Local-global compatibility for <span class="mathjax-formula">$l=p$</span>, I

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ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

THOMASBARNET-LAMB, TOBYGEE, DAVIDGERAGHTY, RICHARDTAYLOR

Local-global compatibility forl=p, I

Tome XXI, n^o1 (2012), p. 57-92.

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Annales de la Facult´e des Sciences de Toulouse Vol. XXI, n 1, 2012 pp. 57–92

Local-global compatibility for

l=p

, I

Thomas Barnet-Lamb⁽¹⁾, Toby Gee⁽²⁾, David Geraghty⁽³⁾, and Richard Taylor⁽⁴⁾

ABSTRACT.— We prove the compatibility of the local and global Lang- lands correspondences at places dividinglfor thel-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GLn over an imaginary CM ﬁeld, under the assumption that the automorphic representations have Iwahori-ﬁxed vectors at places dividingland have Shin-regular weight.

R^ÉSUMÉ.— Nous prouvons la compatibilité entre les correspondances de Langlands locale et globale aux places divisantlpour les représentations galoisiennesl-adiques associèes à des représentations automorphes cus- pidales algébriques et régulières de GLnsur un corps CM qui sont duales de leur conjuguée complexe, sous les hypothèses supplémentaires que ces représentations automorphes ont des vecteurs fixes par un sous-groupe d’Iwahori aux places divisantlet ont un poids régulier au sens de Shin.

(∗) Re¸cu le 08/06/2011, accept´e le 08/12/2011

The second author was partially supported by NSF grant DMS-0841491, the third author was partially supported by NSF grant DMS-0635607 and the fourth author was partially supported by NSF grant DMS-0600716 and by the Oswald Veblen and Simonyi Funds at the IAS.

(1) Department of Mathematics, Brandeis University [email protected]

(2) Department of Mathematics, Northwestern University [email protected]

(3) Princeton University and Institute for Advanced Study [email protected]

(4) Department of Mathematics, Harvard University [email protected]

Article propos´e par Pierre Colmez.

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Introduction

In this paper we prove the compatibility at places dividinglof the local and global Langlands correspondences for thel-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations ofGLnover an imaginary CM field in the special case that the automorphic representations have Iwahori-fixed vectors at places dividingl and have Shin-regular weight. In the sequel to this paper [2] we build on these results to prove the compatibility in general (up to semisimplification in the case of non-Shin-regular weight).

Our main result is as follows (see Theorem 1.2 and Corollary 1.3).

Theorem A. — Let m 2 be an integer, l a rational prime and ı : Ql ∼

→ C. Let F be an imaginary CM ﬁeld and Π a regular algebraic, conjugate self-dual cuspidal automorphic representation ofGLm(A^F). IfΠ has Shin-regular weight andv|l is a place ofF such thatΠ^Iwv ^m,v ={0}, then

ıWD(rl,ı(Π)|^GFv)^F−ss∼= rec(Πv⊗ |det|^(1−m)/2).

In particularWD(rl,ı(Π)|^GFv)is pure.

(See Section 1 for any unfamiliar terminology.) The proof is essentially an immediate application of the methods of [14], applied in the setting of [13]

rather than that of [8], and we refer the reader to the introductions of those papers for the details of the methods that we use. Indeed, if Π is square- integrable at some ﬁnite place, then the result is implicit in [14], although it is not explicitly recorded there. For the convenience of the reader, we make an eﬀort to make our proof as self-contained as possible.

Notation and terminology

We write all matrix transposes on the left; so^tA is the transpose of A.

We letBm⊂GLmdenote the Borel subgroup of upper triangular matrices and Tm⊂GLm the diagonal torus. We let Im denote the identity matrix in GLm. We will sometimes denote the productGLm×GLn byGLm,n.

If M is a ﬁeld, we let M denote a separable closure of M and GM

the absolute Galois group Gal (M /M). Let ldenote the l-adic cyclotomic character

Letpbe a rational prime andK/Q^pa ﬁnite extension. We letO^Kdenote the ring of integers ofK, ℘K the maximal ideal of O^K, k(νK) the residue ﬁeld O^K/℘K, νK : K^× Z the canonical valuation and | |^K : K^× →Q^×

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the absolute value given by |x|^K = #(k(νK))⁻^ν^K^(x). We let| |^1/2K :K^× → R^×>0 denote the unique positive unramiﬁed square root of| |^K. IfK is clear from the context, we will sometimes write| |for| |^K. We let FrobK denote the geometric Frobenius element ofGk(νK)andIK the kernel of the natural surjectionGK Gk(νK). We will sometimes abbreviate Frob_Q_p by Frobp. We let WK denote the preimage of Frob^Z_K under the mapGK Gk(ν(K)), endowed with a topology by decreeing thatIK ⊂WKwith its usual topology is an open subgroup of WK. We let ArtK : K^{× ∼}→W_K^ab denote the local Artin map, normalized to take uniformizers to lifts of FrobK.

Let Ω be an algebraically closed ﬁeld of characteristic 0. A Weil-Deligne representation of WK over Ω is a triple (V, r, N) where V is a ﬁnite dimensional vector space over Ω, r:WK →GL(V) is a representation with open kernel and N : V → V is an endomorphism with r(σ)N r(σ)⁻¹ =

|Art⁻¹_K (σ)|^KN. We say that (V, r, N) is Frobenius semisimple ifris semisimple and we let (V, r, N)^F−ss denote the Frobenius semisimpliﬁcation of (V, r, N) (see for instance Section 1 of [14]) and we let (V, r, N)^ss denote (V, r^ss,0). If Ω has the same cardinality asC, we have the notions of a Weil- Deligne representation beingpure or pure of weightk – see the paragraph before Lemma 1.4 of [14].

We will let recK be the local Langlands correspondence of [8], so that if π is an irreducible complex admissible representation of GLn(K), then recK(π) is a Weil-Deligne representation of the Weil group WK. We will write rec for recK when the choice of K is clear. If ρ is a continuous representation of GK over Ql with l = p then we will write WD(ρ) for the corresponding Weil-Deligne representation ofWK. (See for instance Section 1 of [14].)

Ifm1 is an integer, we let Iwm,K ⊂GLm(O^K) denote the subgroup of matrices which map to an upper triangular matrix inGLm(k(νK)). Ifπis an irreducible admissible supercuspidal representation ofGLm(K) ands1 is an integer we let Sp_s(π) be the square integrable representation ofGLms(K) deﬁned for instance in Section I.3 of [8]. Similarly, ifr:WK →GLm(Ω) is an irreducible representation with open kernel and π is the supercuspidal representation rec⁻_K¹(r), we let Sp_s(r) = recK(Sp_s(π)). If K/K is a ﬁnite extension and if π is an irreducible smooth representation of GLn(K) we will write BCK/K(π) for the base change ofπtoKwhich is characterized by recK(πK) = recK(π)|^WK.

Ifρis a continuous de Rham representation ofGK overQp then we will write WD(ρ) for the corresponding Weil-Deligne representation ofWK (its construction, which is due to Fontaine, is recalled in Section 1 of [14]), and

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ifτ:K →Qpis a continuous embedding of ﬁelds then we will write HTτ(ρ) for the multiset of Hodge-Tate numbers ofρwith respect toτ. Thus HTτ(ρ) is a multiset of dimρintegers. In fact, ifW is a de Rham representation of GK over Qp and ifτ :K →Qp then the multiset HTτ(W) containsiwith multiplicity dim_Q

p(W ⊗^τ,KK(i)) ^G^K. Thus for example HTτ(l) ={−1}. IfF is a number ﬁeld andv a prime ofF, we will often denote FrobFv, k(νFv) and Iwm,Fv by Frobv, k(v) and Iwm,v. If σ : F → Qp or C is an embedding of ﬁelds, then we will writeFσ for the closure of the image ofσ.

IfF/F is a soluble, finite Galois extension and ifπis a cuspidal automorphic representation ofGLm(A^F) we will write BC_F/F(π) for its base change to F, an automorphic representation of GLn(A^F). If R : GF → GLm(Ql) is a continuous representation, we say that R is pure of weightw if for all but finitely many primesvofF,Ris unramified atv and every eigenvalue ofR(Frobv) is a Weil (#k(v))^w-number. (See Section 1 of [14].) If F is an imaginary CM field, we will denote its maximal totally real subfield byF⁺ and letcdenote the non-trivial element of Gal (F/F⁺).

1. Automorphic Galois representations

We recall some now-standard notation and terminology. Let F be an imaginary CM ﬁeld with maximal totally real subﬁeld F⁺. By aRACSDC (regular, algebraic, conjugate self dual, cuspidal) automorphic representation ofGLm(A^F) we mean that

− Π is a cuspidal automorphic representation of GLm(A^F) such that Π_∞has the same inﬁnitesimal character as some irreducible algebraic representation of the restriction of scalars fromF toQofGLm,

− and Π^c∼= Π^∨.

We will say that Π haslevel prime tol(resp.level potentially prime tol) if for allv|l the representation Πv is unramified (resp. becomes unramified after a finite base change).

If Ω is an algebraically closed ﬁeld of characteristic 0 we will write (Z^m)Hom (F,Ω),+for the set ofa= (aτ,i)∈(Z^m)^{Hom (F,Ω)} satisfying

aτ,1. . .aτ,m.

We will write (Z^m)^{Hom (F,Ω)}₀ for the subset of elements a∈ (Z^m)^{Hom (F,Ω)} with

aτ,i+a_{τ◦c,m+1−i}= 0.

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IfF/F is a ﬁnite extension we deﬁneaF ∈(Z^m)^{Hom (F}^,Ω),+ by (aF)τ,i=aτ|F,i.

Following [13] we will be interested, inter alia, in the case that either m is odd; or that m is even and for some τ ∈Hom (F,Ω) and for some odd integeriwe haveaτ,i> aτ,i+1. If either of these conditions hold then we will say that a isShin-regular. (We warn the reader that this is often referred to as ‘slightly regular’ in the literature. However as this notion is strictly stronger than ‘regularity’ we prefer to use the term Shin-regular.)

If a ∈ (Z^m)^{Hom (F,}^C^),+, let Ξa denote the irreducible algebraic representation of GL^{Hom (F,}m ^C⁾ which is the tensor product over τ of the irreducible representations of GLm with highest weights aτ. We will say that a RACSDC automorphic representation Π ofGLm(A^F) hasweightaif Π_∞ has the same inﬁnitesimal character as Ξ^∨_a. Note that in this case a must lie in (Z^m)^{Hom (F,}₀ ^C⁾.

We recall (see Theorem 1.2 of [3]) that to a RACSDC automorphic representation Π ofGLm(A^F) andı:Ql ∼

→Cwe can associate a continuous semisimple representation

rl,ı(Π) : Gal (F /F)−→GLm(Ql)

with the properties described in Theorem 1.2 of [3]. In particular rl,ı(Π)^c∼=rl,ı(Π)^∨⊗^1−m_l .

For v|l a place ofF, the representationrl,ı(Π)|^GFv is de Rham and if τ : F →Ql then

HTτ(rl,ı(π)) ={aıτ,1+m−1, aıτ,2+m−2, ..., aıτ,m}. Ifv |l, then the main result of [4] states that

ıWD(rl,ı(Π)|^GFv)^F⁻^ss∼= rec(Πv⊗ |det|⁽¹⁻^m)/2).

We recall the following result which will prove useful.

Proposition 1.1. — LetΩbe an algebraically closed ﬁeld of characteristic 0 and of the same cardinality as C.

1. Suppose K/Q^p is a ﬁnite extension. Let (V, r, N) and(V, r, N)be pure, Frobenius semisimple Weil-Deligne representations ofWK over Ω. If the representations(V, r^ss)and(V,(r)^ss)are isomorphic, then (V, r, N)∼= (V, r, N).

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2. If F is an imaginary CM ﬁeld and Π is a RACSDC automorphic representation ofGLn(A^F), then for each ı: Ω→^∼ C and each ﬁnite placev of F,ı⁻¹rec(Πv) is pure.

Proof. — The first part follows from Lemma 1.4(4) of [14]. For the second part, Theorem 1.2 of [4] states that Πv is tempered for each finite place v of F. If σ is an automorphism of C, then there is a RACSDC automorphic representation Π = ^σΠ^∞⊗Π_∞ of GLn(A^F) (see Théorème 3.13 of [5]) and we deduce that σΠv is tempered. The second part then follows from this and Lemma 1.4(3) of [14].

We can now state our main results.

Theorem 1.2. — Let m 2 be an integer, l a rational prime and ı : Ql ∼

→C. Let L be an imaginary CM ﬁeld and Π a RACSDC automorphic representation of GLm(A^L). If Πhas Shin-regular weight andv|l is a place of Lsuch that Π^Iwv ^m,v ={0}, then

ıWD(rl,ı(Π)|^GLv)^F−ss∼= rec(Πv⊗ |det|^(1−m)/2).

Before turning to the proof, we ﬁrst record a corollary.

Corollary 1.3. — Let m 2 be an integer, l a rational prime and ı:Ql

→∼ C. LetLbe an imaginary CM ﬁeld andΠa RACSDC automorphic representation of GLm(A^L). If Πhas Shin-regular weight andv|l is a place of LthenWD(rl,ı(Π)|^GLv) is pure.

Proof. — Choose a ﬁnite CM soluble Galois extension F/L such that for each primew|v ofF, BCFw/Lv(Πv)^Iw^m,w ={0}. Then WD(rl,ı(Π)|^GFw) is pure by Theorem 1.2 and Proposition 1.1. Lemma 1.4 of [14] then implies that WD(rl,ı(Π)|^GLv) is pure.

The rest of this paper will be devoted to the proof of Theorem 1.2.

2. Notation and running assumptions

For the convenience of the reader, we recall here the following notation which appears in [13]:

− If Π is a RACSDC automorphic representation ofGLm(A^F) for some integer m 2 and an imaginary CM ﬁeld F, or if Π is an algebraic Hecke character ofA^×M/M^× for a number ﬁeldM, thenRl,ı(Π) denotesrl,ı(Π^∨).

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− If L/F is a ﬁnite extension of number ﬁelds, then RamL/F (resp.

UnrL/F, resp. Spl_L/F) denotes the set of finite places ofF which are ramified (resp. unramified, resp. completely split) in L. We denote by Spl_L/F,_Q the set of rational primes psuch that every place ofF abovepsplits completely inL.

− If F is a number ﬁeld and π is an automorphic representation of GLn/F, then Ram_Q(π) denotes the set of rational primespsuch that there exists a placev|pofF withπv ramiﬁed.

− IfGis a group of the formH(F) forF/Q^pfinite andH/F a reductive group; or H(A^TF) for F a number field,H/F a reductive group and T a finite set of places of F containing all infinite places (where as usual A^TF is defined in the same way as A^F, except that one takes the restricted direct product over the places not inT); or a product of groups of this form, then we let Irr(G) (resp. Irrl(G)) denote the set of isomorphism classes of irreducible admissible representations of Gon C-vector spaces (resp. Ql-vector spaces). We let Groth (G) (resp. Grothl(G)) denote the Grothendieck group of the category of admissibleC-representations (resp.Ql-representations) of G. (See Section I.2 of [8].)

− :Z→ {0,1} is the unique function such that(n)≡n2.

− Φn is the matrix inGLn with (Φn)ij = (−1)ⁱ⁺¹δi,n+1−j.

− If R → S is a homomorphism of commutative rings, RS/R denotes the restriction of scalars functor.

− If π is a representation of a group G with a central character, we denote the central character byψπ.

We now ﬁx the following notations and assumptions which will be in force from Section 3 to Section 6:

− E is a quadratic imaginary ﬁeld;

− F⁺ is a totally real ﬁeld with [F⁺:Q]2;

− F =EF⁺ and RamF/Q ⊂Spl_F/F⁺_,_Q;

− τ:F →Cis an embedding andτE=τ|^E;

− Φ_C= Hom (F,C) and Φ⁺_C= HomE,τE(F,C);

− n3 is an odd integer;

− p∈Spl_E/_Q is a rational prime andu|pis a prime ofE;

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− wis a prime of F above uandw1=w, w2, . . . , wr denote all of the primes ofF aboveu;

− ιp:Qp ∼

→Cis an isomorphism such thatι⁻_p¹◦τ induces the placew;

− lis a rational prime (possibly equal top) andı:Ql ∼

→C. Deﬁne algebraic groupsGn andGⁿ overZby setting

Gn(R) = {(λ, g)∈R^××GLn(O^F⊗_ZR) :gΦntg^c=λΦn}, and Gⁿ(R) = R_O_E_/_Z(Gn×_ZO^E)(R) =Gn(O^E⊗_ZR)

for anyZ-algebra R. ThenGn×ZQandGn×ZQare reductive. We letθ denote the action onGⁿ induced by (1, c) onGn×_ZO^E.

IfRis anE-algebra, thenGn(R) is a subgroup ofR^××GLn(F⊗_QR) = R^××GLn(F⊗ÊR)×GLn(F⊗Ê,cR) and the projection ontoR^××GLn(F⊗Ê R) defines an isomorphism

Gn(R)∼=R^××GLn(F⊗^ER).

It follows that Gn×_QE →^∼ G^m×RF/E(GLn). [Note that here G^m is the multiplicative group, rather thanGⁿ in the casen=m.]

If v ∈ Unr_F/_Q, then Kv := Gn(Z^v) (resp. K^v := Gⁿ(Z^v)) is a hyper- special maximal compact subgroup ofGn(Q^v) (resp.Gⁿ(Q^v)). In this case we say that a representation ofGn(Q^v) (resp.Gⁿ(Q^v)) is unramified if the space of Kv-invariants (resp. K^v-invariants) is non-zero. Furthermore, we define the unramified Hecke algebras Hûr(Gn(Q^v)) andHûr(Gⁿ(Q^v)) with respect toKv andK^v respectively, as in Section 1.1 of [13]. (We note that these areC-algebras.) IfT is a set of places ofQwith{∞} ∪RamF/Q⊂T, we letK^T =

v∈TKv⊂Gn(A^T).

We say that a representation Πv of Gⁿ(Q^v) isθ-stable if Πv◦θ ∼= Πv

and we let Irr^θ⁻^st(Gⁿ(Q^v)) ⊂ Irr(Gⁿ(Q^v)) be the subset of θ-stable representations. For v ∈ Unr_F/_Q, we let Irr^ur(G(Q^v)) ⊂ Irr(G(Q^v)) (resp.

Irrûr(Gⁿ(Q^v)) ⊂ Irr(Gⁿ(Q^v)), resp. Irrûr,θ⁻^st(Gⁿ(Q^v)) ⊂ Gⁿ(Q^v)) denote the subset consisting of unramified (resp. unramified, resp. unramified, θ- stable) representations.

Let # :RF/Q(GLn)→RF/Q(GLn) denote the map g→Φntg^−cΦ⁻¹_n . If v is a rational prime, then

Gⁿ(Q^v) =Gn(E⊗_QQ^v)∼= (E⊗_QQ^v)^××GLn(F⊗_QQ^v).

If (λ, g) ∈ Gⁿ(Q^v), then θ(λ, g) = (λ^c, λ^cg^#). Let Πv ∈ Irr(Gⁿ(Q^v)) and write Πv = ψv⊗Π¹_v with respect to the above decomposition of Gⁿ(Q^v).

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Then Πv is θ-stable if and only if (Π¹_v)^∨ ∼= Π¹_v◦c, and ψΠ¹_v|(E⊗QQ^v⁾^× = ψ^c_v/ψv.

We now recall the existence of local base change maps in the following cases (see Section 4.2 of [13] for details):

− Case 1:Ifv∈Unr_F/_Q, we have a map

BCv : Irr^ur(Gn(Q^v))→Irr^ur,θ⁻^st(Gⁿ(Q^v)).

(Note that the assumption v ∈ Ram_Q(2) in Case 1 of Section 4.2 of [13] plays no role there.) This is induced by a homomorphism of C-algebras BC^∗_v:H^ur(Gn(Qv))→ H^ur(Gn(Qv)).

− Case 2:Ifv∈Spl_F/F⁺_,_Q, we have a map

BCv: Irr(Gn(Q^v))→Irr^θ⁻^st(Gⁿ(Q^v)).

If in additionv∈UnrF/Q, then this map is compatible with the map in Case 1.

In Case 2, the map BCv is described explicitly in [13]. We recall the explicit deﬁnition here, assumingv ∈Spl_E/_Q. Lety|v be a place of E, and regardQv as an E-algebra viaQv ∼

→Ey. We get an isomorphism Gn(Qv)−→^∼ Q^×v ×

x|y

GLn(Fx)

where the product is over all places ofF dividingy. Letπv ∈Irr(Gn(Qv)) and decomposeπv =πv,0⊗πy with respect to the above decomposition of Gn(Q^v). If we decompose

Gⁿ(Q^v) =E_y^××E_y^×c×

x|y

GLn(Fx)×

x|y

GLn(Fx^c)

then BCv(πv) = (ψy, ψy^c,Πy,Πy^c), where

(ψy, ψy^c,Πy,Πy^c) = (πp,0, πp,0(ψπy|Ey^×◦c), πy, π_y^#) andπ^#_y(g) :=πy(Φntg^−cΦ⁻¹_n ). (In particular,π_y^#∼=π_y^∨◦c.)

The discussion above can be carried out equally well in the setting of Ql-representations, and we define Irr^θ_l⁻^st(Gⁿ(Q^v)), Irrûr_l (G(Q^v)) etc. in the obvious fashion. We also define a base change map BCv in Case 1 (resp.

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Case 2) by setting BCv(π) = ı⁻¹BCv(ıπ) for π ∈ Irr^ur_l (Gn(Q^v)) (resp.

π∈Irr(Gn(Q^v))).

Letξ_Cdenote an irreducible algebraic representation ofGnoverC. There is an isomorphismGⁿ(C) =Gn(E⊗_QC)∼=Gn(C)×Gn(C) induced by the isomorphism E⊗_QC →^∼ C×C which sends e⊗z to (τ(e)z, τ^c(e)z). We associate to ξ a θ-stable irreducible algebraic representation Ξ ofGⁿ over Cby setting Ξ :=ξ⊗ξ. Every such Ξ arises in this way.

We also ﬁx the following data:

− V =Fⁿ as anF-vector space;

− ·,·:V ×V →Qis a non-degenerate pairing such that f v1, v2= v1, f^cv2for allv1, v2∈V andf ∈F;

− h: C → EndF(V)⊗Q R is an R-algebra embedding such that the bilinear pairing (V ⊗QR)×(V ⊗QR)→R; (v1, v2)→ v1, h(i)v2is symmetric and positive deﬁnite.

Under the natural isomorphism EndF(V)⊗_QR∼=

σ∈Φ⁺_C Mn(C) we assume thathsends

z→

zIpσ 0 0 zIqσ

σ∈Φ⁺_C

for somepσ, qσ∈Z0 withpσ+qσ =n.

Deﬁne a reductive algebraic groupG/Qby setting

G(R) ={(λ, g)∈R^××GLn(F⊗_QR) :gv1, gv2=λv1, v2for allv1, v2∈V⊗_QR} for each Q-algebra R. Note that Gn is a quasi-split inner form of G. Let ν :G→G^m denote the homomorphism which sends (λ, g) toλ.

By Lemma 5.1 of [13] we can and do assume that·,·andhhave been chosen so that

− G_Q_v is quasi-split for each rational primev;

− for eachσ∈Φ⁺_C, we have (pσ, qσ) = (1, n−1) ifσ=τand (pσ, qσ) = (0, n) otherwise.

As a consequence, we can and do ﬁx an isomorphism G×_QA^∞∼=Gn×_QA^∞.

Using this isomorphism, we will henceforth identify the groupsGn(Q^v) and G(Q^v) for all primes v. LetCG ∈Z^>0 be the integer|ker¹(Q, G)| ·τ(G) in the notation of [13].

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LetT be a (possibly infinite) set of places of Q containing∞ and let Tfin=T−{∞}. Let Γ be a Galois group with its Krull topology, or the Weil group of a local field, or a quotient of such a group. We define an admissible Ql[G(A^T)×Γ]-module to be an admissibleQl[G(A^T)]-moduleRwith a commuting continuous action of Γ (the continuity condition here means that for each compact open subgroupU ⊂G(A^T), the induced map Γ→Aut (RÛ) is continuous for thel-adic topology onRÛ). We let Grothl(G(A^T)×Γ) denote the Grothendieck group of the category of admissibleQl[G(A^T)×Γ]- modules. If R is an admissible Ql[G(A^T)×Γ]-module, we let [R] denote its image in Grothl(G(A^T)×Γ). We let Irrl(G(A^T)×Γ) denote the set of isomorphism classes of irreducible admissibleQl[G(A^T)×Γ]-modules. (See Section I.2 of [8].)

Now suppose that T is ﬁnite, that p∈T and letJ/Q^p be a reductive group. LetG be a topological group which is of the formG(A^Tfin)×Γ, or G(ATfin−{p})×Γ, orG(ATfin−{p})×J(Q^p). Let [X]∈Grothl(G(A^T)×G) and write [X] = _πT,ρn(π^T⊗ρ)·[π^T⊗ρ] wheren(π^T⊗ρ)∈Zandπ^T (resp.

ρ) runs through Irrl(G(A^T)) (resp. Irrl(G)). For a givenπ^T ∈Irrl(G(A^T)), we let

[X][π^T] =

ρ

n(π^T ⊗ρ)·[π^T⊗ρ]∈Grothl(G(A^T)×G).

If RamF/Q⊂T and Π^T ∈Irr(Gⁿ(A^T)) is unramiﬁed at allv∈T, then we deﬁne

[X][Π^T] =

π^T

[X][π^T]∈Grothl(G(A^T)×G)

where the sum is over allπ^T ∈Irrl(G(A^T)) withπ^T unramiﬁed at allv∈T and BC^T(ıπ^T) :=⊗v∈TBCv(ıπ_v^T)∼= Π^T.

Finally, suppose G of the form G(A^Tfin−{p})×Γ or G(A^Tfin)×Γ and let R be an admissible Ql[G(A^T)×G]-module. Suppose Ram_F/_Q ⊂ T and Π^T ∈ Irr(Gn(A^T)) is unramiﬁed at all v ∈ T. Let H^ur(G(A^T)) =

⊗v∈THûr(G(Q^v)), a commutative polynomial algebra overCin countably many variables. Similarly, letHûr(Gⁿ(A^T)) =⊗v∈THûr(Gⁿ(Q^v)). Then Π^T corresponds to a maximal idealnofHûr(Gⁿ(A^T)) with residue fieldC. Note that the space ofK^T-invariantsR^K^T is a module overι⁻¹Hûr(G(A^T)). We define

R^K^T{Π^T}:=

m

(R^K^T)ι⁻¹m⊂R^K^T

where m runs over the maximal ideals ofH^ur(G(A^T)) with residue ﬁeldC and which pull back to nunder ⊗^v∈TBC^∗_v. ThenR^K^T{Π^T} is aG-stable direct summand ofR^K^T.

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3. Shimura varieties

In this section we recall some results from [13]. We begin with some deﬁnitions and refer the reader to Section 5 of [13] for more details. Let U be a compact open subgroup of G(A^∞) and deﬁne a functor XU from the category of pairs (S, s), where is S is a connected locally Noetherian F-scheme andsis a geometric point ofS, to the category of sets by sending a pair (S, s) to the set of isogeny classes of quadruples (A, λ, i, η) where

− A/Sis an abelian scheme of dimension [F⁺:Q]n;

− λ:A→A^∨ is a polarization;

− i:F →End (A)⊗_ZQsuch that λ◦i(f) =i(f^c)^∨◦λ;

− ηis aπ1(S, s)-invariantU-orbit of isomorphisms ofF⊗_QA^∞-modules η:V⊗_QA^{∞ ∼}−→V Aswhich take the pairing·,·onV to a (A^∞)^×- multiple of theλ-Weil pairing onV As:=H1(As,A^∞) (see Section 5 of [11]);

− for eachf ∈F there is an equality of polynomials det_O_S(f|LieA) = detE(f|V¹) in the sense of Section 5 of [11] (here V¹ ⊂ V ⊗_QE ⊂ V ⊗_QC is theE-subspace whereh(τE(e)) acts by multiplication by 1⊗efor alle∈E);

− two such quadruples (A, λ, i, η) and (A, λ, i, η) are isogenous if there exists an isogeny A → A taking λ, i, η to γλ, i, η for some γ∈Q^×.

Ifsandsare two geometric points of a connected locally NoetherianF- schemeSthen there is a canonical bijection fromXU(S, s) toXU(S, s). We may therefore think ofXU as a functor from connected locally Noetherian F-schemes to sets and then extend it to a functor from all locally Noetherian F-schemes to sets by settingXU(

iSi) =

iXU(Si). WhenU is suﬃciently small the functorXU is represented by a smooth projective variety XU/F of dimensionn−1. The variety XU is denoted ShU in [13]. LetA^U be the universal abelian variety overXU.

IfU andV are suﬃciently small compact open subgroups ofG(A^∞) and g ∈ G(A^∞) is such that g⁻¹V g ⊂U, then we have a map g : XV → XU

and a quasi-isogenyg^∗ :A^V →g^∗A^U of abelian varieties overXV. In this way we get a right action of the groupG(A^∞) on the inverse system of the XU which extends to an action by quasi-isogenies on the inverse system of theA^U.

Letl be a rational prime and letξbe an irreducible algebraic representation ofGoverQl. Thenξgives rise to a lissel-adic sheafL^ξ on eachXU.

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We let

H^k(X,L^ξ) = lim

−→U

H^k(XU×^FF ,L^ξ).

This is a semisimple admissible representation ofG(A^∞) with a commuting continuous action ofGF and therefore decomposes as

H^k(X,L^ξ) =

π^∞

π^∞⊗R^k_ξ,l(π^∞)

whereπ^∞ runs over Irrl(G(A^∞)) and eachR^k_ξ,l(π^∞) is a ﬁnite dimensional continuous representation ofGF overQl.

We now recall results of Shin on the existence of Galois representations in the cohomology of the Shimura varieties in the following two cases:

3.1. The stable case

Assume we are in the following situation:

− Π¹ is a RACSDC automorphic representation ofGLn(A^F);

− ψ:A^×E/E^×→C^×is an algebraic Hecke character such thatψΠ¹|_A^×_E = ψ^c/ψ;

− Π :=ψ⊗Π¹(an automorphic representation ofGⁿ(A)∼=GL1(A^E)× GLn(A^F)) is Ξ-cohomological for some irreducible algebraic representation Ξ ofGⁿ/C;

− Ram_Q(Π)⊂Spl_F/F+,Q.

Then Ξ is θ-stable and so comes from some irreducible algebraic representation ξ_C of Gn over C as in Section 2. Let ξ = ı⁻¹ξ_C, regarded as a representation of G/Ql. Let R^l(Π) denote the set of π^∞ ∈Irrl(G(A^∞)) such that

− R^k_ξ,l(π^∞)= (0) for some k;

− π^∞is unramiﬁed at allv∈Ram_F/_Q∪Ram_Q(Π) and BC^∞(ıπ^∞) = Π^∞.

(We note that BCv(ıπv) is deﬁned for allv |∞.) LetR^k_ξ,l(Π) =

π^∞∈Rl(Π)R^k_ξ,l(π^∞).

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Now, let T ⊃ {∞} be a ﬁnite set of places of Q with RamF/Q ∪ Ram_Q(Π)⊂Tﬁn⊂Spl_F/F⁺_,_Q. Note that

H^k(X,L^ξ)^K^T{Π^T} ∼=

π^∞

πTfin⊗R_ξ,l^k (π^∞)⊂H^k(X,L^ξ)

where the sum is over all π^∞ = π^T ⊗πTfin where π^T is unramiﬁed and BC(ıπ^T)∼= Π^T. We then deﬁne an admissible Ql[Gⁿ(A^Tfin)×GF]-module

BCTfin(H^k(X,L^ξ)^K^T{Π^T}) :=

π^∞

BCTfin(πTfin)⊗R^k_ξ,l(π^∞), whereπ^∞ runs over the same set.

Theorem 3.1. —

1. Ifπ^∞∈ R^l(Π)thenR^k_ξ,l(π^∞)= (0)if and only ifk=n−1.

2. We have

BCTfin(Hⁿ⁻¹(X,L^ξ)^K^T{Π^T}) = (ı⁻¹ΠTfin)⊗R_ξ,lⁿ⁻¹(Π).

3. We have

Rⁿ⁻¹_ξ,l (Π)^ss∼=Rl,ı(Π¹)^C^G⊗Rl,ı(ψ)|^GF.

Proof. — The ﬁrst part follows from Corollary 6.5 of [13]. The second part follows from the proof of Corollary 6.4 ofop. cit.. The third part follows from the proof of Corollary 6.8 ofop. cit.(Note that the character recl,ıl(ψ) which appears in the proof of this corollary is equal toRl,ı(ψ⁻¹).) 3.2. The endoscopic case

We now assume we are in the following situation:

− m1, m2 are positive integers withm1> m2 andm1+m2=n;

− fori= 1,2, Πiis a RACSDC automorphic representation ofGLmi(A^F) with Ram_Q(Πi)⊂Spl_F/F⁺_,_Q;

− 2:A^×E/E^×→C^× is a Hecke character such that – Ram_Q(2)⊂Spl_F/F⁺_,_Q;

– 2|_A^×_Q/Q^×is the quadratic character corresponding to the quadratic extension E/Qby class ﬁeld theory.

− ψ:A^×E/E^×→C^× is an algebraic Hecke character such that

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– (ψΠ1ψΠ2)|_A^×_E = (ψ⊗2^N^(m¹^,m²⁾)^c/(ψ⊗2^N^(m¹^,m²⁾) whereN(m1, m2) = [F⁺:Q](m1(n−m1) +m2(n−m1))/2∈Z;

– Ram_Q(ψ)⊂Spl_F/F⁺_,_Q.

− fori= 1,2, ΠM,i:= Πi⊗(2◦NF/E◦det)^,(n⁻^mⁱ⁾;

− Π :=ψ⊗n−Ind^GL_GLⁿ_m

1×GLm2(ΠM,1⊗ΠM,2) (an automorphic representation ofGⁿ(A)) is Ξ-cohomological for some irreducible algebraic representation Ξ ofGⁿ/C. (We note that the normalized induction is irreducible as ΠM,1and ΠM,2 are unitary.)

As above, we letξbe the irreducible algebraic representation ofGover Ql such that Ξ is associated to ıξ. Let R^l(Π) denote the set of π^∞ ∈ Irrl(G(A^∞)) such that

− R^k_ξ,l(π^∞)= (0) for some k;

− π^∞is unramiﬁed at allv∈RamF/Q∪Ram_Q(Π)∪Ram_Q(2) and BC^∞(ıπ^∞) = Π^∞.

LetR^k_ξ,l(Π) =

π^∞∈Rl(Π)R^k_ξ,l(π^∞).

LetT ⊃ {∞}be a finite set of places ofQwith Ram_F/_Q∪Ram_Q(Π)∪ Ram_Q(2)⊂Tfin⊂Spl_F/F⁺_,_Q. We define

BCTfin(H^k(X,L^ξ)^K^T{Π^T}) exactly as in the previous subsection.

Theorem 3.2. —

1. Ifπ^∞∈ R^l(Π), then R^k_ξ,l(π^∞)= (0) if and only ifk=n−1.

2. We have

BCTfin(H^k(X,L^ξ)^K^T{Π^T}) = (ı⁻¹ΠTfin)⊗Rⁿ_ξ,l⁻¹(Π).

3. There exists an integere2(Π, G)∈ {±1}depending on ΠandGsuch that

(a) Ife2(Π, G) = 1 then

Rⁿ⁻¹_ξ,l (Π)^ss ∼=Rl,ı(Π1)^C^G⊗Rl,ı(ψ2^,(n−m¹⁾| · |^(n−m¹^)/2)|^GF.

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(b) Ife2(Π, G) =−1then

Rⁿ_ξ,l⁻¹(Π)^ss ∼=Rl,ı(Π2)^C^G⊗Rl,ı(ψ2^,(n−m²⁾| · |^(n−m²^)/2)|^GF. Proof. — The ﬁrst and second parts follow respectively from Corollary 6.5 and the proof of Corollary 6.4 of [13]. The third part follows from the proofs of Corollaries 6.8 and 6.10 of op. cit. (Alternative 3.a corresponds to the case when e1 =e2 in the notation of op. cit., while alternative 3.b corresponds to the case whene1=−e2. Note however that by Corollary 6.5 ofop. cit.,e1= (−1)ⁿ⁻¹= 1. We therefore take e2(Π, G) =e2.)

4. Integral models

We now proceed to introduce integral models for the varietiesXU and to deduce various results on these models, following the arguments of Section 3 of [14]. Recall that we have ﬁxed an isomorphismG×_QA^∞∼=Gn×_QA^∞. Sincep∈Spl_E/_Q, we have an isomorphism

G(Q^p)∼=Q^×p × r i=1

GLn(Fwi) and we decomposeG(A^∞) as

G(A^∞) =G(A^∞^,p)×Q^×p × r i=1

GLn(Fwi).

Ifm= (m2, . . . , mr)∈Z^r−10 , set U_p^w(m) =

r i=2

ker(GLn(O^F,wⁱ)→GLn(O^F,wⁱ/w^m_i ⁱ))⊂ r i=2

GLn(Fwi).

We consider the following compact open subgroups ofG(Q^p):

Ma(m) = Z^×p ×GLn(O^F,w)×U_p^w(m) Iw(m) = Z^×p ×Iwn,w×U_p^w(m).

Fix an m as above. If U^p ⊂G(A^∞,p) is a compact open subgroup, we let U0=U^p×Ma(m) andU =U^p×Iw(m). Fori= 1, . . . , r, let Λi⊂V⊗^FFwi

be aGLn(O^F_wi)-stable lattice.

For each suﬃciently smallU^p as above, an integral model ofXU0 over O^F,wis constructed in Section 5.2 of [13] (note thatXU0 is denoted ShU^p(-m)

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in [13] with m@ = (0, m2, . . . , mr)). We denote this integral model also by XU0. It represents a functor XU0 from the category of locally Noetherian O^F,w-schemes to sets which, as in the characteristic 0 case, is initially deﬁned on the category of pairs (S, s) where S is a connected locally Noetherian O^F,w-scheme and sis a geometric point of S. It sends a pair (S, s) to the set of equivalence classes of (r+ 3)-tuples (A, λ, i, η^p,{αi}^ri=2) where

− A/Sis an abelian scheme of dimension [F⁺:Q]n;

− λ:A→A^∨is a prime-to-ppolarization;

− i:O^F →End (A)⊗_ZZ^(p)such thatλ◦i(f) =i(f^c)^∨◦λ;

− η^p is a π1(S, s)-invariant U^p-orbit of isomorphisms of F ⊗_QA^∞^,p- modulesη^p:V ⊗QA^∞^,p−→^∼ V^pAswhich take the pairing·,·onV to a (A^∞^,p)^×-multiple of the λ-Weil pairing onV^pAs;

− for eachf ∈ O^F there is an equality of polynomials det_O_S(f|LieA) = detE(f|V¹) in the sense of Section 5 of [11];

− for 2ir,αi : (w_i⁻^mⁱΛi/Λi)S ∼

−→A[w^m_i ⁱ] is an isomorphism of S-schemes withO^F,wi-actions;

and

− two such tuples (A, λ, i, η^p,{αi}^ri=2) and (A, λ, i,(η^p),{α_i}^ri=2) are equivalent if there is a prime-to-pisogenyA→Atakingλ,i,η^pand αi toγλ,i, (η^p) andα_i for someγ∈Z^×(p).

The scheme XU0 is smooth and projective over O^F,w. As U^p varies, the inverse system of theXU0’s has an action ofG(A^∞^,p).

Given a tuple (A, λ, i, η^p,{αi}^ri=2) overS as above, we letGÂ=A[w^∞], a Barsotti-TateO^F,w-module overS. Ifpis locally nilpotent onS, thenGÂ has dimension 1 and is compatible (which means that the two actions of O^F,w on LieGÂ (coming from the structural morphismS→SpecO^F,w and fromi:O^F →End (A)⊗_ZZ(p)) coincide). We letAÛ0 denote the universal abelian scheme overXU0, and we letG=GAU0.

LetXU0 denote the special ﬁbre XU0×OF,wk(w) ofXU0, and for 0 h n−1, let X^[h]_U₀ denote the reduced closed subscheme of XU0 whose closed geometric pointssare those for which the maximal ´etale quotient of G^shasO^F,w-height at mosth. Let

X^(h)_U₀ =X^[h]_U₀−X^[h_U₀⁻^1]

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(where we setX^[_U⁻₀^1]=∅). ThenX⁽⁰⁾_U₀ is non-empty. [We exhibit anF^ppoint ofX⁽⁰⁾_U₀. Consider thep-adic type (F, η) overF whereηw= 1/(n[k(w) :F^p]) andηwi= 0 fori >1. It corresponds to an isogeny class of abelian varieties with F-action over F^p. Let (A, i)/F^p be an element of this isogeny class.

Then

− Ahas dimension [F⁺:Q]n;

− C0 = End⁰_F(A) is the division algebra with centre F which is split outsideww^c and has Hasse invariant 1/n atw;

− and thep-divisible groupA[w^∞] has pure slope 1/(n[Fw:Q^p]), while A[w^∞_i ] is ´etale fori >1.

(See section 5.2 of [8].) Just as in the proof of Lemma V.4.1 of [8] one shows that there is a polarization λ0 :A→A^∨ and anF-vector spaceW0

of dimension n together with a non-degenerate alternating form , ⁰ : W0×W0→Qsuch that

− λ0◦i(a) =i(ca)^∨◦λ0for alla∈F;

− ax, y⁰=x,(ca)y⁰ for alla∈F andx, y∈W0;

− V^pA∼=W0⊗_QA^∞^,p as A^∞F^,p-modules with alternating pairings de- ﬁned up to (A^∞^,p)^×-multiples (the pairing onV^pAbeing theλ0-Weil pairing);

− W0⊗QR∼=V ⊗QRasF⊗QR-modules with alternating pairings up toR^×-multiples.

(In fact W0 will be the Betti cohomology of a certain lift of (A, i) to characteristic 0.) LetG0 denote the denote the algebraic group ofF-linear au- tomorphisms of W0 that preserve , ⁰ up to scalar multiples, and let φ0∈H¹(Q, G0) represent the diﬀerence between (W0, , ⁰) and (V, , ).

So in fact

φ0∈ker(H¹(Q, G0)−→H¹(R, G0)).

Let‡0 denote theλ0Rosati involution onC0 and deﬁne an algebraic group H₀^AV/Qby

H₀^AV(R) ={g∈(C0⊗_QR)^×:gg^‡⁰ ∈R^×}. There is a natural isomorphism

H₀^AV×_QA^∞^,p−→^∼ G0×_QA^∞^,p

coming from the isomorphismV^pA∼=W0⊗_QA^∞,p. As in Lemma V.3.1 of [8] the polarizations of A which induce complex conjugation on i(F) are parametrized by

ker(H¹(Q, H₀^AV)−→H¹(R, H₀^AV)).