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Ann. Scient. c. Norm. Sup.,4e srie, t. 38, 2005, p. 505 551.

GALOIS REPRESENTATIONS MODULO p ANDCOHOMOLOGY OF HILBERT MODULAR VARIETIES

BY MLADEN DIMITROV

ABSTRACT. The aim of this paper is to extend some arithmetic results on elliptic modular forms to thecase of Hilbert modular forms. Among these results let us mention: control of the image of Galois representations modulo p,

Hidas congruence criterion outside an explicit set of primes, freeness of the integral cohomology of a Hilbert modular variety over certain local components of theHecke algebra and Gorenstein property of these local algebras.

We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galoisrepresentations and our main tool is the action of inertia groups at primes above p. In order to determine thisaction, we compute the HodgeTate (resp. FontaineLaffaille) weights of the p-adic (resp. modulo p) talecohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the workof Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometricconstructions of Tilouine and the author. 2005 Elsevier SAS

RSUM. Le but de cet article est de gnraliser certains rsultats arithmtiques sur les formesmodulaires elliptiques au cas des formes modulaires de Hilbert. Parmi ces rsultats citons :

dtermination de limage de reprsentations galoisiennes modulo p, critre de congruence de Hida en dehors dun ensemble explicite de premiers, libert de la cohomologie entire de la varit modulaire de Hilbert sur certaines composantes locales

de lalgbre de Hecke et la proprit de Gorenstein de celles-ci.Ltude des proprits arithmtiques des formes modulaires de Hilbert se fait travers leurs reprsenta-

tions galoisiennes modulo p et loutil principal est laction des groupes dinertie aux premiers au-dessusde p. Cette action est dtermine par le calcul des poids de HodgeTate (resp. FontaineLaffaille) de la co-homologie tale p-adique (resp. modulo p) de la varit modulaire de Hilbert. La partie cohomologique decet article est inspire par le travail de Mokrane, Polo et Tilouine sur la cohomologie des varits modulairesde Siegel et repose sur des constructions gomtriques de Tilouine et lauteur. 2005 Elsevier SAS

Contents

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5060.1 Galois image results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5070.2 Cohomological results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5070.3 Arithmetic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5080.4 Explicit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

1 Hilbert modular forms and varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5101.1 Analytic Hilbert modular varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

ANNALES SCIENTIFIQUES DE LCOLE NORMALE SUPRIEURE

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506 M. DIMITROV

1.2 Analytic Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5101.3 HilbertBlumenthal Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.4 Hilbert modular varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5121.5 Geometric Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

1.6 Toroidal compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5141.7 q-expansion and Koecher Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5141.8 The minimal compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5151.9 Toroidal compactifications of KugaSato varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

1.10 Hecke operators on modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5151.11 Ordinary modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5171.12 Primitive modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5171.13 External and Weyl group conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5171.14 EichlerShimuraHarder isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

2 HodgeTate weights of Hilbert modular varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5192.1 Motivic weight of the cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5192.2 The BernsteinGelfandGelfand complex overQ . . . . . . . . . . . . . . . . . . . . . . . . . . . 5202.3 HodgeTate decomposition ofH(MQp,Vn(Qp)) . . . . . . . . . . . . . . . . . . . . . . . . 5202.4 Hecke operators on the cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5222.5 HodgeTate weights of IndQF in the crystalline case . . . . . . . . . . . . . . . . . . . . . . .5232.6 HodgeTate weights of in the crystalline case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5242.7 FontaineLaffaille weights of in the crystalline case . . . . . . . . . . . . . . . . . . . . . . . . 525

3 Study of the images of and IndQF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5263.1 Lifting of characters and irreducibility criterion for . . . . . . . . . . . . . . . . . . . . . . . . . 5263.2 The exceptional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5273.3 The dihedral case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5283.4 The image of is large . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5293.5 The image ofIndQF is large . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .530

4 Boundary cohomology and congruence criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5334.1 Vanishing of certain local components of the boundary cohomology . . . . . . . . . . . . . . . 5334.2 Definition of periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5354.3 Computation of a discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

4.4 Shimuras formula forL(Ad0

(f),1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5364.5 Construction of congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5385 FontaineLaffaille weights of Hilbert modular varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

5.1 The BGG complex overO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5395.2 The BGG complex for distributions algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3 BGG complex for crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

6 Integral cohomology over certain local components of the Hecke algebra . . . . . . . . . . . . . . . . 5446.1 The key lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5446.2 Localized cohomology of the Hilbert modular variety . . . . . . . . . . . . . . . . . . . . . . . . 5456.3 On the Gorenstein property of the Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5466.4 An application to p-adic ordinary families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .549Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .550

0. Introduction

Let F be a totally real number field of degree d, ring of integers o and different d. Denote byF the Galois closure ofF in Q and by JF the set of all embeddings ofF into QC.We fix an ideal n o and we put = NF/Q(nd).

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COHOMOLOGY OF HILBERT MODULAR VARIETIES 507

For a weight k =

JFk Z[JF] as in Definition 1.1 we put k0 = max{k | JF}. If

is a Hecke character ofF of conductor dividing n and type 2 k0 at infinity, we denote bySk(n, ) the corresponding space of Hilbert modular cuspforms (see Definition 1.3).

Let f

Sk(n, ) be a newform, that is, a primitive normalized eigenform. For all ideals a

o,we denote by c(f, a) the eigenvalue of the standard Hecke operator Ta on f.

Let p be a prime number and let p :Q Qp be an embedding.Denote by E a sufficiently large p-adic field with ring of integers O, maximal ideal Pand

residue field .

0.1. Galois image results

The absolute Galois group of a field L is denoted by GL. By results of Taylor [40,41] andBlasius and Rogawski [1] there exists a continuous representation = f,p :GF GL2(E)which is absolutely irreducible, totally odd, unramified outside np and such that for each primeideal v ofo, not dividing pn, we have:

tr(Frobv)= pc(f, v), det(Frobv)= p(v)NF/Q(v),where Frobv denotes a geometric Frobenius at v.

By taking a Galois stable O-lattice, we define = f,p mod P: GF GL2(), whose semi-simplification is independent of the particular choice of a lattice.

The following proposition is a generalization to the Hilbert modular case of results ofSerre [37] and Ribet [35] on elliptic modular forms (see Propositions 3.1, 3.8 and 3.17).

PROPOSITION 0.1. (i) For all but finitely many primes p,(Irr) is absolutely irreducible.(ii) Iff is not a theta series, then for all but finitely many primes p,(LI) there exists a powerqofp such thatSL2(Fq) im() GL2(Fq).(iii) Assume thatf is not a twist by a character of any of its internal conjugates and is not a

theta series. Then for all but finitely many primes p,(LIInd) there exist a powerqofp, a partition JF =

iI J

iF and for all JiF an element

i,Gal(Fq/Fp) such that ( = = i, = i,) andIndQF : GF SL2(Fq)JF factorsas a surjection GF SL2(Fq)Ifollowed by the map (Mi)iI (Mi,i )iI,JiF, where Fdenotes the compositum of F and the fixed field of(IndQF )1(SL2(Fq)JF).0.2. Cohomological results

Let Y/Z[ 1 ] be the Hilbert modular variety of level K1(n) (see Section 1.4). Consider the

p-adic tale cohomology H(YQ,Vn(Qp)), where Vn(Qp) denotes the local system of weightn =

JF(k 2) N[JF] (see Section 2.1). By a result of Brylinski and Labesse [3] the

subspace Wf :=ao ker(Ta c(f, a)) ofHd(YQ,Vn(Qp)) is isomorphic, as GF-module andafter semi-simplification, to the tensor induced representation

IndQF .

Assume that(I) p does not divide .Then Y has smooth toroidal compactifications over Zp (see [10]). For each J JF, we put

|p(J)| =J(k0 m 1) +JF\J m, where m = (k0 k)/2 N. By applyinga method of Chai and Faltings [15, Chapter VI] one can prove (see [11, Theorem 7.8,Corollary 7.9]).


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508 M. DIMITROV

THEOREM 0.2. Assume thatp does not divide . Then(i) the Galois representation Hj (YQ,Vn(Qp)) is crystalline atp and its HodgeTate weights

belong to the set{|p(J)|, J JF, |J|j}, and(ii) the HodgeTate weights ofWf are given by the multiset

{|p(J)

|, J

JF

}.

For our main arithmetic applications we need to establish a modulo p version of the abovetheorem. This is achieved under the following additional assumption:

(II) p 1 >JF(k 1).The integer

JF

(k 1) is equal to the difference |p(JF)| |p()| between the largest andsmallest HodgeTate weights of the cohomology of the Hilbert modular variety. We use (I) and(II) in order to apply FontaineLaffailles Theory [17] as well as Faltings Comparison Theoremmodulo p [14]. By adapting to the case of Hilbert modular varieties some techniques developedby Mokrane, Polo and Tilouine [31,33] for Siegel modular varieties, such as the constructionof an integral BernsteinGelfandGelfand complex for distribution algebras, we compute theFontaineLaffaille weights ofH(YQ,Vn()) (see Theorem 5.13).

0.3. Arithmetic results

Consider the O-module of interior cohomology Hd! (Y,Vn(O)), defined as the image ofHdc (Y,Vn(O)) in Hd(Y,Vn(E)). Let T = O[Ta, a o] be the full Hecke algebra acting on it,and let T T be the subalgebra generated by the Hecke operators outside a finite set of placescontaining those dividing np. Denote by m the maximal ideal ofT corresponding to f and p andput m = m T.

THEOREM 0.3. Assume that the conditions (I) and(II) from Section 0.2 hold.(i) If(Irr) holds, d(p 1) > 5

JF

(k 1) and(MW) the middle weight |p(JF)|+|p()|2 =

d(k01)2

does not belong to {|p(J)|, J JF}, thenthe local componentH(Y,Vn(O))m of the boundary cohomology vanishes, and the Poincarpairing Hd! (Y,Vn(

O))m

Hd! (Y,Vn(

O))m

Ois a perfect duality.

(ii) If(LIInd ) holds, then H(Y,Vn(O))m = Hd(Y,Vn(O))m is a free O-module of finiterank and its Pontryagin dual is isomorphic to Hd(Y,Vn(E/O))m .

The proof involves a localglobal Galois argument. The first part is proved in Theorem 4.4using Lemma 4.2(ii) and a theorem of Pink [32] on the tale cohomology of a local systemrestricted to the boundary ofY. The second part is proved in Theorem 6.6 using Lemma 6.5 andthe computation of the FontaineLaffaille weights of the cohomology from Theorem 5.13. Thetechnical assumptions are needed in the lemmas. Since the conclusion of Lemma 6.5 is strongerthen the one of Lemma 4.2(ii) we see that the results of Theorems 0.3(i) and A (see below)remain true under the assumptions (I), (II) and (LIInd ).

Let L(Ad0(f), s) be the imprimitive adjoint L-function of f and let (Ad0(f), s) bethe corresponding Euler factor (see Section 4.4). We denote by f C/O any twocomplementary periods defined by the EichlerShimuraHarder isomorphism (see Section 4.2).

THEOREM A (Theorem 4.11). Let f and p be such that (I), (Irr) and (MW) hold, and

p 1 > max(1, 5d )

JF(k 1). Assume that p( (Ad

0(f),1)L(Ad0(f),1)

+ff

) P. Then thereexists another normalized eigenform g Sk(n, ) such that f g (modP), in the sense thatc(f, a) c(g, a) (mod P) for each ideal a o.

The proof follows closely the original one given by Hida [21] in the elliptic modular case, anduses Theorem 0.3(i) as well as a formula of Shimura relating L(Ad0(f), 1) to the Petersson


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inner product off (see (19)). Let us note that Ghate [18] has obtained a very similar result whenthe weight k is parallel. A converse for Theorem A is provided by the second part of the following

THEOREM B (Theorem 6.7). Letf andp be such that(I), (II) and(LIInd ) hold. Then

(i) H(Y,Vn())[m] = Hd(Y,Vn())[m] is a -vector space of dimension 2d.(ii) H(Y,Vn(O))m = Hd(Y,Vn(O))m is free of rank2d overTm.(iii) Tm is Gorenstein.

By [30] it is enough to prove (i), which is a consequence of Theorem 0.3(ii) and the q-expansion principle Section 1.7. This theorem is due, under milder assumptions, to Mazur [30]for F = Q and k = 2, and to Faltings and Jordan [16] for F = Q. The Gorenstein property isproved by Diamond [8] when F is quadratic and k = (2, 2) under the assumptions (I), (II) and(Irr). We expect that Diamonds approach via intersection cohomology could be generalizedin order to prove the Gorenstein property ofTm under the assumptions (I), (II) and (LI) (seeLemma 4.2(i) and Remark 4.3).

When f is ordinary at p (see Definition 1.13) we can replace the assumptions (I) and (II)of Theorems A and B by the weaker assumptions that p does not divide NF/Q(d) and that

k (modp 1) satisfies (II) (see Corollary 6.10). The proof uses Hidas families of p-adicordinary Hilbert modular forms. We prove an exact control theorem for the ordinary part ofthe cohomology of the Hilbert modular variety, and give a new proof of Hidas exact controltheorem for the ordinary Hecke algebra (see Proposition 6.9).

Theorems A and B prove that the congruence ideal associated to the O-algebra homomorphismT O, Ta p(c(f, a)) is generated by p( (Ad

0(f),1)L(Ad0(f),1)

+ff

). In a subsequent paper [12]

we relate it to the fitting ideal of the BlochKato Selmer group associated to Ad0() E/O. Aninteresting question is whether f are the periods involved in the BlochKato conjecture for the

motive Ad0(f) constructed by Blasius and Rogawski [1] (see the work of Diamond, Flach andGuo [9] for the elliptic modular case).

0.4. Explicit results

By a classical theorem of Dickson, if (Irr) holds but (LI) fails, then the image of inPGL2() should be isomorphic to a dihedral, tetrahedral, octahedral or icosahedral group. Usingthis fact as well as Proposition 3.1, Section 3.2, Propositions 3.5 and 3.13 we obtain the followingcorollary to Theorems A and B.

Denote by o+ (respectively on,1) the group of totally positive (respectively congruent to 1

modulo n) units ofo.

COROLLARY 0.4. Let be any element ofo+ on,1.(i) Assume d = 2 andk = (k0, k0 2m1), with m1 = 0. If

p NF/Q

(m1 1)(k0m11 1)

and p 1 > 4(k0 m1 1) then Theorem A holds. If additionally the image of inPGL2() is not a dihedral group then Theorem B also holds.

(ii) Assume d = 3, id = JF and k = (k0, k0 2m1, k0 2m2), with 0 < m1 + m2 =k01

2. If

p NF /Q (()m1 m2()m1 m2+1k0()m1+1k0 m2 ()k0m11 m2+1k0)


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510 M. DIMITROV

andp 1 > 53 (3k0 2m1 2m2 3) then Theorem A holds. If additionally the imageof in PGL2() is not a dihedral group then Theorem B also holds.

1. Hilbert modular forms and varieties

We define the algebraic groups D/Q = ResFQ Gm, G/Q = Res

FQ GL2 and G

/Q = G D Gm,

where the fiber product is relative to the reduced norm map : G D. The standard Borelsubgroup ofG, its unipotent radical and its standard maximal torus are denoted by B, U and T,respectively. We identify D D with T, by (u, ) u 00 u1.1.1. Analytic Hilbert modular varieties

Let D(R)+ (respectively G(R)+) be the identity component of D(R) = (F R) (re-spectively of G(R)). The group G(R)+ acts by linear fractional transformations on the spaceHF = {z FC | im(z) D(R)+}. We have HF = HJF , where H = {z C | im(z) > 0}is the Poincars upper half-plane (the isomorphism being given by

z

(()z)JF ,

for F, z C). We consider the unique group action of G(R) on the space HF extend-ing the action of G(R)+ and such that, on each copy of H the element

1 0

0 1

acts by

z z. We put i = (1, . . . ,1 ) HF, K+ = StabG(R)+(i) = SO2(F R)D(R) andK = StabG(R)(i) = O2(FR)D(R).

We denote by Z=lZl the profinite completion ofZ and we puto =Z o =v ov , wherev runs over all the finite places of F. Let A (respectivelyAf) be the ring of adles (respectivelyfinite adles) ofQ. We consider the following open compact subgroup ofG(Af):

K1(n) =

a bc d

G(Z) | d 1 n, c n.

The adlic Hilbert modular variety of level K1(n) is defined as

Yan = Y1(n)an = G(Q)\G(A)/K1(n)K+.

By the Strong Approximation Theorem, the connected components ofYan are indexed by thenarrow ideal class group Cl+F = D(A)/D(Q)D(Z)D(R)+ ofF. For each fractional ideal c ofFwe put c = c1d1. We define the following congruence subgroup ofG(Q):

1(c, n) =

a bc d

G(Q)

o c

cdn o

| ad bc o+, d 1 (mod n)

.

Put Man = M1(c, n)an = 1(c, n)\HF. Then we have Y1(n)an

h+Fi=1 M1(ci, n)

an, where theideals ci, 1 i h

+F, form a set of representatives of Cl

+F.

Put HF = HFP1(F). The minimal compactification M an ofMan is defined as M an =1(c, n)\HF. It is an analytic normal projective space whose boundary M an\Man is a finiteunion of closed points, called the cusps ofMan.

The same way, by replacing G by G, we define 11(c, n), M1,an = M11 (c, n)

an and M1,an.

1.2. Analytic Hilbert modular forms

For the definition of the C-vector space of Hilbert modular forms we follow [24].


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DEFINITION 1.1. An element k =

JFk Z[JF] is called a weight. We always

assume that the ks are 2 and have the same parity. We put k0 = max{k | JF},n0 = k0 2, t =

JF

, n =

JF

n = k 2t and m =

JF

m = (k0t k)/2.

For z HF, = a bc d we put jJ(, z) = c zJ + d D(C), wherezJ =

z, J,z, JF\J.

DEFINITION 1.2. The space Gk,J(K1(n)) of adlic Hilbert modular forms of weight k,level K1(n) and type J JF at infinity is the C-vector space of the functions g : G(A) Csatisfying the following three conditions:

(i) g(axy) = g(x) for all a G(Q), y K1(n) and x G(A).(ii) g(x) = ()k+mtjJ(, i)kg(x), for all K+ and x G(A).For all x G(Af) define gx : HF C, by z ()tkmjJ(, i)kg(x), where G(R)+

is such that z =

i. By (ii) gx does not depend on the particular choice of .

(iii) gx is holomorphic at z, for J and anti-holomorphic at z, for JF\J (when F =Qan extra condition of holomorphy at cusps is needed).

The space Sk,J(K1(n)) of adlic Hilbert modular cuspforms is the subspace of Gk,J(K1(n))consisting of functions satisfying the following additional condition:

(iv)

U(Q)\U(A) g(ux) du = 0 for all x G(A) and all additive Haar measures du on U(A).The conditions (i) and (ii) of the above definition imply that for all g Gk,J(K1(n)) there

exists a Hecke character ofF of conductor dividing n and of type n0t at infinity, such thatfor all x G(A) and for all z D(Q)D(Z)D(R), we have g(zx) = (z)1g(x).

DEFINITION 1.3. Let be a Hecke character of F of conductor dividing n and of typen0t at infinity. The space Sk,J(n, ) (respectively Gk,J(n, )) is defined as the subspace ofSk,J(K1(n)) (respectively Gk,J(K1(n))) of elements g satisfying g(zx) = (z)1g(x) for all

x G(A) and for all z D(A). When J = JF this space is denoted by Sk(n, ) (respectivelyby Gk(n, )).

Since the characters of the ideal class group ClF = D(A)/D(Q)D(Z)D(R) ofF form a basisof the complex valued functions on this set, we have:

Gk,J

K1(n)

=

Gk,J(n, ), Sk,J

K1(n)

=

Sk,J(n, )(1)

where runs over the Hecke characters ofF of conductor dividing n and infinity type n0t. Let be a congruence subgroup ofG(Q). We recall the classical definition:

DEFINITION 1.4. The space Gk,J(;C) of Hilbert modular forms of weight k, level and

type J JF at infinity is the C-vector space of the functions g : HF C which are holomorphicat z, for J, anti-holomorphic at z, for JF\J, and such that for every we haveg((z)) = ()tkmjJ(, z)kg(z).

The space Sk,J(;C) of Hilbert modular cuspforms is the subspace ofGk,J(;C), consistingof functions vanishing at all cusps.

Put xi =

i 00 1

, where i is the idle associated to the ideal ci, 1 i h

+F. The map

g (gxi)1ih+F (see Definition 1.2) induces isomorphisms:


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512 M. DIMITROV

Gk,J

K1(n)

1ih+F

Gk,J

1(ci, n);C

,

(2)Sk,JK1(n) 1ih+F Sk,J1(ci, n);C.

Let d(z) =

JFy2 dx dy be the standard Haar measure on HF.

DEFINITION 1.5. (i) The Petersson inner product of two cuspforms g, h Sk,J(K1(n)) is given by the formula

(g, h)K1(n) =

h+Fi=1

1(ci,n)\HF

gi(z)hi(z)yk d(z),

where (gi)1ih+F(respectively (hi)1ih+F

) is the image ofg (respectively h) under theisomorphism (2).

(ii) The Petersson inner product of two cuspforms g, h Sk,J(n, ) is given by

(g, h)n =

G(Q)\G(A)/D(A)K1(n)K

+

g(x)h(x)(x)n0

Ad(x).

1.3. HilbertBlumenthal Abelian varieties

A sheaf over a scheme Swhich is locally free of rank one over o OS, is called an invertibleo-bundle on S.

DEFINITION 1.6. A HilbertBlumenthal Abelian variety (HBAV) over a Z[ 1NF/Q(d)

]-scheme

S is an Abelian scheme : A S of relative dimension d together with an injectiono

End(A/S), such that A/S :=

1

A/S

is an invertible o-bundle on S.

Let c be a fractional ideal of F and c+ be the cone of totally positive elements in c. Givena HBAV A/S, the functor assigning to a S-scheme X the set A(X) o c is representable byanother HBAV, denoted by A o c. Then o End(A/S) yields c Homo(A, A o c). Thedual of a HBAV A is denoted by At.

DEFINITION 1.7. (i) A c-polarization on a HBAV A/S is an o-linear isomorphism : A o c At, such

that under the induced isomorphism Homo(A, A o c) = Homo(A, At) elements of c(respectively c+) correspond exactly to symmetric elements (respectively polarizations).

(ii) A c-polarization class is an orbit ofc-polarizations under o+.

Let (Gm d1)[n] be the reduced subscheme ofGm d1, defined as the intersection ofthe kernels of multiplications by elements of n. Its Cartier dual is isomorphic to the finite group

scheme o/n.DEFINITION 1.8. A n-level structure on a HBAV A/S is an o-linear closed immersion

: (Gm d1)[n] A of group schemes over S.1.4. Hilbert modular varieties

We consider the contravariant functor M1 (respectively M) from the category ofZ[ 1

]-schemes to the category of sets, assigning to a scheme S the set of isomorphism classes


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of triples (A,,) (respectively (A, , )) where A is a HBAV over S endowed with a c-polarization (respectively a c-polarization class ) and a n-level structure . Assume thefollowing condition:

(NT) n does not divide 2, nor 3, nor NF/Q(d).Then 1(c, n) is torsion free, and the functor M1 is representable by a quasi-projective,

smooth, geometrically connected Z[ 1

]-scheme M1 = M11 (c, n) endowed with a universalHBAV :A M1. By definition, the sheaf A/M1 = 1A/M1 is an invertible o-bundleon M1. Consider the first de Rham cohomology sheaf H1dR(A/M1) = R1A/M1 on M1.The Hodge filtration yields an exact sequence:

0 A/M1 H1dRA/M1 A/M1 cd1 0.

Therefore H1dR(A/M1) is locally free of rank two over o OM1 .The functor M admits a coarse moduli space M = M1(c, n), which is a quasi-projective,

smooth, geometrically connected Z[ 1 ]-scheme. The finite group o+/o

2n,1 acts properly and

discontinuously on M1 by [] : (A,,)/S (A,,)/S and the quotient is given by M.This group acts also on A/M1 and on H1dR(A/M1) by acting on the de Rham complex A/M1([] acts on A/M1 by

1/2[]).

These actions are defined over the ring of integers of the number field F(1/2, o+).Let o be the ring of integers of F(1/2, o+). For every Z[ 1 ]-scheme S we put

S = S Spec

o

1

.

The sheaf of o+/o2n,1-invariants of A/M1 (respectively ofH1dR(A/M1)) is locally free of

rank one (respectively two) over o O

M and is denoted by (respectivelyH

1

dR

).

We put Y = Y1(n) =h+F

i=1 M1(ci, n) and Y1 = Y11 (n) =

h+Fi=1 M

11 (ci, n), where the ideals

ci, 1 i h+F, form a set of representatives ofCl

+F.

1.5. Geometric Hilbert modular forms

Under the action of o, the invertible o-bundle on M decomposes as a direct sum of linebundles , JF. For every k =

k Z[JF] we define the line bundle k =

k

on M.One should be careful to observe, that the global section ofk on Man is given by the cocycle

()k/2j(, z)k , meanwhile we are interested in finding a geometric interpretation of thecocycle

()tkmj(, z)k used in Definition 1.4.

The universal polarization class endows H1dR with a perfect symplectic o-linear pairing.Consider the invertible o-bundle := 2oOMH1dR on M. Note that (k + m t) k2 = n02 t.

DEFINITION 1.9. Let R be an o[ 1

]-algebra. A Hilbert modular form of weight k, level1(c, n) and coefficients in R is a global section ofk n0t/2 over MSpec(Z[ 1 ]) Spec(R).We denote by Gk(1(c, n); R) = H0(MSpec(Z[ 1 ]) Spec(R), k n0t/2) the R-module ofthese Hilbert modular forms.


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1.6. Toroidal compactifications

The toroidal compactifications of the moduli space ofc-polarized HBAV with principal levelstructure have been constructed by Rapoport [34]. Several modifications need to be made in order

to treat the case ofn-level structure. These are described in [10, Theorem 7.2].Let be a smooth 11(c, n)-admissible collection of fans (see [10, Definition 7.1]). Then,

there exists an open immersion ofM1 into a proper and smooth Z[ 1 ]-scheme M1 = M1, called

the toroidal compactification ofM1 with respect to . The universal HBAV : A M1 extendsuniquely to a semi-Abelian scheme : G M1. The group scheme G is endowed with an actionofo and its restriction to M1\M1 is a torus. Moreover, the sheaf

G/M1ofG-invariants sections

of1G/M1

is an invertible o-bundle on M1 extending A/M1 .

The scheme M1\M1 is a divisor with normal crossings and the formal completion of M1along this divisor can be completely determined in terms of (see [10, Theorem 7.2]). For thesake of simplicity, we will only describe the completion of M1 along the connected componentofM1\M1 corresponding to the standard cusp at . Let be the fan corresponding to thecusp at

. It is a complete, smooth fan of c

+ {0}

, stable by the action ofo2n,1

, and containinga finite number of cones modulo this action. Put R = Z[q, c] and S = Spec(R) =Gm c. Associated to the fan , there is a toroidal embedding S S (it is obtainedby gluing the affine toric embeddings S S, = Spec(Z[q, c ]) for ). LetS be the formal completion ofS along S\S. By construction, the formal completionofM1 along the connected component of M1\M1 corresponding to the standard cusp at isisomorphic to S/o

2n,1.

Assume that is 1(c, n)-admissible (for the cusp at it means that is stable under theaction ofo+). Then the finite group o

+/o

2n,1 acts properly and discontinuously on M

1 and the

quotient M = M is a proper and smooth Z[ 1 ]-scheme, containing M as an open subscheme.Again by construction, the formal completion of M along the connected component of M\Mcorresponding to the standard cusp at is isomorphic to S/o+.

The invertible o-bundle G/M1 on M1

descends to an invertible o-bundle on M

, extending. We still denote this extension by . For each k Z[JF] this gives us an extension of k to aline bundle on M, still denoted by k .

1.7. q-expansion and Koecher Principles

IfF=Q the Koecher Principle states that

H0

M Spec(R), k n0t/2= H0M Spec(R), k n0t/2.(3)For a proof we refer to [10, Theorem 8.3]. For simplicity, we will only describe the q-expansion

at the standard (unramified) cusp at . For every and every o[ 1 ]-algebra R, the pull-back of to S Spec(R) is canonically isomorphic to o OS R. Thus

H0S Spec(R)/o+, k n0t/2=

c+{0}

aq | a R, au2 = ukk+mta, (u, ) on,1 o+

.

This construction associates to each g Gk(1(c, n); R) an element g =

c+{0}a(g)q

,called the q-expansion ofg at the cusp at . The element a0(g) R is the value ofg at the cuspat .


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PROPOSITION 1.10. LetR be a o[ 1 ]-algebra.(i) (q-expansion Principle) Gk(1(c, n); R) R[[q, c+ {0}]], g g is injective.

(ii) If there exists g Gk(1(c, n); R) such thata0(g) = 0, then k+mt 1 is a zero-divisorin R, for all

o+.

1.8. The minimal compactification

There exists a projective, normal Z[ 1

]-scheme M1, containing M1 as an open densesubscheme and such that the scheme M1\M1 is finite and tale over Z[ 1 ]. Moreover, foreach toroidal compactification M1 of M1 there is a natural surjection M1 M1 inducingthe identity map on M1. The scheme M1 is called the minimal compactification of M1. Theaction ofo+/o

2n,1 on M

1 extends to an action on M1 and the minimal compactification M ofM is defined as the quotient for this action. In general M1 M is not tale.

We summarize the above discussion in the following commutative diagram:

G

M1 M

M1 M

A M1 M

1.9. Toroidal compactifications of KugaSato varieties

Let s be a positive integer. Let s : As M1 be the s-fold fiber product of :A M1 and()s : G

s M1 be the s-fold fiber product of : G M1.Let be a (o c) 11(c, n)-admissible, polarized, equidimensional, smooth collection of

fans, above the 11(c, n)-admissible collection of fans of Section 1.6. Using FaltingsChaismethod [15], the main result of [11, Section 6] is the following: there exists an open immersion ofaA

s into a projective smooth Z[ 1 ]-schemeA

s =

As, and a proper, semi-stable homomorphisms :As M1 extending s :As M1 and such that As\As is a relative normal crossing divisor

above M1\M1. Moreover, As contains Gs as an open dense subscheme and Gs acts on Asextending the translation action ofAs on itself.

The sheafH1log-dR(A/M1) = R11A/M1(dlog ) is independent of the particular choiceof above and is endowed with a filtration:

0 G/M1

H1log-dRA/M1

G/M1 cd1 0.

It descends to a sheafH1log-dR on M which fits in the following exact sequence:

0

H1log-dR

cd1

0.

1.10. Hecke operators on modular forms

Let Z[K1(n)\G(Af)/K1(n)] be the free Abelian group with basis the double cosets ofK1(n)in G(Af). It is endowed with algebra structure, where the product of two basis elements is givenby:

K1(n)xK1(n) K1(n)yK1(n)=

i

K1(n)xiyK1(n)

,(4)


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516 M. DIMITROV

where [K1(n)xK1(n)] =

i K1(n)xi. For g Sk,J(K1(n)) we put:

g|[K1(n)xK1(n)]() =i

g

x1i

.

This defines an action of the algebra Z[K1(n)\G(Af)/K1(n)] on Sk,J(K1(n)) (respectivelyon Gk,J(K1(n))). Since this algebra is not commutative when n = o, we will define acommutative subalgebra. Consider the semi-group:

(n) =

a b

c d

G(Af) M2(o) | dv ov , cv nv for all v dividing n

.

The abstract Hecke algebra of level K1(n) is defined as Z[K1(n)\(n)/K1(n)] endowed withthe convolution product (4). This algebra has the following explicit description.

For each ideal a o we define the Hecke operator Ta as the finite sum of double cosets[K1(n)xK1(n)] contained in the set

{x

(n)

|(x)o = a

}. In the same way, for an ideal a

o

which is prime to n, we define the Hecke operator Sa by the double coset for K1(n) containingthe scalar matrix of the idle attached to the ideal a.

For each finite place v ofF, we have Tv = K1(n)

v 00 1

K1(n) and for each v not dividing n

we have Sv = K1(n)

v 00 v

K1(n), where v is an uniformizer ofFv .

Then, the abstract Hecke algebra of level K1(n) is isomorphic to the polynomial algebra inthe variables Tv , where v runs over the prime ideals ofF, and the variables S1v , where v runsover the prime ideals ofF not dividing n. The action of Hecke algebra obviously preserves thedecomposition (1) and moreover, Sv acts on Sk,J(n, ) as the scalar (v).

Let T(C) be the subalgebra ofEndC(Sk,J(K1(n))) generated by the operators Sv for vnand Tv for all v (we will see in Section 1.13 that T(C) does not depend on J).

The algebra T(C) is commutative, but not semi-simple in general. Nevertheless, for vn theoperators Sv and Tv are normal with respect to the Petersson inner product (see Definition 1.5).

Denote by T

(C) the subalgebra ofT(C) generated by the Hecke operators outside a finite set ofplaces containing those dividing n. The algebra T(C) is semi-simple, that is to say Sk,J(K1(n))has a basis of eigenvectors for T(C).

We will now describe the relation between Fourier coefficients and eigenvalues for theHecke operators. By (2) we can associate to g Sk(K1(n)) a family of classical cuspformsgi Sk(1(ci, n);C), where ci are representatives of the narrow ideal class group Cl+F.

Each form gi is determined by its q-expansion at the cusp of M1(ci, n)an. For eachfractional ideal a = ci, with F+ , we put c(g, a) = ma(gi). By Section 1.7 for each o+,we have a = k+mta and therefore the definition ofc(g, a) does not depend on the choice of (nor on the particular choice of the ideals ci; see [20, IV.4.2.9]).

DEFINITION 1.11. We say that g Sk(K1(n)) is an eigenform, if it is an eigenvector forT(C). In this case g Sk(n, ) for some Hecke character , called the central characterofg.We say that an eigenform g is normalizedifc(g, o) = 1.

LEMMA 1.12 ([24, Proposition 4.1, Theorem 5.2], [20, (4.64)]). If g Sk(K1(n)) is anormalized eigenform, then the eigenvalue ofTa on g is equal to the Fourier coefficientc(g, a).

The pairing T(C) Sk(K1(n)) C, (T, g) c(g|T, o) is a perfect duality.A consequence of this lemma and the q-expansion Principle (see Section 1.7) is the Weak

Multiplicity One Theorem stating that two normalized eigenforms having the same eigenvaluesare equal.


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1.11. Ordinary modular forms

When the weight k is non-parallel, the definition of the Hecke operators should be slightlymodified. We put T0,v = mv Tv and S0,v =

2mv Sv (see [24, Section 3]; in the applications

our base ring will be the p-adic ring O which satisfies the assumptions of this reference).The advantage of the Hecke operators T0,v and S0,v is that they preserve in an optimal way

the O-integral structures on the space of Hilbert modular forms and on the cohomology of theHilbert modular variety.

DEFINITION 1.13. A normalized Hilbert modular eigenform is ordinary at p if for allprimes p ofF dividing p, the image by p of its T0,p-eigenvalue is a p-adic unit.

1.12. Primitive modular forms

For each n1 dividing n and divisible by the conductor of , and for all n2 dividing nn11 we

consider the linear map

Sk(n1, )

Sk(n, ), g

g

|n2,

where g|n2 is determined by the relation c(a, g|n2) = c(an12 , g).We define the subspace Soldk (n, ) of Sk(n, ) as the subspace generated by the images of

all these linear maps. This space is preserved by the Hecke operators outside n. We define thespace Snewk (n, ) of the primitive modular forms as the orthogonal of S

oldk (n, ) in Sk(n, )

with respect to the Petersson inner product (see Definition 1.5). Since the Hecke operatorsoutside n are normal for the Petersson inner product, the direct sum decomposition Sk(n, ) =Snewk (n, ) Soldk (n, ) is preserved by T(C). The Strong Multiplicity One Theorem, due toMiyake in the Hilbert modular case, asserts that if f Snewk (n, ) is an eigenform for T(C),then it is an eigenform for T(C).

A normalized primitive eigenform is called a newform.

1.13. External and Weyl group conjugates

For an element Aut(C) we define the external conjugate ofg Sk(K1(n)) as the uniqueelement g Sk(K1(n)) satisfying c(g, a) = c(g, a) for each ideal a ofo.

We identify {1}JF with the Weyl group K/K+ of G by sending J = (1J, 1JF\J) tocJK+, where for all JF, det(cJ,) < 0 if and only if J. The length ofJ is |J|.

We have an action of the Weyl group on the space of Hilbert modular forms. More precisely, Jacts as the double class [K1(n)cJK1(n)] and maps bijectively Sk(K1(n)) onto Sk,JF\J(K1(n)).The action ofJ commutes with the action of the Hecke operators. For an element g Sk(K1(n))we put gJ = JF\J g.1.14. EichlerShimuraHarder isomorphism

Let R be an O-algebra and let Vn(R) be the polynomial ring over R in the variables(X, Y)JF which are homogeneous of degree n in (X, Y). We have a pairing (perfect

ifn0! is invertible in R) , : Vn(R) Vn(R) R, given by(5) 0jn

ajXnjYj ,

0jn

bj Xnj Yj

=

0jn

(1)j aj bnj

n

j

,

where

n

j

=

JF

nj

.


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The transcendental decomposition (9) has an algebraic interpretation, via the so-called BGGcomplex, that we will describe in the next section.

2. HodgeTate weights of Hilbert modular varieties

The aim of this section is to determine the HodgeTate weights of the p-adic talecohomology of a Hilbert modular variety H(MQp ,Vn(Qp)) as well as those of the p-adic Galoisrepresentation associated to a Hilbert modular form. In all this section we assume

(I) p does not divide = NF/Q(nd).The proof relies on Faltings Comparison Theorem [14] relating the tale cohomology of M

with coefficients in the local system Vn(Qp) to the de Rham logarithmic cohomology of thecorresponding vector bundle Vn over a smooth toroidal compactification M ofM. The HodgeTate weights are given by the jumps of the Hodge filtration of the associated de Rham complex.These are computed, following [15], using the so-called BernsteinGelfandGelfand complex(BGG complex).

Instead of using Faltings Comparison Theorem, one can apply Tsujis results to the talecohomology with constant coefficients of the KugaSato variety As (s-fold fiber product of theuniversal Abelian variety A above the fine moduli space M1 associated to M; see [11, Section 6]for the construction of toroidal compactifications ofAs).

For each subset J ofJF we put p(J) =

J(k0 m 1) +

JF\Jm Z[JF] and

for each a =

JFa Z[JF] we put |a| =

JF

a Z.

2.1. Motivic weight of the cohomology

Consider the smooth sheaf R1Qp on M1, where :A M1 is the universal HBAV. It

corresponds to a representation of the fundamental group of M1 in G(Qp). By composing thisrepresentation with the algebraic representation Vn ofG of highest weight n (see Section 1.14),we obtain a smooth sheaf on M1 (thus on Y1). It descends to a smooth sheaf on Y, denoted by

Vn(Qp).Let Wf =

ao ker(Ta c(f, a)) be the subspace of Hd(YQ,Vn(Qp)) corresponding to the

Hilbert modular newform f Sk(n, ). Put s =

(n + 2m) = dn0.

PROPOSITION 2.1. Wf is pure of weight d + s, that is to say for all prime l p the

eigenvalues of the geometric Frobenius Frobl atl are Weil numbers of absolute value ld+s2 .

Proof. Since f is cuspidal Wf Hd! (YQ,Vn(Qp)). We recall that YQ is a disjoint union ofits connected components MQ = M1(ci, n)Q, where the cis form a set of representatives ofCl

+F.

Let c be one of the cis and M1 = M11 (c, n). For = , c we have

H0

o+/o

2n,1, H

d

M1

Q,Vn(Qp)

= Hd

MQ,Vn(Qp)

,

and therefore, it is enough to prove that Hd! (M1Q

,Vn(Qp)) is pure of weight d + s. We use

Delignes method [4]. Let : A M1 be the universal Abelian variety (see Section 1.4). ThesheafVn(Qp) corresponds to the representation

JF

Symn detm of the group G andcan therefore be cut out by algebraic correspondences in (R1Qp)

s. Let s : As M1 be theKugaSato variety. By the Kunneths formula we have

Hd!

M1Q

,

R1Qps Hd! M1Q, RssQp Hd+s! AsQ,Qp Hd+sAsQ,Qp,


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520 M. DIMITROV

where the middle inclusion comes from the degeneration of the Leray spectral sequenceEi,j2 = H

i(M

1Q

, RjsQp) Hi+j (AsQ,Qp) for = , c (see [4]). The proposition is thena consequence of the Weil conjectures for the eigenvalues of the Frobenius, proved byDeligne [5]. 2

2.2. The BernsteinGelfandGelfand complex over Q

In this and the next sections we describe, following Faltings [13], an algebraic construction ofthe transcendental decomposition of the Betti cohomology described in (9).

All the objects in this section are defined over a characteristic zero field splitting G.Let g, b, tand u denote the Lie algebras ofG, B, T and U, respectively. Consider the canonical

splitting g = b u. Let U(g), U(b) be the enveloping algebras ofg and b, respectively.The aim of this section is to write down a resolution ofVn of the type:

0 Vn U(g) U(b) Kn,

where the Kjn are finite-dimensional semi-simple b-modules, with explicit simple components.

We start by the case n = 0. If we put Kj0 =j

(g/b) we obtain the so-called bar-resolutionof V0. Since

i(g/b) is a b-module with trivial u-action we deduce that Kj0 =

W with

running over the weights ofB that are sums ofj distinct negative roots.By tensoring this resolution with Vn we obtain Koszuls complex:

0 Vn U(g) U(b) j

(g/b) Vn|b

,(10)

which is a resolution ofVn by b-modulesi(g/b) Vn|b, not semi-simple in general.

The BGG complex that we are going to define is a direct factor of Koszuls complex cut bythe action of the center U(g)G ofU(g).

Denote by n the weight n character ofU(g)G. It is a classical result that

LEMMA 2.2. n = if, and only if, there exists J JF such that = J(n + t) t.By taking the n-part of the bar resolution (10) ofVn we obtain a complex:

0 Vn U(g) U(b) Kn, with Kin =

JJF,|J|=i

WJ(n+t)t,(11)

which is still a resolution ofVn, as a direct factor of a resolution. We call this resolution the BGGcomplex.

2.3. HodgeTate decomposition ofH(MQp,Vn(Qp))

In this paragraph we summarize the results of [11, Section 7]. The algebraic groups G, B, Tand D of Section 1 have models over Z, denoted by the same letters. For a scheme S, we putS = S Spec(o[ 1 ]).

By Section 1.9 we can extend the vector bundles and H1dR to M. Only the constructiondepends on a choice of a toroidal compactification :A M1 of : A M1.

The sheafMD = IsomoOM

(, o OM) is a D-torsor over M (for the Zariski topology).We have a functor FD from the category of algebraic representations of D to the category of


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vector bundles on M which are direct sums of invertible bundles. To an algebraic representation

W ofD, FD associates the fiber product W:= MDD W.

The sheafMB = IsomfiloO

M(H1log-dR, (oOM)2) is a B-torsor over M. We have a functor

FB from the category of algebraic representations ofB to the category of filtered vector bundleson M whose graded are sums of invertible bundles. To an algebraic representation V ofB, FBassociates the fiber product V:= MB

B V.A representation ofG (respectively T) can be considered as a representation ofB by restriction

(respectively by making U act trivially). Thus, we may define a filtered vector bundle Vn onM associated to the algebraic representation Vn ofG, and an invertible bundle Wn,n0 on Massociated to the algebraic representation ofT = D D, given by (u, ) unm.

The sheafMG = IsomoOM

(H1log-dR, (oOM)2) is a G-torsor over M. We have a functorFG from the category of algebraic representations ofG to the category of flat vector bundles onM (that is vector bundles endowed with an integrable quasi-nilpotent logarithmic connection).

To any algebraic representation V of B, FG associates the fiber product V := MGG V.

For j N, we put Hjlog-dR(M

, V) = Rj

(V

M(dlog )), where : M

Spec(o

[1 ])

denotes the structural homomorphism.By Faltings Comparison Theorem [14], the GQp -representation H(M1Qp ,Vn(Qp)) is crys-

talline, hence de Rham, and we have a canonical isomorphism

H

M1Qp

,Vn(Qp) BdR = Hlog-dRM1/Qp ,Vn BdR.

By [11, Section 7] the Hodge to de Rham spectral sequence

Ei,j1 = Hi+j

M1/Qp , griVn M1(dlog )= Hi+jlog-dRM1/Qp ,Vn,

degenerates at E1 (the filtration being the tensor product of the two Hodge filtrations). In orderto compute the jumps of the resulting filtration we introduce the BGG complex:

K in =

JJF,|J|=i

WJ(n+t)t,n0 .

The fact that K n is a complex follows from (11) and from the following isomorphism (see [15,Proposition VI.5.1])

HomU(g)

U(g) U(b) W1

,

U(g) U(b) W2 Diff.Op.(W2,W1).(12)

Define a filtration on Kn by Fili Kn =JJF,|p(J)|i

WJ(n+t)t,n0 .The image of Koszuls complex (10) by the contravariant functor W Wis equal to the de

Rham complex. Since the BGG complex is a direct (filtered) factor of the Koszuls complex, weobtain:

THEOREM 2.3 [11, Theorem 7.8]. (i) There is a quasi-isomorphism of filtered complexes

Kn Vn M1(dlog ).


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522 M. DIMITROV

(ii) The spectral sequence given by the Hodge filtration

Ei,j1 = JJF,|p(J)|=iHi+j|J|

M1/Qp , WJ(n+t)t,n0

= Hi+jlog-dRM1/Qp , Vn

degenerates atE1.(iii) For all j d, the HodgeTate weights of the p-adic representation Hj (M1

Qp,Vn(Qp))

belong to the set{|p(J)|, |J|j}.

2.4. Hecke operators on the cohomology

We describe the standard Hecke operator Ta as a correspondence on Y1. We are indebted toM. Kisin for pointing us out that the usual definition of Hecke operators on Y extends to Y1 (see[29, 1.91.11]). Note that the corresponding Hecke action on analytic modular forms for G

(see Section 1.10) is not easy to write down, because the double class for the Hecke operator Tv

does not belong to G

(Af), unless v is inert in F.Recall that Y11 (n) =

h+Fi=1 M

11 (ci, n), where c1, . . . , ch+F

form a set of representatives ofCl+F.

Assume that cia and cj have the same class in Cl+F. Consider the contravariant functor M1a

from the category ofZ[ 1

]-schemes to the category of sets, assigning to a scheme S the set ofisomorphism classes of quintuples (A,,,C,) where (A,,)/S is a ci-polarized HBAVwith n-level structure, C is a closed subscheme of A[a] which is o-stable, disjoint from(Gm d1) and locally isomorphic to the constant group scheme o/a over S, and is ano2n,1-orbit of isomorphisms (cia, (cia)+)

(cj , cj+).We have a projection M1a M1, (A,,,C,) (A,,) which is relatively representable

by 1 : M1a (ci, n) M11 (ci, n). We have also a projection 2 : M1a (ci, n) M11 (cj , n) comingfrom (A,,,C,) (A/C,, ), where is the composed map of and A A/Cand is a cj -polarization ofA/C (defined via and ).

Put Y1a =h+F

i=1 M1a (ci, n). As ci cj cia is a permutation of Cl+F, we get two finite

projections 1, 2 : Y1a Y1:

A

Aaa

A

Y1 Y1a21

Y1

From this diagram we obtain 2H1dR 1H1dR. Therefore, for every algebraic representationV ofG, we have 2V 1V. By composing this morphism by 1 and taking the trace, weobtain

V

1

2

V

1

1

V

V. This gives an action ofTa on H(Y1,

V).

The same way, the above diagram gives 2 1 and 2 1 . Therefore, for eachalgebraic representation W ofT, we get 2W 1W. In order to define the action of Ta onHilbert modular forms, we need to modify the last arrow: we decompose Was (W 2t)2tand we define 2(W2t) 1(W2t) as above and 22t 12t via the KodairaSpencer isomorphism 1Y1 2 o dc1 as in [29, 1.11]. Thus we obtain W 12W 1

1W Wand an action ofTa on H(Y1, W).

In particular, we obtain an action of Ta on the space H0(Y1, k n0t/2) of geometricHilbert modular forms for G. As it has been observed in [29, 1.11.8] over C this action is given


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by the projection

1

[o+: on,1] []o+/on,1

[] : H0

Y1, k n0t/2

H0

Y, k n0t/2

,

followed by the usual Hecke operator on the space of Hilbert modular forms (see Section 1.10).

2.5. HodgeTate weights of IndQF in the crystalline case

We first recall the notion of induced representation. Let V0 be a vector space over a field L,and let 0 : GF GL(V0) be a linear representation. The induced representation IndQF 0 of0from F to Q is by definition the L-vector space

HomGF(GQ, V0) :=

0 : GQ V0 | x GF, y GQ, 0(yx) = 0

x1

0(y)

,

where y GQ acts on 0 HomGF(GQ, V0) by y 0() = 0(y1).For any fixed decomposition

GQ =JF GF, the map 0 (0()) gives an isomor-phism between HomGF(GQ, V0) and the direct sum V (where each V is isomorphic to V0).

Via this identification, the action ofGQ on

V is given by:IndQF 0

(y)

(v)

=

0

1yy

(vy )

,

where y1 yGF. In fact (0()) y(0(y1)) = (0(1yy)(0(y))).Keeping the same notations we define, following Yoshida [44], the tensor induced representa-

tion

IndQF 0 :GQ GL(V) as:IndQF 0

(y)

v

=

0

1yy

(vy ).

Remark 2.4. For each y GQ the map y is a permutation of JF and it is trivial if,and only if, y GF. Therefore, for each y GF, we have ( IndQF 0)(y) = 0(1y).Moreover for all y, y GQ we have (y)y = yy .

DEFINITION 2.5. The internal conjugate g ofg Sk,J(n, ) by JF is defined as theunique element g Sk,J((n), ) satisfying c(g, a) = c(g, (a)) for each ideal a of o,where k =

k

and (a) = ((a)). It is a Hilbert modular form on (F).

If = f,p by the previous remark we have (

IndQF )(y) =

f(y) for all y GF.Brylinski and Labesse [3] have shown (see [40] for this formulation):

THEOREM 2.6 (BrylinskiLabesse). The restrictions to G

F

of the two GQ-representationsWf and

IndQF have the same characteristic polynomial.

COROLLARY 2.7 [11, Corollary 7.9]. (i) The spectral sequence given by the Hodge filtration

Ei,j1 =

JJF,|p(J)|=i

Hi+j|J|

M/Qp , WJ(n+t)t,n0

= Hi+jlog-dR(M/Qp , Vn)

degenerates atE1 and is Hecke equivariant.(ii) The HodgeTate weights ofWf are given by the multiset{|p(J)|, J JF}.


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524 M. DIMITROV

Proof. (i) By taking the invariants of the Hodge filtration of Vn M1

(dlog ) by theGalois group of the tale covering M1 M we obtain a filtration of the complex Vn

M(dlog ) on M, still called the Hodge filtration. The same way, we define the BGG complex

over M by taking the invariants of the BGG complex over M1. The associated spectral sequenceis given by the invariants of the spectral sequence of Theorem 2.3(ii). We have now to see that itis Hecke equivariant.

The Hecke operator Ta extends to a correspondence on Y1. One way to define it is totake the schematic closure of Ta Y1 Y1 in Y1 Y1. Another way is to take a toroidalcompactification Y1a ofY

1a over the toroidal compactification Y1 ofY

1.Hence Ta acts on H(Y1,W) and on H(Y1, V). Moreover, the Tas commute. In fact

they commute on the right-hand side of Theorem 2.3(ii), because this side is independent ofthe toroidal compactification by Faltings Comparison Theorem. Since the spectral sequence ofTheorem 2.3(ii) degenerates at E1, they also commute on the left-hand side.

(ii) We have WJ(n+t)t,n0 = J(n+t)+t p(J). It follows from Theorem 2.3 (as in [15,Theorem 5.5] and [31, Section 2.3]) that the jumps of the Hodge filtration are among |p(J)|,J

J

F.

Moreover gr|p(J)| Hd(M/Qp , Vn M(dlog ) ) = Hd|J|(M/Qp , J(n+t)+t p(J)).It is enough to see that the Qp-vector space H

d|J|(YQp , J(n+t)+t p(J))[f] is of

dimension 1 for all J JF.By the existence of a BGG complex over Q giving by base change the BGG complexes over

Qp and C, we have a Hecke-equivariant isomorphism

Hd|J|

YQp , J(n+t)+t p(J)Qp C= HdYan,Vn(C)[J c].

For all J JF, the f-part ofHd(Yan,Vn(C))[Jc] is equal to Hd! (Yan,Vn(C))[Jc, f]and is therefore one dimensional by (8). 2

Remark 2.8. (1) We proved that Wf is pure of weight d(k0 1). The set of its HodgeTate weights isstable by the symmetry h d(k0 1) h, since |p(JF\J)| = d(k0 1) |p(J)|. Thissymmetry is induced by the Poincar duality Wf Wf Qp(d(k0 1)).

(2) IfF is a real quadratic field and denotes the non-trivial automorphism of F, then theHodgeTate weights ofWf are given by m, k0 m 1, k0 + m 1, 2k0 m 2.

2.6. HodgeTate weights of in the crystalline case

The embedding p :Q Qp allows us to identify JF with HomQalg.(F,Qp). For each primep ofF dividing p, we put JF,p = HomQpalg.(Fp,Qp). Thus we get a partition JF =

p JF,p.

Let Dp (respectively Ip) be a decomposition (respectively inertia) subgroup ofGF at p.The following result is due to Wiles ifk is parallel, and to Hida in the general case.

THEOREM 2.9 (Wiles [43], Hida [23]). Assume thatf is ordinary atp (see Definition 1.13).Then |Dp is reducible and:

(ORD) |Ip

p 0 p

,

where p (respectively p) is obtained by composing the class field theory map Ip op with themap op Qp , x

JF,p

(x)m (respectively x JF,p (x)(k0m1)).4e SRIE TOME 38 2005 N 4

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Breuil [2] has shown that ifp > k0 and p does not divide , then is crystalline at each primep ofF dividing p, with HodgeTate weights between 0 and k0 1.

COROLLARY 2.10. Assume p > k0 and that p does not divide . Then for each prime

p of F dividing p, |Dp is crystalline with HodgeTate weights the 2[Fp : Qp] integers (m,k0 m 1)JF,p .Proof. Assume first that n = 0. Let K be a CM quadratic extension of F, splitting all the

primes p ofF dividing p. Blasius and Rogawski [1] have constructed a pure motive over KwithHodge weights (m, k0 m 1)JF , whose p-adic realization is isomorphic to the restrictionof to GK. This shows that |Dp is de Rham for all p, and crystalline for p big enough.

By Faltings Comparison Theorem the Hodge weights of this motive correspond viap :Q Qp to the HodgeTate weights of its p-adic realization, which are the same as theHodgeTate weights of at primes p dividing p. This proves the corollary for n = 0.

Ifn = 0 (or more generally ifk is parallel) we can complete the proof using the following

LEMMA 2.11. Leta andb be two positive integers and let(a)JF (respectively (b)JF)

be integers satisfying 0

2a < a (respectively 0

2b < b). Assume that the following twomultisets are equalJ

a +

JF\J

(a a), J JF

=

J

b +

JF\J

(b b), J JF

.

Then a = b and we have equality of multisets {a, JF} = {b, JF}.Using this lemma together with Theorem 2.6 and Corollary 2.7(ii) we obtain, up to

permutation, the HodgeTate weights of at primes p dividing p. In particular, we know exactlythe HodgeTate weights of when k is parallel. 2

2.7. FontaineLaffaille weights of in the crystalline case

Our aim is to find the weights of |Ip for p dividing p. If f is ordinary at p we know byTheorem 2.9 that |Dp is reducible and by a simple reduction modulo Pwe obtain the weightsof|Ip .

PROPOSITION 2.12. Assume p > k0 and thatp does not divide . Then is crystalline ateach p dividing p with FontaineLaffaille weights (m, k0 m 1)JF,p .

Proof. It follows from FontaineLaffailles theory [17] and from the computation of HodgeTate weights of|Dp from Section 2.6.

Consider a Galois stable lattice O2 in the crystalline representation , as well as the sub-lattice P2. The representation is equal to the quotient of these two lattices. It is crystalline,as a sub-quotient of a crystalline representation. Its weights are determined by the associated

filtered FontaineLaffaille module. Since the FontaineLaffaille functor is exact, this module isgiven by the quotient of FontaineLaffailles filtered modules associated to the two lattices. Bycompatibility of the filtrations on these two lattices, and by the condition p > k0, the graded ofthe quotient have the right dimension. 2

COROLLARY 2.13. Letp be a prime ofF above p. Then

|Ip

p 0 p

,


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Assume now that k = k0t is parallel and that for all J JF, there exists o+, 1 n,such that p does not divide the non-zero integer NF/Q(p(J) 1). The same arguments as aboveshow that the restriction to

p|p o

p of the character gal (respectively gal) Cl

+F,np is

trivial (respectively given by the (1k0)th power of the norm). By the following lemma (appliedto P = Cl+F,np and Q = (p|p op )/{ o+ | 1 n}) there exists a unique character gal(respectively gal):Cl

+F,np O lifting gal (respectively gal) and whose restriction to

p|p op is trivial (respectively given by the (1 k0)th power of the norm).

LEMMA 3.3. Let P be an Abelian group and Q be a subgroup, such that the factorgroup P/Q is finite. LetP : P andQ : Q O be two characters such that we haveP|Q = Q modp. Then, there exists a unique character P : P O, whose restriction to Qis Q and such thatP modp = P.

For x AF, we put (x) := gal(x) and (x) := gal(x)xkp xk. Then (respectively )is a Hecke character ofF, of conductor dividing n and infinity type 0 (respectively (1 k0)t). Itis crucial to observe that there are only finitely many such and .

Assume now that for infinitely many primes p, is reducible. Then there exist Heckecharacters and as above, such that for infinitely many primes p we have s.s. gal gal(mod P). Hence for each prime v of F not dividing n we have c(f, v) (v) + (v)(mod P) for infinitely many Ps and hence c(f, v) = (v) + (v). By the CebotarevDensity Theorem we obtain s.s. = . This contradicts the irreducibility of. 2

3.2. The exceptional case

The aim of this paragraph is to find a bound for the primesp such that pr((GF)) is isomorphicto one of the groups A4, S4 or A5. We will only use the fact that the elements of these groupsare of order at most 5.

Assume that pr((

GF)) = A4, S4 or A5. By Corollary 2.13 there exist

{1

},

JF,

such that for all p |p and for any generator x ofFph , where h = |JF,p|, the elementGal(Fph/Fp)

(x)(k1) Fph

belongs to pr((Ip)) and is therefore of order at most 5 (if(ORD) holds we may assume that = 1 for all ). Denote by 1, . . . , h the elements ofJF,p. Then

1(k1 1) + 2p(k2 1) + + hph1(kh 1) Z/ph 1

is of order 5, hence 5((k1 1) +p(k2 1) + +ph1(kh 1))ph 1.

If we replace the generator x by xp

, xp2

, . . . , xph1

and then sum these inequalities we find5JF,p(k 1) |JF,p|(p 1). Hence 5JF(k 1) d(p 1).We conclude that pr((GF)) cannot be isomorphic to A4, S4 or A5 if

d(p 1) > 5

JF

(k 1).

Note that this assumption follows from (II) ifd 5.


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528 M. DIMITROV

3.3. The dihedral case

In this paragraph we study the case when pr((GF)) is isomorphic to the dihedral group D2r ,where r 3 is an integer prime to p. Let Cr be the cyclic subgroup of order r of D2r . Sincepr1(Cr) is a commutative group containing only semi-simple elements (p does not divide r), itis diagonalizable. Since pr1(D2r\Cr) is contained in the normalizer ofpr1(Cr), it is containedin the set of anti-diagonal matrices.

Let : D2r {1} be the sign map and let K be the fixed field of ker( pr ). Theextension K/F is quadratic and unramified outside np.

Let c be the non-trivial element of the Galois group Gal(K/F). Since is absolutelyirreducible, but |GK is not, there exists a character gal : GK distinct from its Galoisconjugate cgal and such that |GK = gal cgal.

LEMMA 3.4. Letp be a prime ofF dividing p. Assume that there exists JF,p such thatp = 2k 1. Then the fieldK is unramified atp, andgal is crystalline at primes P ofKabovep of weights (m, k0 m 1)JF,p .

Proof. IfK/F ramifies at p then (Ip) would contain at least one anti-diagonal matrix andthe basis vectors would not be eigen for (Ip). But the group (Ip) has a common eigenvector.Hence, the elements ofpr((Ip)) would be of order 2. Using the computations of Section 3.2and p > k0, we find that for all JF,p we have 2(k 1) =p 1.

By Corollary 2.13, gal cgal is crystalline at P of weights (m, k0 m 1)JF,p and(gal

cgal)|IP =

1k0|IP

. 2

Let O be the ring of integers of K, and O its profinite completion. Denote by On,1 thesubgroup ofO of elements 1 (mod n). Then On,1 is a product of its p-partP|p OP and itspart outside p, denoted by

O

(p)n,1 .

By the global class field theory, the Galois group of the maximal n-ramified (respectively np-

ramified) Abelian extension ofK is isomorphic to ClK,n := A

K/KOn,1K (respectively to

ClK,np :=AK/K

O(p)n,1 K). We have the following exact sequence:1 P|p

OP

/

O | 1 n ClK,np ClK,n 1.(14)PROPOSITION 3.5. (i) Assume that for all JF, p = 2k 1 and thatpr((GF)) is dihedral. LetK/F be the

quadratic extension defined above. Then one of the following holds: K is CM and there exists a Hecke character of K of conductor of norm dividing

n1K/F and infinity type (m, k0 1 m)JF such that IndFK (modP),

K is not CM and we can choose places of K above each

JF such that for all

O, 1 n the prime p divides NK/Q(JF ()m(c())k0m1 1).(ii) Assume that f is not a theta series. Then for all but finitely many primes p the group

pr((GF)) is not dihedral.Remark 3.6.

(i) The primes p for which the congruence IndFK (modP) may occur should becontrolled by the value at 1 of the L-function associated to the CM character /c (inthe elliptic case it is proved by Hida [22] and Ribet [36]; see also Theorems A and B).


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(ii) We would like to thank E. Ghate for having pointed us out the possible existence ofdihedral primes for non-CM fields K. It would be interesting to explore the conversestatement, that is to say to try to construct for a given prime p dividing the above norms, anewform f such that pr((

GF)) is dihedral.

Proof. (i) By (14) and the above lemma, we have gal : ClK,np whose restrictiontoP|p O

P is given by the reduction modulo p of an algebraic character x xk , where

k =

JFm + (k0 m 1) c, for some choice of places ofK above JF.

We observe that the character x xk is trivial on o+, whereas it is only trivial modulo p on{ O | 1 n}. The case when K is not CM follows immediately.

Assume now that K is CM. In this case { o+ | 1 n} is a finite index subgroup of{ O | 1 n}. Since ker(O ) does not contain elements of finite order, the abovecharacter is trivial on { O | 1 n}.

By Lemma 3.3 (applied to P = ClK,np and Q = (P|p O

P)/{ O | 1 n}) there

exists a lift gal : ClK,np O whose restriction to

P|p O

P is given by x xk .

We put (x) := gal(x)xk

p xk. Then is a Hecke character ofK as desired.(ii) There are finitely many fields K as above. For those K that are not CM it is enough to

choose O, 1 n, of infinite order in O/o.For each of the CM fields K that are only finitely many characters as above. Therefore, if

pr((GF)) is dihedral for infinitely many primes p, then there would exist K and as above,such that the congruence IndFK (modP) happens for infinitely many Ps. Hence f wouldbe equal to the theta series associated to . 2

3.4. The image of is large

THEOREM 3.7 (Dickson). (i) An irreducible subgroup ofPSL2() of order divisible byp is conjugated inside PGL2()

to PSL2(Fq) or to PGL2(Fq), for some powerqofp.

(ii) An irreducible subgroup ofPSL2() of order prime to p is either dihedral, or isomorphicto one of the groups A4, S4 orA5.

As an application of this theorem, Propositions 3.1, 3.5 and Section 3.2 we obtain the following

PROPOSITION 3.8. Assume that f Sk(n, ) is a newform, which is not a theta series.Then for all but finitely many primes p, the image of is large in the following sense:

(LI) there exists a powerqofp such thatSL2(Fq) im() GL2(Fq).Let F be the compositum of F and of the subfield ofQ fixed by the Galois group ker(n0).

The extension F /F is Galois and unramified atp, since F is unramified atp and is of conductorprime to p. Therefore G

F

is a normal subgroup ofGF containing the inertia subgroups Ip, forall p dividing p.

We putD

= det((GF) ) = (Fp )1k0 .

PROPOSITION 3.9. Assume (LI). Then there exists a powerqofp such that, either

(GF) = G L2(Fq)D := x GL2(Fq) | det(x) D, orUU(GF) = Fq2 GL2(Fq)D := x Fq2 GL2(Fq) | det(x) D.

Proof. We first show that pr((GF)) is still irreducible of order divisible by p. By (LI) thegroup pr((GF)) is isomorphic to PSL2(Fq) or PGL2(Fq). The group pr((GF)) is a non-trivialANNALES SCIENTIFIQUES DE LCOLE NORMALE SUPRIEURE

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530 M. DIMITROV

normal subgroup ofpr(im()) (because it contains pr((Ip)) and p > k0; see Corollary 2.13).Since PSL2(Fq) is a simple group of index 2 in the group PGL2(Fq), we deduce

PSL2(Fq)

pr (GF) pr (GF) PGL2(Fq).LEMMA 3.10. Let H be a group of center Z and let pr : H H/Z the canonical

projection. Let P and Q be two subgroups of H such that pr(P) pr(Q). Assume moreoverthatQ does not have non-trivial Abelian quotients. Then P Q.

It follows from this lemma that (GF) SL2(Fq), hence GL2(Fq)

D (GF) GL2(Fq)D.Since [( GL2(Fq))D : GL2(Fq)D] 2 we are done. 2

Let Fq2\Fq be such that 2 Fq . Then (Fq2 GL2(Fq))D = GL2(Fq)D ( GL2(Fq))D

and hence tr((Fq2 GL2(Fq))D) = Fq Fq . Therefore, the Fp-algebra generated by the traces ofthe elements of(Fq2 GL2(Fq))

D is Fq2 , while pr((Fq2 GL2(Fq))

D) PGL2(Fq). This reflectsthe existence of a congruence with a form having inner twists.

3.5. The image ofIndQF is large

We assume in this paragraph that (LI) holds.By Proposition 3.9 there exists a power qofp such that pr((GF)) = PSL2(Fq) or PGL2(Fq).

Consider the representation pr(IndQF ) : GF PGL2(Fq)JF . An automorphism of the simplegroup PSL2(Fq) is a composition of a conjugation by an element of PGL2(Fq) with anautomorphism ofFq . By a lemma of Serre (see [35]), there exist a partition JF =

iI J

iF

and for all i I, JiF, an element i, Gal(Fq/Fp) such that

pr

SL2(Fq)I prIndQF (GF) prGL2(Fq)I,

where = (i)iI : GL2(Fq)I GL2(Fq)JF is given by i(Mi) = (Mi,i )JiF .Keeping these notations, we introduce the following assumption on the image ofIndQF :(LIInd) the condition (LI) holds and i I, , JiF (= = i, = i,).We now introduce a genericity assumption on the weight k.

DEFINITION 3.11. We say that the weight k Z[JF] is non-induced, if there do not exist astrict subfield F ofF and a weight k Z[JF ] such that for each JF, k = k|F .

Remark 3.12. Define k =

J

Fk Z[J

F] by putting k = k|F , for all J

F. The

group GQ acts on Z[JF] by k =JF k k =JF k . It is easy to see thatk Z[JF] is non-induced if, and only if, { GQ | k = k} equals GF.

PROPOSITION 3.13. Assume that(LI) holds andk is non-induced. Assume moreover thatfor all = JF, p = k + k 1. Then (LIInd) holds.

Proof. Let 1, 2 GQ be such that for all y GF we have pr((11 y1)) = pr((12 y2)).We have to prove that 11 2 GF. For i = 1, 2, let i(y) = (1i yi).


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Let P be a prime ideal of F above a prime ideal p of F dividing p. By Corollary 2.13 wehave i| s.s.IP = i i, where i and i are two tame characters whose product equals 1k0 andwhose sum has FontaineLaffaille weights (m, k0 m 1)JF,pi .

Since IP GF and pr 1 = pr 2 on GF, we have 1/1 = 2/2. By varying P we deducethat for all JF, k = k11 2 (here we use that p > k0 and p = k + k 1). Since k is non-

induced, it follows from Remark 3.12 that 11 2 GF. 2The following corollary generalizes a result of Ribet [35] on the image of a Galois

representation associated to a family of classical modular forms, to the case of the family ofinternal conjugates of a Hilbert modular form.

COROLLARY 3.14. Assume that(LI) holds andk is non-induced. Assume moreover thatp > 2k0 is totally split in F. Then,

GL2(Fq)

JF

D IndQF (G

F

)

(G

F

)JF

D

, where D =

Fp

1k0

.

Put

H(Fq) =

iI

GL2(Fq)

D:=

(Mi)iI

iI

GL2(Fq) | D, i, det(Mi) =

.

LEMMA 3.15. Assume that(LI) holds andp > 2k0. Then,(i) for all p dividing p, (Ip) is contained (possibly after conjugation by an element of

GL2(Fq)) either in the Borel subgroup ofGL2(Fq), or in the non-split torus ofGL2(Fq).The second case cannot occur iff is ordinary atp.

(ii) IndQF (Ip) (H(Fq)).

Proof. (i) By Corollary 2.13 | s.s.Ip = pp, where p, p : Ip F

p are two tame characters

of level h :=

|JF,p

|or 2

|JF,p

|whose product equals 1k0 and whose sum has Fontaine

Laffaille weights (m, k0 m 1)JF,p .Let xh be a generator ofF

ph

, and let and be the characters ofFph

deduced from p and p.Since by (LI) the traces of the elements of(GF) are in FqFq (see Section 3.4), we deducethat ((xh) + (xh))2 Fq and hence (xh)2 + (xh)2 Fq .

If (xh)2, (xh)2 Fq , then it is easy to see that (xh), (xh) Fq (we use p > k0 andp = 2k 1). In this case Ip fixes a Fq-rational line and therefore (Ip) is contained in a Borelsubgroup ofGL2(Fq).

Otherwise (xh)2 and (xh)2 are conjugated by the non-trivial element of Gal(Fq2/Fq),hence (xh)2 = (xh)2q . Since p > 2k0, we have (xh) = (xh)q and so (xh) + (xh)q Fq .Hence tr((Ip)) Fq and (Ip) GL2(Fq). In this case (Ip) is contained in a non-split torusofGL2(Fq). If f is ordinary at p, then the FontaineLaffaille weights of p are strictly smallerthan those ofp, and therefore the second case cannot occur.

(ii) The determinant condition D being satisfied, all we have to check is the following: for alli I and , JiF the character

Ip {1}, y

1i,yi,1

1i,yi,

is trivial. This follows, as in the proof of Proposition 3.13, from the fact that p > 2k0. 2

LEMMA 3.16. Assume that(LI) holds. Then (H(Fq)) IndQF (GF).ANNALES SCIENTIFIQUES DE LCOLE NORMALE SUPRIEURE

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532 M. DIMITROV

Proof. We have seen in the beginning of this paragraph that

pr

SL2(Fq)

I

pr

IndQF (G

F

)

.

By Lemma 3.10, we deduce that (PSL2(Fq)I) IndQF (GF).Since (H(Fq)) = (SL2(Fq)I)Ind

QF (Ip), we are done. 2

PROPOSITION 3.17. Assume that (LI) holds but (LIInd) fails for some p > 2k0(respectively for infinitely many primes p). Then, there exist JF, = idF and a finiteorder Hecke character of F of conductor dividing NF/Q(n) such that for all primes v ofF not dividing NF/Q(n)p we have c(f, v) (v)c(f, v) (mod P) (respectively c(f, v) =(v)c(f, v)).

Proof. Since (LI) holds but (LIInd) fails, there exist 1, 2 GQ such that

:= 12 1|F = idF

and such that pr (12 y1) = pr (12 y2), for all y GF. Since GF is a normal subgroupof GF, the above relation holds for every y GF. Therefore, there exists a charactergal :GF such that for all y GF, f(y) = gal(y)f(y). Since p > 2k0, the sameargument as in the proof of Proposition 3.13 shows that gal is unramified at primes dividing p.By Lemma 3.3, gal can then be lifted to a finite order Hecke character of F of conductordividing NF /Q(n). By evaluating at Frobv , for every prime vNF/Q(n)p of F, we obtain

c(f, v) (v)c(f, v) (mod P).

By the determinant relation = 2gal, there are finitely many such characters . Therefore,if(LIInd) fails for infinitely many primes p, then the above congruence will be an equality. 2

COROLLARY 3.18. Assume thatF is a Galois field of odd degree and the central character off is trivial (F = F). Assume moreover thatf is not a theta series and that(LIInd) doesnot hold for infinitely many primes p. Then, there exist a subfieldF F and a Hilbert modularform f on F, such that the base change off to F is a twist off by a quadratic character ofconductor dividing NF/Q(n).

Proof. As in the proof of Proposition 3.17 there exist a quadratic character of F ofconductor dividing NF/Q(n) and idF = Gal(F/Q) such that f = gal = f. LetF F (respectively Fi F) be the fixed field of (respectively of ker(i)). We know thatF/F is a cyclic extension ofodddegree h. Let F =

hi=1 Fi. Then we have

Gal(F/F) =

(u1, . . . , uh) {1}h |

h

i=1ui = 1

i | 0 i h 1

,

where acts on (u1, . . . , uh) by cyclic permutation. When h = 3 the group Gal(F/F) isisomorphic to A4.

The representation |GF is invariant by Gal(F/F), but Langlands Cyclic Descent doesnot apply directly because the order of Gal(F/F) is even. Consider the quadratic character= 2 h1 . Then the GF-representation gal is invariant by the group Gal(F/F),so extends to a representation ofGF . By applying Langlands Cyclic Descent to f we obtainf as desired. 2


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4. Boundary cohomology and congruence criterion

We recall that f Sk(n, ) is supposed to be a Hilbert modular newform.

DEFINITION 4.1. We say that a normalized eigenform g Sk(n, ) is congruent tof modulo P if their respective eigenvalues for the Hecke operators (that is their Fouriercoefficients) are congruent modulo P.

We say that a prime P is a congruence prime for f if there exists a normalized eigenformg Sk(n, ) distinct from f and congruent to f modulo P.

One expects that, as in the elliptic modular case (carried out by Hida [21,22] and Ribet [36]),the congruence primes for f are controlled by the value at 1 of the adjoint L-function off. Suchresults have been obtained by Ghate [18] when k is parallel.

Following [21,18] and using a vanishing result of the boundary cohomology of a Hilbertmodular variety we obtain a new result in this direction (see Theorems 4.11 and 6.7(ii)).

4.1. Vanishing of certain local components of the boundary cohomology

We introduce the following condition:(MW) the middle weight |p(JF)|+|p()|

2= d(k01)

2does not belong to {|p(J)|, J JF}.

This condition is automatically satisfied when the motivic weight d(k0 1) is odd, or whend = 2 and k is non-parallel.

LEMMA 4.2. Let0 be a representation ofGF on a finite-dimensional -vector space W.Assume that for every y GF, the characteristic polynomial of( IndQF )(y) annihilates 0(y).

(i) If(I), (II) and(LI) hold, then for all h Z the weights h andd(k0 1) h occur withthe same multiplicity in each GF-irreducible subquotient of0.

(ii) If (I), (Irr) and (MW) hold and p 1 > max(1, 5d )

JF(k 1), then each GF-

irreducible subquotient of0 contains at least two different weights for the action of thetame inertia atp.

Proof. We may assume that 0 is irreducible.(i) By Lemmas 3.15(ii) and 3.16 we have IndQF (Ip) (H(Fq)) IndQF (GF). Let T be

the torus ofH(Fq) containing the image of the tame inertia, and N be the normalizer ofT inH(Fq). The image ofN/T = {1}I by is the subgroup of the Weyl group N/T = {1}JFofG containing the elements which are constant on the partition JF =

iI J

iF. In particular,

the longest Weyl element JF belongs to the image ofN/T.

Let x W be an eigenvector for the action ofT. By the annihilation condition, there exists asubset Jx JF, such that Ip acts on x by the weight |p(Jx)|.

Let yJF GF be such that IndQF (yJF) = JF mod T. Then 0(yJF)(x) is of weight|p(JxJF)| = d(k0 1)|p(Jx)|. Therefore, for each h Z, 0(yJF) gives a bijection betweenthe eigenspaces for the tame inertia of weight h and d(k0 1) h.

(ii) If (LI) holds, then the statement follows from (i) and (MW). Otherwise, by Proposi-

tion 3.8 the group pr((GF)) is dihedral. Since F is totally real, pr((GF)) is also dihedral (seeSection 3.3).Denote by N the normalizer of the standard torus T in G. Put N = IndQF (GF) N() and

T = N T(). Then N/T is a subgroup of the Weyl group {1}JF = N/T ofG.As we have seen in Section 3.3, the representation IndQF is tamely ramified at p and the

image of the inertia group Ip is contained in T.Let x W be an eigenvector for the action ofT. By the annihilation condition, there exists

a subset Jx JF, such that Ip acts on x by the weight |p(Jx)|. For every element J N/T,


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534 M. DIMITROV

(J JF), let yJ GF be such that IndQF (yJ) = J mod T. Then 0(yJ)(x) is of weight|p(JxJ)|. It remains to show that the |p(JxJ)| are not all equal when J runs over theelements of N/T. Note that, for all JF, the -projection N/T {1} is a surjectivehomomorphism (because the group pr(f(

GF)) is also dihedral). Therefore, we have:JN/T

p(JxJ)= |N/T|d(k0 1)2

.

The statement now follows from the (MW) assumption. 2

Remark 4.3. The first part of the previous lemma is a generalization from the quadratic tothe arbitrary degree case of the key lemma in [8]. This lemma is false in general under the onlyassumptions (I), (II) and (Irr) when d 3. In fact, consider the following construction in thecubic case: let L be a Galois extension ofQ of group A4, such that the cubic subfield F fixed bythe Klein group is totally real; let Kbe a quadratic extension ofF in L and consider a theta seriesf of weight (2, 2, 2) attached to a Hecke character of K; then the tensor induced representation IndQF has two irreducible four-dimensional subquotients of HodgeTate weights (0, 2, 2, 2)and (1, 1, 1, 3).

As in the introduction, let T T denote the subalgebra generated by the Hecke operatorsoutside a finite set of places containing those dividing np.

THEOREM 4.4. Assume that(I), (Irr) and(MW) hold, and

p 1 > max

1,5

d

JF

(k 1).

Denote by m the maximal ideal ofT corresponding to f andp and putm = m T. Then(i) the m

-torsion of the boundary cohomology H(Y,Vn())[m

] vanishes,(ii) the Poincar pairing Hd! (Y,Vn(O))m Hd! (Y,Vn(O))m O is a perfect duality offree O-modules of finite rank,

(iii) H(Y,Vn(O))m = Hc (Y,Vn(O))m = H! (Y,Vn(O))m .

Proof. (i) Consider the minimal compactification YQj

YQ

i YQ

. The Hecke correspon-dences extend to Y

Q. By the Betti-tale comparison isomorphism, we identify (in a Hecke-

equivariant way) the following two long exact cohomology sequences:

Hrc (Y,Vn()) Hr(Y,Vn()) Hr(Y,Vn())

Hr(YQ , j!Vn()) Hr(YQ , jVn()) Hr(YQ , iRjVn())

Consider the GQ-module Wr = Hr(YQ , iRjVn()). We have to show that Wr[m] = 0.By the Cebotarev Density Theorem and the congruence relations at totally split primes of F,

we can apply Lemma 4.2 to Wr[m]. Therefore each GF-irreducible subquotient ofWr[m] has

at least two different weights for the action of the tame inertia at p. So it is enough to show thateach GQ-irreducible subquotient ofWr is pure (= contains a single weight for the action of the


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tame inertia at p). Since YQ

is zero-dimensional, the spectral sequence

H

Y

Q, iRjVn()

= H

Y

Q, iRjVn()

shows that Wr = H0(YQ , iRrjVn()).Since H0(Y

Q, iRrjVn()) is a subquotient of H0(Y

1,

Q, iRrjVn()) it is enough to

show that each GQ-irreducible subquotient of this last is pure.This will be done using a result of Pink [32]. We replaced Y by Y1, since the group G does

not satisfy the conditions of this reference, while G satisfies them.Consider the decomposition T = Dl Dh, according to

u 00 u1

=

u 00 u1

00 1

.

Put 1 = 11(c, n). By [32, Theorem 5.3.1], the restriction of the tale sheaf iRrjVn(Fp) to a

cusp C = ofY1,

Q is the image by the functor of Pink of the 11 B/11 DlU-module

a+b=r

Ha

11 Dl, Hb

11 U,Vn(Fp)

.

Under the assumption (II), a modulo p version of a theorem of Kostant (see [33])gives an isomorphism of T-module Hb(11 U,Vn(Fp)) =

|J|=b WJ(n+t)t,n0 . By

decomposing WJ(n+t)t = WlJ(n+t)t,n0

WhJ(n+t)t,n0 according to T = Dl Dh, weget

Ha

11 Dl, Hb

11 U,Vn(Fp)

=|J|=b Ha

11 Dl, WlJ(n+t)t,n0 W

hJ(n+t)t,n0

,

where Galois acts only on the second factors of the right-hand side.Therefore H0(Y

Q, iRrjVn(Fp)) is a direct sum of subspaces H0(C, WhJ(n+t)t,n0(Fp)),

|J| r, each containing a single FontaineLaffaille weight, namely the weight |p(J)|.(ii) Since the Poincar duality is perfect over E, it is enough to show that the m-localization

of natural map Hd(O)/Hd! (O) Hd(E)/Hd! (E) is injective. For this, it is sufficient to showthat Hd(O)m := Hd(M ,Vn(O))m is torsion free, which would follow from the vanishingof Hd1 (E/O)m . We have a surjection Hd1 ()m Hd1 (E/O)m [], where is anuniformizer ofO. Finally, by (i) and Nakayamas lemma, Hd1 ()m = 0.

(iii) The vanishing ofH()m gives the vanishing ofH(O)m = 0. 2

4.2. Definition of periods

By taking the subspace ao ker(Ta c(f, a)) of (8) we obtainJ :CfJ

Hd!

Yan,Vn(C)

[J, f].

Fix an isomorphism C = Qp compatible with p. We recall that Hd! (Yan,Vn(O)) denotesthe image of the natural map Hdc (Y

an,Vn(O)) Hd(Yan,Vn(C)). Since O is principal, theO-module Lf,J := Hd! (Yan,Vn(O))[J, f] is free of rank 1. We fix a basis (f, J) ofLf,J.


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536 M. DIMITROV

DEFINITION 4.5. For each J JF we define the period (f, J) = J(fJ)(f,J) C/O. Wefix J0 JF and put +f = (f, J0) and f = (f, JF\J0).

Remark 4.6. The periods f

differ from the ones originally introduced by Hida in [21].Hidas periods put together all the external conjugates of f. Our slightly different definition ismotivated by the congruence criterion that we want to show (Theorem 4.11). Since we c

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