GALOIS REPRESENTATIONS MODULO P and COHOMOLOGY of HILBERT MODULAR VARIETIES

Ann. Scient. Éc. Norm. Sup., 4e série, t. 38, 2005, p. 505 à 551.

GALOIS REPRESENTATIONS MODULO p AND COHOMOLOGY OF HILBERT MODULAR VARIETIES

BY MLADEN DIMITROV

ABSTRACT. – The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let us mention: • control of the image of Galois representations modulo p, • Hida’s congruence criterion outside an explicit set of primes, • freeness of the integral cohomology of a Hilbert modular variety over certain local components of the Hecke algebra and Gorenstein property of these local algebras. We study the arithmetic properties of Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of inertia groups at primes above p. In order to determine this action, we compute the Hodge–Tate (resp. Fontaine–Laffaille) weights of the p-adic (resp. modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of Siegel modular varieties and builds upon geometric constructions of Tilouine and the author.  2005 Elsevier SAS

RÉSUMÉ. – Le but de cet article est de généraliser certains résultats arithmétiques sur les formes modulaires elliptiques au cas des formes modulaires de Hilbert. Parmi ces résultats citons : • détermination de l’image de représentations galoisiennes modulo p, • critère de congruence de Hida en dehors d’un ensemble explicite de premiers, • liberté de la cohomologie entière de la variété modulaire de Hilbert sur certaines composantes locales de l’algèbre de Hecke et la propriété de Gorenstein de celles-ci. L’étude des propriétés arithmétiques des formes modulaires de Hilbert se fait à travers leurs représenta- tions galoisiennes modulo p et l’outil principal est l’action des groupes d’inertie aux premiers au-dessus de p. Cette action est déterminée par le calcul des poids de Hodge–Tate (resp. Fontaine–Laffaille) de la cohomologie étale p-adique (resp. modulo p) de la variété modulaire de Hilbert. La partie cohomologique de cet article est inspirée par le travail de Mokrane, Polo et Tilouine sur la cohomologie des variétés modulaires de Siegel et repose sur des constructions géométriques de Tilouine et l’auteur.  2005 Elsevier SAS

Contents

0 Introduction ...... 506 0.1 Galoisimageresults...... 507 0.2 Cohomological results ...... 507 0.3 Arithmeticresults...... 508 0.4 Explicitresults...... 509 1 Hilbert modular forms and varieties ...... 510 1.1 Analytic Hilbert modular varieties ...... 510

1.2 Analytic Hilbert modular forms ...... 510 1.3 Hilbert–BlumenthalAbelianvarieties...... 512 1.4 Hilbert modular varieties ...... 512 1.5 Geometric Hilbert modular forms ...... 513 1.6 Toroidal compactifications ...... 514 1.7 q-expansionandKoecherPrinciples...... 514 1.8 Theminimalcompactification...... 515 1.9 ToroidalcompactificationsofKuga–Satovarieties...... 515 1.10 Hecke operators on modular forms ...... 515 1.11 Ordinary modular forms ...... 517 1.12 Primitive modular forms ...... 517 1.13 ExternalandWeylgroupconjugates...... 517 1.14 Eichler–Shimura–Harderisomorphism...... 517 2 Hodge–Tate weights of Hilbert modular varieties ...... 519 2.1 Motivic weight of the cohomology ...... 519 2.2 The Bernstein–Gelfand–Gelfand complex over Q ...... 520 • ⊗ Q V Q 2.3 Hodge–Tate decomposition of H (M p, n( p)) ...... 520 2.4 Hecke operators on the cohomology ...... 522 ⊗ Q 2.5 Hodge–Tate weights of IndF ρ in the crystalline case ...... 523 2.6 Hodge–Tate weights of ρ in the crystalline case ...... 524 2.7 Fontaine–Laffaille weights of ρ¯ in the crystalline case ...... 525 Q 3 Study of the images of ρ¯ and IndF ρ¯ ...... 526 3.1 Lifting of characters and irreducibility criterion for ρ¯ ...... 526 3.2 Theexceptionalcase...... 527 3.3 Thedihedralcase...... 528 3.4 The image of ρ¯ is“large”...... 529 Q 3.5 The image of IndF ρ¯ is“large”...... 530 4 Boundary cohomology and congruence criterion ...... 533 4.1 Vanishing of certain local components of the boundary cohomology ...... 533 4.2 Definition of periods ...... 535 4.3 Computationofadiscriminant...... 536 4.4 Shimura’s formula for L(Ad0(f), 1) ...... 536 4.5 Construction of congruences ...... 538 5 Fontaine–Laffaille weights of Hilbert modular varieties ...... 539 5.1 The BGG complex over O ...... 539 5.2 TheBGGcomplexfordistributionsalgebras...... 541 5.3 BGGcomplexforcrystals...... 542 6 Integral cohomology over certain local components of the Hecke algebra ...... 544 6.1 Thekeylemma...... 544 6.2 Localized cohomology of the Hilbert modular variety ...... 545 6.3 OntheGorensteinpropertyoftheHeckealgebra...... 546 6.4 An application to p-adic ordinary families ...... 547 Listofsymbols...... 549 Acknowledgements...... 549 References...... 550

0. Introduction

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

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 507 For a weight k = k τ ∈ Z[J ] as in Definition 1.1 we put k =max{k | τ ∈ J }.If τ∈JF τ F 0 τ F ψ is a Hecke character of F of conductor dividing n and type 2 − k0 at infinity, we denote by Sk(n,ψ) the corresponding space of Hilbert modular cuspforms (see Definition 1.3). Let f ∈ Sk(n,ψ) 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. Q → Q Let p be a prime number and let ιp : p be an embedding. Denote by E a sufficiently large p-adic field with ring of integers O, maximal ideal P and residue field κ.

0.1. Galois image results

The absolute Galois group of a field L is denoted by GL. By results of Taylor [40,41] and Blasius and Rogawski [1] there exists a continuous representation ρ = ρf,p : GF → GL2(E) which is absolutely irreducible, totally odd, unramified outside np and such that for each prime ideal v of o, not dividing pn,wehave: tr ρ(Frobv) = ιp c(f,v) , det ρ(Frobv) = ιp ψ(v) NF/Q(v), where Frobv denotes a geometric Frobenius at v. By taking a Galois stable O-lattice, we define ρ¯ = ρf,p mod P : GF → GL2(κ), 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 of Serre [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, (Irrρ¯)¯ρ is absolutely irreducible. (ii) If f is not a theta series, then for all but finitely many primes p, × (LIρ¯) there exists a power q of p such that SL2(Fq) ⊂ im(¯ρ) ⊂ κ GL2(Fq). (iii) Assume that f 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, i ∈ i (LIInd¯ρ) there exist a power q of p, a partition JF = i∈I JF and for all τ JF an element Q J σ ∈Gal(F /F ) such that (τ = τ =⇒ σ = σ ) and Ind ρ¯: G → SL (F ) F factors i,τ q p i,τ i,τ F F 2 q G F I → σi,τ as a surjection SL2( q) followed by the map (Mi)i∈I (M ) ∈ ∈ i , where F F i i I,τ JF Q − 1 F JF denotes the compositum of F and the fixed field of (IndF ρ¯) (SL2( q) ).

0.2. Cohomological results

Let Y Z 1 be the Hilbert modular variety of level K1(n) (see Section 1.4). Consider the / [ ∆ ] • V Q V Q p-adic étale cohomology H (YQ, n( p)), where n( p) denotes the local system of weight n = ∈ (kτ − 2)τ ∈ N[JF ] (see Section 2.1). By a result of Brylinski and Labesse [3] the τ JF subspace W := ker(T − c(f,a)) of Hd(Y , V (Q )) is isomorphic, as G -module and f a⊂o a Q n p F Q after semi-simpliﬁcation, to the tensor induced representation IndF ρ. Assume that (I) p does not divide ∆. Z ⊂ Then Yhas smooth toroidal compactiﬁcations over p (see [10]). For each J JF , we put |p(J)| = (k − m − 1) + m , where m =(k − k )/2 ∈ N. By applying τ∈J 0 τ τ∈JF\J τ τ 0 τ a method of Chai and Faltings [15, Chapter VI] one can prove (see [11, Theorem 7.8, Corollary 7.9]).

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 508 M. DIMITROV

THEOREM 0.2. – Assume that p does not divide ∆. Then j V Q (i) the Galois representation H (YQ, n( p)) is crystalline at p and its Hodge–Tate weights belong to the set {|p(J)|,J⊂ JF , |J| j}, and (ii) the Hodge–Tate weights of Wf are given by the multiset {|p(J)|,J⊂ JF }. For our main arithmetic applications we need to establish a modulo p version of the above theorem. This is achieved under the following additional assumption: (II) p − 1 > (k − 1). τ∈JF τ The integer (k − 1) is equal to the difference |p(J )|−|p(∅)| between the largest and τ∈JF τ F smallest Hodge–Tate weights of the cohomology of the Hilbert modular variety. We use (I) and (II) in order to apply Fontaine–Laffaille’s Theory [17] as well as Faltings’ Comparison Theorem modulo p [14]. By adapting to the case of Hilbert modular varieties some techniques developed by Mokrane, Polo and Tilouine [31,33] for Siegel modular varieties, such as the construction of an integral Bernstein–Gelfand–Gelfand complex for distribution algebras, we compute the • Fontaine–Laffaille weights of H (YQ, Vn(κ)) (see Theorem 5.13).

0.3. Arithmetic results

O d V O Consider the -module of interior cohomology H! (Y, n( )) , deﬁned as the image of d V O d V T O ⊂ Hc (Y, n( )) in H (Y, n(E)).Let = [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 places containing those dividing np. Denote by m the maximal ideal of T corresponding to f and ιp and put 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 (k − 1) and ρ¯ τ∈JF τ | | | ∅ | − p(JF ) + p( ) d(k0 1) {| | ⊂ } (MW) the middle weight 2 = 2 does not belong to p(J) ,J JF , then • V O the local component H∂ (Y, n( ))m of the boundary cohomology vanishes, and the Poincaré d V O × d V O →O pairing H! (Y, n( ))m H! (Y, n( ))m is a perfect duality. • d (ii) If (LIInd ρ¯) holds, then H (Y,Vn(O))m =H (Y,Vn(O))m is a free O-module of finite d rank and its Pontryagin dual is isomorphic to H (Y,Vn(E/O))m . The proof involves a “local–global” Galois argument. The first part is proved in Theorem 4.4 using Lemma 4.2(ii) and a theorem of Pink [32] on the étale cohomology of a local system restricted to the boundary of Y . The second part is proved in Theorem 6.6 using Lemma 6.5 and the computation of the Fontaine–Laffaille weights of the cohomology from Theorem 5.13. The technical assumptions are needed in the lemmas. Since the conclusion of Lemma 6.5 is stronger then 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 ρ¯). Let L∗(Ad0(f),s) be the imprimitive adjoint L-function of f and let Γ(Ad0(f),s) be ∈ C× O× the corresponding Euler factor (see Section 4.4). We denote by Ωf / any two complementary periods defined by the Eichler–Shimura–Harder isomorphism (see Section 4.2).

THEOREM A (Theorem 4.11). – Let f and p be such that (I), (Irrρ¯) and (MW) hold, and ∗ 5 Γ(Ad0(f),1)L (Ad0(f),1) p − 1 > max(1, ) (k − 1). Assume that ι ( + − ) ∈P. Then there d τ∈JF τ p Ωf Ωf exists another normalized eigenform g ∈ Sk(n,ψ) such that f ≡ g (mod P), in the sense that c(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, and uses Theorem 0.3(i) as well as a formula of Shimura relating L∗(Ad0(f), 1) to the Petersson

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 509 inner product of f (see (19)). Let us note that Ghate [18] has obtained a very similar result when the weight k is parallel. A converse for Theorem A is provided by the second part of the following

THEOREM B (Theorem 6.7). – Let f and p be such that (I), (II) and (LIInd ρ¯) hold. Then • d d (i) H (Y,Vn(κ))[m]=H (Y,Vn(κ))[m] is a κ-vector space of dimension 2 . • d d (ii) H (Y,Vn(O))m =H (Y,Vn(O))m is free of rank 2 over Tm. (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 is proved by Diamond [8] when F is quadratic and k =(2, 2) under the assumptions (I), (II) and (Irrρ¯). We expect that Diamond’s approach via intersection cohomology could be generalized in order to prove the Gorenstein property of Tm under the assumptions (I), (II) and (LIρ¯) (see Lemma 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 (mod p − 1) satisfies (II) (see Corollary 6.10). The proof uses Hida’s families of p-adic ordinary Hilbert modular forms. We prove an exact control theorem for the ordinary part of the cohomology of the Hilbert modular variety, and give a new proof of Hida’s exact control theorem for the ordinary Hecke algebra (see Proposition 6.9). Theorems A and B prove that the congruence ideal associated to the O-algebra homomorphism ∗ T →O → Γ(Ad0(f),1)L (Ad0(f),1) , Ta ιp(c(f,a)) is generated by ιp( + − ). In a subsequent paper [12] Ωf Ωf we relate it to the fitting ideal of the Bloch–Kato Selmer group associated to Ad0(ρ) ⊗ E/O.An interesting question is whether Ωf are the periods involved in the Bloch–Kato conjecture for the motive Ad0(f) constructed by Blasius and Rogawski [1] (see the work of Diamond, Flach and Guo [9] for the elliptic modular case).

0.4. Explicit results

By a classical theorem of Dickson, if (Irrρ¯) holds but (LIρ¯) fails, then the image of ρ¯ in PGL2(κ) should be isomorphic to a dihedral, tetrahedral, octahedral or icosahedral group. Using this fact as well as Proposition 3.1, Section 3.2, Propositions 3.5 and 3.13 we obtain the following corollary to Theorems A and B. × × Denote by o+ (respectively on,1) the group of totally positive (respectively congruent to 1 modulo n) units of o. × ∩ × COROLLARY 0.4. – Let be any element of o+ on,1. (i) Assume d =2and k =(k0,k0 − 2m1), with m1 =0 .If m1 k0−m1−1 p ∆ NF/Q ( − 1)( − 1)

and p − 1 > 4(k0 − m1 − 1) then Theorem A holds. If additionally the image of ρ¯ in PGL2(κ) is not a dihedral group then Theorem B also holds. (ii) Assume d =3, id = τ ∈ JF and k =(k0,k0 − 2m1,k0 − 2m2), with 0

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 510 M. DIMITROV

− 5 − − − and p 1 > 3 (3k0 2m1 2m2 3) then Theorem A holds. If additionally the image of ρ¯ in PGL2(κ) is not a dihedral group then Theorem B also holds.

1. Hilbert modular forms and varieties

F G F ∗ × G We deﬁne the algebraic groups D/Q =ResQ m, G/Q =ResQ GL2 and G/Q = G D m, where the ﬁber product is relative to the reduced norm map ν : G → D. The standard Borel subgroup of G, its unipotent radical and its standard maximal torus are denoted by B, U and T , × → u 0 respectively. We identify D D with T ,by(u, ) 0 u−1 . 1.1. Analytic Hilbert modular varieties

× Let D(R)+ (respectively G(R)+) be the identity component of D(R)=(F ⊗ R) (respectively of G(R)). The group G(R)+ acts by linear fractional transformations on the space ∼ JF HF = {z ∈ F ⊗ C | im(z) ∈ D(R)+}.WehaveHF = H , where H = {z ∈ C | im(z) > 0} ⊗ → is the Poincaré’s upper half-plane (the isomorphism being given by ξ z (τ(ξ)z)τ∈JF , for ξ ∈ F , z ∈ C). We consider the unique group action of G(R) on the space HF extend- R −10 ing the action of G( √)+ and such√ that, on each copy of H the element 01 acts by →− − − ∈ + ⊗ R R z z¯. We put i =( 1,..., 1) HF , K∞ =StabG(R)+ (i)=SO2(F )D( ) and ⊗ R R K∞ =StabG(R)(i)=O 2(F )D( ). Z Z Z Z ⊗ We denote by = l l the profinite completion of and we put o = o = v ov, where v runs over all the finite places of F .LetA (respectively Af ) be the ring of adèles (respectively finite adèles) of Q. We consider the following open compact subgroup of G(Af ): ab K (n)= ∈ G(Z) | d − 1 ∈ n,c∈ n . 1 cd

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

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

By the Strong Approximation Theorem, the connected components of Y an are indexed by the + A Q Z R narrow ideal class group ClF = D( )/D( )D( )D( )+ of F . For each fractional ideal c of F we put c∗ = c−1d−1. We deﬁne the following congruence subgroup of G(Q): ∗ ab oc × Γ (c, n)= ∈ G(Q) ∩ | ad − bc ∈ o ,d≡ 1 (mod n) . 1 cd cdn o +

h+ an an \ an F an Put M = M1(c, n) =Γ1(c, n) HF . Then we have Y1(n) i=1 M1(ci, n) , where the + + ideals ci, 1 i hF , form a set of representatives of ClF . ∗ P1 ∗ an an ∗ an Put HF = HF (F ). The minimal compactification M of M is defined as M = \ ∗ ∗ an\ an Γ1(c, n) HF . It is an analytic normal projective space whose boundary M M is a finite union of closed points, called the cusps of M an. ∗ 1 1,an 1 an 1∗,an The same way, by replacing G by G , we define Γ1(c, n), M = M1 (c, n) and M .

1.2. Analytic Hilbert modular forms

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

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 511 ∈ Z DEFINITION 1.1. – An element k = τ∈J kτ τ [JF ] is called a weight. We always F { | ∈ } assume that the kτ ’s are 2 and have the same parity. We put k0 =max kτ τ JF , n = k − 2, t = τ, n = n τ = k − 2t and m = m τ =(k t − k)/2. 0 0 τ∈JF τ∈JF τ τ∈JF τ 0 ∈ ab · J ∈ C For z HF , γ = cd we put jJ (γ,z)=c z + d D( ), where ∈ J zτ ,τ J, zτ = z¯τ ,τ∈ JF\J.

DEFINITION 1.2. – The space Gk,J (K1(n)) of adélic 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) → C satisfying the following three conditions: (i) g(axy)=g(x) for all a ∈ G(Q), y ∈ K1(n) and x ∈ G(A). k+m−t −k + (ii) g(xγ)=ν(γ) jJ (γ,i) g(x), for all γ ∈ K∞ and x ∈ G(A). t−k−m k For all x ∈ G(Af ) deﬁne gx : HF → C,byz → ν(γ) jJ (γ,i) g(xγ), where γ ∈ G(R)+ is such that z = γ · i. By (ii) gx does not depend on the particular choice of γ. (iii) gx is holomorphic at zτ ,forτ ∈ J and anti-holomorphic at zτ ,forτ ∈ JF\J (when F = Q an extra condition of holomorphy at cusps is needed). The space Sk,J (K1(n)) of adélic Hilbert modular cuspforms is the subspace of Gk,J (K1(n)) consisting of functions satisfying the following additional condition: ∈ A A (iv) U(Q)\U(A) g(ux) du =0for all x G( ) and all additive Haar measures du on U( ).

The conditions (i) and (ii) of the above deﬁnition imply that for all g ∈ Gk,J (K1(n)) there exists a Hecke character ψ of F of conductor dividing n and of type −n0t at inﬁnity, such that for all x ∈ G(A) and for all z ∈ D(Q)D(Z)D(R),wehaveg(zx)=ψ(z)−1g(x).

DEFINITION 1.3. – Let ψ be a Hecke character of F of conductor dividing n and of type −n0t at infinity. The space Sk,J (n,ψ) (respectively Gk,J (n,ψ)) is defined as the subspace of −1 Sk,J (K1(n)) (respectively Gk,J (K1(n))) of elements g satisfying g(zx)=ψ(z) g(x) for all x ∈ G(A) and for all z ∈ D(A). When J = JF this space is denoted by Sk(n,ψ) (respectively by Gk(n,ψ)). Since the characters of the ideal class group ClF = D(A)/D(Q)D(Z)D(R) of F form a basis of the complex valued functions on this set, we have: (1) Gk,J K1(n) = Gk,J (n,ψ),Sk,J K1(n) = Sk,J (n,ψ) ψ ψ where ψ runs over the Hecke characters of F of conductor dividing n and infinity type −n0t.Let Γ be a congruence subgroup of G(Q). We recall the classical definition:

DEFINITION 1.4. – The space Gk,J (Γ; C) of Hilbert modular forms of weight k,levelΓ and type J ⊂ JF at inﬁnity is the C-vector space of the functions g : HF → C which are holomorphic at zτ ,forτ ∈ J, anti-holomorphic at zτ ,forτ ∈ JF\J, and such that for every γ ∈ Γ we have t−k−m k g(γ(z)) = ν(γ) jJ (γ,z) g(z). The space Sk,J (Γ; C) of Hilbert modular cuspforms is the subspace of Gk,J (Γ; C), consisting of functions vanishing at all cusps. ηi 0 + Put xi = 01 , where ηi is the idèle associated to the ideal ci, 1 i hF .Themap → + g (gxi ) (see Deﬁnition 1.2) induces isomorphisms: 1 i hF

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 512 M. DIMITROV Gk,J K1(n) Gk,J Γ1(ci, n); C , + 1 i hF (2) Sk,J K1(n) Sk,J Γ1(ci, n); C . 1ih+ F Let dµ(z)= y−2 dx dy be the standard Haar measure on H . τ∈JF τ τ τ F DEFINITION 1.5. – (i) The Petersson inner product of two cuspforms g,h ∈ Sk,J (K1(n)) is given by the formula

h+ F k (g,h)K1(n) = gi(z)hi(z)y dµ(z), i=1 Γ1(ci,n)\HF

where (gi) + (respectively (hi) + ) is the image of g (respectively h) under the 1 i hF 1 i hF isomorphism (2). (ii) The Petersson inner product of two cuspforms g,h ∈ Sk,J (n,ψ) is given by − n0 (g,h)n = g(x)h(x) ν(x) A dµ(x).

+ G(Q)\G(A)/D(A)K1(n)K∞

1.3. Hilbert–Blumenthal Abelian varieties

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

DEFINITION 1.6. – A Hilbert–Blumenthal Abelian variety (HBAV)over a Z[ 1 ]-scheme NF/Q(d) S is an Abelian scheme π : A → S of relative dimension d together with an injection → 1 o End(A/S), such that ωA/S := π∗ΩA/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.Given aHBAVA/S, the functor assigning to a S-scheme X the set A(X) ⊗o c is representable by another HBAV, denoted by A ⊗o c. Then o → End(A/S) yields c → Homo(A, A ⊗o c).The dual of a HBAV A is denoted by At.

DEFINITION 1.7. – ∼ t (i) A c-polarization on a HBAV A/S is an o-linear isomorphism λ : A ⊗o c −→ A , such ∼ t that under the induced isomorphism Homo(A, A ⊗o c) = Homo(A, A ) elements of c (respectively c+) correspond exactly to symmetric elements (respectively polarizations). ¯ × (ii) A c-polarization class λ is an orbit of c-polarizations under o+. −1 −1 Let (Gm ⊗ d )[n] be the reduced subscheme of Gm ⊗ d , deﬁned as the intersection of the kernels of multiplications by elements of n. Its Cartier dual is isomorphic to the ﬁnite group scheme o/n.

DEFINITION 1.8. – A µn-level structure on a HBAV A/S is an o-linear closed immersion −1 α :(Gm ⊗ d )[n] → A of group schemes over S.

1.4. Hilbert modular varieties

We consider the contravariant functor M1 (respectively M) from the category of Z 1 [ ∆ ]-schemes to the category of sets, assigning to a scheme S the set of isomorphism classes

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 513 of triples (A, λ, α) (respectively (A, λ,¯ α)) where A is a HBAV over S endowed with a c- ¯ polarization λ (respectively a c-polarization class λ) and a µn-level structure α. Assume the following condition: (NT) n does not divide 2, nor 3, nor NF/Q(d). 1 Then Γ1(c, n) is torsion free, and the functor M is representable by a quasi-projective, Z 1 1 1 smooth, geometrically connected [ ∆ ]-scheme M = M1 (c, n) endowed with a universal A→ 1 1 HBAV π : M . By definition, the sheaf ωA/M 1 = π∗ΩA/M 1 is an invertible o-bundle 1 H1 A 1 1 • 1 on M . Consider the first de Rham cohomology sheaf dR( /M )=R π∗ΩA/M 1 on M . The Hodge filtration yields an exact sequence: → →H1 A 1 → ∨ ⊗ −1 → 0 ωA/M 1 dR /M ωA/M 1 cd 0.

H1 A 1 ⊗O Therefore dR( /M ) is locally free of rank two over o M 1 . The functor M admits a coarse moduli space M = M1(c, n), which is a quasi-projective, Z 1 × ×2 smooth, geometrically connected [ ∆ ]-scheme. The finite group o+/on,1 acts properly and discontinuously on M 1 by []:(A, λ, α)/S → (A, λ, α)/S and the quotient is given by M. H1 A 1 • This group acts also on ωA/M 1 and on dR( /M ) by acting on the de Rham complex ΩA/M 1 −1/2 ∗ ([] acts on ωA/M 1 by [] ). 1/2 ∈ × These actions are defined over the ring of integers of the number field F ( , o+). 1/2 ∈ × Z 1 Let o be the ring of integers of F ( , o+). For every [ ∆ ]-scheme S we put 1 S = S × Spec o . ∆

× ×2 H1 A 1 The sheaf of o+/on,1-invariants of ωA/M 1 (respectively of dR( /M )) is locally free of ⊗O H1 rank one (respectively two) over o M and is denoted by ω (respectively dR). + + hF 1 1 hF 1 We put Y = Y1(n)= i=1 M1(ci, n) and Y = Y1 (n)= i=1 M1 (ci, n), where the ideals + + ci, 1 i hF , form a set of representatives of ClF .

1.5. Geometric Hilbert modular forms

Under the action of o, the invertible o-bundle ω on M decomposes as a direct sum of line ⊗ ∈ ∈ Z k kτ bundles ωτ , τ JF . For every k = τ kτ τ [JF ] we define the line bundle ω = τ ωτ on M . One should be careful to observe, that the global section of ωk on M an is given by the cocycle γ → ν(γ)−k/2j(γ,z)k, meanwhile we are interested in finding a geometric interpretation of the cocycle γ → ν(γ)t−k−mj(γ,z)k used in Definition 1.4. ¯ H1 The universal polarization class λ endows dR with a perfect symplectic o-linear pairing. ∧2 H1 − − k n0 Consider the invertible o-bundle ν := ⊗O on M . Note that (k + m t) = t. o M dR 2 2

1 DEFINITION 1.9. – Let R be an o [ ∆ ]-algebra. A Hilbert modular form of weight k,level k −n0t/2 Γ1(c, n) and coefﬁcients in R is a global section of ω ⊗ ν over M × Z 1 Spec(R). Spec( [ ∆ ]) 0 k −n0t/2 We denote by Gk(Γ1(c, n); R)=H (M × Z 1 Spec(R),ω ⊗ ν ) the R-module of Spec( [ ∆ ]) these Hilbert modular forms.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 514 M. DIMITROV

1.6. Toroidal compactiﬁcations

The toroidal compactifications of the moduli space of c-polarized HBAV with principal level structure have been constructed by Rapoport [34]. Several modifications need to be made in order to treat the case of µn-level structure. These are described in [10, Theorem 7.2]. 1 Let Σ be a smooth Γ1(c, n)-admissible collection of fans (see [10, Definition 7.1]). Then, 1 Z 1 1 1 there exists an open immersion of M into a proper and smooth [ ∆ ]-scheme M = MΣ, called the toroidal compactification of M 1 with respect to Σ. The universal HBAV π : A→M 1 extends uniquely to a semi-Abelian scheme π¯ : G → M 1. The group scheme G is endowed with an action 1\ 1 of o and its restriction to M M is a torus. Moreover, the sheaf ωG/M 1 of G-invariants sections 1 1 of π¯∗Ω is an invertible o-bundle on M extending ωA 1 . G/M 1 /M The scheme M 1\M 1 is a divisor with normal crossings and the formal completion of M 1 along this divisor can be completely determined in terms of Σ (see [10, Theorem 7.2]). For the sake of simplicity, we will only describe the completion of M 1 along the connected component of M 1\M 1 corresponding to the standard cusp at ∞.LetΣ∞ ∈ Σ be the fan corresponding to the ∞ ∗ ∪{ } ×2 cusp at . It is a complete, smooth fan of c+ 0 , stable by the action of on,1, and containing ξ a finite number of cones modulo this action. Put R∞ = Z[q ,ξ ∈ c] and S∞ =Spec(R∞)= ∗ ∞ Gm ⊗ c . Associated to the fan Σ , there is a toroidal embedding S∞ → SΣ∞ (it is obtained ξ ∞ by gluing the affine toric embeddings S∞ → S∞,σ =Spec(Z[q ,ξ∈ c ∩ σˇ]) for σ ∈ Σ ). Let ∧ ∞ ∞\ SΣ∞ be the formal completion of SΣ along SΣ S∞. By construction, the formal completion of M 1 along the connected component of M 1\M 1 corresponding to the standard cusp at ∞ is ∧ ×2 isomorphic to SΣ∞ /on,1. ∞ Assume that Σ is Γ1(c, n)-admissible (for the cusp at ∞ it means that Σ is stable under the × × ×2 1 action of o+). Then the finite group o+/on,1 acts properly and discontinuously on M and the Z 1 quotient M = MΣ is a proper and smooth [ ∆ ]-scheme, containing M as an open subscheme. Again by construction, the formal completion of M along the connected component of M\M ∞ ∧ × corresponding to the standard cusp at is isomorphic to SΣ∞ /o+. 1 The invertible o-bundle ωG/M 1 on M descends to an invertible o-bundle on M , extending k ω. We still denote this extension by ω. For each k ∈ Z[JF ] this gives us an extension of ω to a line bundle on M , still denoted by ωk.

1.7. q-expansion and Koecher Principles

If F = Q the Koecher Principle states that − − (3) H0 M × Spec(R),ωk ⊗ ν n0t/2 =H0 M × Spec(R),ωk ⊗ ν n0t/2 .

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 σ ∈ Σ∞ and every o[ 1 ]-algebra R, the pull- ∧ ∆ × ⊗O ∧ ⊗ back of ω to SΣ∞ Spec(R) is canonically isomorphic to o S ∞ R. Thus Σ ∧ × − 0 × k ⊗ n0t/2 H SΣ∞ Spec(R)/o+,ω ν ξ | ∈ k k+m−t ∀ ∈ × × × = aξq aξ R, au2ξ = u aξ, (u, ) on,1 o+ .

ξ∈c+∪{0} ξ This construction associates to each g ∈ G (Γ (c, n); R) an element g∞ = a (g)q , k 1 ξ∈c+∪{0} ξ called the q-expansion of g at the cusp at ∞. The element a0(g) ∈ R is the value of g at the cusp at ∞.

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1 PROPOSITION 1.10. – Let R be a o [ ∆ ]-algebra. ξ (i) (q-expansion Principle) Gk(Γ1(c, n); R) → R[[q ,ξ∈ c+ ∪{0}]],g→ g∞ is injective. k+m−t (ii) If there exists g ∈ Gk(Γ1(c, n); R) such that a0(g) =0 , then − 1 is a zero-divisor ∈ × in R, for all o+. 1.8. The minimal compactiﬁcation

Z 1 1∗ 1 There exists a projective, normal [ ∆ ]-scheme M , containing M as an open dense 1∗\ 1 Z 1 subscheme and such that the scheme M M is finite and étale over [ ∆ ]. Moreover, for each toroidal compactification M 1 of M 1 there is a natural surjection M 1 → M 1∗ inducing the identity map on M 1. The scheme M 1∗ is called the minimal compactification of M 1.The × ×2 1 1∗ ∗ action of o+/on,1 on M extends to an action on M and the minimal compactification M of M is defined as the quotient for this action. In general M 1∗ → M ∗ is not étale. We summarize the above discussion in the following commutative diagram:

π¯ G M 1 M

M 1∗ M ∗

π A M 1 M

1.9. Toroidal compactiﬁcations of Kuga–Sato varieties

s 1 1 Let s be a positive integer. Let πs : A → M be the s-fold fiber product of π : A→M and s 1 1 (¯π)s : G → M be the s-fold fiber product of π¯ : G → M . ⊕ 1 Let Σ be a (o c) Γ1(c, n)-admissible, polarized, equidimensional, smooth collection of 1 fans, above the Γ1(c, n)-admissible collection of fans Σ of Section 1.6. Using Faltings–Chai’s method [15], the main result of [11, Section 6] is the following: there exists an open immersion of a As into a projective smooth Z[ 1 ]-scheme As = As , and a proper, semi-stable homomorphism ∆ Σ s 1 s 1 s s πs : A → M extending πs : A → M and such that A \A is a relative normal crossing divisor above M 1\M 1. Moreover, As contains Gs as an open dense subscheme and Gs acts on As extending the translation action of As on itself. 1 1 1 • The sheaf H (A/M )=R π1∗Ω (dlog ∞) is independent of the particular choice log-dR A/M 1 of Σ above Σ and is endowed with a filtration: 0 → ω →H1 A/M 1 → ω∨ ⊗ cd−1 → 0. G/M 1 log-dR G/M 1

H1 It descends to a sheaf log-dR on M which ﬁts in the following exact sequence:

→ →H1 → ∨ ⊗ −1 → 0 ω log-dR ω cd 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 of K1(n) in G(Af ). It is endowed with algebra structure, where the product of two basis elements is given by: (4) K1(n)xK1(n) · K1(n)yK1(n) = K1(n)xiyK1(n) , i

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 516 M. DIMITROV ∈ where [K1(n)xK1(n)] = i K1(n)xi.Forg Sk,J (K1(n)) we put: | · · −1 g [K1(n)xK1(n)]( )= g xi . i

This defines an action of the algebra Z[K1(n)\G(Af )/K1(n)] on Sk,J (K1(n)) (respectively on Gk,J (K1(n))). Since this algebra is not commutative when n = o, we will define a commutative subalgebra. Consider the semi-group: ab ∆(n)= ∈ G(A ) ∩ M (ô) | d ∈ o×,c ∈ n for all v dividing n . cd f 2 v v v v

The abstract Hecke algebra of level K1(n) is defined as Z[K1(n)\∆(n)/K1(n)] endowed with the 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) | ν(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) containing the scalar matrix of the idèle attached to the ideal a. v 0 For each finite place v of F,wehaveTv = K1(n) 01K1(n) and for each v not dividing n we have S = K (n) v 0 K (n), where is an uniformizer of F . v 1 0 v 1 v v Then, the abstract Hecke algebra of level K1(n) is isomorphic to the polynomial algebra in ±1 the variables Tv, where v runs over the prime ideals of F , and the variables Sv , where v runs over the prime ideals of F not dividing n. The action of Hecke algebra obviously preserves the decomposition (1) and moreover, Sv acts on Sk,J (n,ψ) as the scalar ψ(v). Let T(C) be the subalgebra of EndC(Sk,J (K1(n))) generated by the operators Sv for v n and Tv for all v (wewillseeinSection1.13thatT(C) does not depend on J). The algebra T(C) is commutative, but not semi-simple in general. Nevertheless, for v n the operators Sv and Tv are normal with respect to the Petersson inner product (see Definition 1.5). Denote by T(C) the subalgebra of T(C) generated by the Hecke operators outside a finite set of places 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 the Hecke operators. By (2) we can associate to g ∈ Sk(K1(n)) a family of classical cuspforms ∈ C + gi Sk(Γ1(ci, n); ), where ci are representatives of the narrow ideal class group ClF . an Each form gi is determined by its q-expansion at the cusp ∞ of M1(ci, n) . For each ∈ × m ∈ × fractional ideal a = ciξ, with ξ F+ , we put c(g,a)=ξ aξ(gi). By Section 1.7 for each o+, k+m−t we have aξ = aξ and therefore the definition of c(g,a) does not depend on the choice of ξ (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 for T(C). In this case g ∈ Sk(n,ψ) for some Hecke character ψ, called the central character of g. We say that an eigenform g is normalized if c(g,o)=1.

LEMMA 1.12 ([24, Proposition 4.1, Theorem 5.2], [20, (4.64)]). – If g ∈ Sk(K1(n)) is a normalized eigenform, then the eigenvalue of Ta on g is equal to the Fourier coefﬁcient c(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 eigenvalues are 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 slightly −m −2m modified. We put T0,v = v Tv and S0,v = v 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 the Hilbert modular variety.

DEFINITION 1.13. – A normalized Hilbert modular eigenform is ordinary at p if for all primes p of F dividing p, the image by ιp of its T0,p-eigenvalue is a p-adic unit.

1.12. Primitive modular forms

−1 For each n1 dividing n and divisible by the conductor of ψ, and for all n2 dividing nn1 we consider the linear map

Sk(n1,ψ) → Sk(n,ψ),g→ g|n2, | | −1 where g n2 is determined by the relation c(a,g n2)=c(an2 ,g). old We define the subspace Sk (n,ψ) of Sk(n,ψ) as the subspace generated by the images of all these linear maps. This space is preserved by the Hecke operators outside n. We define the new old space Sk (n,ψ) of the primitive modular forms as the orthogonal of Sk (n,ψ) in Sk(n,ψ) with respect to the Petersson inner product (see Definition 1.5). Since the Hecke operators outside n are normal for the Petersson inner product, the direct sum decomposition Sk(n,ψ)= new ⊕ old T C Sk (n,ψ) Sk (n,ψ) is preserved by ( ). The Strong Multiplicity One Theorem, due to ∈ new T C Miyake in the Hilbert modular case, asserts that if f Sk (n,ψ) is an eigenform for ( ), 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 σ ∈ Aut(C) we deﬁne the external conjugate of g ∈ Sk(K1(n)) as the unique σ σ σ element g ∈ Sk(K1(n)) satisfying c(g , a)=c(g,a) for each ideal a of o. {± }JF + − We identify 1 with the Weyl group K∞/K∞ of G by sending J =( 1J , 1JF\J ) to + cJ K∞, where for all τ ∈ JF , det(cJ,τ ) < 0 if and only if τ ∈ J. The length of J is |J|. We have an action of the Weyl group on the space of Hilbert modular forms. More precisely, J acts as the double class [K1(n)cJ K1(n)] and maps bijectively Sk(K1(n)) onto Sk,JF\J (K1(n)). The action of J commutes with the action of the Hecke operators. For an element g ∈ Sk(K1(n)) · we put gJ = JF\J g. 1.14. Eichler–Shimura–Harder isomorphism

Let R be an O-algebra and let Vn(R) be the polynomial ring over R in the variables

(Xτ ,Yτ )τ∈JF which are homogeneous of degree nτ in (Xτ ,Yτ ). We have a pairing (perfect if n0! is invertible in R)

(5) , : Vn(R) × Vn(R) → R, given by n−j j n−j j j n ajX Y , bjX Y = (−1) ajbn−j , j 0 j n 0 j n 0 j n n n where = τ . j jτ τ∈JF

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 518 M. DIMITROV nτ ⊗ mτ The R-module Vn(R) realizes the algebraic representation Vn = τ (Sym det ) of JF G(R). We endow Vn(R) with an action of (M2(O) ∩ GL2(E)) given by m −1 t γ.P (Xτ ,Yτ )τ∈JF = ν(γ) P det(γ)γ (Xτ ,Yτ )τ∈JF .

Let Vn(R) be the sheaf of continuous (thus locally constant) sections of

+ + an G(Q)\G(A) × Vn(R)/K1(n)K∞ → G(Q)\G(A)/K1(n)K∞ = Y ,

+ where y ∈ K1(n)K∞ acts on Vn(R) via its p-part yp. For each y ∈ ∆(n) the map [y]:G(A) × Vn(R) → G(A) × Vn(R), (x, v) → (xy, yp.v) is a homomorphism of sheaves. This induces an action of the Hecke operator [K1(n)yK1(n)] on d an V d an V H (Y , n(R)) preserving the cuspidal cohomology Hcusp(Y , n(R)). an Van · − The action of J on (M , n ) given by J ((zJ ,zJF\J ),v)=(( z¯J ,zJF\J ),v) induces an d an Van action of the Weyl group on H (Y , n ) commuting with the Hecke action. d an V C d an V C By Harder [19] we know that if n =0then H! (Y , n( )) = Hcusp(Y , n( )). d an V × d an V → By (5) we have a Poincaré pairing , :H (Y , n(R)) Hc (Y , n(R)) R. 01 Let η be the idèle corresponding to the ideal n and let ι = −η 0 be the Atkin–Lehner involution. By putting [x, y]=x, ιy we obtain a new pairing d an V × d an V → (6) [ , ]:H! Y , n(R) H! Y , n(R) R, which is Hecke-equivariant. We call it the twisted Poincaré pairing. Now we state the Eichler–Shimura–Harder isomorphism:

THEOREM 1.14 (Hida [25]). – If n =0 , then there exists an isomorphism: ∼ d an V C (7) δ : Sk,J (n,ψ) = H! Y , n( ) ,

ψ J⊂JF where ψ runs over the Hecke characters of conductor dividing n and type −n0t at inﬁnity. This isomorphism is equivariant for the actions of the Hecke algebra and the Weyl group.

JF For each J ⊂ JF let ˆJ : {±1} →{±1} be the unique character of the Weyl group sending τ τ =(−1τ , 1 ) to 1,ifτ ∈ J, and to −1 if τ ∈ JF\J. The restriction of the Eichler–Shimura– Harder isomorphism (7) to Sk,J (n,ψ), followed by the projection on the (ψ,ˆJ )-part yields a Hecke equivariant isomorphism ∼ d V C (8) δJ : Sk,J (n,ψ) = H! Y, n( ) [ψ,ˆJ ].

Remark 1.15. – We have a direct sum decomposition: d an d an (9) H M , Vn(C) = H M , Vn(C) [J ⊗ c].

J⊂JF where c denotes the complex conjugation on the coefﬁcients. This decomposition is ﬁner than the usual Hodge decomposition, whose graded pieces are given by (0 a d): a d an d an gr H M , Vn(C) = H M , Vn(C) [J ⊗ c].

J⊂JF ,|J|=a

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

2. Hodge–Tate weights of Hilbert modular varieties

The aim of this section is to determine the Hodge–Tate weights of the p-adic étale • cohomology of a Hilbert modular variety H (MQ , Vn(Q )) as well as those of the p-adic Galois p p representation 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 étale cohomology of M with coefficients in the local system Vn(Qp) to the de Rham logarithmic cohomology of the corresponding vector bundle Vn over a smooth toroidal compactification M of M. The Hodge– Tate 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 Bernstein–Gelfand–Gelfand complex (BGG complex). Instead of using Faltings’ Comparison Theorem, one can apply Tsuji’s results to the étale cohomology with constant coefficients of the Kuga–Sato variety As (s-fold fiber product of the universal Abelian variety A above the fine moduli space M 1 associated to M; see [11, Section 6] s for the construction of toroidal compactifications of A ). For each subset J of J we put p(J)= (k − m − 1)τ + m τ ∈ Z[J ] and F τ∈J 0 τ τ∈JF\J τ F for each a = a τ ∈ Z[J ] we put |a| = a ∈ Z. τ∈JF τ F τ∈JF τ 2.1. Motivic weight of the cohomology

1 Q 1 A→ 1 Consider the smooth sheaf R π∗ p on M , where π : M is the universal HBAV. It 1 Q corresponds to a representation of the fundamental group of M in G( p). By composing this representation with the algebraic representation Vn of G of highest weight n (see Section 1.14), we obtain a smooth sheaf on M 1 (thus on Y 1). It descends to a smooth sheaf on Y , denoted by V Q n( p). − d V Q Let Wf = a⊂o ker(Ta c(f,a)) be the subspace of H (YQ, n( p)) corresponding to the ∈ Hilbert modular newform f Sk(n,ψ). Put s = τ (nτ +2mτ )=dn0.

PROPOSITION 2.1. – Wf is pure of weight d + s, that is to say for all prime l p∆ the d+s eigenvalues of the geometric Frobenius Frobl at l are Weil numbers of absolute value l 2 . ⊂ d V Q Proof. – Since f is cuspidal Wf H! (YQ, n( p)). We recall that YQ is a disjoint union of + its connected components MQ = M1(ci, n)Q, where the ci’s form a set of representatives of ClF . 1 1 ∗ ∅ Let c be one of the ci’s and M = M1 (c, n).For = ,cwe have 0 × ×2 d 1 V Q d V Q H o+/on,1, H∗ MQ, n( p) =H∗ MQ, n( p) ,

d 1 V Q and therefore, it is enough to prove that H! (MQ, n( p)) is pure of weight d + s.Weuse 1 Deligne’s method [4]. Let π : A→M be the universal Abelian variety (see Section 1.4). The sheaf V (Q ) corresponds to the representation Symnτ ⊗ detmτ of the group G∗ and n p τ∈JF 1 Q ⊗s As → 1 can therefore be cut out by algebraic correspondences in (R π∗ p) .Letπs : M be the Kuga–Sato variety. By the Kunneth’s formula we have

⊗ d 1 1 Q s ⊂ d 1 s Q ⊂ d+s As Q ⊂ d+s As Q H! MQ, R π∗ p H! MQ,R πs∗ p H! Q, p H Q, p ,

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 520 M. DIMITROV where the middle inclusion comes from the degeneration of the Leray spectral sequence i,j i 1 j Q ⇒ i+j As Q ∗ ∅ E2∗ =H∗(MQ,R πs∗ p) H∗ ( Q, p) for = ,c (see [4]). The proposition is then a consequence of the Weil conjectures for the eigenvalues of the Frobenius, proved by Deligne [5]. 2

2.2. The Bernstein–Gelfand–Gelfand complex over Q

In this and the next sections we describe, following Faltings [13], an algebraic construction of the 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, t and u denote the Lie algebras of G, B, T and U, respectively. Consider the canonical splitting g = b ⊕ u−.LetU(g), U(b) be the enveloping algebras of g and b, respectively. The aim of this section is to write down a resolution of Vn of the type:

← ← ⊗ • 0 Vn U(g) U(b) Kn,

j where the Kn are ﬁnite-dimensional semi-simpleb-modules, with explicit simple components. j j We start by the case n =0. If we put K0 = (g/b) we obtain the so-called bar-resolution i j of V0. Since (g/b) is a b-module with trivial u-action we deduce that K0 = Wµ with µ running over the weights of B that are sums of j distinct negative roots. By tensoring this resolution with Vn we obtain Koszul’s complex: j (10) 0 ← Vn ← U(g) ⊗U(b) (g/b) ⊗ Vn|b ,

i which is a resolution of Vn by b-modules (g/b) ⊗ Vn|b, not semi-simple in general. The BGG complex that we are going to deﬁne is a direct factor of Koszul’s complex cut by the action of the center U(g)G of U(g). G Denote by χn the weight n character of U(g) . It is a classical result that

LEMMA 2.2. – χn = χµ if, and only if, there exists J ⊂ JF such that µ = J (n + t) − t.

By taking the χn-part of the bar resolution (10) of Vn we obtain a complex: ← ← ⊗ • i (11) 0 Vn U(g) U(b) Kn, with Kn = WJ (n+t)−t,

J⊂JF ,|J|=i which is still a resolution of Vn, as a direct factor of a resolution. We call this resolution the BGG complex.

• ⊗ Q V Q 2.3. Hodge–Tate decomposition of H (M p, n( p))

In this paragraph we summarize the results of [11, Section 7]. The algebraic groups G, B, T and D of Section 1 have models over Z, denoted by the same letters. For a scheme S, we put × 1 S = S Spec(o [ ∆ ]). H1 By Section 1.9 we can extend the vector bundles ω and dR to M . Only the construction depends on a choice of a toroidal compactiﬁcation π¯ : A→M 1 of π : A→M 1. ⊗O The sheaf MD =Isomo⊗O (ω, o M ) is a D -torsor over M (for the Zariski topology). M We have a functor FD from the category of algebraic representations of D to the category of

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 521 vector bundles on M which are direct sums of invertible bundles. To an algebraic representation D W of D , FD associates the fiber product W := MD × W . fil 1 2 H ⊗O The sheaf MB =Isomo⊗O ( log-dR, (o M ) ) is a B -torsor over M . We have a functor M FB from the category of algebraic representations of B to the category of filtered vector bundles on M whose graded are sums of invertible bundles. To an algebraic representation V of B , FB B associates the fiber product V := MB × V . A representation of G (respectively T ) can be considered as a representation of B by restriction (respectively by making U act trivially). Thus, we may define a filtered vector bundle Vn on W M associated to the algebraic representation Vn of G, and an invertible bundle n,n0 on M associated to the algebraic representation of T = D × D, given by (u, ) → unm. 1 2 H ⊗O The sheaf MG =Isomo⊗O ( log-dR, (o M ) ) is a G -torsor over M . We have a functor M FG from the category of algebraic representations of G to the category of flat vector bundles on M (that is vector bundles endowed with an integrable quasi-nilpotent logarithmic connection). ∇ G To any algebraic representation V of B , FG associates the fiber product V := MG × V . j j ∇ • 1 For j ∈ N, we put H (M , V)=R θ∗(V ⊗ Ω (dlog ∞)), where θ : M → Spec(o [ ]) log-dR M ∆ denotes the structural homomorphism. • 1 GQ V Q By Faltings’ Comparison Theorem [14], the p -representation H (MQ , n( p)) is crys- p talline, hence de Rham, and we have a canonical isomorphism • 1 V Q ⊗ ∼ H• 1 V ⊗ H MQ , n( p) BdR = log dR M Q , n BdR. p - / p

By [11, Section 7] the Hodge to de Rham spectral sequence i,j i+j 1 i • i+j 1 E =H M Q , gr Vn ⊗ Ω (dlog ∞) =⇒H M Q , Vn , 1 / p M 1 log-dR / p degenerates at E1 (the filtration being the tensor product of the two Hodge filtrations). In order to compute the jumps of the resulting filtration we introduce the BGG complex: K i W n = J (n+t)−t,n0 .

J⊂JF ,|J|=i

K • The fact that n is a complex follows from (11) and from the following isomorphism (see [15, Proposition VI.5.1]) (12) HomU(g) U(g) ⊗U(b) W1 , U(g) ⊗U(b) W2 → Diff.Op.(W2, W1). • i • Define a filtration on K by Fil K = W − . n n J⊂JF ,|p(J)|i J (n+t) t,n0 The image of Koszul’s complex (10) by the contravariant functor W → W is equal to the de Rham complex. Since the BGG complex is a direct (filtered) factor of the Koszul’s complex, we obtain:

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

• K → V ⊗ Ω• (dlog ∞). n n M 1

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 522 M. DIMITROV

(ii) The spectral sequence given by the Hodge ﬁltration i,j i+j−|J| 1 i+j 1 E = H M Q , W (n+t)−t,n =⇒H M Q , Vn 1 / p J 0 log-dR / p J⊂JF ,|p(J)|=i

degenerates at E1. j 1 V Q (iii) For all j d, the Hodge–Tate weights of the p-adic representation H (MQ , n( p)) p belong to the set {|p(J)|, |J| j}.

2.4. Hecke operators on the cohomology

1 We describe the standard Hecke operator Ta as a correspondence on Y . We are indebted to M. Kisin for pointing us out that the usual definition of Hecke operators on Y extends to Y 1 (see [29, §1.9–1.11]). 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 . + 1 hF 1 + Recall that Y1 (n)= i=1 M1 (ci, n), where c1,...,c + form a set of representatives of ClF . hF + M1 Assume that cia and cj have the same class in ClF . Consider the contravariant functor a Z 1 from the category of [ ∆ ]-schemes to the category of sets, assigning to a scheme S the set of isomorphism classes of quintuples (A, λ, α, C, β) where (A, λ, α)/S is a ci-polarized HBAV with µn-level structure, C is a closed subscheme of A[a] which is o-stable, disjoint from −1 α(Gm ⊗ d ) and locally isomorphic to the constant group scheme o/a over S, and β is an ×2 −→∼ on,1-orbit of isomorphisms (cia, (cia)+) (cj, cj+). M1 →M1 → We have a projection a , (A, λ, α, C, β) (A, λ, α) which is relatively representable 1 → 1 1 → 1 by π1 : Ma (ci, n) M1 (ci, n). We have also a projection π2 : Ma (ci, n) M1 (cj, n) coming from (A, λ, α, C, β) → (A/C, λ,α), where α is the composed map of α and A → A/C and λ is a cj -polarization of A/C (defined via λ and β). h+ 1 F 1 → + Put Ya = i=1 Ma (ci, n).Asci cj cia is a permutation of ClF , we get two finite 1 → 1 projections π1,π2 : Ya Y :

A Aa A

π πa π π π 1 1 1 2 1 Y Ya Y

∗H1 → ∗H1 From this diagram we obtain π2 dR π1 dR. Therefore, for every algebraic representation ∗V∇ → ∗V∇ V of G,wehaveπ2 π1 . By composing this morphism by π1∗ and taking the trace, we V∇ → ∗V∇ → ∗V∇ →V∇ • 1 V∇ obtain π1∗π2 π1∗π1 . This gives an action of Ta on H (Y , ). ∗ → ∗ ∗ → ∗ The same way, the above diagram gives π2 ω π1 ω and π2 ν π1 ν. Therefore, for each ∗W→ ∗W algebraic representation W of T , we get π2 π1 . In order to deﬁne the action of Ta on Hilbert modular forms, we need to modify the last arrow: we decompose W as (W⊗ω−2t)⊗ω2t −2t −2t 2t 2t and we deﬁne π2∗(Wω ) → π1∗(Wω ) as above and π2∗ω → π1∗ω via the Kodaira– 1 2 ⊗ −1 W→ ∗W→ Spencer isomorphism ΩY 1 ω o dc as in [29, §1.11]. Thus we obtain π1∗π2 ∗W→W • 1 W π1∗π1 and an action of Ta on H (Y , ). 0 1 k −n0t/2 In particular, we obtain an action of Ta on the space H (Y ,ω ⊗ ν ) of geometric Hilbert modular forms for G∗. As it has been observed in [29, 1.11.8] over C this action is given

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 523 by the projection 1 − − []· :H0 Y 1,ωk ⊗ ν n0t/2 → H0 Y,ωk ⊗ ν n0t/2 , [o×: o× ] + n,1 ∈ × × [] o+ /on,1 followed by the usual Hecke operator on the space of Hilbert modular forms (see Section 1.10). ⊗ Q 2.5. Hodge–Tate weights of IndF ρ in the crystalline case

We first recall the notion of induced representation. Let V0 be a vector space over a field L, G → Q and let ρ0 : F GL(V0) be a linear representation. The induced representation IndF ρ0 of ρ0 from F to Q is by definition the L-vector space G G → |∀ ∈G ∈G −1 HomGF ( Q,V0):= φ0 : Q V0 x F ,y Q,φ0(yx)=ρ0 x φ0(y) , −1 ∈GQ ∈ G GQ · · · where y acts on φ0 Hom F ( ,V0) by y φ0( )=φ0(y ). For any fixed decomposition GQ = τ˜G ,themapφ → (φ (˜τ)) gives an isomor- τ∈JF F 0 0 τ G GQ phism between Hom F ( ,V0) and the direct sum τ Vτ (where each Vτ is isomorphic to V0). G Via this identification, the action of Q on τ Vτ is given by: Q −1 IndF ρ0 (y) (vτ )τ = ρ0 τ˜ yτ˜y (vτy ) τ ,

−1 y −1 −1 where y τ˜ ∈ τ˜yGF . In fact (φ0(˜τ))τ →(φ0(y τ˜))τ =(ρ0(˜τ yτ˜y)(φ0(˜τy)))τ . Keeping the same notations we deﬁne, following Yoshida [44], the tensor induced representa- Q G → ⊗ tion IndF ρ0 : Q GL( τ Vτ ) as: Q −1 IndF ρ0 (y) vτ = ρ0 τ˜ yτ˜y (vτy ). τ τ ∈G → Remark 2.4. – For each y Q the map τ τy is a permutation of JF and it is trivial if, ∈G ∈G Q −1 and only if, y . Therefore, for each y ,wehave( IndF ρ0)(y)= τ ρ0(˜τ yτ˜). F F Moreover for all y,y ∈GQ we have (τy)y = τyy .

DEFINITION 2.5. – The internal conjugate gτ of g ∈ Sk,J (n,ψ) by τ ∈ JF is deﬁned as the ∈ unique element gτ Skτ ,J τ (τ(n),ψτ ) satisfying c(gτ , a)=c(g,τ(a)) for each ideal a of o, τ where k = τ kττ τ and ψτ (a)=ψ(τ(a)). It is a Hilbert modular form on τ(F ). Q If ρ = ρ by the previous remark we have ( Ind ρ)(y)= ρ (y) for all y ∈G . f,p F τ fτ F Brylinski and Labesse [3] have shown (see [40] for this formulation):

THEOREM 2.6 (Brylinski–Labesse). – The restrictions to G of the two GQ-representations F Q Wf and IndF ρ have the same characteristic polynomial. COROLLARY 2.7 [11, Corollary 7.9]. – (i) The spectral sequence given by the Hodge ﬁltration i,j i+j−|J| i+j E = H M Q , W (n+t)−t,n =⇒H (M Q , Vn) 1 / p J 0 log-dR / p J⊂JF ,|p(J)|=i

degenerates at E1 and is Hecke equivariant. (ii) The Hodge–Tate weights of Wf are given by the multiset {|p(J)|,J⊂ JF }.

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• Proof. – (i) By taking the invariants of the Hodge filtration of Vn ⊗ Ω (dlog ∞) by the M 1 1 Galois group of the étale covering M → M we obtain a filtration of the complex Vn ⊗ Ω• (dlog ∞) on M , still called the Hodge filtration. The same way, we define the BGG complex M over M by taking the invariants of the BGG complex over M 1. The associated spectral sequence is given by the invariants of the spectral sequence of Theorem 2.3(ii). We have now to see that it is Hecke equivariant. 1 The Hecke operator Ta extends to a correspondence on Y . One way to define it is to 1 1 1 1 take the schematic closure of Ta ⊂ Y × Y in Y × Y . Another way is to take a toroidal 1 1 1 1 compactification Ya of Ya over the toroidal compactification Y of Y . • 1 • 1 ∇ Hence Ta acts on H (Y , W) and on H (Y , V ). Moreover, the Ta’s commute. In fact they commute on the right-hand side of Theorem 2.3(ii), because this side is independent of the toroidal compactification by Faltings’ Comparison Theorem. Since the spectral sequence of Theorem 2.3(ii) degenerates at E1, they also commute on the left-hand side. − W J (n+t)+t ⊗ p(J) (ii) We have J (n+t)−t,n0 = ω ν . 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 ⊂ JF . |p(J)| d • d−|J| −J (n+t)+t p(J) Moreover gr H (M Q , Vn ⊗ Ω (dlog ∞)) = H (M Q ,ω ⊗ ν ). / p M / p d−|J| −J (n+t)+t p(J) It is enough to see that the Q -vector space H (Y Q ,ω ⊗ ν )[f] is of p p dimension 1 for all J ⊂ JF . By the existence of a BGG complex over Q giving by base change the BGG complexes over Q C p and , we have a Hecke-equivariant isomorphism d−|J| −J (n+t)+t p(J) d an H Y Q ,ω ⊗ ν ⊗Q C =H Y , Vn(C) [J ⊗ c]. p p

⊂ d an V C ⊗ d an V C ⊗ For all J JF ,thef-part of H (Y , n( ))[J c] is equal to H! (Y , n( ))[J c, 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 Hodge–Tate weights is stable by the symmetry h → d(k0 − 1) − h, since |p(JF\J)| = d(k0 − 1) −|p(J)|.This × → Q − − symmetry is induced by the Poincaré duality Wf Wf p( d(k0 1)). (2) If F is a real quadratic ﬁeld and τ denotes the non-trivial automorphism of F , then the Hodge–Tate weights of Wf are given by mτ ,k0 − mτ − 1,k0 + mτ − 1, 2k0 − mτ − 2.

2.6. Hodge–Tate weights of ρ in the crystalline case

Q → Q Q The embedding ιp : p allows us to identify JF with HomQ−alg.(F, p). For each prime Q − Q p of F dividing p, we put JF,p =Hom p alg.(Fp, p). Thus we get a partition JF = p JF,p. Let Dp (respectively Ip) be a decomposition (respectively inertia) subgroup of GF at p. The following result is due to Wiles if k is parallel, and to Hida in the general case.

THEOREM 2.9 (Wiles [43], Hida [23]). – Assume that f is ordinary at p (see Deﬁnition 1.13).

Then ρ|Dp is reducible and: ε˜p ∗ (ORD) ρ| ∼ , Ip ˜ 0 δp where δ˜ (respectively ε˜ ) is obtained by composing the class ﬁeld theory map I → o× with the p p p p × × − − − − map o → Q , x → τ(x) mτ (respectively x → τ(x) (k0 mτ 1)). p p τ∈JF,p τ∈JF,p

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Breuil [2] has shown that if p>k0 and p does not divide ∆, then ρ is crystalline at each prime p of F dividing p, with Hodge–Tate weights between 0 and k0 − 1.

COROLLARY 2.10. – Assume p>k0 and that p does not divide ∆. Then for each prime Q p of F dividing p, ρ|Dp is crystalline with Hodge–Tate weights the 2[Fp : p] integers (mτ , − − k0 mτ 1)τ∈JF,p . Proof. – Assume ﬁrst that n =0 .LetK be a CM quadratic extension of F , splitting all the primes p of F dividing p. Blasius and Rogawski [1] have constructed a pure motive over K with − − Hodge weights (mτ ,k0 mτ 1)τ∈JF , whose p-adic realization is isomorphic to the restriction G of ρ to K . This shows that ρ|Dp is de Rham for all p, and crystalline for p big enough. By Faltings’ Comparison Theorem the Hodge weights of this motive correspond via Q → Q ιp : p to the Hodge–Tate weights of its p-adic realization, which are the same as the Hodge–Tate weights of ρ at primes p dividing p. This proves the corollary for n =0 . If n =0(or more generally if k is parallel) we can complete the proof using the following

LEMMA 2.11. – Let a and b be two positive integers and let (aτ )τ∈JF (respectively (bτ )τ∈JF ) be integers satisfying 0 2aτ

Then a = b and we have equality of multisets {aτ ,τ∈ JF } = {bτ ,τ∈ JF }. Using this lemma together with Theorem 2.6 and Corollary 2.7(ii) we obtain, up to permutation, the Hodge–Tate weights of ρ at primes p dividing p. In particular, we know exactly the Hodge–Tate weights of ρ when k is parallel. 2

2.7. Fontaine–Laffaille weights of ρ¯ in the crystalline case

Our aim is to ﬁnd the weights of ρ¯|Ip for p dividing p.Iff is ordinary at p we know by P Theorem 2.9 that ρ|Dp is reducible and by a simple reduction modulo we obtain the weights of ρ¯|Ip .

PROPOSITION 2.12. – Assume p>k0 and that p does not divide ∆. Then ρ¯ is crystalline at − − each p dividing p with Fontaine–Laffaille weights (mτ ,k0 mτ 1)τ∈JF,p . Proof. – It follows from Fontaine–Laffaille’s theory [17] and from the computation of Hodge–

Tate weights of ρ|Dp from Section 2.6. Consider a Galois stable lattice O2 in the crystalline representation ρ, as well as the sub- lattice P2. The representation ρ¯ 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 Fontaine–Laffaille module. Since the Fontaine–Laffaille functor is exact, this module is given by the quotient of Fontaine–Laffaille’s filtered modules associated to the two lattices. By compatibility of the filtrations on these two lattices, and by the condition p>k0, the graded of the quotient have the right dimension. 2

COROLLARY 2.13. – Let p beaprimeofF above p. Then ∗ ∼ εp ρ¯|Ip , 0 δp

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→ F× | | | | where εp,δp : Ip p are two tame characters of level JF,p or 2 JF,p , whose product equals the (1−k0)th power of the modulo p cyclotomic character and whose sum has Fontaine–Laffaille − − weights (mτ ,k0 mτ 1)τ∈JF,p .

Q 3. Study of the images of ρ¯ and IndF ρ¯

In all this section we assume that p>k0 and p does not divide 6∆. G → F× → Let ω : Q p be the modulo p cyclotomic character and let pr : GL2(κ) PGL2(κ) be the canonical projection. We recall that ρ¯ =¯ρf,p.

3.1. Lifting of characters and irreducibility criterion for ρ¯

PROPOSITION 3.1. – (i) For all but finitely many primes p (Irrρ¯) holds, that is ρ¯ is absolutely irreducible. ⊂ ∈ × − ∈ (ii) Assume that k is non-parallel. If for all J JF there exists o+, 1 n, such that p p(J) does not divide the non-zero integer NF/Q( − 1), then (Irrρ¯) holds. ∅ ∈ × Remark 3.2. – Assume that k = k0t is parallel and that for all J JF , there exists o+, p(J) − 1 ∈ n such that p does not divide the non-zero integer NF/Q( − 1). Then we expect ρ¯ to be absolutely irreducible, unless p divides the constant term of an Eisenstein series of weight k and level dividing n, that is the numerator of the value at 1 − k0 of the L-function of a finite order Hecke character of F of conductor dividing n (see [16, §3.2] for the case F = Q). Proof. – Since ρ¯ is totally odd, if it is irreducible, then it is absolutely irreducible. Assume that s.s. ⊕ G → × ρ¯ is reducible: ρ¯ = ϕgal ϕgal. The characters ϕgal,ϕgal : F κ are unramified outside ¯ −1 − np and ϕgalϕgal =det(¯ρ)=ψgalω (recall that ψ is a Hecke character of infinity type n0t). Denote by ô× the subgroup of ô× of elements ≡ 1 (mod n). Then ô× is a product of its p-part n,1 n,1 × ×(p) p|p op and its part outside p, denoted by ôn,1 . By the global class field theory, the Galois group of the maximal n-ramified (respectively np∞- + A× × × R ramified) Abelian extension of F is isomorphic to ClF,n = F /F ôn,1D( )+ (respectively + + A× × ×(p) R ClF,np∞ := lim← ClF,npr = F /F ôn,1 D( )+). We choose the convention in which an uniformizer corresponds to a geometric Frobenius. We have the following exact sequence → × ∈ × | − ∈ → + → + → (13) 1 op / o+ 1 n ClF,np∞ ClF,n 1. p|p

By Corollary 2.13, for each | ⊕ p p, ϕgal ϕgal − − is crystalline at p of weights (mτ ,k0 mτ 1)τ∈JF,p . ∈ × − ∈ By (13) for every o+, 1 n we have the following equality in κ: mτ or (k0−mτ −1) p(J) 1=ϕgal()= ϕgal,p()= τ() = ,

p|p p|p τ∈JF,p

p(J) for some subset J ⊂ JF . By the assumption p>k0,ifk is non-parallel, then =1 for all J ⊂ JF . Thus we obtain (ii) and (i) when k is non-parallel.

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∅ ∈ × − ∈ Assume now that k = k0t is parallel and that for all J JF , there exists o+, 1 n, p(J) − such that p does not divide the non-zero integer NF/Q( 1). The same arguments as above × + → × show that the restriction to p|p op of the character ϕgal (respectively ϕgal) ClF,np∞ κ is − trivial (respectively given by the (1 k0)th power of the norm). By the following lemma (applied + × { ∈ × | − ∈ } to P =ClF,np∞ and Q =( p|p op )/ o+ 1 n ) there exists a unique character ϕ˜gal + →O× (respectively ϕ˜gal):ClF,np∞ lifting ϕgal (respectively ϕgal) 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 factor × × group P/Q is finite. Let ϕP : P → κ and ϕ˜Q : Q →O be two characters such that we have × ϕP |Q =˜ϕQ mod p. Then, there exists a unique character ϕ˜P : P →O , whose restriction to Q is ϕ˜Q and such that ϕ˜P mod p = ϕP . ∈ A× −k k For x F , we put ϕ(x):=˜ϕgal(x) and ϕ (x):=˜ϕgal(x)xp x∞. Then ϕ (respectively ϕ ) is a Hecke character of F , of conductor dividing n and infinity type 0 (respectively (1 − k0)t). It is crucial to observe that there are only finitely many such ϕ and ϕ. Assume now that for infinitely many primes p, ρ¯ is reducible. Then there exist Hecke s.s. ≡ ⊕ characters ϕ and ϕ as above, such that for infinitely many primes p we have ρ¯ ϕgal ϕgal (mod P). Hence for each prime v of F not dividing n we have c(f,v) ≡ ϕ(v)+ϕ (v) (mod P) for infinitely many P’s and hence c(f,v)=ϕ(v)+ϕ (v). By the Cebotarev Density Theorem we obtain ρ s.s. = ϕ ⊕ ϕ. This contradicts the irreducibility of ρ. 2

3.2. The exceptional case

The aim of this paragraph is to ﬁnd a bound for the primes p such that pr(¯ρ(GF )) is isomorphic to one of the groups A4, S4 or A5. We will only use the fact that the elements of these groups are of order at most 5. ∼ Assume that pr(¯ρ(GF )) = A4,S4 or A5. By Corollary 2.13 there exist τ ∈{±1}, τ ∈ JF , | F× | | such that for all p p and for any generator x of ph , where h = JF,p , the element − × τ (kτ 1) ∈ F τ(x) ph ∈ F F τ Gal( ph / p) belongs to pr(¯ρ(Ip)) and is therefore of order at most 5 (if (ORD) holds we may assume that τ =1for all τ). Denote by τ1,...,τh the elements of JF,p. Then − − ··· h−1 − ∈ Z h − τ1 (kτ1 1) + τ2 p(kτ2 1) + + τh p (kτh 1) / p 1

− − ··· h−1 − h − is of order 5, hence 5((kτ1 1) + p(kτ2 1) + + p (kτh 1)) p 1. p p2 ph−1 If we replace the generator x by x ,x ,...,x and then sum these inequalities we ﬁnd 5 (k − 1) |J |(p − 1). Hence 5 (k − 1) d(p − 1). τ∈JF,p τ F,p τ∈JF τ We conclude that pr(¯ρ(GF )) cannot be isomorphic to A4, S4 or A5 if d(p − 1) > 5 (kτ − 1).

τ∈JF

Note that this assumption follows from (II) if d 5.

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3.3. The dihedral case

In this paragraph we study the case when pr(¯ρ(GF )) is isomorphic to the dihedral group D2r, where r 3 is an integer prime to p.LetCr be the cyclic subgroup of order r of D2r. Since −1 pr (Cr) is a commutative group containing only semi-simple elements (p does not divide r), it −1 −1 is diagonalizable. Since pr (D2r\Cr) is contained in the normalizer of pr (Cr), it is contained in the set of anti-diagonal matrices. Let ε : D2r →{±1} be the sign map and let K be the fixed field of ker(ε ◦ pr ◦ρ¯).The extension K/F is quadratic and unramified outside np. Let c be the non-trivial element of the Galois group Gal(K/F). Since ρ¯ is absolutely G → × irreducible, but ρ¯|GK is not, there exists a character ϕgal : K κ distinct from its Galois c ⊕ c conjugate ϕgal and such that ρ¯|GK = ϕgal ϕgal.

LEMMA 3.4. – Let p be a prime of F dividing p. Assume that there exists τ ∈ JF,p such that p =2 kτ − 1. Then the ﬁeld K is unramiﬁed at p, and ϕgal is crystalline at primes P of K above − − p of weights (mτ ,k0 mτ 1)τ∈JF,p .

Proof. – If K/F ramiﬁes at p then ρ¯(Ip) would contain at least one anti-diagonal matrix and the basis vectors would not be eigen for ρ¯(Ip). But the group ρ¯(Ip) has a common eigenvector. Hence, the elements of pr(¯ρ(Ip)) would be of order 2. Using the computations of Section 3.2 and p>k0, we ﬁnd that for all τ ∈ JF,p we have 2(kτ − 1) = p − 1. ⊕ c − − By Corollary 2.13, ϕgal ϕgal is crystalline at P of weights (mτ ,k0 mτ 1)τ∈JF,p and c 1−k0 (ϕgalϕ )| = ω . 2 gal IP |IP

Let O be the ring of integers of K, and O its profinite completion. Denote by O × the n,1 × ≡ × × subgroup of O of elements 1 (mod n). Then On,1 is a product of its p-part P|p OP and its ×(p) part outside p, denoted by On,1 . By the global class field theory, the Galois group of the maximal n-ramified (respectively np∞- A× × × × ramified) Abelian extension of K is isomorphic to ClK,n := K /K On,1K∞ (respectively to × × ×(p) × ∞ A ClK,np := K /K On,1 K∞). We have the following exact sequence: × × → ∈ | − ∈ → ∞ → → (14) 1 OP / O 1 n ClK,np ClK,n 1. P|p

PROPOSITION 3.5. – (i) Assume that for all τ ∈ JF , p =2 kτ − 1 and that pr(¯ρ(GF )) is dihedral. Let K/F be the quadratic extension defined above. Then one of the following holds: • K is CM and there exists a Hecke character ϕ of K of conductor of norm dividing −1 F − − ∈ ≡ P n∆K/F and infinity type (mτ ,k0 1 mτ )τ JF such that ρ IndK ϕ (mod ), • ∈ K is not CM and we can choose places τ˜ ofK above each τ JF such that for all × m k −m −1 ∈ O , − 1 ∈ n the prime p divides Q( τ˜() τ τ˜(c()) 0 τ − 1). NK/ τ∈JF (ii) Assume that f is not a theta series. Then for all but finitely many primes p the group pr(¯ρ(GF )) is not dihedral. Remark 3.6. – ≡ F P (i) The primes p for which the congruence ρ IndK ϕ (mod ) may occur should be controlled by the value at 1 of the L-function associated to the CM character ϕ/ϕc (in the 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 of dihedral primes for non-CM fields K. It would be interesting to explore the converse statement, that is to say to try to construct for a given prime p dividing the above norms, a newform f such that pr(¯ρ(GF )) is dihedral. × ∞ → Proof. – (i) By (14) and the above lemma, we have ϕgal :ClK,np κ whose restriction to O× is given by the reduction modulo p of an algebraic character x → xk˜, where P|p P k˜ = m τ˜ +(k − m − 1)˜τ ◦ c, for some choice of places τ˜ of K above τ ∈ J . τ∈JF τ 0 τ F → k˜ × We observe that the character x x 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× → κ×) does not contain elements of finite order, the above × character is trivial on { ∈ O | − 1 ∈ n}. × × By Lemma 3.3 (applied to P =Cl ∞ and Q =( O )/{ ∈ O | − 1 ∈ n}) there K,np P|p P × × ˜ ∞ →O → k exists a lift ϕ˜gal :ClK,np whose restriction to P|p OP is given by x x . −k˜ k˜ We put ϕ(x):=˜ϕgal(x)xp x∞. Then ϕ is a Hecke character of K 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 ϕ as above. Therefore, if pr(¯ρ(GF )) is dihedral for infinitely many primes p, then there would exist K and ϕ as above, ≡ F P P such that the congruence ρ IndK ϕ (mod ) happens for infinitely many ’s. Hence f would be equal to the theta series associated to ϕ. 2

3.4. The image of ρ¯ is “large”

THEOREM 3.7 (Dickson). – (i) An irreducible subgroup of PSL2(κ) of order divisible by p is conjugated inside PGL2(κ) to PSL2(Fq) or to PGL2(Fq), for some power q of p. (ii) An irreducible subgroup of PSL2(κ) of order prime to p is either dihedral, or isomorphic to one of the groups A4, S4 or A5. 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,ψ) is a newform, which is not a theta series. Then for all but finitely many primes p, the image of ρ¯ is large in the following sense: × (LIρ¯) there exists a power q of p such that SL2(Fq) ⊂ im(¯ρ) ⊂ κ GL2(Fq). Let F be the compositum of F and of the subfield of Q fixed by the Galois group ker(ψω¯ n0 ). The extension F/F is Galois and unramified at p, since F is unramified at p and ψ is of conductor prime to p. Therefore G is a normal subgroup of G containing the inertia subgroups I ,for F F p all p dividing p. × − We put D =det(¯ρ(G )) = (F )1 k0 . F p

PROPOSITION 3.9. – Assume (LIρ¯). Then there exists a power q of p such that, either ρ¯(G )=GL (F )D := x ∈ GL (F ) | det(x) ∈D , or F 2 q 2 q D UUρ¯(G )= F× GL (F ) := x ∈ F× GL (F ) | det(x) ∈D . F q2 2 q q2 2 q Proof. – We ﬁrst show that pr(¯ρ(G )) is still irreducible of order divisible by p.By(LI ) the F ρ¯ group pr(¯ρ(G )) is isomorphic to PSL (F ) or PGL (F ). The group pr(¯ρ(G )) is a non-trivial F 2 q 2 q F

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 530 M. DIMITROV normal subgroup of pr(im(¯ρ)) (because it contains pr(¯ρ(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 PSL (F ) ⊂ pr ρ¯(G ) ⊂ pr ρ¯(G ) ⊂ PGL (F ). 2 q F F 2 q

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 moreover that Q does not have non-trivial Abelian quotients. Then P ⊃ Q. It follows from this lemma that ρ¯(G ) ⊃ SL (F ), hence F 2 q

D κ× GL (F ) ⊃ ρ¯(G ) ⊃ GL (F )D. 2 q F 2 q

× D D Since [(κ GL2(Fq)) :GL2(Fq) ] 2 we are done. 2 ∈ F \F 2 ∈ F F× F D F D F D Let θ q2 q be such that θ q. Then ( q2 GL2( q)) =GL2( q) (θ GL2( q)) F× F D F ∪ F F and hence tr(( q2 GL2( q)) )= q θ q. Therefore, the p-algebra generated by the traces of F× F D F F× F D ⊂ F the elements of ( q2 GL2( q)) is q2 , while pr(( q2 GL2( q)) ) PGL2( q). This reﬂects the existence of a congruence with a form having inner twists.

Q 3.5. The image of IndF ρ¯ is “large”

We assume in this paragraph that (LIρ¯) holds. By Proposition 3.9 there exists a power q of p such that pr(¯ρ(G )) = PSL (F ) or PGL (F ). F 2 q 2 q Q G → F JF Consider the representation pr(IndF ρ¯): PGL2( q) . An automorphism of the simple F F F group PSL2( q) is a composition of a conjugation by an element of PGL2( q)with an F i automorphism of q. By a lemma of Serre (see [35]), there exist a partition JF = i∈I JF ∈ ∈ i ∈ F F and for all i I, τ JF , an element σi,τ Gal( q/ p) such that

Q pr φ SL (F )I ⊂ pr Ind ρ¯(G ) ⊂ pr φ GL (F )I , 2 q F F 2 q

i I JF i σi,τ where φ =(φ )i∈I :GL2(Fq) → GL2(Fq) is given by φ (Mi)=(M ) ∈ i . i τ JF Q Keeping these notations, we introduce the following assumption on the image of IndF ρ¯: ∀ ∈ ∀ ∈ i ⇒ (LIInd¯ρ) the condition (LIρ¯) holds and i I, τ,τ JF (τ = τ = σi,τ = σi,τ ). 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 a ∈ Z ∈ strict subﬁeld F of F and a weight k [JF ] such that for each τ JF , kτ = k | . τ F Remark 3.12. – Deﬁne k˜ = k τ˜ ∈ Z[J ] by putting k = k | , for all τ˜ ∈ J .The τ˜∈J τ˜ F τ˜ τ˜ F F F τ˜ group GQ acts on Z[J ] by k˜ = k τ˜ → k˜ = k τ˜. It is easy to see that F τ˜∈J τ˜ τ˜∈J τ˜τ˜ F F τ˜ k ∈ Z[JF ] is non-induced if, and only if, {τ˜ ∈GQ | k˜ = k˜ } equals GF .

PROPOSITION 3.13. – Assume that (LIρ¯) holds and k is non-induced. Assume moreover that ∈ − for all τ = τ JF , p = kτ + kτ 1. Then (LIInd¯ρ) holds. −1 −1 Proof. – Let τ˜ , τ˜ ∈GQ be such that for all y ∈G we have pr(¯ρ(˜τ yτ˜ )) = pr(¯ρ(˜τ yτ˜ )). 1 2 F 1 1 2 2 −1 ∈G −1 We have to prove that τ˜1 τ˜2 F .Fori =1, 2,letρ¯i(y)=¯ρ(˜τi yτ˜i).

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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 we − have ρ¯ | s.s. = ε ⊕ δ , where ε and δ are two tame characters whose product equals ω1 k0 and i IP i i i i − − ∈ whose sum has Fontaine–Laffaille weights (mτ ,k0 mτ 1)τ JF,pτi . Since I ⊂G and pr ◦ ρ¯ =pr◦ ρ¯ on G ,wehaveε /δ = ε /δ . By varying P we deduce P F 1 2 F 1 1 2 2 ˜ ˜ that for all τ˜ ∈ J, kτ˜ = k −1 (here we use that p>k0 and p = kτ + kτ − 1). Since k is non- F τ˜τ˜1 τ˜2 −1 ∈G 2 induced, it follows from Remark 3.12 that τ˜1 τ˜2 F . The 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 of internal conjugates of a Hilbert modular form.

COROLLARY 3.14. – Assume that (LIρ¯) holds and k is non-induced. Assume moreover that p>2k0 is totally split in F . Then, D Q D × 1−k GL (F )JF ⊂ Ind ρ¯(G ) ⊂ ρ¯(G )JF , where D = F 0 . 2 q F F F p

Put D H(Fq)= GL2(Fq) := (Mi)i∈I ∈ GL2(Fq) |∃δ ∈D, ∀i, det(Mi)=δ . i∈I i∈I

LEMMA 3.15. – Assume that (LIρ¯) holds and p>2k0. Then, (i) for all p dividing p, ρ¯(Ip) is contained (possibly after conjugation by an element of GL2(Fq)) either in the Borel subgroup of GL2(Fq), or in the non-split torus of GL2(Fq). The second case cannot occur if f is ordinary at p. Q ⊂ F (ii) IndF ρ¯(Ip) φ(H( q)). × Proof. – (i) By Corollary 2.13 ρ¯| s.s. = ε ⊕δ , where ε ,δ : I → F are two tame characters Ip p p p p p p 1−k0 of level h := |JF,p| or 2|JF,p| whose product equals ω and whose sum has Fontaine– − − Laffaille weights (mτ ,k0 mτ 1)τ∈JF,p . F× F× Let xh be a generator of ph , and let ε and δ be the characters of ph deduced from εp and δp. Since by (LI ) the traces of the elements of ρ¯(G ) are in F θF (see Section 3.4), we deduce ρ¯ F q q 2 2 2 that (ε(xh)+δ(xh)) ∈ Fq and hence ε(xh) + δ(xh) ∈ Fq. 2 2 ∈ F× ∈ F× If ε(xh) ,δ(xh) q , then it is easy to see that ε(xh),δ(xh) q (we use p>k0 and p =2 kτ − 1). In this case Ip ﬁxes a Fq-rational line and therefore ρ¯(Ip) is contained in a Borel subgroup of GL2(Fq). 2 2 Otherwise ε(xh) and δ(xh) are conjugated by the non-trivial element of Gal(Fq2 /Fq), 2 2q q q ∈ F× hence ε(xh) = δ(xh) . Since p>2k0,wehaveε(xh)=δ(xh) and so ε(xh)+δ(xh) q . Hence tr(¯ρ(Ip)) ⊂ Fq and ρ¯(Ip) ⊂ GL2(Fq). In this case ρ¯(Ip) is contained in a non-split torus of GL2(Fq).Iff is ordinary at p, then the Fontaine–Laffaille weights of δp are strictly smaller than those of εp, and therefore the second case cannot occur. (ii) The determinant condition D being satisﬁed, all we have to check is the following: for all ∈ ∈ i i I and τ,τ JF the character −1 −1 −1 →{± } → Ip 1 ,yρ¯ σ˜i,τ yσ˜i,τ ρ¯ σ˜i,τ yσ˜i,τ is trivial. This follows, as in the proof of Proposition 3.13, from the fact that p>2k0. 2 Q LEMMA 3.16. – Assume that (LI ) holds. Then φ(H(F )) ⊂ Ind ρ¯(G ). ρ¯ q F F

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

Q pr φ SL (F )I ⊂ pr Ind ρ¯(G ) . 2 q F F

Q By Lemma 3.10, we deduce that φ(PSL (F )I ) ⊂ Ind ρ¯(G ). 2 q F F F F I Q 2 Since φ(H( q)) = φ(SL2( q) )IndF ρ¯(Ip), we are done.

PROPOSITION 3.17. – Assume that (LIρ¯) holds but (LIInd¯ρ) fails for some p>2k0 (respectively for inﬁnitely many primes p). Then, there exist τ ∈ J , τ =id and a ﬁnite F F order Hecke character ε of F of conductor dividing NF/Q(n) such that for all primes v of F not dividing NF/Q(n)p we have c(fτ ,v) ≡ ε(v)c(f,v) (mod P) (respectively c(fτ ,v)= ε(v)c(f,v)). ∈G Proof. – Since (LIρ¯) holds but (LIInd¯ρ) fails, there exist τ˜1, τ˜2 Q such that

−1 | τ :=τ ˜2 τ˜1 F =idF and such that prρ ¯(˜τ −1yτ˜ )=pr¯ρ(˜τ −1yτ˜ ), for all y ∈G . Since G is a normal subgroup 2 1 2 2 F F of G , the above relation holds for every y ∈G . Therefore, there exists a character F F ε : G → κ× such that for all y ∈G , ρ¯ (y)=ε (y)¯ρ (y). Since p>2k ,thesame gal F F fτ gal f 0 argument as in the proof of Proposition 3.13 shows that εgal is unramiﬁed at primes dividing p. By Lemma 3.3, εgal can then be lifted to a ﬁnite order Hecke character ε of F of conductor dividing NF/Q(n). By evaluating at Frobv, for every prime v NF/Q(n)p of F , we obtain

c(fτ ,v) ≡ ε(v)c(f,v) (mod P).

2 ¯ By the determinant relation ψτ = εgalψ, there are finitely many such characters ε. Therefore, 2 if (LIInd¯ρ) fails for infinitely many primes p, then the above congruence will be an equality. COROLLARY 3.18. – Assume that F is a Galois field of odd degree and the central character ψ of f is trivial (F = F ). Assume moreover that f is not a theta series and that (LIInd¯ρ) does not hold for infinitely many primes p. Then, there exist a subfield F F and a Hilbert modular form f on F , such that the base change of f to F is a twist of f by a quadratic character of conductor dividing NF/Q(n). Proof. – As in the proof of Proposition 3.17 there exist a quadratic character ε of F of ∈ Q ⊗ conductor dividing NF/Q(n) and idF = τ Gal(F/ ) such that ρfτ = εgal ρ = ρf⊗ε.Let ⊂ ⊃ F F (respectively Fi F ) be the fixed field of τ (respectively of ker(ετ i )). We know that h F/F is a cyclic extension of odd degree h.LetF = i=1 Fi. Then we have h h i Gal(F /F )= (u1,...,uh) ∈{±1} | ui =1 τ | 0 i h − 1 , i=1

where τ acts on (u1,...,uh) by cyclic permutation. When h =3the group Gal(F /F ) is isomorphic to A4. |G The representation ρ F is invariant by Gal(F /F ), but Langlands Cyclic Descent does not apply directly because the order of Gal(F /F ) is even. Consider the quadratic character δ = ε · ετ 2 ···ετ h−1 . Then the GF -representation δgal ⊗ ρ is invariant by the group Gal(F/F ), so extends to a representation of GF . By applying Langlands Cyclic Descent to δ ⊗ f we obtain f as desired. 2

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4. Boundary cohomology and congruence criterion

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

DEFINITION 4.1. – We say that a normalized eigenform g ∈ Sk(n,ψ) is congruent to f modulo P if their respective eigenvalues for the Hecke operators (that is their Fourier coefﬁcients) are congruent modulo P. WesaythataprimeP is a congruence prime for f if there exists a normalized eigenform g ∈ Sk(n,ψ) 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 of f. Such results have been obtained by Ghate [18] when k is parallel. Following [21,18] and using a vanishing result of the boundary cohomology of a Hilbert modular 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: | | | ∅ | − p(JF ) + p( ) d(k0 1) {| | ⊂ } (MW) the middle weight 2 = 2 does not belong to p(J) ,J JF . This condition is automatically satisﬁed when the motivic weight d(k0 − 1) is odd, or when d =2and k is non-parallel.

LEMMA 4.2. – Let ρ be a representation of G on a ﬁnite-dimensional κ-vector space W . 0 F Q Assume that for every y ∈G , the characteristic polynomial of ( Ind ρ¯)(y) annihilates ρ (y). F F 0 (i) If (I), (II) and (LIρ¯) hold, then for all h ∈ Z the weights h and d(k0 − 1) − h occur with the same multiplicity in each G -irreducible subquotient of ρ . F 0 (ii) If (I), (Irr ) and (MW) hold and p − 1 > max(1, 5 ) (k − 1), then each G - ρ¯ d τ∈JF τ F irreducible subquotient of ρ0 contains at least two different weights for the action of the tame inertia at p.

Proof. – We may assume that ρ0 is irreducible. Q ⊂ F ⊂ Q G (i) By Lemmas 3.15(ii) and 3.16 we have IndF ρ¯(Ip) φ(H( q)) IndF ρ¯( ).LetT be F the torus of H(Fq) containing the image of the tame inertia, and N be the normalizer of T in F ∼ {± }I {± }JF H( q). The image of N /T = 1 by φ is the subgroup of the Weyl group N/T = 1 i of G containing the elements which are constant on the partition JF = i∈I JF . In particular, the longest Weyl element JF belongs to the image of N /T . Let x ∈ W be an eigenvector for the action of T . By the annihilation condition, there exists a subset Jx ⊂ JF , such that Ip acts on x by the weight |p(Jx)|. ∈G Q Let yJF be such that IndF ρ¯(yJF )=JF mod T . Then ρ0(yJF )(x) is of weight | | F − −| | ∈ Z p(Jx∆JF ) = d(k0 1) p(Jx) . Therefore, for each h , ρ0(yJF ) gives a bijection between the eigenspaces for the tame inertia of weight h and d(k0 − 1) − h. (ii) If (LIρ¯) holds, then the statement follows from (i) and (MW). Otherwise, by Proposi- tion 3.8 the group pr(¯ρ(G )) is dihedral. Since F is totally real, pr(¯ρ(G )) is also dihedral (see F F Section 3.3). Q Denote by N the normalizer of the standard torus T in G. Put N =Ind ρ¯(G ) ⊂ N(κ) and F F T = N ∩ T (κ). Then N /T is a subgroup of the Weyl group {±1}JF = N/T of G. Q As we have seen in Section 3.3, the representation IndF ρ¯ 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 of T . 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 ,

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 534 M. DIMITROV

Q (J ⊂ J ), let y ∈G be such that Ind ρ¯(y )= mod T . Then ρ (y )(x) is of weight F J F F J J 0 J |p(Jx∆J)|. It remains to show that the |p(Jx∆J)| are not all equal when J runs over the elements of N /T . Note that, for all τ ∈ JF ,theτ-projection N /T →{±1} is a surjective homomorphism (because the group pr(¯ρ (G )) is also dihedral). Therefore, we have: fτ F − d(k0 1) p(Jx∆J) = |N /T | . 2 J ∈N /T

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 to the arbitrary degree case of the key lemma in [8]. This lemma is false in general under the only assumptions (I), (II) and (Irrρ¯) when d 3. In fact, consider the following construction in the cubic case: let L be a Galois extension of Q of group A4, such that the cubic subfield F fixed by the Klein group is totally real; let K be a quadratic extension of F in L and consider a theta series f of weight (2, 2, 2) attached to a Hecke character of K; then the tensor induced representation Q IndF ρ has two irreducible four-dimensional subquotients of Hodge–Tate weights (0, 2, 2, 2) and (1, 1, 1, 3).

As in the introduction, let T ⊂ T denote the subalgebra generated by the Hecke operators outside a ﬁnite set of places containing those dividing n p.

THEOREM 4.4. – Assume that (I), (Irrρ¯) and (MW) hold, and 5 p − 1 > max 1, (k − 1). d τ τ∈JF

Denote by m the maximal ideal of T corresponding to f and ιp and put m = m ∩ T . Then • V (i) the m -torsion of the boundary cohomology H∂ (Y, n(κ))[m ] vanishes, d V O × d V O →O (ii) the Poincaré pairing H! (Y, n( ))m H! (Y, n( ))m is a perfect duality of free O-modules of ﬁnite rank, • • • V O V O V O (iii) H (Y, n( ))m =Hc (Y, n( ))m =H! (Y, n( ))m .

→j ∗ ←i ∗ Proof. – (i) Consider the minimal compactiﬁcation YQ YQ ∂YQ . The Hecke correspon- ∗ dences extend to YQ . By the Betti-étale comparison isomorphism, we identify (in a Hecke- equivariant way) the following two long exact cohomology sequences:

··· r V r V r V ··· Hc(Y, n(κ)) H (Y, n(κ)) H∂ (Y, n(κ))

r ∗ V r ∗ V r ∗ ∗ V ··· H (YQ ,j! n(κ)) H (YQ ,j∗ n(κ)) H (∂YQ ,i Rj∗ n(κ)) ···

G r r ∗ ∗ V r Consider the Q-module W∂ =H (∂YQ ,i Rj∗ n(κ)). We have to show that W∂ [m ]=0. By the Cebotarev Density Theorem and the congruence relations at totally split primes of F , we can apply Lemma 4.2 to W r[m]. Therefore each G -irreducible subquotient of W r[m] has ∂ F ∂ at least two different weights for the action of the tame inertia at p. So it is enough to show that G r each Q-irreducible subquotient of W∂ 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 • ∗ ∗ • V ⇒ • ∗ ∗ V H ∂YQ ,i R j∗ n(κ) = H ∂YQ ,i Rj∗ n(κ)

r 0 ∗ ∗ r V shows that W∂ =H (∂YQ ,i R j∗ n(κ)). 0 ∗ ∗ r V 0 1,∗ ∗ r V Since H (∂YQ ,i R j∗ n(κ)) is a subquotient of H (∂YQ ,i R j∗ n(κ)) 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 Y 1, 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 0 u 0 0 = . 0 u−1 0 u−1 01

1 1 ∗ r V F Put Γ =Γ1(c, n). By [32, Theorem 5.3.1], the restriction of the étale sheaf i R j∗ n( p) to a C ∞ 1,∗ −1 1 ∩ −1 1 ∩ cusp = γ of YQ is the image by the functor of Pink of the γ Γ γ B/γ Γ γ DlU- module a −1 1 b −1 1 H γ Γ γ ∩ Dl, H γ Γ γ ∩ U, Vn(Fp) . a+b=r Under the assumption (II), a modulo p version of a theorem of Kostant (see [33]) b −1 1 ∩ V F gives an isomorphism of T -module H (γ Γ γ U, n( p)) = |J|=b WJ (n+t)−t,n0 .By l h decomposing W − = W ⊗ W according to T = D × D ,we J (n+t) t J (n+t)−t,n0 J (n+t)−t,n0 l h get a −1 1 b −1 1 H γ Γ γ ∩ Dl, H γ Γ γ ∩ U, Vn(Fp) = Ha γ−1Γ1γ ∩ D ,Wl ⊗ W h , l J (n+t)−t,n0 J (n+t)−t,n0 |J|=b where Galois acts only on the second factors of the right-hand side. 0 ∗ ∗ r 0 h Therefore H (∂Y ,i R j∗V (F )) is a direct sum of subspaces H (C,W (F )), Q n p J (n+t)−t,n0 p |J| r, each containing a single Fontaine–Laffaille weight, namely the weight |p(J)|. (ii) Since the Poincaré duality is perfect over E, it is enough to show that the m-localization d O d O → d d of natural map H ( )/H! ( ) H (E)/H! (E) is injective. For this, it is sufﬁcient to show d O d V O that H∂ ( )m := H (∂M, n( ))m is torsion free, which would follow from the vanishing d−1 d−1 d−1 O O of H∂ (E/ )m . We have a surjection H∂ (κ)m H∂ (E/ )m [], where is an d−1 O uniformizer of . Finally, by (i) and Nakayama’s lemma, H∂ (κ)m =0. • • O 2 (iii) The vanishing of H∂ (κ)m gives the vanishing of H∂ ( )m =0. 4.2. Deﬁnition of periods − By taking the subspace a⊂o ker(Ta c(f,a)) of (8) we obtain C −→∼ d an V C δJ : fJ H! Y , n( ) [ˆJ ,f].

C ∼ Q d an V O Fix an isomorphism = p compatible with ιp. We recall that H! (Y , n( )) denotes d an V O → d an V C O the image of the natural map Hc (Y , n( )) H (Y , n( )). Since is principal, the O d an V O -module Lf,J := H! (Y , n( )) [ˆJ ,f] is free of rank 1.Weﬁxabasisη(f,J) of Lf,J.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 536 M. DIMITROV

⊂ δJ (fJ ) ∈ C× O× DEFINITION 4.5. – For each J JF we define the period Ω(f,J)= η(f,J) / . We ⊂ + − \ fix J0 JF and put Ωf =Ω(f,J0) and Ωf =Ω(f,JF J0). ± Remark 4.6. – The periods Ωf differ from the ones originally introduced by Hida in [21]. Hida’s periods put together all the external conjugates of f. Our slightly different definition is motivated by the congruence criterion that we want to show (Theorem 4.11). Since we can prove the perfectness of the twisted Poincaré pairing only for certain local components of the middle d an V O degree cohomology H! (Y , n( )) and in general f and its external conjugates do not belong to the same local component, we have to separate them in the definition of the period.

4.3. Computation of a discriminant

O The aim of this paragraph is to compute the discriminant disc(Lf ) of the -lattice L := Hd(Y an, V (O))[f]= L , with respect to the twisted Poincaré pairing [ , ] f ! n J⊂JF f,J deﬁned in (6). We follow [18, Section 6]. We have disc(Lf ) = det(([η(f,J),η(f,J )])J,J ⊂JF ). ∈ ∈ d an V C · − · By [18, (41)], for every τ JF and x, y H! (Y , n( )) we have [τ x, y]= [x, τ y]. The embedding O → C that we have ﬁxed gives an embedding τ0 : F→ C. 0[η(f,J),η(f,JF\J)] disc(Lf )= [η(f,JF\J),η(f,J)] 0 τ ∈J⊂J 0 F 2 [δJ (f),δ \ (f)] = − JF J , Ω(f,J)Ω(f,JF\J) τ0∈J⊂JF d · d c d and [δJ (f),δJF\J (f)] = 2 JF δ(f),ι δ(f) =2 W (f) JF δ(f),δ(f ) =2 W (f)(f,f)K1(n), where f c is the complex conjugate of f and W (f) is the complex constant of the functional equa- tion of the standard L-function of f. By [9, Lemma 2.13] W (f) ∈O×. Therefore the following equality holds in E×/O×: (f,f) 2 ⊕ K1(n) (15) disc(Lf,J0 Lf,JF\J0 )= + − . Ωf Ωf

4.4. Shimura’s formula for L(Ad0(f), 1)

For a prime v of F we deﬁne αv and βv by: ψ(v) N Q(v), if v n, α + β = c(f,v),αβ = F/ v v v v 0, if v | n.

The naive adjoint L-function of f is deﬁned by the Euler product: (n) 0 − −1 −s − −s (16) L Ad (f),s = 1 αvβv NF/Q(v) 1 NF/Q(v) vn × − −1 −s −1 1 βvαv NF/Q(v) . c −s In [38] Shimura studies the series D(f,f ,s)= a⊂o c(f,a)c(f,a) NF/Q(a) and shows that it has the following Euler product (see [26, Lemma 7.2]):

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 537 c −2s −s −1 D f,f ,s = 1 − αvβvαvβv NF/Q(v) 1 − αvαv NF/Q(v) v −s −1 −s −1 × 1 − αvβv NF/Q(v) 1 − βvαv NF/Q(v) −s −1 × 1 − βvβv NF/Q(v) . Using the fact that for all v n c(f,v)=ψ(v)c(f,v) a direct computation gives: (n) (n) c − (n) (n) 0 (17) ζF (2s)D f,f ,s+ k0 1 = ζF (s)L Ad (f),s , where D(n)(f,fc,s) is obtained from D(f,fc,s) by removing the Euler factors for v|n.

THEOREM 4.7 (Shimura [38, (2.31), Proposition 4.13]). – Let f ∈ Sk(n,ψ) be a newform. Then c − d−1 |k| −1 × ×2 −1 Ress=1 D f,f ,s+ k0 1 =2 (4π) Γ(kτ ) RF o+ : o µ (f,f)K1(n), ∈ τ JF 2 −2 3/2 N Q(n) (1−N Q(v) ) \ 2 NF/Q(d) ζF (2) F/ v|n F/ where µ = µ(Γ1(c, n) HF )= d × ×2 . × × . π [o+ : o ] [o : on,1] By a direct computation: (n) c − (18) ζF (2) Ress=1 D f,f ,s+ k0 1 Res ζ(n)(s)(4π)|k|πd[o× : o×2][o× : o× ](f,f) s=1 F + n,1 K1(n) = + − . 2∆h Q(n) Γ(k ) (1 − Q(v) 1) F NF/ τ∈JF τ v|n NF/ We deﬁne the imprimitive adjoint L-function L∗(Ad0(f),s) by completing the naive adjoint L-function L(n)(Ad0(f),s) deﬁned in (16), in order to have the relation: ∗ 0 (n) c (n) 0 c L Ad (f),s D f,f ,s+ k0 − 1 = L Ad (f),s D f,f ,s+ k0 − 1 . ∗ 0 (n) 0 ∗ 0 | By [26, (7.7)] we have L (Ad (f),s)=L (Ad (f),s) v|n Lv(Ad (f),s), where for v n  −s −1  (1 − NF/Q(v) ) , if f is a principal series and minimal at v, ∗ 0 − −s−1 −1 Lv Ad (f),s =  (1 NF/Q(v) ) , if f is special and minimal at v, 1, otherwise.

Following Deligne [6] we associate to L∗(Ad0(f),s) an Euler factor: 0 −(s+1)/2 1−kτ −s Γ Ad (f),s = π Γ (s +1)/2 (2π) Γ(s + kτ − 1).

τ∈JF

Finally, by (17) and (18), there exists a ∈ Z such that: 2a (19) Γ Ad0(f), 1 L∗ Ad0(f), 1 = (f,f) . ∆ n

Remark 4.8. – Consider the adjoint L-function L(Ad0(ρ),s) attached to the three-dimensional 0 GF -representation Ad (ρ) on trace zero 2 × 2 matrices. By compatibility between local and global Langlands correspondence L(Ad0(ρ),s) is equal to the adjoint L-function L(Ad0(f),s) associated to the automorphic representation attached to f. Nevertheless L(Ad0(f),s) may differ from L∗(Ad0(f),s) at some places v dividing n (see [26, (7.3c)]).

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 538 M. DIMITROV

4.5. Construction of congruences

LEMMA 4.9. – Let V1 and V2 be two ﬁnite-dimensional E-vector spaces and let L be a j O-lattice in V = V1 ⊕ V2.Forj =1, 2, put Lj = L ∩ Vj and denote L the projection of L in Vj following the above direct sum decomposition. Then: j (i) Lj ⊂ L are two lattices of Vj , and Lj is a direct factor in L. (ii) we have isomorphisms of ﬁnite O-modules:

1 ∼ ∼ 2 L /L1 ←− L/L1 ⊕ L2 −→ L /L2.

This ﬁnite O-module is called the congruence module, and is denoted by C0(L; V1,V2). The following proposition follows from Deligne–Serre [7, Lemma 6.11] and will be used to construct congruences:

PROPOSITION 4.10. – Keep the notations of Lemma 4.9. Let T be a commutative O-algebra consisting of endomorphisms of V , preserving the lattice L and the direct sum decomposition V1 ⊕ V2.Forj =1, 2, denote by Tj the image of T in End(Vj ). Assume that C0(L; V1,V2) is non-zero and that its support contains P. T 1 ⊗ Let m1 be maximal ideal 1 of residue field κ1, such that L /L1 T1 κ1 is non zero, and ¯ denote by θ1 : T1 → κ1 the corresponding character. Then there exists a discrete valuation ring O of maximal ideal P (with P ∩O= P), residue field κ ⊃ κ1 and whose fraction field E is a finite extension of E, and there exists a character ¯ θ2 : T2 →O such that for each T ∈T, θ1(T ) ≡ θ2(T ) (mod P ). T T −1 Proof. – For j =1, 2, denote by πj the projection of onto j . Then m = π1 (m1) is a maximal ideal of T of residue field κ1. Put m2 = π2(m). Since the isomorphism of Lemma 4.9(ii) is T -equivariant, we have 1 ⊗ T ∼ ⊕ ⊗ T ∼ 2 ⊗ T L /L1 T1 ( 1/m1) = L/(L1 L2) T ( /m) = L /L2 T2 ( 2/m2).

1 ⊗ T T By assumption (L /L1) T1 ( 1/m1) is non-zero. Therefore m2 is a maximal ideal of 2 of ¯ residue ﬁeld κ1 and the corresponding character θ2 : T2 → κ1 ﬁts in the following commutative diagram: T 1 θ¯1

T κ1

θ¯ T2 2

Since T2 is a (ﬁnite) ﬂat O-algebra, there exists a prime ideal P2, contained in m2 and such that P2 ∩O=0. The reduction modulo P2 gives a character θ2 of T2 as in the statement. 2

THEOREM 4.11 (Theorem A). – Let f and p be such that (I), (Irrρ¯) and (MW) hold, and ∗ 5 Γ(Ad0(f),1)L (Ad0(f),1) p − 1 > max(1, ) (k − 1).Ifι ( + − ) ∈P, then P is a congruence d τ∈JF τ p Ωf Ωf prime for f.

d an d an V O ± ⊂ V ± Proof. – Let L =H! (Y , n( ))m [ ˆJ0 ,ψ] V =H! (Y , n(E))m [ ˆJ0 ,ψ] and d an V ± ∩ ⊕ V1 =H! (Y , n(E))[ ˆJ0 ,f]. Then L1 = L V1 = Lf,J0 Lf,JF\J0 (see Section 4.2). By (15) the twisted Poincaré pairing [ , ] is non-degenerate on V1.LetV2 be the orthogonal subspace of V1 in V .

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d an V O ± By Theorem 4.4(ii) the twisted Poincaré pairing is perfect on H! (Y , n( ))m [ ˆJ0 ].After rescaling by the factor (f,f)n (coming from congruences obtained by varying the central (f,f)K1(n) character) it restricts to a perfect pairing on L. Then, by [21, (4.6)] we have 2 1 (f,f)n L : L1 = disc(L1). (f,f)K1(n)

1 Using now (15), (19), and the assumption on P we obtain that the O-module L /L1 is non- zero and P belongs to its support. By Lemma 4.9 the same holds for the congruence module C0(L; V1,V2). By Proposition 4.10 and the duality between T(C) and Sk(n,ψ) there exists another normalized eigenform g ∈ Sk(n,ψ) congruent to f. Hence P is a congruence prime for f. 2

5. Fontaine–Laffaille weights of Hilbert modular varieties

In this section all the objects are over O. The aim is to establish a modulo p version of Theorem 2.3 under the assumptions that p does not divide ∆ and p − 1 > |n| + d.

5.1. The BGG complex over O

Koszul’s complex. The Koszul’s complex of the trivial G-module O is given by

2 ···→UO(g) ⊗ g → UO(g) ⊗ g → UO(g) →O→0. O

− Since g = b ⊕ u ,theO[b]-module g/b is a direct factor in g and we have a homomorphism of ⊗ • → ⊗ • B-modules UO(g) O g UO(g) UO (b) O(g/b). Thus, we deduce another complex

• ⊗ →O→ UO(g) UO (b) (g/b) 0, O

• denoted by SO(g, b). More generally, for a free O-module V endowed with an action of UO(g), we consider the • complex SO(g, b) ⊗ V endowed with the diagonal action of UO(g). For every UO(b)-module W which is free over O, there is a canonical isomorphism of UO(g)-modules ⊗ ⊗ ∼ ⊗ ⊗ | UO(g) UO (b) W V = UO(g) UO (b) (W V b).

Therefore we obtain another complex • ⊗ ⊗ | → → UO(g) UO (b) (g/b) V b V 0, O

• • denoted by SO(g, b,V). In the case where V = Vn we denote it by SO(g, b,n).

Verma modules. For each weight µ ∈ Z[JF ], we deﬁne a UO(g)-module VO(µ):= ⊗ O UO(g) UO (b) Wµ( ), called the Verma module of weight µ.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 540 M. DIMITROV

LEMMA 5.1. – Let W be a B-module, free of ﬁnite rank over O, whose weights are smaller than (p − 1)t. Then, there exists a ﬁltration of B-modules 0=W0 ⊂ W1 ⊂···⊂Wr = W such ∈ Z ∼ O that for every 1 i r there exists µi [JF ] such that Wi/Wi+1 = Wµi ( ). Moreover the O Wµi ( ), 1 i r, are the irreducible factors of theT -module W . ∼ r O In particular, if U acts trivially on W , then W = i=1 Wµi ( ).

Proof. – Let µ1 be a maximal weight of W (for the partial order given by the positive roots of G) and let v ∈ W be a O-primitive vector of weight µ1.LetW be the UO(b)-submodule ∼ O ⊗ generated by v. Then W = Wµ1 ( ) and W κ is irreducible, because µ1 is smaller than (p − 1)t (and W is free of rank 1). Since W is free over O we have an exact sequence of B-modules → O → ⊗ → ⊗ 0 Tor1 (W/W ,κ) W κ W κ. Since W ⊗ κ is irreducible and v is primitive, the last arrow is injective. Therefore

O Tor1 (W/W ,κ)=0, that is W/W is free over O. The lemma follows then by induction. 2 i LEMMA 5.2. – The module SO(g, b,n) has a finite filtration by UO(g)-submodules whose ∈ i i graded pieces are of the form VO(µ), µ Ω (n), where Ω (n) is the set of weights of the i t-module O(g/b) ⊗ Vn(O)|b. − | | • ⊗ O | Proof. – Since p 1 > n + d the previous lemma applies to O(g/b) Vn( ) b. This gives i a filtration 0=W0 ⊂ W1 ⊂···⊂Wr = O(g/b) ⊗ Vn(O)|b whose graded pieces are Wµ(O), ∈ i ⊗ • 2 µ Ω (n). Since UO(g) is UO(b)-free, the functor UO(g) UO (b) is exact.

Central characters. Let UO(g) → UO(t) be the projection coming from the Poincaré– − Birkhoff–Witt decomposition UO(g)=UO(t) ⊕ (u UO(g)+UO(g)u). We take its restriction G to the invariants for the adjoint action θ : UO(g) → UO(t). Note that UF (t) identiﬁes with ∼ p the algebra of regular functions on HomO(t, Fp) = Fp[JF ] (a Laurent polynomial algebra). The JF Weyl group {±1} of G acts on it by (J · P )(µ)=P (J (µ + t) − t). The following result is analogous to the theorem of Harish–Chandra:

THEOREM 5.3 (Jantzen [28]). – θ induces an algebra isomorphism Fp

J U (g)G → U (t){±1} F . Fp Fp

For every µ ∈ Z[JF ] and every O-algebra R, we denote by dµR : tR → R the corresponding G character and by χµ,R = dµR ◦ θR the composed map UR(g) → UR(t) → R. This deﬁnition is compatible with the O-algebra homomorphisms. G If V is a UR(g)-module generated by a vector v of weight µ and annihilated by u, then UR(g) acts over V by χ . Put χ = χ O and χ = χ . µ,R µ,p µ, µ,p µ,Fp ⊂ − − ∈ COROLLARY 5.4. – If χn,p = χµ,p, then there exists J JF such that µ (J (n + t) t) pZ[JF ]. In particular, if µ is smaller than (p − 1)t, then we have µ = J (n + t) − t. ∈ i PROPOSITION 5.5. – Let µ Ω (n) (see Lemma 5.2). Then χn,p = χµ,p if, and only if, there exists a subset J ⊂ JF containing i elements and such that µ = J (n + t) − t.

Proof. – By the corollary, it remains to show that for J ⊂ JF ,wehaveJ (n + t) − t ∈ i | | Ω (n) if, and only if, J = i. By Lemma 5.2, we have to show that WJ (n+t)−t,n0 (E)

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 541 i ⊗ | | | occurs in E(g/b) Vn(E) t (with multiplicity one) if and only if J = i. The weights of i ⊗ | − ⊂ E(g/b) Vn(E) t are of the form J (n + t) t + ν, where J JF is a subset containing i elements and ν is a weight of Vn(E). Therefore J (n + t) − t = J (n + t) − t + ν and so n = J (ν)+J (J (t)) − t. Since n is a maximal weight of Vn(E), we deduce that J = J . 2

i Decomposition with respect to central characters. By Lemma 5.2, SO(g, b,n) admits a i ﬁnite ﬁltration by UO(g)-submodules with graded of the form VO(µ), µ ∈ Ω (n). Therefore • SO(g, b,n) is annihilated by a power of the ideal I := µ∈Ω•(n) ker(χµ,p) of the commutative G • ring UO(g) . In fact, it would follow from Proposition 5.5 that SO(g, b,n) is annihilated by I itself. We have the following commutative algebra result:

LEMMA 5.6. – Let P1,...,Pr be ideals of a commutative ring R such that P1 ...Pr =0 and for all i = j, Pi + Pj = R. Then each R-module W admits a direct sum decomposition Pi Pi { ∈ | } W = 1ir W , with W = m W Pim =0 . G ∈ • Consider the maximal ideals (p, ker(χµ,p)) = ker(χµ,p) of UO(g) , where µ Ω (n).Let ∈ • χ1 = χn,p, χ2,...,χr be the set of distinct characters among χµ,p, µ Ω (n). Put Pi = ker(χµ,p). By the above lemma we get a decomposition χµ,p=χi

r • • Pi SO(g, b,n)= SO(g, b,n) i=1

O O which is a direct sum, because the differentials are U (g)-equivariant. Moreover, V (µ)χn,p = O O V (µ) if χµ,p = χn,p, and V (µ)χn,p =0otherwise. From here and from Proposition 5.5 we get: • • HEOREM T 5.7. – The complex SO(g, b,n)χn,p is a direct factor in SO(g, b,n) and we have 0 O i SO(g, b,n)χn,p = Vn( ). For each i 1, SO(g, b,n)χn,p has a ﬁltration whose graded are given by the VO(J (n + t) − t) where J ⊂ JF , |J| = i (with multiplicity one).

5.2. The BGG complex for distributions algebras

Let UO(G) be the distribution O-algebra over G. For each G-module V , free over O,we define the complex • ← ←U ⊗ ⊗ | 0 V O(G) UO (B) (g/b) V b , O • and denote it by SO(G, B, V ). In the case where V = Vn(O) we denote this complex by • SO(G, B, n). • Remark 5.8. – The complex SO(G, B, V ) is not exact. It will become exact after applying the Grothendieck linearization functor to the associated complex of vector bundles over the Hilbert modular variety. ∈ Z V U ⊗ O For all µ [JF ], we define the Verma module O(µ)= O(G) UO (B) Wµ( ) (see − | | i ∈ Z Section 5.1). We recall that, since p 1 > n + d, Ω (n) is the set of µ [JF ] such that i Wµ(O) is an irreducible subquotient of O(g/b) ⊗ Vn(O)|b. Lemma 5.2 translates as: • LEMMA 5.9. – The module SO(G, B, n) has a finite filtration by UO(G)-submodules whose i successive quotients are given by VO(µ), with µ ∈ Ω (n).

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G Since UO(g) ⊂UO(G) ⊂ UE(g), the center UO(g) of UO(g) is contained in the center of UO(G). Consider the central characters χ = χ O and χ = χ (see Section 5.1). µ,p µ, µ,p µ,Fp If W is a UO(G)-module generated by a vectorv of weight µ and annihilated by u, then G UO(g) acts on W by the character χµ,p. Put I = µ∈Ω•(n) ker(χµ,p). By the last lemma the O S• G ﬁnite -module O(G, B, n) is a R := UO(g) /I-module. Let χ1 = χn,p, χ2,...,χr be the distinct algebra homomorphisms from R in Fp.For1 j r, we put S• ∈S• O(G, B, n)χj = x O(G, B, n) ker(χµ,p) x =0 . ∈ • µ Ω (n),χµ,p=χj

The same way as in Theorem 5.7 we obtain a decomposition:

r S• S• (20) O(G, B, n)= O(G, B, n)χj . j=1 i ∼ THEOREM 5.10. – S (G, B, n) VO( (n + t) − t). O χn,p = J⊂JF ,|J|=i J i Proof. – Assume ﬁrst n =0. Since u is Abelian, U acts trivially on O(g/b) and Lemma 5.2 i ∼ gives (g/b) W − (O). Since UO(G) is free over UO(B) we obtain: O = J⊂JF ,|J|=i J (t) t i i ∼ S S VO − O(G, B, 0) = O(G, B, 0)χ0,p = J (t) t .

J⊂JF ,|J|=i

For n 0,usingthen =0case, we already have a decomposition: Si ∼ U ⊗ O ⊗ O O(G, B, n) = O(G) UO (B) WJ (t)−t( ) Vn( ) .

J⊂JF ,|J|=i

By (20), the theorem is a consequence of the following lemma, whose proof follows directly from the one of Proposition 5.5. ∼ EMMA UO ⊗U − O ⊗ O VO − L 5.11. – ( (G) O (B) (WJ (t) t( ) Vn( )))χn,p = (J (n + t) t).

5.3. BGG complex for crystals

r+1 Our reference is [31, Section 4]. For every integer r 0 we put Sr =Spec(Z/p ).Fora Z 1 × [ ∆ ]-scheme X, we put Xr = X Sr. crys We have an equivalence of categories between the category of crystals over (X0/Sr)log and the category of O -modules M which are locally free and endowed with integrable, quasi- Xr 1 unipotent connection with logarithmic poles ∇ : M→M⊗O Ω (dlog(∞X )). Xr Xr /Sr We have a functor L, called the linearization functor, from the category of sheaves of O -modules to the category of crystals on (X /S )crys. Xr 0 r log By the log-crystalline Poincaré lemma, we have a resolution: • 0 →M→L M⊗O Ω (dlog ∞) . Xr Xr /Sr

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Let W1 and W2 be two B-modules with weights smaller than (p − 1)t. Put Wi = FB(Wi), i =1, 2 (see Section 2.3). By [31, §5.2.4] we have a homomorphism U ⊗ U ⊗ → W W HomUO (G) O(G) UO (B) W1 , O(G) UO (B) W2 Diﬀ.Op.( 2,r, 1,r), which becomes an isomorphism after tensoring with E (see (12)). We apply now the above construction to the toroidal compactiﬁcation of the Hilbert modular variety M and the vector bundle Vn. For every r 0 we have an injective homomorphism of complexes of vector bundles over M r • • K W → V ⊗ ∞ (21) n := J (n+t)−t,n0 n O Ω (dlog ). Mr M r /Sr J⊂JF

PROPOSITION 5.12. – The map (21) is a strict injective homomorphism of ﬁltered complexes.

K• V ⊗ • ∞ By the last proposition L( n) is a direct factor in L( n O Ω (dlog )), which is Mr M r /Sr exact by the Poincaré’s crystalline lemma. Therefore L(K• ) is also exact. Since the functor L is n Hj V ∼ Hj K• exact, we deduce ﬁltered isomorphisms log-dR(M r/Sr, n) = (M r/Sr, n). Recall that p does not divide ∆ and p − 1 > |n| + d. Under this assumption we have

THEOREM 5.13. – The spectral sequence given by the Hodge ﬁltration

i,j i+j−|J| i+j W − ⇒H V E1 = H (M r, J (n+t) t,n0 )= log-dR(M r, n) J⊂JF ,|p(J)|=i degenerates at E1 for r =0: i j j−|J| gr H (M , V )= H (M , W − ). log-dR /Fp n /Fp J (n+t) t,n0 J⊂JF ,|J|j,|p(J)|=i

Proof. – The proof is formally the same as the one of Theorem 2.3(ii), once we have Proposition 5.12. The degeneration of the spectral sequence follows from a result of Il- s 1 lusie [27, Proposition 4.13.] applied to the semi-stable morphism πs : A → M of smooth Zp-schemes. 2

Remark 5.14. – (i) It follows from the same arguments as in Corollary 2.7(i), that the above decomposition is Hecke equivariant, except for the Tp operators, when p divides p. When p is totally split in F , we could use Wedhorn’s results [42] to write Tp as a sum of correspondences and try to adapt to this case the method of [16]. Unfortunately, this approach is not available when p is not totally split in F . In the proof of Theorem 6.7, we will use a different method to prove the Tp-equivariance of the above decomposition after a localization outside p. (ii) The commutativity of the Hecke operators outside p follows from the degeneration at E1 0 as in the proof of Corollary 2.7(i). The last graded piece H (Y,W − ) of the JF (n+t) t,n0 ﬁltration is independent of the toroidal compactiﬁcation by the Koecher Principle (3).

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 544 M. DIMITROV

6. Integral cohomology over certain local components of the Hecke algebra

6.1. The key lemma

r Let q = p and denote by σ1,...,σr the elements of Gal(Fq/Fp). F THEOREM 6.1 (Brauer–Nesbitt, Steinberg [39]). – The group SL2( q) has exactly q irre- r F aj σj ducible representations on finite-dimensional q-vector spaces, namely the j=1(Sym ) , for 0 aj p − 1. F |I| COROLLARY 6.2. – For every finite set I, the group i∈I SL2( q) has exactly q irreducible representations on finite-dimensional Fq-vector spaces, namely the r ai,j σj − Symi , for 0 ai,j p 1. i∈I j=1

In [30] Mazur states the following:

LEMMA 6.3. – Let Φ be a group and let ρ0 be a representation of Φ on a ﬁnite-dimensional Fq-vector space W . Let ρ :Φ→ GL2(Fq) be an absolutely irreducible representation such that ∈ s.s. ⊕···⊕ for all y Φ, the characteristic polynomial of ρ(y) annihilates ρ0(y). Then, ρ0 = ρ ρ and in particular ρ ⊂ ρ0.

The corresponding statement for another group than GL2 is false in general. Here is an 2 example for GL3:takeρ =Sym :GL2(Fq) → GL3(Fq) and ρ0 = det : GL2(Fq) → GL1(Fq). Nevertheless, we have a generalization for the special group: D H(Fq)= GL2(Fq) := (Mi)i∈I ∈ GL2(Fq) ∃δ ∈D, ∀i ∈ I, det(Mi)=δ i∈I i∈I and the particular representation σi,τ F → F → σi,τ ρ1 = Sti : H( q) GL2d ( q), (Mi)i∈I Mi , ∈ ∈ i ∈ ∈ i i I,τ JF i I,τ JF

i where (J )i∈I is a partition of JF and for all i ∈ I, (σi,τ ) ∈ i are two by two distinct elements F τ JF 1 of Gal(Fq/Fp) (St = Sym denotes the standard representation of GL2).

LEMMA 6.4. – Let ρ0 be a representation of H(Fq) on a ﬁnite-dimensional Fq-vector space W , such that for all y ∈ H(Fq) the characteristic polynomial of ρ1(y) annihilates ρ0(y). s.s. ⊕···⊕ Then ρ0 = ρ1 ρ1 (each irreducible subquotient of ρ0 is isomorphic to ρ1).

Proof. – We can assume that ρ0 is absolutely irreducible. Consider the exact sequence → F F → F →Dν → 1 H1( q)= i∈I SL2( q) H( q) 1. By Corollary 6.2, we know that each r a i,j σj irreducible subquotient of ρ0|H1(Fq ) is of the form i∈I ( j=1(Symi ) ), with 0 ai,j p − 1.

The subspace corresponding to the highest weight of the representation ρ0|H1(Fq ) is preserved by the standard torus of H(Fq) and therefore contains an eigenvector x for the action of this torus. Since ρ0 is irreducible, it is generated by x, and therefore ρ0 is isomorphic to a twist r ai,j σj of i∈I ( j=1(Symi ) ) by some power of the character ν (in particular ρ0|H1(Fq ) is also irreducible).

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 545

Since the characteristic polynomial of ρ1 annihilates ρ0, the set of the weights of ρ0 is a subset of the set of the weights of ρ1, and therefore ρ0 = ρ1. 2 Q In Section 3.5 we proved under the assumption (LI ) that Ind ρ¯(G ) contains the image Ind ρ¯ F F i JF of the map φ =(φ )i∈I : H(Fq) → GL2(Fq) . −1 Denote by F the ﬁxed ﬁeld of ρ¯ (φ(H(Fq))).

LEMMA 6.5(Keylemma).– Let ρ be a representation of G on a ﬁnite-dimensional 0 F κ-vector space W . Assume (LI ) and assume that, for every y ∈G , the characteristic Ind ρ¯ F Q polynomial of ( Ind ρ¯)(y) annihilates ρ (y). Then each G -irreducible subquotient of ρ is F 0 F 0 Q isomorphic to IndF ρ¯.

Proof. – It is enough to treat the case where ρ0 is irreducible. The idea is show that the action of G on W is through the algebraic group H(F ) and use Lemma 6.4. F q Q Put ρ¯ =(Ind ρ¯)|G . By the annihilation assumption, the group ρ (ker(¯ρ )) is an unipotent F 0 F p-group and therefore W ker(ρ ¯ ) is non-zero. Moreover the subspace W ker(ρ ¯ ) is preserved by G . F ker(ρ ¯) G Since W is irreducible we get W = W and therefore the action of on W is through F F H( q). Hence there exists a homomorphism ρ0 ﬁtting in the following commutative diagram:

Q ⊗ Ind ρ¯ G F Q GL2d (κ)

⊗ Q IndF ρ¯ G JF ρ F GL2(κ) 1

φ −1◦ G φ ρ¯ F F H( q) ρ0 ρ0

GL(W )

The characteristic polynomial of ρ1 annihilates the representation ρ0. By Lemma 6.4 each H- s.s. irreducible subquotient of W is isomorphic to ρ1, that is to say W = ρ1 as H(Fq)-modules. Since the action of G on both sides is through H(F ), we are done. 2 F q 6.2. Localized cohomology of the Hilbert modular variety Let T ⊂ T be the subalgebra generated by the Hecke operators outside a ﬁnite set of places containing those dividing np. Put m = m ∩ T.

THEOREM 6.6. – Assume f and p satisfy (I), (II) and (LIInd ρ¯). Then • d (i) H (Y,Vn(κ))m =H (Y,Vn(κ))m , • d (ii) H (Y,Vn(O))m =H (Y,Vn(O))m is a free O-module of ﬁnite rank and the O-module • d H (Y,Vn(E/O))m =H (Y,Vn(E/O))m is divisible of ﬁnite corank. d d (iii) H (Y,Vn(O))m × H (Y,Vn(E/O))m →Ois a perfect Pontryagin pairing. Proof. – (i) By Faltings’ Comparison Theorem [14] and Theorem 5.13(i) the integer |p(J)| is not a Fontaine–Laffaille weight of Hr(κ) when r

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 546 M. DIMITROV assumptions of the key Lemma 6.5 are fulﬁlled. We deduce that Hr(κ)[m]=0and therefore by r Nakayama’s lemma H (κ)m =0. The case n>dfollows by Poincaré duality. (ii), (iii) By the long exact cohomology sequence

···→Hr−1(κ) → Hr(O) −→ Hr(O) → Hr(κ) →···,

r and by the vanishing of H (κ)m for r = d, we deduce that (for r = d) the multiplication by an r uniformizer is a surjective endomorphism of H (O)m , so this last vanishes. The same way, by the long exact sequence

···→Hr(−1O/O) → Hr(E/O) −→ Hr(E/O) → Hr+1(−1O/O) →···,

r r r we deduce a surjection H (κ)m H (E/O)m [] for r = d. Since H (E/O)m is a torsion O-module, it vanishes (for r = d). The localization at m of the long exact sequence of O-modules:

···→Hr−1(E/O) → Hr(O) → Hr(E) → Hr(E/O) →···, is concentrated at the three terms of degree r = d. From this we deduce the freeness. 2

6.3. On the Gorenstein property of the Hecke algebra

THEOREM 6.7 (Theorem B). – Let f and p be such that (I), (II) and (LIInd ρ¯) hold. Then • d d (i) H (Y,Vn(κ))[m]=H (Y,Vn(κ))[m] is a κ-vector space of dimension 2 . • d d (ii) H (Y,Vn(O))m =H (Y,Vn(O))m is free of rank 2 over Tm. (iii) Tm is Gorenstein. d Proof. – In this proof we put W =H (YQ, Vn(κ))m. By using an auxiliary level structure as in [8], we can assume that the condition (NT) of Section 1.4 is fulﬁlled. (i) As in the proof of Theorem 6.6(i), by Lemma 6.5 we have an isomorphism of G -modules F ⊕r s.s. Q W [m] = IndF ρ¯ .

It is crucial to observe that I ⊂G . By Theorem 2.6 we have r 1. In order to show that p F r =1we consider the restriction of these representations to Ip. The multiplicity of the maximal Fontaine–Laffaille weight |p(JF )| in the right-hand side is r by Theorem 2.6, Corollary 2.7(ii) and Fontaine–Laffaille’s theory. On the other hand, the multiplicity of |p(JF )| in the left-hand side is equal, by Theorem 5.13, 0 to the dimension of H (Y ⊗ κ, W − )[m]: In fact, by Remark 5.14 all we have to JF (n+t) t,n0 |p(JF )| d check is the Tp-linearity of the Fontaine–Laffaille functor gr on H (YQ, Vn(κ))m .By d Theorem 6.6(ii) the module H (YQ, Vn(O))m is torsion free. Therefore Tp-linearity may be checked after extending the scalars to C, where it follows from the Strong Multiplicity One Theorem (since p is prime to the level n). We owe this idea to Diamond (see [8, Proposition 1]). 0 We will now see that dimκ H (Y ⊗ κ, W − )[m]=1.WehaveW − = JF (n+t) t,n0 JF (n+t) t,n0 ωk ⊗ νn0t/2. So we are led to show that two normalized Hilbert modular forms of weight k, level n and coefﬁcients in κ = Tm/m having the same eigenvalues for all the Hecke operators are equal. One should be careful to observe that the Hecke operators permute the connected components M1(c, n) of the Shimura variety Y = Y1(n) (here the ideal c runs over a set of + representatives of ClF ). We use Hecke relations between Fourier coefﬁcients and eigenvalues for

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 547 the Hecke operators and the q-expansion Principle at the ∞ cusp of each connected component M1(c, n) (see Section 1.7). (ii), (iii) Mazur’s argument in the elliptic modular case remains valid. By Theorem A, the d V O d V O twisted Poincaré pairing (6) on H (Y, n( ))m =Hc (Y, n( ))m is a perfect duality of Tm-modules, so it would be enough to show (ii). 2 ∼ Again using the perfectness of the twisted Poincaré pairing W × W → κ we obtain W = ⊗ ∼ HomTm (W, κ), and so W Tm κ = W/mW = Hom(W [m],κ), and therefore ⊗ dimκ(W Tm k)=dimκ W [m] , which equals 2d, by (i). Then (ii) follows from the following:

LEMMA 6.8. – Let T be a torsion free local O-algebra (T →T ⊗O E) of maximal ideal m and residue ﬁeld κ = T /m. Let M be a ﬁnitely generated T -module such that M⊗O E is free of rank r over T⊗O E.If M⊗T κ is a κ-vector space of dimension r, then M is free of rank r over T .

Proof. – Since M⊗T k is of dimension r, the Nakayama’s lemma gives a surjective homomorphism of T -modules T r M. Denote by I its kernel. We have an exact sequence of O-modules 0 → I →Tr →M→0.

By tensoring it by ⊗OE (or equivalently by ⊗T (T⊗O E)) we obtain another exact sequence

r 0 → I ⊗O E → (T⊗O E) →M⊗O E → 0.

By comparing the dimensions over E we get I ⊗O E =0. Since I is torsion free, I =0. 2

6.4. An application to p-adic ordinary families

For r 1, consider the following open compact subgroups of G(Af ) ∗∗ K pr = u ∈ K (n) | u ≡ mod pr , 0 1 0 ∗ 1 ∗ K pr = u ∈ K (n) | u ≡ mod pr . 11 1 01 r r r Let Y0(p ) (respectively Y11(p )) be the Hilbert modular variety of level K0(p ) (respectively r K11(p )). • r ∗ r r The cohomology group H (Y11(p ), Vn(E/O)) has a natural action of K0(p )/K11(p ) r × r × ∗ × (o/p ) × (o/p ) (we denote by the Pontryagin dual). Therefore the group T (Zp)/o acts on the inductive limit H•(Y (p∞), V (E/O))∗ := lim H•(Y (pr), V (E/O))∗. 11 n → 11 n • ∞ ∗ By Hida’s stabilization lemma, the ordinary part of H (Y11(p ), Vn(E/O)) (that is the part where the Hecke operators T0,p of Definition 1.13 are invertible for all p dividing p)is H• • ∞ O ∗ independent on n. We denote it by ord := Hord(Y11(p ),E/ ) . H• O Z × By the above discussion ord has a structure of a Λ:= [[T ( p)/o ]]-module. It is of finite type, by a theorem of Hida. We also define the p-adic ordinary Hecke Λ-algebra T∞ := lim T (Y (pr)).AsT∞ k,ord ← k,ord 11 k,ord T∞ H• T∞ is independent of k, we denote it by ord. Then ord is a ord-module.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 548 M. DIMITROV

× An arithmetic character of T (Zp)/o is by definition a character whose restriction to an open subgroup is given by an algebraic character. It is immediate that such a character is a product of an algebraic character and a finite order character. An algebraic character of × n −m T (Zp) D(Zp) × D(Zp) trivial on o is necessarily of the form (u, ) → u , where m, n ∈ Z[JF ] and m +2n ∈ Zt. Hence, the general form of an arithmetic character ψ of × n −m T (Zp)/o is (u, ) → u ψ1(u)ψ2(), where ψ1,ψ2 are finite order characters. Every such ψ induces an O-algebra homomorphism Λ →O, whose kernel is denoted by Pψ. T T∞ Let m be a maximal ordinary ideal of and let m∞ be a maximal ideal of ord above m.We T∞ H• T∞ H• denote by m∞ (respectively m∞ ) the localization of ord (respectively of ord)atm∞.

PROPOSITION 6.9. – Let m be such that (I), (II) and (LIInd ρ¯) hold. Then Hd (i) m∞ is free of ﬁnite rank over Λ and we have an exact control: Hd Hd • r V O ∗ ∞ /Pψ ∞ H Y11 p , ψ(E/ ) , m m mr

Hd d T∞ (ii) m∞ is free of rank 2 over m∞ , and T∞ (iii) Hida’s control theorem for the Hecke algebra holds, that is m∞ is a free Λ-algebra of T∞ T∞ T r ﬁnite rank and for every ψ we have m∞ /Pψ m∞ ψ(Y11(p ))mr .

Proof. – (i) The proof is very similar to the one of [31, Theorem 9]. It uses that a Λ-module is free, if it is free of constant rank over O for inﬁnitely many specializations. In our case, it is enough to specialize at weights of the form k +(p − 1)k and verify the exact control criterion using Theorem 6.6. We omit the details, because (i) follows from (ii) and (iii). → T∞ → Hd (ii) Consider Λ m∞ EndO( m∞ ). The specialization at ψ = ψk gives O→T∞ T∞ → Hd Hd m∞ /Pk m∞ EndO m∞ /Pk m∞ .

d V O ∗ d V O ∗ d V O By Theorem 6.6 we have H (Y0(p), n(E/ ))m H (Y, n(E/ ))m H (Y, n( ))m Hd Hd d V O and an exact control: m∞ /Pk m∞ H (Y, n( ))m. d ∞ ∞ d H ⊗T∞ T T H ⊗ From here and from Theorem B we obtain that m∞ m∞ ( m∞ /Pψ m∞ ) m∞ Λ d d d Λ/P is free of rank 2 over T . Hence H ⊗T∞ κ is free of rank 2 over T ⊗T∞ T∞ k m m∞ m∞ m m∞ /Pψ m∞ T∞ Hd κ = κ. Then Lemma 6.8 applies to the m∞ -module m∞ which is ﬁnitely generated over the local algebra Λ. Hd (iii) Since m∞ is a free Λ-module, it admits a direct sum decomposition with respect to the Weyl group action on the Betti cohomology: Hd Hd m∞ = m∞ [J ].

J⊂JF

Hd T∞ T∞ Every m∞ [J ] is free of rank 1 over m∞ and free over Λ. Therefore m∞ is free over Λ and exact control holds. 2

∈ r COROLLARY 6.10. – Let f Sk+(p−1)k (Y0(p )) be a newform and let p be a prime not dividing NF/Q(d), such that p − 1 > (kτ − 1) and (LIInd ρ¯) holds. Then Theorems A and B hold.

◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 549

List of symbols

1 A ...... HBAV,Definition 1.6 K1(n),K1 (n) ...... Section1.1 α ..µn-level structure, Definition 1.8 t • A ...... dualHBAV Kn ...... Section2.2 β ...... Section2.4 A K• R ...... universalHBAVSection1.4 n ...... Section2.3 γ ...... elementofG( ) a ...... idealofo L ...... field Γ1(c, n) ...... Section1.1 − 1 B ...... standardBorelofG m =(k0t k)/2 ....Definition 1.1 Γ1(c, n) ...... Section1.1 1 b ...... LiealgebraofB M,M connected HMV Section 1.4 δ, δJ ...... Section1.14 c, c+ ...... Section1.3 M ...... Section1.4 ∆ ...... NF/Q(nd) 1 c(f,a) ...... Section1.10 M,M ...... Section1.6 δp,εp ...... tamecharactersofIp ∗ 1∗ ClF ...... classgroupSection1.2 M ,M ...... Section1.8 ε ...... finiteordercharacter Cl+ . narrow class group Section 1.1 m . . . maximal ideal of T Section 0.3 ...... unitofF F d ...... degreeofF m . . maximal ideal of T Section 6.2 J ...... Section1.13 F D ...... ResQ Gm n ...... weightofG ˆJ ...... Section1.14 Dp decomposition group Section 2.6 N ...... normalizerofT η ...... idèle d ...... differentofF n ...... levelideal ι ...... involutionSection1.14 D O Q Q ...... Section3.4 ...... ringofintegersofE ιp ...... embeddingof in p E ...... largep-adic field o ...... ringofintegersofF κ ...... residuefieldofO f ...... Hilbertmodular newform o ...... Section1.4 λ ...... c-polarization, Definition 1.7 × × ¯ F ...... totallyrealnumberfield o+, on,1 ...... Section0.4 λ . c-polarization class, Definition 1.7 F ...... GaloisclosureofF p ...... primenumber µ ...... weightofB F ...... beforeLemma6.5 p(J) ...... Section2 ν . . reduced norm G → D Section 1 F ...... Section3.4 P ...... maximalidealofO ν ...... Section1.5 FD, FB, FG . . functors Section 2.3 p ...... primeofF dividing p ξ ...... elementofF g ...... Hilbertmodular form q ...... powerofp π ...... Section1.4 gJ ...... Section1.13 R ...... ring π¯ ...... Section1.6 gτ ....internalconjugateSection2.5 s ...... Sections1.9,2.1 πs, (¯π)s, πs ...... Section1.9 F G ...... ResQ GL2 S ...... basescheme v ...... uniformizerofFv ∗ G ...... G ×D Gm Sk(n,ψ) ...... Definition 1.3 ρ = ρf,p ....p-adic repr. Section 0.1 GL ...... GaloisgroupofL Sa,Ta Hecke operators Section 1.10 ρ¯ =¯ρf,p ....modp repr. Section 0.1 g ...... LiealgebraofG t = ∈ τ ...... Definition 1.1 ρ1 ...... Section6.1 τ JF G ...... Section1.6 T ...... standardtorusofG Σ, Σ ...... Sections1.6,1.9 H ...... aboveLemma3.15 t ...... LiealgebraofT σi,τ ...... Section3.5 + T τ ...... infiniteplaceofF hF ...... Section1.1 ...... HeckealgebraSection0.3 H1 H1 T φ ...... Section3.5 dR, log-dR . . . . Sections 1.4, 1.9 ...... reducedHeckealgebra HF , H ...... Section1.1 U ...... standardunipotentofG ϕ ...... Heckecharacter Ip .....inertiagroupatp Section 2.6 U(b),U(g) ...... Section2.2 χn ...... Section2.2 J ...... subsetofJF u ...... LiealgebraofU ψ . . . Hecke character, Definition 1.3 JF ...... setofinfiniteplaces of F v ...... finiteplaceofF ω ...... modp cycl. char. i ω ...... Section1.4 JF ...... Section3.5 Vn ...... G-modules Section 1.14 k JF,p ...... Section2.6 Vn . local systems Sections 1.14, 2.1 ω ...... Section1.5 k = n +2t ...... Definition 1.1 V, W ...... Section2.3 Ωf ...... periods,Definition 4.5 k0,n0 ...... Definition 1.1 Wµ ...... B-modules Section 2.2 ( , )K1(n) ...... Definition 1.5 K ...... CMfield Wf ...... Section2.1 ( , )n ...... Definition 1.5 + 1 K∞,K∞ ...... Section1.1 Y,Y ...... HMVSection1.4 [ , ] ...... Section1.14

Acknowledgements

I would like to thank J. Bellaïche, D. Blasius, G. Chenevier, L. Dieulefait, E. Ghate, M. Kisin, V. Lafforgue, A. Mokrane, E. Urban, J. Wildeshaus and J.-P. Wintenberger for helpful conversations. This article was completed during my visit at UCLA on an invitation by H. Hida. I would like to thank him heartily for his hospitality and for many inspiring discussions. I am grateful to G. Kings and R. Taylor for their interesting comments on an earlier version of this paper that have much improved it. Finally, I would like to thank J. Tilouine who suggested that I study this problem and supported me during the preparation of this article.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE 550 M. DIMITROV

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◦ 4e SÉRIE – TOME 38 – 2005 – N 4 COHOMOLOGY OF HILBERT MODULAR VARIETIES 551

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(Manuscrit reçu le 16 février 2004; accepté, après révision, le 31 mars 2005.)

Mladen DIMITROV Université Paris 7, UFR de Mathématiques, Case 7012, 2 place Jussieu, 75251 Paris, France E-mail: [email protected]

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE