology 17, (1978), [So2] C. SOULÉ. - Ii-théorie des anneaux d'entiers de corps de nombres et cohomolo- gie étale, Inv. Math. 55, (1979), Amer ASH The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus, Ohio USA Séminaire de Théorie des Nombres paris Introduction. Modular forms and abelian varieties Don BLASIUS* 1.1. In this article we prove a theorem concerning the relation between the Galois representations associated to modular forrns of higher weight and those defined by the étale cohomology of abelian varieties To describe the result, let f be a holomorphic newform of weight k > 2. Let p be a rational prime and let i : for aii. Let r a, be an embedding, fixed once and P, : Gal(Q/S) --+ GL(Sp) be the p-adic representation defined by f (see (2.1) below). Let X, : Gd(Q/Q) -+ Zp be the p-adic cyclotomic character. 1.3 THEOREM. - Suppose that f is not of CM tuve (see (2.2) below). Then there does mt exist any abelian variety A demed over Q such that for sorne Pair (n, j) E Z x Z the GL~(Q,) representation X; is isornorphic to a sub-gal@/q)-module of a,, where H~(A) is the j-th p-adic étale cohornology group of A x Q For example, the representation pa attached to the Ramanujan A func- Uon cannot occur inside any Tate twist (i.e. $-twist) of the cohomology of an abeiian variety. So far as we are aware, this theorem provida the first exam- Ples of motivic Galois representations which do not arise from abelian varieties. On the other hand, it is well known that the Galois representations attached to modular forms of CM type as weli as those attached to holomorphic forms of weight one always occur in the cohomology of abelian varieties.

16 24 D. BLASIUS MODUIAR FORMS AND ABELIAN VARIETIES That such a result as (1.3) rnight hold was conjectured independently by J.-P. Serre. The present argument was found in conversation with A.J. Scholl. 1 thank him warrnly for his interest and help. 1 also thank the Institute for Advanced Study for its hospitality in when this work was done The paper is organized as follows. In the next section we very briefly recail the connection between newforms and Galois representations, and introduce the Tannakian language ((DM]) which is natural for our proof. In the third section we recail some notions of Hodge-Tate theory and compute the group attached by Tannakian duality to a non-cm newform. This result is probably weil-known. In section 4 we prove the main result. In the last section we give some complements and an open question. 2. Preliminaries 2.1. Recall ([Dl) that attached to a holomorphic newform Cp vector spaces. Since pf is is semisimple. Let Gf = ~ut@(w,) be the algebraic group over Cp consisting of the tensor compatible automorphisms of the usual fiber functor w, --+ Cp-vector spaces which sends a representation to its underlying space. is semisimple. G is reductive. is generated by pf, Gf rnay be regarded as a reductive subgroup of GLz(Cp). 3. Hodge-Tate theory Let Dl C Gal(Q/&) be the subgroup consisting of all a for which LO a O L-' extends continuously to C,. If x E Cp we w~ite simply az for the image of z under this extension of a. Let L C Cp be a finite extension of Q,, and let DLtL = Dl n Aut(c,/~) Let V be a finite dimensional vector space over L endowed with a continuous L-linear action of DL,L. For n E Z, let V(n) be the module with underlying space V on which a E Dl,L acts by sending v E V to x;(a)a(v). Let of weight k 2 1 is a p-adic representation pf as in (1.2) such that for al1 but finitely many primes pl ;(a,, = Trace (PAF,, 1) where F,, E G~~(Q/Q) is a Frobenius element for pl. It is well-known that pf is irreducible and adrnits the closure i(tf) of i(tf) as a field of definition, where Tf is the number field generated by the an (n 2 1) We Say that f is of CM me if either of the following equivalent conditions holds : Here u E Dl,L acts on z by z) = a(z). We Say that V is Hodge-Tate if relative to the evident rnap. If W is a Cp [DL] module such that W is isomorphic as a Dl module to one of the form Cp with a field L and a space V as just above, we put a) some conjugate of the image of pf in GL~(~,) is contained in the normalizer of the diagonal matrices. b) there exists a quadratic character E of Gal(G/Q) such that E is isomorphic to pf Let Cp be a completion of g,. be the Tannakian category gen- erated by pf, regarded as a GL2(Cp)-valued representation. Thus, the ob- jects are the tensor products of subquotients of the tensor powers (Pf n)@(p;!)@ni, and the morphisms are just the Galois equivariant maps of these Then is independent of the choice of the auxiliary field L, and we Say that W is Hodge-Tate if the identity (3.2.1) holds for W. Let ww : Z + N be defined by ww(n) = dim% w(,).

17 26 D. BLASIUS MODULAR FORMS AND ABELIAN VARIETIES 27 We cal1 ww the Hodge-Tate tyl?e of W (rel. DL) Now let W/Cp be a finite dimensional Gal(Q/Q)-module which is Hodge- Tate relative to DL. be the Tannakian category generated by W as Gal(Q/Q)-module. Then (3.2.3) defines a canonical functor to the category of graded Cp-vector spaces. Dualizing, if Gw is the automorphism group of the usual fiber functor w, we have a morphism which is defined by the properiy that for V and the representation pv of Gw on w,(v) = V, we have Thus, k is odd and pf X, has finite image. Thus, L( f, s + (k- 1)/2) is an Artin L-function. One concludes by comparing the Artin functional equation of (k-1)/2 s) with that of L( f, s + (k- 1)/2). The compatibility of r-factors forces k = Proof of the theorem Suppose now that there exists an abelian variety A defined over Q such that a representation isomorphic to pf Cp). Such an isomorphism can be extended to a fuuy faithful functor G). with this image and letting GA denote the group defined by Hi Cp by Tannakian dualiiy, we obtain a surjection Especially, recall that pf is Hodge-Tate and let pf : G, -+ GL2(Cp) be the cocharacter defined in this way for the Hodge-Tate module pf ([FI). 3.4 LEMMA. - Suppose that f is not of CM type and has weight k 2 2. Then Gf E GL Proof. Let Gf be the connected component of G f. Then Gf is a reductive subgroup of GL2. Hence it is isomorphic to 1, G, (central in GL2), G,, (non- central in GL2), GE,, 5'12, or GL2. If Gf is isomorphic to G,, noncentrally embedded, then pf is reducible. If Gf is isomorphic to G&, then f is of CM type. If Gf is isomorphic to G, centrally embedded, then there exists a finite extension L of & such that p I L takes values in CP. l2 c GL2(Cp). On the other hand, the Hodge-Tate type of pf is supported on exactly the set {O, k - 1). Thus,if (z) is equivalent to the diagonal matrix diag (1, xk-j). Since Im(p f) C Gi. this case cannot occur, and neither can Gf = (1). Thus, Gf = SL2 or GL2. But A2p is a one-dimensional representation of infinite order. Thus G and hence Gf has a surjective homomorphism to G,. Hence G = Gf = GL2. Q.E.D. 3.6 REMARK : we can conclude this proof globally without resort to Hodge-Tate theory. If Gf 1 G,,. centrally embedded, then for some finite extension L of Q, pf 1 L is the p-adic representation attached to an algebraic Hecke character of L. For L large enough, the square of this character is the (1- k)-th power of x, Let : G,, -+ GA be the cocharacter defined by the Hodge-Tate splitting of Hj Cp. Let y.4 : GA -t GL(H; Cp) be the tautological representation. Then 9.4 O p~ and pf o PA = pf define the Hodge-Tate splittings on Hj Cp and the space of pf, respectively. Note that the Hodge-Tate weight of is supported on {O, 1) and wh1 (0) = whi (1) = dim A Since pf sux-jects GL2, the Lie algebra g~ of GA is of the form : - where 8 is the Lie algebra of the center of g ~ gr, is semisimple, and Dv g'. Also. D ~ factors A as DPA = (P', PZ, pal. O on Note that since D(y O pa) = Dp diagonalizes to the map Dp (z) = Diag((k - 1)~,0),~2 diagonalizes to PZ(Z) = Diag((k - 1)/2 z,(1 - k)/2 z). Let $ be an irreducible component of which is non-trivial on the given se2 factor of BA. Then $ is isomorphic to a representation of the form with irreducible representations $' : gr -+ GL(X),$2 : sez + GL(Y) and : 8 + Z = C,. In particular, if a,,l3, and y denote respectively eigenvalues of

18 28 D. BLASIUS MODULAR FORMS AND ABELIAN VARIETIES 29 $'opl(l), $20p2(1), and $30pd(l). then cu+p+y is aneigenvalue of D(p~op~)(l). But D (~A O pa)(l) has only the two eigenvalues 1 and O. Hence G2 O pz has 2 eigenvalues whereas 4' O p' and $ O p each have one eigenvalue. Thus, $2 is the standard 2 dimensional representation of s12. and the eigenvalues which occur are require that the weight of the holomorphic newform be i) at least 2 at each infi- nite place of F, and ii) greater that 2 for at least one such place. The details are left to the reader. See [BR] for the result that the pf in this case are Hodge-Tate Let Hc be the Tannakian categoiy made from HQ by replacing Hom(Xl Y) with Y) 8 C. Let Mc(f) E Hc be the element defined by Since E is contained in {O, 1). we see at once that k = 1 or k = 2. Q.E.D The above proof, while elementary, is not the most general. In particular, one may also conclude in the following ways : ll or 1) by using the obvious fact that AdG, O p~ can only have coweights 2) by arguing more generally, as in [D2], that for any Hodge-Tate p : Gal(g/~) -t C;L,(C,) with associated group GO, and cocharacter p,,, the action AdG; O p, can only have weights among -1,0, and 1 (if p occurs inside some twist of the cohomology of an abelian variety). 5. Complements 5.1. Let L be a finite extension of Q. Then the conclusion of the theorem holds with pf replaced by pf I L and abelian varieties over Q replaced by those over L Let Ha be the Tannakian category generated by all irreducible Gal(a/ Q)- modules W of Hodge-Tate type ww where ww (n) # O implies w w(n * j) # O for some j with 1 5 j < a. Then the above argument shows that pf doesn't belong to Ha provided i) f is not of CM type and ii) the weight of f is at least a + 2. Thus, the category of representations attached to modular forms cannot be generated by any category of representations whose Hodge-Tate types belong to a finite list. where MB(f) is the Betti realization of the Grothendieck motive attached to f by Scholl ([Schl). and let be the associated group. 5.7 CONJECTURE. - IJ f is not of CM type and has weight k 2 2, then Gc.r 7 GL2. If this conjecture is true, then Mc( f ) is not contained in the subcategoiy of Hc generated by the H"S of abelian varieties. This follows by an evident variant of our argument above. This conjecture is, of course, simply the assertion that MB( f) is not a Hodge-structure of CM type. However, 1 don't know any example for which it is known to hold. ('1 p. 23 : Partially supporteci by NSF grant No # DMS Manuscrit reçu le 28 septembre The theorem has little to do with modular forms; the pf can be replaced by any 2-dimensional, irreducible non-cm p whose Hodge-Tate type does not belong to H2. However, modular forms provide the only known supply of such representations The result extends easily to the p-adic representations associated to holomorphic Hilbert modular forrns attached to a totally real field F. Here we only

19 D. BLASIUS Séminaire de Théorie des Nombres Paris REFERENCES [BR] D. BLASIUS and J. ROCAWSKI. - Galois representations for Hilbert modular forms, Bull. AMS. July [Dl P. DELICNE. - Formes modulaires et représentations 1-adique Séminaire Bour- baki 355, Springer- Verlag, New York SLN 179, (février 1969) [D2] P. DELIGNE. - La conjecture de Weil pour les surfaces K3, hv. Math. 15, (19721, [FI G. FALTINCS. - Hodge-Tate structures and modular forms. Math. Ann 278, (1987) [DM] P. DELICNE and J. MILNE. - Tannakian categories. In : Hodge Cycles, motives and Shimura varieties, Springer-Verlag Introduction Sommes de Moosterman généralisées : l'équation fonctionnelle Michel CARPENTIER Soit VT la variété définie sur F, par l'équation 3;' x... x 3n = Ç, ou gl,..., g, sont des entiers naturels premiers entre eux dans leur ensemble. Dans un travail précédent [2], nous avions construit la cohomologie p-adique associée aux sommes exponentielles : Don BLASIUS University of California, Los Angeles (U.C.L.A.) CALIFOFNA U.S.A. où XI,..., xn sont des caractères multiplicatifs et O un caractère additif non trivial de F,. Cette construction nous avait permis d'obtenir de bonnes estimations pour le polygone de Newton de la fonction L(&, 0, X; T) associée. Le but du présent travail est double : il s'agit à la fois d'étudier la variation de la cohomologie en fonction du paramètre 2, et de calculer l'équation fonctionnelle de la fonction L lorsque p > 5: nous verrons que ces deux question sont étroitement liées. Le résultat principal est alors le suivant : L'application y H qn-l /y est une bijection de l'ensemble des zéros réciproques de L(VF, O, X ; T)(-')" sur ceux de L(1.2, O-', X-l; T)(-')". La première partie est consacrée à l'étude de la cohomologie du point de vue de sa structure cristalline. Soient r le p.p.c.m. des ordres des caractère x;, " G = n Z/,kin et H C G le sous-groupe cyclique engendré par (rgl,..., rg,). i=l la cohomologie se décompose en une somme directe W, = $ W,(E). Dans EEGIH

20 32 M. CARPENTIER chaque composante W,(Z), la déformation est commandée par une équation différentielle de type hypergéométrique, possédant un point singulier régulier en O, et un point singulier irrégulier à l'infini. Grâce à un argument adapté de 171, nous montrons au 5 3 que la partie uniforme des solutions au voisinage de l'infini converge p-adiquement dans le disque D(m, 1-). Appelons Ç(") la matrice de la connexion agissant sur W, (a)(dans une base appropriée), et soit T= O 1 1 O : O 1 1 O... O- - Le choix du caractère additif O est subordonné à celui d'une racine T de l'équation XP-l = -p. Si l'on dénote par le suffixe "-T" le fait de changer T en son opposé dans la définition des objets correspondants, on a alors la relation : 1. L'équation de déformation Soit O un caractère additif non trivial de F,. Soient XI,..., xn des caractères multiplicatifs de IF: et soit r le p.p.c.m. de leurs ordres. Soient l 1 d, =rk, i {l,..., n} Soit R un corps algébriquement clos et complet contenant Q,. Soit T la racine de l'équation A?-' de? E ff,, onait = -p telle que, si x E R est le représentant de Teichmüller O(F) = exp T (X - xq). Soient tl,..., t,, des indéterminées. Si a = (al,...,an) E Zn on utilisera la notation multi-indicielle ta = trl... trn. Soit H = H(t) = x(c1t$ cnt$) où ci E 0, c:-l = 1. Si i E (1,..., n - 1) on pose De plus, chaque W, (ai) est muni naturellement d'une structure de Frobenius forte : si c(") est une matrice de Frobenius pour w("), il en va de même pour (-75)* TC-, S. Nous utilisons le principe d'unicité du Frobenius (161) pour montrer l'existence d'une constante Ii telle que Si cu E Z1' on pose La dernière partie est consacrée au calcul de la constante I< : les coefficients diagonaux de la matrice da) évalués en x = O sont, à des puissances de T près, des produits des valeurs de la fonction gamma p-adique qui interviennent dans la formule de Gross-Koblitz. On en déduit que K-l = pn-l, d'où la forme précise de l'équation fonctionnelle : Nous tenons à remercier B. Dwork qui nous a suggéré ce travail et nous en a fourni les idées essentielles (voir en particulier [6]), ainsi que S. Sperber. ~(a) = J(a) - Ns(a). Soient h et c deux nombres réels, b 2 O. On définit : L(h, c) = {V = aen" A(a)talA(a) E R et ord A(a) 2 bj(a) + c),