Eichler-Shimura relations and semisimplicity of étale cohomology of quaternionic Shimura varieties

Jan Nekovář

Abstract: we show that the non CM part of ℓ-adic étale cohomology of any compact quaternionic Shimura variety with coefficients in any automorphic local system is a semisimple Galois representation. If the local system has weight k = (k_{1}, …, k_{d}) with all k_{i} of the same parity, the full ℓ-adic étale cohomology is semisimple. For Hilbert modular varieties, analogous results are proved for ℓ-adic intersection cohomology of the Baily-Borel compactification. The proof combines a representation-theoretical criterion of semisimplicity with Eichler-Shimura relations for partial Frobenius morphisms.

Résumé : on montre que l’action galoisienne sur la partie sans multiplication complexe de la cohomologie étale d’un faisceau ℓ-adique lisse automorphe sur une variété de Shimura quaternionique compacte est semisimple. Si le poids du faisceau s’écrit k = (k_{1}, …, k_{d}), où les k_{i} ont la même parité, toute la cohomologie étale est semi-simple. Les mêmes résultats sont montrés pour la cohomologie d’intersection ℓ-adique de la compactification de Baily-Borel des variétés modulaires de Hilbert. La preuve utilise un critère abstrait de semi-simplicité et les relations d’Eichler-Shimura pour les morphismes de Frobenius partiels.

0. Introduction

(0.1) General conventions and notation. The characteristic polynomial of an endomorphism u of a finite-dimensional vector space over a field k will be denoted by P_{u}(X) = det(X · id − u) ∈ k[X]. If k ⊂ K are fields and X is a k-vector subspace of a K-vector space Y, we denote by K · X the K-vector subspace of Y generated by X. We abbreviate ⊗_{Z} as ⊗. For an abelian group A we let Â = A ⊗ Ẑ. We denote by A and A_{k}, respectively, the ring of adeles of Q and of a number field k.

Throughout the article we fix an isomorphism C ⟶^{∼} \overline{Q}_{ℓ}. For any algebraic object (−) defined over a subfield of C we denote by (−)_{ℓ} its base change to \overline{Q}_{ℓ}. Let \overline{Q} be the algebraic closure of Q in C. The reciprocity map of class field theory is normalised by letting uniformisers correspond to geometric Frobenius elements Fr(P). All representations and characters are assumed to be continuous with respect to the natural topologies involved. A representation of a profinite group is called strongly irreducible if its restriction to every open subgroup is irreducible. By an automorphic representation we mean an irreducible automorphic representation. (0.2) Let us recall basic facts about decomposition of singular and étale cohomology of compact Shimura varieties. As in the classical case of cuspidal cohomology of modular curves, everything boils down to the fact that the Hecke operators act on the space of cuspidal automorphic forms in a selfadjoint way (up to a twist).

Let (G, 𝒳 ) be a (pure) Shimura datum. A rational representation ξ : G_{C} ⟶ GL(N)_{C} (whose restriction to the centre Z_{C} satisfies an appropriate condition) gives rise, for each sufficiently small open compact subgroup K ⊂ G(Q̂), to a locally constant sheaf of complex vector spaces ℒ_{ξ} on the complex manifold Sh_{K}(G, 𝒳 )^{an} = G(Q)\(𝒳 × G(Q̂)/K). (0.3) If, in addition, the derived group G^{der} is anisotropic, then Sh_{K}(G, 𝒳 )^{an} is compact and its cohomology H^{∗}(Sh_{K}(G, 𝒳 )^{an}, ℒ_{ξ}) is described in terms of relative Lie algebra cohomology.

Write 𝒳 = G(R)/K_{∞}, where K_{∞} is the stabiliser of a fixed base point in 𝒳 , and denote, for any G(R)-module V, by V_{0} the subspace of K_{∞}-finite vectors in V. There is a canonical isomorphism

H^{i}(Sh_{K}(G, 𝒳 )^{an}, ℒ_{ξ}) = H^{i}(𝔤, K_{∞}; C^{∞}(G(Q)\G(A)/K) ⊗ ξ) = H^{i}(𝔤, K_{∞}; C^{∞}(G(Q)\G(A)/K)_{0} ⊗ ξ),

where 𝔤 = Lie(G(R)) [BW, VII.2.7]. It gives rise to a G(Q̂)-equivariant isomorphism

The author’s research was supported in part by the grant ANR-BLAN-0114

H^{i}(Sh(G, 𝒳 )^{an}, ℒ_{ξ}) = \underrightarrow{lim}_{K} H^{i}(Sh_{K}(G, 𝒳 )^{an}, ℒ_{ξ}) = H^{i}(𝔤, K_{∞}; C^{∞}(G(Q)\G(A))_{0} ⊗ ξ).  (0.3.1)

For every character ω : Z(Q)\Z(A) ⟶ C^{×} fix a character ω′ : G(Q)\G(A) ⟶ R_{+}^{×} such that ω′|_{Z(A)} = |ω|. The space G(Q)Z(A)\G(A) is compact (since G^{der} is anisotropic) and the completion L^{2}(G, ω) of

C^{∞}(G, ω) = {f ∈ C^{∞}(G(Q)\G(A)) | f(gz) = ω(z)f(g) ∀z ∈ Z(A)}

with respect to the norm

|f|^{2} = ∫_{G(Q)Z(A)\G(A)}(ω′(g)^{−1}|f(g)|)^{2} dg

is a unitary representation of G(A) under the action (g ∗ f)(h) = ω′(g)^{−1}f(hg). This representation decomposes as a discrete Hilbert sum L^{2}(G, ω) = ˆ⨁ m(π′) π′ of unitary automorphic representations π′ of G(A) with finite multiplicities m(π′).

Each π′ has central character ω_{π′} = ω/|ω| and gives rise to an automorphic representation π = ω′π′ = π_{∞} ⊗ π^{∞} of G(A) = G(R) × G(Q̂) with central character ω_{π} = ω.

Matsushima’s formula [BW, Thm. VII.5.2] yields a G(Q̂)-equivariant isomorphism

H^{i}(𝔤, K_{∞}; C^{∞}(G, ω)_{0} ⊗ ξ) = ⨁_{π=π_{∞}⊗π^{∞}} m(π) H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ⊗ π^{∞}, m(π) = m((ω′)^{−1}π),

hence, after putting the contributions of all ω together, a G(Q̂)-equivariant isomorphism

H^{i}(Sh(G, 𝒳 )^{an}, ℒ_{ξ}) = ⨁_{π=π_{∞}⊗π^{∞}} m(π) H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ⊗ π^{∞},  (0.3.2)

where π runs through automorphic representations of G(A) and m(π) is the multiplicity of the unitary representation π′ = (ω′)^{−1}π in L^{2}(G, ω_{π}). (0.4) The Shimura variety Sh(G, 𝒳 ) is defined over its reflex field E = E(G, 𝒳 ) ⊂ \overline{Q} ⊂ C. For sufficiently small K, the representation ξ_{ℓ} : G_{\overline{Q}_{ℓ}} ⟶ GL(N)_{\overline{Q}_{ℓ}} gives rise to a lisse \overline{Q}_{ℓ}-sheaf ℒ_{ξ,ℓ} on Sh_{K}(G, 𝒳 ) and

the comparison theorem between analytic and étale cohomology defines a G(Q̂)-equivariant isomorphism

H_{et}^{i}(Sh(G, 𝒳 )⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) = \underrightarrow{lim}_{K} H_{et}^{i}(Sh_{K}(G, 𝒳 )⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) ≃ H^{i}(Sh(G, 𝒳 )^{an}, ℒ_{ξ})_{ℓ} = ⨁_{π^{∞}} V^{i}(π^{∞})⊗π^{∞},

(0.4.1)

where V^{i}(π^{∞}) = Hom_{G(_{Q})}(π^{∞}, H_{et}^{i}), π^{∞} runs through irreducible unitarisable smooth representations of

G(Q̂) for which there exists a unitarisable irreducible (𝔤, K_{∞})-module π_{∞} such that H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ≠ 0 and π_{∞} ⊗ π^{∞} is an automorphic representation of G(A). If ξ is irreducible (more generally, if it admits a central character ω_{ξ}), then the condition H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ≠ 0 (“π_{∞} being cohomological in degree i for ξ”) implies a compatibility of central characters ω_{ξ}|_{Z(R)} = ω_{π_{∞}}^{−1}.

The \overline{Q}_{ℓ}-vector space V^{i}(π^{∞}) has finite dimension

dim V^{i}(π^{∞}) = ∑_{π=π_{∞}⊗π^{∞}} m(π) dim H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ)

and the natural Galois action of Γ_{E} = Gal(\overline{Q}/E) on étale cohomology (which commutes with the action of G(Q̂)) gives rise to a representation

Γ_{E} ⟶ Aut_{G(_{Q})}(V^{i}(π^{∞}) ⊗ π^{∞}) = Aut_{\overline{Q}_{ℓ}}(V^{i}(π^{∞}))

(the last equality follows from dim End_{G(_{Q})}(π^{∞}) = 1, which holds by a variant of Schur’s Lemma).

(0.5) There is a huge industry based on pioneering work of Langlands and Kottwitz (with first steps due to Ihara in the case of Shimura curves) whose aim is to determine the isomorphism class of the semisimplification of the Galois representation V^{i}(π^{∞}). The main steps in this approach – which is still far from being completed in full generality – are the following:

(0.5.1) a construction of a canonical integral model S_{K,𝔭} of Sh_{K}(G, 𝒳 ) over O_{E,𝔭}, for (almost) all finite primes

𝔭 of E at which Sh_{K}(G, 𝒳 ) has good reduction; (0.5.2) a group-theoretical description of the set of \overline{k(𝔭)}-rational points of the special fibre of S_{K,𝔭}; (0.5.3) a comparison of the expression for

∑_{i=0}^{2 dim}(−1)^{i} Tr (Fr(𝔭)^{n} | H_{et}^{i}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}))

obtained from the previous two steps (via the Lefschetz formula) with terms occurring in the stable trace formula.

(0.6) In the present article we use a much more elementary method, based on Eichler-Shimura relations, to obtain information about the Galois representations V^{i}(π^{∞}). The study of Eichler-Shimura relations in this context has a long history. What is relevant to us is the approach of Faltings and Chai [FC, ch. VII], generalised by Wedhorn [W]. Our results are, naturally, weaker than those obtained by the Langlands-Kottwitz method, with one notable exception: in favourable cases we are able to show that V^{i}(π^{∞}) is a semisimple representation of Γ_{E}. (0.7) An Eichler-Shimura relation (a “congruence relation”) is a statement about compatibility of the commuting actions of Γ_{E} and the spherical part of the Hecke algebra \overline{Q}[G(Q̂)//K] on H_{et}^{i}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}). In vague terms, the relation states that the (geometric) Frobenius Fr(𝔭) is a root of a certain Hecke polynomial. (0.8) In the classical case when G = GL(2)_{Q}, K ⊇ K(N) (N > 2) and ξ is the (k-2)-th symmetric power of the standard representation (k ≥ 2), the Shimura variety Y = Sh_{K}(G, 𝒳 ) is an open modular curve of level dividing N and the classical Eichler-Shimura relation states that

∀p ∤ ℓN  Fr(p)^{2} − T_{p} Fr(p) + pS_{p} = 0  (0.8.1)

on H^{i} = H_{et}^{i}(Y ⊗_{Q} \overline{Q}, ℒ_{ξ,ℓ}), where T_{p} and S_{p} denote the Albanese (= covariant) action of the double cosets

[K ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) K] and [K ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) K], respectively.

The spherical Hecke algebra T = Q[T_{p}, pS_{p}]_{p∤N} acts semisimply on cuspidal cohomology

H_{!}^{1} = H_{et}^{1}(X ⊗_{Q} \overline{Q}, j_{∗}ℒ_{ξ,ℓ}),

where j : Y ↪ X = Y ∪ {cusps}. For any cuspidal Hecke eigenform ϕ ∈ S_{k}(N, χ) of weight k for T

ϕ|_{T_{p}} = λ_{p}ϕ, ϕ|_{pS_{p}} = p^{k−1}χ(p)ϕ,

the analogue of V^{i}(π^{∞}) from (0.4.1)

ρ := H_{!}^{1}[ϕ − eigenspace for T]

is a non-zero representation of Γ_{Q}. Let ρ_{ϕ} : Γ_{Q} ⟶ GL_{2}(\overline{Q}_{ℓ}) be the Galois representation attached to ϕ. Its characteristic polynomial is characterised by the fact that

∀p ∤ ℓN  P_{ρ_{ϕ}(Fr(p))}(X) = X^{2} − λ_{p}X + p^{k−1}χ(p).  (0.8.2)

The Eichler-Shimura relation (0.8.1) implies that

∀p ∤ ℓN  P_{ρ_{ϕ}(Fr(p))}(ρ(Fr(p))) = 0,  (0.8.3)

hence

∀g ∈ Γ_{Q}  P_{ρ_{ϕ}(g)}(ρ(g)) = 0,  [“P_{ρ_{ϕ}}(ρ) = 0”]  (0.8.4)

by the Čebotarev density theorem. Of course, ρ ≃ ρ_{ϕ}^{⊕m} for some m ≥ 1 (by a variant of (0.4.1) or by Theorem 3.7 below) and ρ_{ϕ} is constructed as ρ for a suitable choice of K (the formula (0.8.2) is deduced from (0.8.1) and Poincaré duality). (0.9) We are interested in those Shimura varieties for which G = R_{F/Q}(H) is the restriction of scalars of a (connected reductive) algebraic group H defined over a totally real number field F. In this case the corresponding analytic objects decompose according to the decomposition of F ⊗R ⟶^{∼} ∏_{v|∞} F_{v} = ∏_{v|∞} R:

G(R) = ∏_{v|∞} H_{v}(R),  𝒳 = ∏_{v|∞} 𝒳_{v},  K_{∞} = ∏_{v|∞} K_{∞,v},  𝔤 = ∏_{v|∞} 𝔥_{v},  π_{∞} = ⨂_{v|∞} π_{v}.

If, in addition, ξ = ⨂_{v|∞} ξ_{v} with ξ_{v} : (H_{v})_{C} ⟶ GL(N_{v})_{C}, the Künneth formula

H^{∗}(𝔤, K_{∞}; π_{∞} ⊗ ξ) = ⨂_{v|∞} H^{∗}(𝔥_{v}, K_{∞,v}; π_{v} ⊗ ξ_{v})

combined with (0.3.1) implies that the analytic cohomology H^{∗}(Sh_{K}(G, 𝒳 )^{an}, ℒ_{ξ}) admits a natural decomposition as a finite direct sum of tensor products ⨂_{v|∞}.

(0.10) Such a “weak Künneth decomposition” is expected to be of a motivic origin (see the discussion in [NS,§6]). In particular, there should be a finite extension E′/E depending only on (G, 𝒳 ) and F for which the Galois representation H_{et}^{∗}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) restricted to Γ_{E′} is isomorphic to a direct sum of tensor products ⨂_{v|∞} of representations of Γ_{E′}. Equivalently, each V (π^{∞}) = ⨁_{i} V^{i}(π^{∞}) in (0.4.1) should be of this form.

This behaviour is expected to reflect geometry of the integral models S_{K,𝔭} from (0.5.1) at those primes 𝔭 of E (of good reduction) which split completely in E′/E. For such primes the Frobenius morphism on the special fibre should be a product of partial Frobenius morphisms, for which a refinement of (0.5.2) and (0.5.3) should be valid. See 5.16 and the Appendix for a discussion of this phenomenon in a special case. (0.11) Eichler-Shimura relations in this context lead to identities of the form

P_{(ρ_{1}⊗···⊗ρ_{r})(Fr(𝔭))}(ρ(Fr(𝔭))) = 0

and

∀g ∈ Γ_{E′}  P_{(ρ_{1}⊗···⊗ρ_{r})(g)}(ρ(g)) = 0  [“P_{ρ_{1}⊗···⊗ρ_{r}}(ρ) = 0”]  (0.11.1)

which generalise (0.8.3-4). Above, ρ = V^{i}(π^{∞}) ⊗ (π^{∞})^{K}|_{Γ_{E′}} and ρ_{1}, …, ρ_{r} are certain representations of Γ_{E′} attached to π.

Note that the formulation (0.11.1) of Eichler-Shimura relations requires the knowledge of the existence of the Galois representations ρ_{i}. (0.12) It is natural to consider the relation (0.11.1) in the following abstract context. Let ρ_{1}, …, ρ_{r} be irreducible finite-dimensional representations of Γ (which can be a group, a profinite group, a Lie algebra, an algebraic group etc.). (Q1) If a finite-dimensional representation ρ of Γ satisfies P_{ρ_{1}⊗···⊗ρ_{r}}(ρ) = 0, is it true that

ρ^{ss} ⊆ (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})^{⊕m})^{ss}  (0.12.1)

for some m ≥ 1? Note that ρ_{1} ⊗ ⋯ ⊗ ρ_{r} is automatically semisimple if the field of coefficients is of characteristic zero.

(Q2) If (0.12.1) holds, under what additional assumptions is ρ semisimple (i.e., when is ρ = ρ^{ss})? In fact, the discussion in 0.10 leads naturally to the following, more precise version of (Q2).

(Q2’) Let Γ be a profinite group containing a dense subset Σ such that, for each g ∈ Σ, ρ(g) = u_{1} ⋯ u_{r}, where u_{i} commute with each other and P_{ρ_{i}(g)}(u_{i}) = 0 for all i = 1, …, r (u_{i} corresponds to the action of a partial Frobenius morphism). Under what additional assumptions is ρ semisimple?

If the polynomials P_{ρ_{i}(g)}(X) (g ∈ Σ) have distinct roots, then each u_{i} is semisimple, and so is their product ρ(g) = u_{1} ⋯ u_{r}, for all g ∈ Σ. It is then natural to ask the following question. (Q3) If a finite-dimensional representation ρ of Γ satisfies P_{ρ_{1}⊗···⊗ρ_{r}}(ρ) = 0 and if ρ(Γ) contains many

semisimple elements, under what additional assumptions is ρ semisimple? (0.13) Boston, Lenstra and Ribet [BLR] showed that both questions (Q1) and (Q2) have a positive answer if r = 1 and ρ_{1} is a two-dimensional absolutely irreducible representation of a group Γ. Their result and its applications were inspired by [Mz, Prop. 14.2]. Dimitrov [Di, Lemma 6.5] considered a variant of question (Q1) for certain two-dimensional representations ρ_{1}, …, ρ_{r} : Γ ⟶ GL_{2}(F_{q}).

For certain higher-dimensional representations ρ_{1} : Γ ⟶ GL_{n}(\overline{F}_{p}) of a finite group Γ which have a sufficiently large image, Emerton and Gee [EG, Sect. 4] showed that (Q1) for r = 1 has a positive answer. (0.14) It may be helpful to keep in mind the following two toy models. (0.15) A toy model for (Q1). If Γ is the Lie algebra sl(2) (over a field of characteristic zero), every finite-dimensional representation of Γ is semisimple and the irreducible representations V_{n} = Sym^{n}(V_{1}) (dim(V_{n}) = n + 1) are indexed by their highest weights n ∈ N. As

{weights of V_{n}} = {n, n − 2, …, 2 − n, −n}

(all weights occurring with multiplicity one), we have

P_{V_{m}}(V_{n}) = 0 ⟺ {weights of V_{n}} ⊂ {weights of V_{m}} ⟺ n ≤ m, n ≡ m (mod 2).

In particular, question (Q1) has a negative answer even for r = 1 (namely, for ρ_{1} = V_{n}, n > 1). On the other hand, the Clebsch-Gordan formula

V_{n} ⊗ V_{1} = V_{n+1} ⊕ V_{n−1}

implies that

P_{V_{1}^{⊗r}}(V_{n}) = 0 ⟺ {weights of V_{n}} ⊂ {weights of V_{1}^{⊗r}} ⟺ n ≤ r, n ≡ r (mod 2) ⟺ V_{n} ⊂ V_{1}^{⊗r}.

In other words, question (Q1) has a positive answer if ρ_{1} = ⋯ = ρ_{r} = V_{1}, for any r ≥ 1.

The key point is that V_{1} is a minuscule representation of sl(2). This is generalised in Proposition 1.6 to representations of arbitrary split reductive Lie algebras. In Proposition 3.10 we deduce from this result an affirmative answer to question (Q1) for representations ρ_{1}, …, ρ_{r} of a profinite group which have a large image. (0.16) A toy model for (Q3). Assume that Γ is a profinite group, r = 2, ρ_{1} = ρ_{2}, dim(ρ_{1}) = 2, ρ_{1}(Γ) ⊂ GL_{2}(\overline{Q}_{ℓ}) is big (for example, contains a conjugate of an open subgroup of SL_{2}(Z_{ℓ})) and the semisimplification of ρ is of the form ρ^{ss} ≃ ρ_{1}^{⊗2} = Sym^{2}(ρ_{1}) ⊕ ∧^{2}(ρ_{1}).

We have (possibly after dualising if necessary) an exact sequence of representations of Γ

0 ⟶ Sym^{2}(ρ_{1}) ⟶ ρ ⟶ ∧^{2}(ρ_{1}) ⟶ 0.  (0.16.1)

After choosing a splitting of (0.16.1) we can write ρ in a matrix form as

ρ(g) = ( \begin{matrix}Sym^{2}(ρ_{1}(g)) & ∗ \\ 0 & det(ρ_{1}(g)) \end{matrix} ) , ∗ = ( \begin{matrix}∗ \\ ∗ \\ ∗ \end{matrix} ) = c(g) det(ρ_{1}(g)),

where c ∈ Z^{1}(Γ, Hom(∧^{2}(ρ_{1}), Sym^{2}(ρ_{1}))) is a 1-cocycle describing the extension (0.16.1).

If the matrix ρ_{1}(g) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) is diagonal, then

ρ(g) = ( \begin{matrix}a^{2} & & & x \\ & ab & & y \\ & & b^{2} & z \\ & & & ab \end{matrix} ) ,

which means that

ρ(g) is semisimple ⟺ y = 0.

In particular, the condition “ρ(g) is semisimple for many g ∈ Γ” imposes many non-trivial constraints on the cocycle c. A natural guess would be that these constraints force c to be a coboundary (which implies that ρ is isomorphic to ρ_{1}^{⊗2}, and therefore is semisimple). This is indeed the case. A general version of this argument (which uses in a crucial way the fact that Sym^{2}(ρ_{1}(g)) and ∧^{2}(ρ_{1}(g)) have a common eigenvalue) is given in Theorem 2.4 below (note that our toy model corresponds to Example 2.5 for sl(2) and the adjoint representation).

(0.17) In 0.16 we considered a special case of (Q3) when we knew in advance that (Q1) had a positive answer (a fairly general result in this direction is proved in Theorem 3.3 below). In his thesis at Université Pierre et Marie Curie, K. Fayad [Fa] shows that (Q2’) has a positive answer in many cases when (Q1) does not.

(0.18) In this article we consider questions (Q1), (Q2) and (Q3) for representations of a profinite group Γ with coefficients in \overline{Q}_{ℓ}. Our main abstract results proved in §3 involve a passage to Lie algebras and an application of the general results on Lie algebra representations proved in §1 and §2. This means that the assumptions are far from being optimal. In §4 we consider the simplest possible case of induced representations, when the methods of §3 do not apply.

(0.19) In §5 we combine the results of §3 and §4 with Eichler-Shimura relations on quaternionic Shimura varieties Sh(G, 𝒳 ), for which G = D^{×} is the multiplicative group of a quaternion algebra D over a totally real number field F. As mentioned in 0.6, we recover only a weak form of the results obtained by the Langlands-Kottwitz method (Theorem 5.18, Theorem 5.20(1),(2)).

However, we are able to show (Theorem 5.20(3)) that the Galois action on the full (cuspidal) étale cohomology with coefficients in ℒ_{ξ,ℓ} of these Shimura varieties is semisimple, using the Eichler-Shimura relations proved in §5 and §6 of the Appendix. This result is new already in the special case of cuspidal cohomology of Hilbert modular varieties.

In §6 we study étale cohomology of closely related quaternionic Shimura varieties Sh(G^{∗}, 𝒳 ^{∗}), where G^{∗} is the subgroup of D^{×} consisting of elements whose reduced norm lies in Q^{×}. For local systems ℒ_{ξ^{∗},ℓ} of non-motivic weight (i.e., for those whose Tate twists do not extend to Sh(G, 𝒳 )) we show an analogous semisimplicity result, but only for the non CM part of the cohomology.

Partial results on semisimplicity of the Galois action on certain subspaces of non-endoscopic étale cohomology of unitary Shimura varieties are proved in K. Fayad’s thesis [Fa]. These results are further generalised in [FaN].

(0.20) The general formalism of Eichler-Shimura relations on essentially PEL Shimura varieties at split primes is discussed in the Appendix.

(0.21) The author would like to thank A. Genestier, M. Harris and B.C. Ngô for pointing out (independently) the importance of partial Frobenius morphisms, to S. Bijakowski, P. Boyer, K. Buzzard, G. Chenevier, L. Clozel, C. Cornut and B. Stroh for helpful discussions, to L. Clozel and K. Fayad for discovering several inaccuracies in earlier versions of this article, and to the referee for helpful comments and suggestions.

Much of this work was carried out during the author’s stay at the Fields Institute in Toronto in spring 2012. He would like to thank the Institute for invitation and for providing excellent working conditions.

1. Lie algebras

(1.1) Notation and conventions (see [LIE, Ch. VI, VIII]). Let 𝔤 be a split semisimple Lie algebra over a field k ⊃ Q. A choice of a Cartan subalgebra 𝔥 ⊂ 𝔤 determines the following objects: the set of roots R ⊂ 𝔥^{∗}, the root decomposition

𝔤 = 𝔥 ⊕ ⨁_{α∈R} 𝔤^{α}, 𝔤^{α} = k · X_{α},

the root lattice Q = Σ_{α∈R} Zα ⊂ 𝔥^{∗}, the dual root system R^{∨} = {α^{∨} | α ∈ R} ⊂ 𝔥 satisfying ⟨α^{∨}, α⟩ = 2 for all α ∈ R, the coroot lattice Q^{∨} = Σ_{α^{∨}∈R^{∨}} Zα^{∨} ⊂ 𝔥 (= the root lattice of R^{∨}), the weight lattice P ⊃ Q (the Z-dual of Q^{∨}) and the coweight lattice P^{∨} ⊃ Q^{∨}.

A choice of a Weyl chamber C ⊂ 𝔥^{∗} is equivalent to a decomposition R = R_{+} ∪ R_{−}, where R_{+} = {α ∈ R | ⟨α^{∨}, C⟩ ≥ 0} (resp. R_{−} = −R_{+}) is the set of positive (resp. negative roots). Such a decomposition also determines the set Δ ⊂ R of simple roots (which forms a Z-basis of Q), a decomposition

𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}, 𝔫_{±} = ⨁_{α∈R_{±}} 𝔤^{α},

the monoid of dominant weights P_{++} = P ∩ \overline{C} ⊂ P_{+} = P ∩ ∑_{α∈R_{+}} Q_{≥0} α = P ∩ ∑_{α∈Δ} Q_{≥0} α ⊂ P [LIE, Ch. VI, §1, no. 6], submonoids Q_{++} = P_{++} ∩ Q ⊂ Q_{+} = P_{+} ∩ Q = ∑_{α∈R_{+}} Z_{≥0} α = ∑_{α∈Δ} Z_{≥0} α ⊂ Q and a partial order on x + Q (for any x ∈ P) given by λ ≤ μ ⟺ μ − λ ∈ Q_{+}. Set Q_{−} = −Q_{+} ⊂ Q.

The universal envelopping algebra

U(𝔫_{±}) = ⨁_{μ∈Q_{±}} U(𝔫_{±})_{μ}

has a natural grading deg(X_{α_{1}} ⋯ X_{α_{r}}) = α_{1} + ⋯ + α_{r} by Q_{±}. (1.2) More generally, if 𝔤 is a split reductive Lie algebra over k (of finite dimension), then 𝔤 = 𝔷(𝔤)⊕𝒟𝔤 is a direct sum of the centre 𝔷(𝔤) with the derived Lie algebra 𝒟𝔤 = [𝔤, 𝔤], which is split semisimple. Any Cartan subalgebra 𝔥 ⊂ 𝔤 is of the form 𝔥 = 𝔷(𝔤) ⊕ 𝔥′, where 𝔥′ is a Cartan subalgebra of 𝒟𝔤. The “root lattice” of (𝔤, 𝔥) is the subgroup of 𝔥^{∗} generated by the weights of 𝔥 occurring in the adjoint representation of 𝔤. It is a lattice in Ker(𝔥^{∗} ⟶ 𝔷(𝔤)^{∗}) and it corresponds to the root lattice of 𝒟𝔤 under the natural isomorphism between the latter space and 𝔥^{′∗}.

A (non-zero) finite-dimensional simple 𝔤-module V will be called minuscule if its highest weight – considered as a weight of 𝔥′ – is a minuscule weight of the semisimple algebra 𝒟𝔤 (in other words, the weights of 𝔥′ occurring in V form a single orbit under the action of the Weyl group of (𝒟𝔤, 𝔥′); see [LIE, Ch. VIII, §7, no. 3]).

(1.3) Proposition. Let 𝔤 be as in 1.1, let V be a non-zero simple 𝔤-module of finite dimension. Write V = ⊕_{μ}V (μ), where V (μ) denotes the subspace of weight μ ∈ 𝔥^{∗} with respect to the action of 𝔥. Let M ⊂ End(V) be a non-zero 𝔤-submodule. Then:

(1) The weight zero subspace M(0) = M^{𝔥} ⊂ End(V)(0) = End(V)^{𝔥} = ⊕_{μ} End(V (μ)) is non-zero.

(2) More precisely, if λ ∈ P_{++} is the highest weight of V, then

Im (M(0) ⟶ End(V (λ)) ≠ 0.

Proof. (1) We can suppose that M is simple. All the weights of V are contained in λ − Q_{+}, which implies that the weights of End(V) (in particular, the highest weight λ_{M} of M) belong to Q. As λ_{M} ∈ Q_{++} ⊂ Q_{+}, it follows from [LIE, Ch. VIII, §7, Prop. 5] that 0 ∈ P_{++} ∩ (λ_{M} − Q_{+}) occurs as a weight in M.

(2) Each element m ∈ M(0) is of the form m = (m_{μ}), where m_{μ} ∈ End(V (μ)). According to (1), the set S = {μ | ∃ m ∈ M(0) m_{μ} ≠ 0} ⊂ λ − Q_{+} is not empty. Let μ be a maximal element of S (i.e., ∀ν ∈ Q_{+} ∖ {0} μ + ν ∉ S). If μ = λ, then we are done. Assume that μ ≠ λ and fix m ∈ M(0) such that m_{μ} ≠ 0. Recall that X ∈ 𝔤 acts on f ∈ End(V) by X ∗ f = X ∘ f − f ∘ X. This implies that, for any positive roots α_{1}, …, α_{r}, β_{1}, …, β_{s} ∈ R_{+} (r, s ≥ 1) such that α_{1} + ⋯ + α_{r} = β_{1} + ⋯ + β_{s} = ν (ν ∈ Q_{+}), the element m′ = X_{α_{1}} ∗ ⋯ ∗ X_{α_{r}} ∗ Y_{β_{1}} ∗ ⋯ ∗ Y_{β_{s}} ∗ m ∈ M(0) satisfies

m′_{μ+ν} = ±X_{α_{1}} ∘ ⋯ ∘ X_{α_{r}} ∘ m_{μ} ∘ Y_{β_{s}} ∘ ⋯ ∘ Y_{β_{1}},

by the maximality of μ ∈ S. As m′_{μ+ν} = 0 (again by the maximality of μ), we deduce that

∀ν ∈ Q_{+} ∖ {0} U(𝔫_{+})_{ν} ∘ m_{μ} ∘ U(𝔫_{−})_{−ν} = 0 ∈ End(V (μ + ν)).

Taking ν = λ − μ and using the equality U(𝔫_{−})_{μ−λ}V (λ) = V (μ) we obtain that

Im(m_{μ}) ⊂ N(μ) := ⋂_{z∈U(𝔫_{+})_{λ−μ}} Ker (z : V (μ) ⟶ V (λ)) .

An easy induction shows that N(μ′) = 0 for all μ′ ∈ λ − Q_{+}: indeed, N(λ) = 0 by definition, and if μ′ ≠ λ but N(μ′ + α) = 0 for all α ∈ Δ, then X_{α} N(μ′) ⊂ N(μ′ + α) = 0 for all such α, which means that N(μ′) ⊂ V (μ′) ∩ {highest weight vectors} = V(μ′) ∩ V(λ) = 0. In particular, N(μ) = 0, which implies that m_{μ} = 0, contrary to our assumption μ ∈ S. This contradiction shows that μ = λ, as claimed. (1.4) Both statements of Proposition 1.3 still hold if we merely assume that 𝔤 is a reductive Lie algebra (of finite dimension) and V is a (non-zero) simple 𝔤-module. In this case 𝔤 = 𝔷(𝔤) ⊕ 𝒟𝔤, each element of 𝔷(𝔤) acts on V by a scalar, and the action of 𝔤 on End(V) factors through 𝒟𝔤. (1.5) There is an analogue of Proposition 1.3 (in the form 1.4) in which 𝔤 is replaced by a split connected reductive group G over k, 𝔥 by a split maximal torus T ⊂ G and V by a non-zero irreducible rational representation of G. The Lie algebra 𝔤 of G then acts on V and End(V), the weight subspaces for T and its Lie algebra 𝔥 (which is a Cartan subalgebra of 𝔤) coincide and G-submodules of End(V) are the same as 𝔤-submodules.

(1.6) Proposition. Let (𝔤, 𝔥) be as in 1.2, let M be a finite direct sum of finite tensor products of onedimensional or minuscule simple 𝔤-modules. If N ≠ 0 is a simple 𝔤-module with the property that each weight of 𝔥 occurring in N occurs in M, then N is isomorphic to a submodule of M.

Proof. Let ν be the highest weight of N. By assumption, ν occurs in M, hence in M′ = M_{1} ⊗ ⋯ ⊗ M_{r} ⊂ M, where each 𝔤-module M_{i} is one-dimensional or minuscule, which means that the weights occurring in M_{i} form a single orbit of the Weyl group W of (𝔤, 𝔥). If we denote by μ_{i} the highest weight of M_{i}, then ν = w_{1}(μ_{1}) + ⋯ + w_{r}(μ_{r}) for some w_{i} ∈ W, since ν occurs in M′. The statement of the conjecture of Parthasarathy, Ranga Rao and Varadarajan (proved in [Ku] and [Ma]) then implies that there is an injective morphism of 𝒟𝔤-modules f : N ↪ M′. The centre 𝔷(𝔤) acts on N (resp. on M′) by a single weight equal to ν|_{𝔷(𝔤)} (resp. to ∑_{i=1}^{r} μ_{i}|_{𝔷(𝔤)}). These two elements of 𝔷(𝔤)^{∗} coincide (again, since ν occurs in M′), which means that f is a morphism of 𝔤-modules. (1.7) Let V be a finite-dimensional vector space over a field k ⊃ Q and 𝔤 ⊂ End_{k}(V) a k-Lie subalgebra. As in [LIE, Ch. VII, §5, no. 3], denote by 𝔫_{V} (𝔤) the set of all elements of the radical of 𝔤 that are nilpotent in End_{k}(V). It is a nilpotent ideal of 𝔤 containing the intersection of the radical with 𝒟𝔤.

Recall that 𝔤 is a decomposable linear Lie algebra [LIE, Ch. VII, §5, Def. 1] if both the semisimple and the nilpotent part of every element of 𝔤 belong to 𝔤. The following facts will be used in §2.

(1.8) Proposition. (1) [LIE, Ch. VII, §5, Thm. 2] The Lie algebra 𝔤 ⊂ End_{k}(V) is decomposable ⟺ some ( ⟺ each) Cartan subalgebra of 𝔤 is decomposable ⟺ the radical of 𝔤 is decomposable.

(2) [LIE, Ch. VII, §5, Thm. 1] If 𝔤 is generated as a k-Lie algebra by a subset S ⊂ 𝔤 such that every X ∈ S is either semisimple or nilpotent in End_{k}(V), then 𝔤 is decomposable.

(3) [LIE, Ch. VII, §5, Prop. 7] If 𝔤 is decomposable, then there exists a Lie subalgebra 𝔪 ⊂ 𝔤, reductive in End_{k}(V) (in particular, 𝔪 is a reductive Lie algebra acting semisimply on V), such that 𝔤 = 𝔪 ⋉ 𝔫_{V} (𝔤).

(4) [LIE, Ch. VII, §3, Ex. 16] The set of elements of 𝔤 that are semisimple in End_{k}(V) is Zariski dense in 𝔤 ⟺ some ( ⟺ each) Cartan subalgebra of 𝔤 is commutative and consists of elements that are semisimple in End_{k}(V).

(1.9) Corollary. If the set of elements of 𝔤 that are semisimple in End_{k}(V) is Zariski dense in 𝔤, then:

(1) 𝔤 is decomposable; (2) 𝔤 = 𝔪 ⋉ 𝔫_{V} (𝔤) for a suitable reductive Lie subalgebra 𝔪 ⊂ 𝔤 acting semisimply on V;

(3) There exists a flag {0} = V_{0} ⊊ V_{1} ⊊ ⋯ ⊊ V_{s} = V of 𝔤-submodules such that 𝔫_{V} (𝔤) acts trivially (and 𝔪 semisimply) on gr(V) = ⨁_{i=1}^{s} V_{i}/V_{i−1} (one such flag is V_{0} = {0} and V_{i+1} = {v ∈ V | 𝔫_{V} (𝔤) v ∈ V_{i}}). The isomorphism class of the semisimple 𝔤/𝔫_{V} (𝔤)-module gr(V) does not depend on the choice of {V_{i}}.

2. Semisimplicity criteria for Lie algebra representations

(2.1) Proposition. Let 𝔤_{1}, …, 𝔤_{m} be simple Lie algebras (of finite dimension) over a field k ⊃ Q. Let 𝔤 ⊂ 𝔤_{1} × ⋯ × 𝔤_{m} be a Lie subalgebra which projects surjectively on each factor 𝔤_{i}. Then there exist:

• a partition I = {1, …, m} = I_{1} ∪^{·} ⋯ ∪^{·} I_{n} (for non-empty subsets I_{j} ⊂ I),

• for each j ∈ J = {1, …, n} a Lie algebra 𝔤^{(j)},

• for each j ∈ {1, …, n} and each i ∈ I_{j} an isomorphism of Lie algebras f_{ji} : 𝔤^{(j)} ⟶^{∼} 𝔤_{i}, such that

𝔤 = Im (∏_{j∈J} 𝔤^{(j)} ⟶^{Δ} ∏_{j∈J} (𝔤^{(j)})^{I_{j}} ⟶^{f} ∏_{j∈J} ∏_{i∈I_{j}} 𝔤_{i} = ∏_{i∈I} 𝔤_{i}) ,

where Δ = (Δ_{j})_{j∈J}, each Δ_{j} : 𝔤^{(j)} ⟶ (𝔤^{(j)})^{I_{j}} is the diagonal map and f = (f_{j})_{j∈J} is a Lie algebra isomorphism with components f_{j} = (f_{ji})_{i∈I_{j}}.

Proof. This is well-known. If we denote by p_{i} the projection map 𝔤 ↪ 𝔤_{1} × ⋯ × 𝔤_{m} ⟶ 𝔤_{i}, then the image p_{i}(𝔫) of any abelian ideal 𝔫 ⊂ 𝔤 is an abelian ideal of 𝔤_{i}, hence p_{i}(𝔫) = 0 for all i, which implies that 𝔫 = 0 and 𝔤 is semisimple, thus 𝔤 = 𝔤^{(1)} × ⋯ × 𝔤^{(n)} for simple Lie algebras 𝔤^{(j)}. For each j ∈ J = {1, …, n} the set

I_{j} = {i ∈ I | p_{i}(𝔤^{(j)}) ≠ 0} = {i ∈ I | p_{i} induces an isomorphism 𝔤^{(j)} ⟶^{∼} 𝔤_{i}}

is non-empty. If j ≠ j′ and i ∈ I_{j} ∩ I_{j′}, then

∀X ∈ 𝔤^{(j)} ∀X′ ∈ 𝔤^{(j′)} [f_{ji}(X), f_{j′i}(X′)] = p_{i}([X, X′]) = 0 ∈ 𝔤_{i},

hence [𝔤_{i}, 𝔤_{i}] = 0, which is not true. This contradiction implies that I_{j} ∩ I_{j′} = ∅. As ⋃_{j} I_{j} = I, the sets I_{1}, …, I_{n} form a partition of I. The rest of the proposition follows from the previous discussion.

(2.2) Proposition. Let 𝔤_{1}, …, 𝔤_{m} be reductive Lie algebras (of finite dimension) over an algebraically closed field k ⊃ Q. For each i ∈ I = {1, …, m} let M_{i} be a non-zero simple 𝔤_{i}-module of finite dimension. If 𝔤 ⊂ 𝔤_{1} × ⋯ × 𝔤_{m} is a Lie subalgebra which projects surjectively on each factor 𝔤_{i}, then:

• 𝔤 is reductive, 𝔤 = 𝔷(𝔤) ⊕ 𝒟𝔤;

• each element of 𝔷(𝔤) acts on M = M_{1} ⊠ ⋯ ⊠ M_{m} by a scalar;

• 𝒟𝔤_{i} = ∏_{t} 𝔤_{i,t} and M_{i} = ⊠_{t}M_{i,t}, where 𝔤_{i,t} is a simple Lie algebra and M_{i,t} is a simple 𝔤_{i,t}-module;

• applying Proposition 2.1 to 𝒟𝔤 ⊂ ∏_{i,t} 𝔤_{i,t} and replacing each 𝔤_{i,t} (resp. M_{i,t}) for (i, t) ∈ I_{j} by 𝔤^{(j)} (resp. by the 𝔤^{(j)}-module N_{i,t} = f_{j,(i,t)}^{∗}(M_{i,t}), where f_{j,(i,t)} : 𝔤^{(j)} ⟶^{∼} 𝔤_{i,t} is the isomorphism from Proposition 2.1), we have

𝒟𝔤 = Im (∏_{j∈J} 𝔤^{(j)} ⟶^{Δ=(Δ_{j})} ∏_{j∈J} (𝔤^{(j)})^{I_{j}} = ∏_{j∈J} ∏_{(i,t)∈I_{j}} 𝔤_{i,t} = ∏_{(i,t)} 𝔤_{i,t});

• if we identify 𝒟𝔤 with ∏_{j∈J} 𝔤^{(j)} via Δ, then the 𝒟𝔤-module M is isomorphic to ⊠_{j∈J}M^{(j)}, where M^{(j)} is the 𝔤^{(j)}-module

M^{(j)} = ⨂_{(i,t)∈I_{j}} N_{i,t};

• if 𝔥^{(j)} ⊂ 𝔤^{(j)} is a Cartan subalgebra, then 𝔥 = ∏_{j∈J} 𝔥^{(j)} is a Cartan subalgebra of 𝒟𝔤 and 𝔷(𝔤) ⊕ 𝔥 is a Cartan subalgebra of 𝔤. All weights of 𝔷(𝔤) ⊕ 𝔥 occurring in M lie in μ + λ + Q, where Q denotes the root lattice of 𝒟𝔤, μ ∈ 𝔷(𝔤)^{∗} is the weight by which 𝔷(𝔤) acts on M, λ = ∑_{j∈J} ∑_{(i,t)∈I_{j}} λ_{i,t} and λ_{i,t} ∈ 𝔥^{(j)∗} is the highest weight of the 𝔤^{(j)}-module N_{i,t};

• if each M_{i,t} is either one-dimensional or a minuscule 𝔤_{i,t}-module, then each M^{(j)} is a tensor product (possibly empty) of minuscule 𝔤^{(j)}-modules.

• if each M_{i} is a faithful 𝔤_{i}-module, then the following are equivalent: M is a simple 𝔤-module ⟺ 𝒟𝔤 = 𝒟𝔤_{1} × ⋯ × 𝒟𝔤_{r} ⟺ ∀i ≠ j p_{ij}(𝒟𝔤) = 𝒟𝔤_{i} × 𝒟𝔤_{j}, where p_{ij} denotes the projection onto the i-th and the j-th factors.

Proof. We have, for each i ∈ I, 𝔤_{i} = 𝔷(𝔤_{i}) ⊕ 𝒟𝔤_{i}, where 𝒟𝔤_{i} = [𝔤_{i}, 𝔤_{i}] is semisimple. Decomposing each 𝒟𝔤_{i} = ∏_{t} 𝔤_{i,r} into a product of simple Lie algebras and applying Proposition 2.1 to 𝒟𝔤 ⊂ 𝒟𝔤_{1} × ⋯ × 𝒟𝔤_{m}, we deduce that 𝒟𝔤 is semisimple, hence 𝔤 is reductive and 𝔤 = 𝔷(𝔤)⊕𝒟𝔤. As each element of p_{i}(𝔷(𝔤)) = 𝔷(𝔤_{i}) acts on M_{i} by a scalar, the same holds for the action of each element of 𝔷(𝔤) on M. The remaining statements follow from Proposition 2.1 (and the fact that M^{(j)} is a simple 𝔤^{(j)}-module ⟺ |I_{j}| = 1, by [Dy2, Thm. 3.2]).

(2.3) Corollary. For any surjective morphism of Lie algebras f : 𝔤 ⟶ \overline{𝔤}, the weights of any Cartan subalgebra \overline{𝔥} ⊂ \overline{𝔤} on \overline{M} = M/Ker(f)M lie in one coset modulo the root lattice of (\overline{𝔤}, \overline{𝔥}).

Proof. This follows from the corresponding statement for 𝔤 and M, the fact that Ker(f) ⊂ 𝔤 is a Lie ideal and that 𝔤 is isomorphic to Ker(f) × \overline{𝔤}.

(2.4) Theorem. Let k ⊃ Q be a complete non-discrete non-archimedean field (for example, k = Q_{ℓ}) and V a non-zero \overline{k}-vector space of finite dimension. If 𝔤 ⊂ End\overline{k}(V) is a k-Lie subalgebra of finite dimension (over k) such that (H1) 𝔤 contains a dense set (in the topology induced by the non-archimedean norm on k) of elements that are semisimple in End\overline{k}(V), then:

(1) \overline{𝔤} = \overline{k} · 𝔤 ⊂ End\overline{k}(V) is a decomposable linear Lie algebra over \overline{k}.

(2) Let \overline{𝔪} ⊂ \overline{𝔤} (\overline{𝔪} ≃ \overline{𝔤}/𝔫_{V} (\overline{𝔤})), \{0\} = V_{0} ⊊ V_{1} ⊊ ⋯ ⊊ V_{s} = V and gr(V) be as in Corollary 1.9 (over the base field \overline{k}). Fix a Cartan subalgebra \overline{𝔥} ⊂ \overline{𝔪}. Assume that the following condition holds: (H2) all weights of \overline{𝔥} occurring in gr(V) lie in one coset of the root lattice of (\overline{𝔪}, \overline{𝔥}). Then 𝔫_{V} (\overline{𝔤}) = 0, \overline{𝔤} = \overline{𝔪} is a reductive Lie algebra and V is a semisimple \overline{𝔤}-module.

(2.5) Example. Let 𝔤_{0} be a reductive Lie algebra over k = \overline{k} and M a faithful simple finite-dimensional 𝔤_{0}-module. If we identify the semidirect product 𝔤 = M ⋊ 𝔤_{0} with

\{( \begin{matrix}X & m \\ 0 & 0 \end{matrix} ) | X ∈ 𝔤_{0} ⊂ End(M), m ∈ M\} ⊂ End(M ⊕ k),

then 𝔤 will be a decomposable linear Lie algebra, 𝔫_{V} (𝔤) = M and 𝔪 = 𝔤_{0}. The flag {0} = V_{0} ⊊ V_{1} = M ⊕0 ⊊ V_{2} = V = M ⊕ k is as in Corollary 1.9(3). If 𝔤_{0} = 𝔰l_{2} and M is the standard two-dimensional representation (resp. the adjoint representation), then (H1) is satisfied and (H2) is not (resp. (H2) is satisfied and (H1) is not). Proof of Theorem 2.4. (1) There exist a finite subextension k′/k of \overline{k}/k and a k′-vector space V′ ⊂ V of finite dimension such that V′ ⊗_{k′} \overline{k} = V and 𝔤 ⊂ End_{k′}(V′). Assumption (H1) implies that the set of all elements of 𝔤 that are semisimple in End_{k′}(V′) is Zariski dense in 𝔤, hence in

𝔤 ⊗_{k} \overline{k} ⊂ End_{k′}(V′) ⊗_{k} \overline{k} = ⨁_{σ:k′⟶\overline{k}} End\overline{k}(V′ ⊗_{k′,σ} \overline{k}).

Taking the projection onto the factor corresponding to the inclusion of k′ into \overline{k}, we deduce that \overline{𝔤} ⊂ End\overline{k}(V) satisfies the assumptions of Corollary 1.9 (over \overline{k}), which proves (1).

(2) The arguments in the proof of (1) show that we can replace k by \overline{k}, 𝔤 by \overline{𝔤} and the non-archimedean topology in (H1) by the Zariski topology. In other words, it is enough to prove part (2) of the following theorem.

(2.6) Theorem. Let V be a non-zero vector space of finite dimension over an algebraically closed field k ⊃ Q. If \overline{𝔤} ⊂ End_{k}(V) is a k-Lie subalgebra such that (H1-ZAR) \overline{𝔤} contains a Zariski dense set of elements that are semisimple in End_{k}(V), then:

(1) \overline{𝔤} ⊂ End_{k}(V) is a decomposable linear Lie algebra.

(2) Let \overline{𝔫} = 𝔫_{V} (\overline{𝔤}) and fix \overline{𝔪} ⊂ \overline{𝔤} (\overline{𝔪} ≃ \overline{𝔤}/\overline{𝔫}), a Cartan subalgebra \overline{𝔥} ⊂ \overline{𝔪} and \{0\} = V_{0} ⊊ V_{1} ⊊ ⋯ ⊊ V_{s} = V as in Corollary 1.9. Assume that the following condition holds: (H2) all weights of \overline{𝔥} occurring in gr(V) lie in one coset of the root lattice of (\overline{𝔪}, \overline{𝔥}). Then \overline{𝔫} = 0, \overline{𝔤} = \overline{𝔪} is a reductive Lie algebra and V is a semisimple \overline{𝔤}-module.

Proof. (1) See the proof of Theorem 2.4(1).

(2) There is nothing to prove if s = 1. Assume that s = 2. The Lie algebra \overline{𝔤} = \overline{𝔪}⋉\overline{𝔫} satisfies [\overline{𝔫}, \overline{𝔫}] = 0 (since s = 2) and the normaliser of the abelian subalgebra \overline{𝔥} × \overline{𝔫}\overline{𝔥} in \overline{𝔤} is equal to \overline{𝔥} ⋉ 𝔫̃, where 𝔫̃ = \{X ∈ \overline{𝔫} | \overline{𝔥} · X ∈ \overline{𝔫}\overline{𝔥}\} = \overline{𝔫}\overline{𝔥} (the last equality follows from the fact that \overline{𝔥} acts semisimply on gr(V) = V_{1} ⊕ V/V_{1}, hence also on Hom_{k}(V/V_{1}, V_{1}) ⊃ \overline{𝔫}). In other words, \overline{𝔥} × \overline{𝔫}\overline{𝔥} is an abelian Cartan subalgebra of \overline{𝔤}. Proposition 1.8(4) implies, therefore, that all elements of \overline{𝔫}\overline{𝔥} act semisimply on V, hence \overline{𝔫}\overline{𝔥} = 0. The equality \overline{𝔫} = 0 then follows from the fact (used already in the proof of Proposition 1.3(1)) that 0 occurs as a weight of \overline{𝔥} in any non-zero simple \overline{𝔪}-module whose highest weight lies in the root lattice of (\overline{𝔪}, \overline{𝔥}) (in particular, in any non-trivial simple \overline{𝔪}-submodule of \overline{𝔫} ⊂ End_{k}(gr(V))). This shows that \overline{𝔤} is equal to \overline{𝔪}, which is a reductive Lie algebra acting semisimply on V. This finishes the proof if s = 2.

Assume now that s > 2. We are going to prove the statement by induction on d = dim(V). The cases d = 1, 2 have already been treated. Assume that d > 2 and that the statement has been proved for all pairs (V, \overline{𝔤}) with dim < d. Consider the \overline{𝔤}-submodule V′ = V_{s−1} ⊊ V and the Lie algebra \overline{𝔤}′ = Im(res : \overline{𝔤} ⟶ End_{k}(V′)). Note that the pair \overline{𝔤}′ ⊂ End_{k}(V′) satisfies (H1-ZAR). The image of the radical of \overline{𝔤} under the restriction map res is contained in the radical of \overline{𝔤}′, which implies that res(\overline{𝔫}) ⊂ \overline{𝔫}′ = 𝔫_{V′}(\overline{𝔤}′). As a result, res induces a surjective morphism of reductive Lie algebras r : \overline{𝔪} ⟶^{∼} \overline{𝔤}/\overline{𝔫} ↠ \overline{𝔤}′/\overline{𝔫}′. Fix a \overline{𝔤}′-stable flag \{0\} = V′_{0} ⊊ V′_{1} ⊊ ⋯ ⊊ V′_{s′} = V′ such that \overline{𝔫}′ acts trivially on gr(V′) = ⊕_{i=1}^{s′}V′_{i}/V′_{i−1}. All weights of \overline{𝔥} ⊂ \overline{𝔪} on the semisimple \overline{𝔤}′/\overline{𝔫}′-module gr(V′) lie in one coset modulo the root lattice of (\overline{𝔪}, \overline{𝔥}), thanks to (H2) and the fact that gr(V) ≃ gr(V′) ⊕ V/V′ as \overline{𝔪}-modules. After choosing a decomposition \overline{𝔪} ⟶^{∼} Ker(r) × \overline{𝔤}′/\overline{𝔫}′, we obtain the same statement for an appropriate Cartan subalgebra of \overline{𝔤}′/\overline{𝔫}′, which means that the flag \{V′_{i}\} and the Lie algebra \overline{𝔤}′ satisfy (H2). As dim(V′) < d, the induction hypothesis implies that \overline{𝔫}′ = 0, \overline{𝔤}′ is a reductive Lie algebra and V′ = gr(V′) is a semisimple \overline{𝔤}′-module. This means that \{0\} ⊊ V′ ⊊ V is a flag of \overline{𝔤}-submodules of length s = 2 satisfying the assumptions of Theorem 2.6(2). However, the case s = 2 was already treated, which concludes the proof.

3. Semisimplicity criteria for representations of profinite groups

(3.1) Let Γ be a profinite group, V a non-zero vector space of finite dimension over \overline{Q}_{ℓ} and ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) a representation (continuous, according to the convention from 0.1).

In this situation ρ(Γ) is a compact subgroup of Aut_{\overline{Q}_{ℓ}}(V), which implies that there exists a finite extension E of Q_{ℓ} contained in \overline{Q}_{ℓ} and an E-structure V_{E} ⊂ V (an E-vector subspace such that V_{E} ⊗_{E} \overline{Q}_{ℓ} = V) for which ρ(Γ) ⊂ Aut_{E}(V_{E}). According to a non-archimedean version of Lie’s theorem [LIE, Ch. III, §8, no. 2, Thm. 2], ρ(Γ) is a (compact) Lie group of finite dimension over Q_{ℓ}. In particular, the profinite topology on ρ(Γ) coincides with the topology induced by the ℓ-adic valuation on \overline{Q}_{ℓ}. The Lie algebra Lie(ρ(Γ)) ⊂ End_{E}(V_{E}) ⊂ End_{\overline{Q}_{ℓ}}(V) is a Q_{ℓ}-Lie algebra of finite dimension.

(3.2) The following properties of ρ are equivalent:

3.2.1. ρ is semisimple.

3.2.2. There exists an open subgroup U ⊂ Γ such that ρ|_{U} is semisimple.

3.2.3. V is a semisimple \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} Lie(ρ(Γ))-module.

This implies that the semisimplification ρ^{ss} of ρ satisfies ρ^{ss}|_{U} = (ρ|_{U})^{ss}, for any open subgroup U ⊂ Γ.

(3.3) Theorem. Let Γ be a profinite group, V, W_{1}, …, W_{r} non-zero vector spaces of finite dimension over \overline{Q}_{ℓ} and ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V), ρ_{i} : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(W_{i}) representations. Assume that the following conditions hold: (A) Each ρ_{i} is strongly irreducible. (B) The semisimplification ρ^{ss} of ρ is isomorphic to a subrepresentation of (ρ_{1} ⊗⋯⊗ρ_{r})^{⊕m}, for some m ≥ 1. (C) There exists an open subgroup Γ′ ⊂ Γ for which ρ(Γ′) contains a dense subset consisting of semisimple elements of Aut_{\overline{Q}_{ℓ}}(V). Then the representation ρ is semisimple, ρ = ρ^{ss}.

Proof. Assumption (B) (resp. (C)) for ρ implies the corresponding condition for any subquotient of ρ. It is sufficient, therefore, to consider only the case when V is an extension of two irreducible representations of Γ. In addition, we can replace Γ by any of its open subgroups, thanks to 3.2. This implies that we can assume, after shrinking Γ and passing to another subquotient if necessary, that V sits in an exact sequence of \overline{Q}_{ℓ}[Γ]-modules

0 ⟶ V_{1} ⟶ V ⟶ V/V_{1} ⟶ 0,

in which both V_{1} and V/V_{1} are strongly irreducible.

We want to show that Theorem 2.4 (for s = 2) applies to the Q_{ℓ}-Lie algebra 𝔤 = Lie(ρ(Γ)) ⊂ End_{\overline{Q}_{ℓ}}(V) and the \overline{Q}_{ℓ}-Lie algebra \overline{𝔤} = \overline{Q}_{ℓ} · 𝔤 ⊂ End_{\overline{Q}_{ℓ}}(V). Firstly, both V_{1} and V/V_{1} are irreducible \overline{𝔤}-modules. Secondly, condition (H1) in Theorem 2.4 is a consequence of (C). It follows that \overline{𝔤} is a decomposable Lie subalgebra of End_{\overline{Q}_{ℓ}}(V). We distinguish two cases.

Case (a): there exists a \overline{𝔤}-submodule 0 ⊊ W ⊊ V such that W ≠ V_{1}. Irreducibility of V_{1} and V/V_{1} implies that W ∩ V_{1} = 0 and W + V_{1} = V, hence W ≃ V/V_{1} is a complementary \overline{𝔤}-submodule to V_{1}; thus V = V_{1} ⊕ W ≃ V ⊕ V/V_{1} is a semisimple \overline{𝔤}-module, which implies that ρ is semisimple.

Case (b): V_{1} is the only proper non-zero \overline{𝔤}-submodule of V. This uniqueness implies that the flag 0 ⊂ V_{1} ⊂ V_{2} = V is as in Proposition 1.9(3): the nilpotent ideal 𝔫_{V} (\overline{𝔤}) ⊂ \overline{𝔤} acts trivially on gr(V) = V_{1} ⊕ V/V_{1}. Moreover, both V_{1} and V/V_{1} are simple \overline{𝔤}/𝔫_{V} (\overline{𝔤})-modules.

It remains to check condition (H2). Consider the representation ρ_{0} = ρ_{1}⊕⋯⊕ρ_{r} : Γ ⟶ ∏_{i=1}^{r} Aut_{\overline{Q}_{ℓ}}(W_{i}). Assumption (B) implies that Ker(ρ_{0}) ⊂ Ker(ρ^{ss}), which yields surjective morphisms ρ_{0}(Γ) ↠ ρ^{ss}(Γ) and Lie(ρ_{0}(Γ)) ↠ Lie(ρ^{ss}(Γ)). The \overline{Q}_{ℓ}-Lie subalgebra

\overline{Q}_{ℓ} ⊗_{Q_{ℓ}} Lie(ρ_{0}(Γ)) ⊂ 𝔤_{1} × ⋯ × 𝔤_{r}, 𝔤_{i} = \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} Lie(ρ_{i}(Γ)),

and the 𝔤_{i}-modules M_{i} = W_{i} satisfy the assumptions of Proposition 2.2 (for k = \overline{Q}_{ℓ}), thanks to (A). Corollary 2.3 tells us that all weights of a fixed Cartan subalgebra of \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} Lie(ρ^{ss}(Γ)) on gr(V) ⊂ (W_{1} ⊗⋯⊗W_{r})^{⊕m} lie in one coset of the root lattice of \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} Lie(ρ^{ss}(Γ)). The action of this Lie algebra on gr(V) factors through Im(\overline{𝔤} ⟶ End_{\overline{Q}_{ℓ}}(gr(V))), which is a quotient of the reductive Lie algebra \overline{𝔤}/𝔫_{V} (\overline{𝔤}). This implies, as in the proof of Corollary 2.3, that condition (H2) in Theorem 2.4 is satisfied. Applying Theorem 2.4, we conclude that V = V_{1} ⊕ W ≃ V ⊕ V/V_{1} is a semisimple \overline{𝔤}-module, as in case (a) (more precisely, the above argument shows that case (b) does not occur).

(3.4) Condition (C) in Theorem 3.3 is satisfied if there is a dense subset Σ ⊂ Γ′ with the following property: for each g ∈ Σ there exist polynomials P_{1}, …, P_{r} ∈ \overline{Q}_{ℓ}[X] without multiple roots and pairwise commuting elements u_{1}, …, u_{r} ∈ Aut_{\overline{Q}_{ℓ}}(V (g)), where V (g) ⊃ V is a finite dimensional vector space over \overline{Q}_{ℓ} (depending on g) such that P_{i}(u_{i}) = 0 for all i = 1, …, r, V is stable under u_{1} ⋯ u_{r} and ρ(g) = u_{1} ⋯ u_{r}|_{V}.

(3.5) For representations V occurring in cohomology of Shimura varieties (see §5 and A5 below) the group Γ is the Galois group of a suitable extension of the reflex field, the set Σ consists of Frobenius elements, P_{i}(X) = P_{ρ_{i}(g)}(X) is the characteristic polynomial of ρ_{i}(g) and u_{i} is induced by a partial Frobenius morphism acting on the special fibre of an integral model of the Shimura variety in question. In our abstract context, a sufficient (and also necessary) condition for these characteristic polynomials to be without multiple roots is as follows.

(3.6) Proposition. In the situation of Theorem 3.3, assume that condition (A) holds. For each i = 1, …, r denote by \overline{𝔤}_{i} = \overline{Q}_{ℓ} · Lie(ρ_{i}(Γ)) the image of 𝔤_{i} in End_{\overline{Q}_{ℓ}}(W_{i}). This is a reductive Lie algebra whose centre is contained in \overline{Q}_{ℓ} · id. If, for each i = 1, …, r, a fixed Cartan subalgebra \overline{𝔥}_{i} of \overline{𝔤}_{i} acts on W_{i} without multiplicities, then there exists an open subgroup Γ_{0} ⊂ Γ and an open dense subset U_{0} ⊂ Γ_{0} such that for all g ∈ U_{0} and all a ≥ 1 the characteristic polynomials P_{ρ_{i}(g^{a})}(X) (i = 1, …, r) are without multiple roots.

Proof. Using the notation of the proof of Theorem 3.3, set \overline{𝔤}_{0} = \overline{Q}_{ℓ}·Lie(ρ_{0}(Γ)) ⊂ \overline{𝔤}_{1}×⋯×\overline{𝔤}_{r}; then p_{i}(\overline{𝔤}_{0}) = \overline{𝔤}_{i}. For each i = 1, …, r, the polynomial function Δ_{i} : End_{\overline{Q}_{ℓ}}(W_{i}) ⟶ \overline{Q}_{ℓ} given by the discriminant of the characteristic polynomial is not identically equal to zero on \overline{𝔤}_{i}, which implies that ⋂_{i=1}^{r} p_{i}^{−1}(Δ_{i}^{−1}(\overline{Q}_{ℓ} ∖ \{0\})) is a dense Zariski open in \overline{𝔤}_{0}, hence its intersection U with 𝔤_{0} := Lie(ρ_{0}(Γ)) is a dense Zariski open in 𝔤_{0}. Fix a sufficiently small Z_{ℓ}-module of finite type T_{0} ⊂ 𝔤_{0} stable under the Lie bracket such that the exponential map induces a homeomorphism between T_{0} and ρ_{0}(Γ_{0}) (for an open subgroup Γ_{0} ⊂ Γ) and that ρ_{0}(Γ_{0}) acts trivially on T/2ℓT, for some ρ_{0}(Γ_{0})-stable \overline{Z}_{ℓ}-lattice T ⊂ ⊕_{i}W_{i}. The image exp(T_{0} ∩ U) is open and dense in ρ_{0}(Γ_{0}) (for the profinite topology), which implies that U_{0} = ρ_{0}^{−1}(exp(T_{0} ∩ U)) is open and dense in Γ_{0}. By construction, for each g ∈ U_{0} and each i = 1, …, r, the polynomial P_{ρ_{i}(g)}(X) has distinct roots λ_{1}, …, λ_{d_{i}} contained in 1 + 2ℓ\overline{Z}_{ℓ}, which implies that the powers λ_{1}^{a}, …, λ_{d_{i}}^{a} are also distinct, since 1 + 2ℓ\overline{Z}_{ℓ} contains no non-trivial roots of unity.

(3.7) Theorem. In the situation of Theorem 3.3 with r = 1, assume that conditions (A), (B) from Theorem 3.3 and (C’) below hold. (C’) There exists an open subgroup Γ′ ⊂ Γ such that P_{ρ_{1}(g)}(ρ(g)) = 0 ∈ End_{\overline{Q}_{ℓ}}(V) holds for all elements g of a certain dense subset of Γ′. Then the representation ρ is semisimple (and isomorphic to ρ_{1}^{⊕n}, for some n ≥ 1).

Proof. As any submodule of ρ_{1}^{⊕m} is isomorphic to ρ_{1}^{⊕m′}, we can assume that ρ^{ss} = ρ_{1}^{⊕m}. By induction it is sufficient to consider the case m = 2:

0 ⟶ W_{1} ⟶ V ⟶ W_{1} ⟶ 0.  (3.7.1)

We can assume that Γ = Γ′, by 3.2. Condition (C’) then implies, by continuity, that P_{ρ_{1}(g)}(ρ(g)) = 0 for all g ∈ Γ.

Denote by G ⊂ Aut_{\overline{Q}_{ℓ}}(V) (resp. G_{1} ⊂ Aut_{\overline{Q}_{ℓ}}(W_{1})) the Zariski closure of ρ(Γ) (resp. ρ_{1}(Γ)). These are (the sets of \overline{Q}_{ℓ}-valued points of) affine algebraic groups over \overline{Q}_{ℓ} such that

G = \{g = ( \begin{matrix}g_{1} & u \\ 0 & g_{1} \end{matrix} ) | g_{1} ∈ G_{1}, u ∈ N\} = N ⋊ G_{1},  (3.7.2)

for a suitable \overline{Q}_{ℓ}-linear splitting of the exact sequence (3.7.1) and a \overline{Q}_{ℓ}-vector subspace N ⊂ End_{\overline{Q}_{ℓ}}(W_{1}). Our aim is to show that N = 0. We have, again by continuity,

∀g = ( \begin{matrix}g_{1} & u \\ 0 & g_{1} \end{matrix} ) ∈ G P_{g_{1}}(g) = 0.  (3.7.3)

Condition (A) implies that the connected component of the identity G_{1}^{∘} acts irreducibly on W_{1}, hence G_{1}^{∘} is a connected reductive group over \overline{Q}_{ℓ} and its centre acts on W_{1} by a character. Fix a maximal torus T ⊂ G_{1}^{∘}. If N ≠ 0, Proposition 1.3(2) (in its version 1.5) states that there is a weight λ : T ⟶ G_{m} which occurs in W_{1} with multiplicity one and n ∈ N^{T} with non-zero image n_{λ} under the composite map

N^{T} ⊂ End_{\overline{Q}_{ℓ}}(W_{1})^{T} = ⨁_{μ} End_{\overline{Q}_{ℓ}}(W_{1}(μ)) ⟶ End_{\overline{Q}_{ℓ}}(W_{1}(λ)).

Choose t ∈ T such that μ(t) ≠ λ(t) for all weights μ ≠ λ of T occurring in W_{1}. The element

g = ( \begin{matrix}t & n \\ 0 & t \end{matrix} ) ∈ G

satisfies

P_{t}(X) = ∏_{μ}(X − μ(t))^{dim W_{1}(μ)},  P_{t}(g) = ( \begin{matrix}0 & n′ \\ 0 & 0 \end{matrix} ) ,  n′_{λ} = n_{λ} ∏_{μ≠λ}(λ(t) − μ(t))^{dim W_{1}(μ)} ≠ 0,

which is impossible, by (3.7.3). This contradiction shows that N = 0, hence ρ = ρ_{1}^{⊕2}, as claimed. (3.8) Questions. (1) Does Theorem 3.3 still hold if we relax assumption (A) by merely requiring each ρ_{i} to be irreducible and without multiplicities after restricting to open subgroups of Γ?

(2) Is there a common generalisation of Theorem 3.7 and the special case 3.4 of Theorem 3.3? (3.9) We are now going to show that a variant of condition (B) of Theorem 3.3 (in its version 3.4) is a consequence of the other assumptions, provided the representations ρ_{i} have sufficiently large image,

(3.10) Proposition. Assume that the representations ρ and ρ_{i} from Theorem 3.3 satisfy the following conditions. (A’) Each W_{i} is a direct sum of simple modules for the \overline{Q}_{ℓ}-Lie algebra \overline{Q}_{ℓ}·Lie(ρ_{i}(Γ)) (which is then reductive); each of these simple modules is one-dimensional or minuscule. (C’) There exist an integer a ≥ 1, an open subgroup Γ′ ⊂ Γ and a dense subset Σ ⊂ Γ′ such that

∀g ∈ Σ P_{(ρ_{1}⊗···⊗ρ_{r})(g^{a})}(ρ(g^{a})) = 0 ∈ End_{\overline{Q}_{ℓ}}(V).

Then: (1) There is an open subgroup U ⊂ Γ′ such that ρ^{ss}|_{U} = (ρ|_{U})^{ss} is isomorphic to a subrepresentation of (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{U}^{⊕m}, for some m ≥ 1.

(2) If ρ_{1} ⊗ ⋯ ⊗ ρ_{r} is strongly irreducible and a = 1 or a = 2, then every irreducible constituent of ρ^{ss}|_{Γ′} is isomorphic to (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{Γ′} ⊗ σ, for some character σ : Γ′ ⟶ {±1} satisfying σ^{a} = 1.

(3) If a = r = 1 and W_{1} is a minuscule representation of \overline{Q}_{ℓ} · Lie(ρ_{1}(Γ)), then ρ is semisimple and ρ|_{Γ′} is isomorphic to ρ_{1}|_{Γ′}^{⊕n}, for some n ≥ 1.

Proof. Thanks to 3.2, we can (and will) assume that Γ′ = Γ. By continuity, (C’) implies that

∀g ∈ Γ  P_{(ρ_{1}⊗···⊗ρ_{r})(g^{a})}(ρ(g^{a})) = 0 ∈ End_{\overline{Q}_{ℓ}}(V).  (3.10.1)

(1) We proceed in several steps. Step 1: It is enough to consider the case when ρ is irreducible. Furthermore, after shrinking Γ if necessary we can assume that ρ is strongly irreducible. This means that V is a simple \overline{𝔤}-module, where 𝔤 = Lie(ρ(Γ)) and \overline{𝔤} = \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} 𝔤. As in the proof of Theorem 2.4 we deduce that 𝔤 (resp. \overline{𝔤}) is a reductive Lie algebra over Q_{ℓ} (resp. over \overline{Q}_{ℓ}) and each element of the centre of \overline{𝔤} acts on V by a scalar. Consider the representation ρ_{0} = ρ_{1} ⊕ ⋯ ⊕ ρ_{r} : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(W_{0}), where W_{0} = W_{1} ⊕ ⋯ ⊕ W_{r} (cf. the proof of Theorem 3.3).

The subgroup ρ(Ker(ρ_{0})) ⊂ Aut_{\overline{Q}_{ℓ}}(V) is a compact Lie group of finite dimension over Q_{ℓ}. The formula (3.10.1) implies that (ρ(g^{a}) − 1)^{N} = 0 for all g ∈ Ker(ρ_{0}) (where N = dim(W_{1} ⊗ ⋯ ⊗ W_{r})), which means that the Lie ideal 𝔞 = \overline{Q}_{ℓ} · Lie(ρ(Ker(ρ_{0}))) in \overline{Q}_{ℓ} · 𝔤 = Im(\overline{𝔤} ⟶ End_{\overline{Q}_{ℓ}}(V)) consists of nilpotent elements, hence is a nilpotent Lie ideal [LIE, Ch. I, §4, no. 2, Cor. 3], and so 𝔞 = 0, since \overline{Q}_{ℓ} · 𝔤 is reductive. It follows that ρ(Ker(ρ_{0})) is a finite subgroup of Aut_{\overline{Q}_{ℓ}}(V), hence Ker(ρ_{0}) ∩ Ker(ρ) is an open subgroup of Ker(ρ_{0}). The surjection Γ/(Ker(ρ_{0}) ∩ Ker(ρ)) ↠ Γ/Ker(ρ) yields a canonical surjection f : 𝔤_{0} = Lie(ρ_{0}(Γ)) ↠ 𝔤. We consider V and all W_{i} as irreducible representations of the reductive \overline{Q}_{ℓ}-algebra \overline{𝔤}_{0} = \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} 𝔤_{0}. Step 2: The Q_{ℓ}-Lie algebra 𝔤_{0} ⊂ 𝔤_{1} × ⋯ × 𝔤_{r} (where 𝔤_{i} = Lie(ρ_{i}(Γ))) satisfies p_{i}(𝔤_{0}) = 𝔤_{i} for all i = 1, …, r. After shrinking Γ if necessary we can assume that, for each g ∈ Γ, all eigenvalues of ρ(g) and ρ_{i}(g) (i = 1, …, r) are contained in 1 + 2ℓ \overline{Z}_{ℓ}. After taking ℓ-adic logarithms ρ_{i}(Γ) ⟶ Lie(ρ_{i}(Γ)) (and similarly for ρ(Γ)) we deduce from (3.10.1) that

∀X = (X_{1}, …, X_{r}) ∈ 𝔤_{0} ⊂ 𝔤_{1} × ⋯ × 𝔤_{r} P_{(X_{1},...,X_{r})|W_{1}⊗···⊗W_{r}}(f(X)) = 0 ∈ End_{\overline{Q}_{ℓ}}(V),

where (X_{1}, …, X_{r}) ∈ End_{\overline{Q}_{ℓ}}(W_{1})×⋯×End_{\overline{Q}_{ℓ}}(W_{r}) acts on W_{1} ⊗⋯⊗W_{r} by ∑_{i} 1⊗⋯⊗1⊗X_{i} ⊗1⊗⋯⊗1. As 𝔤_{0} is Zariski dense in \overline{𝔤}_{0}, we obtain that

∀X = (X_{1}, …, X_{r}) ∈ \overline{𝔤}_{0} ⊂ \overline{𝔤}_{1} × ⋯ × \overline{𝔤}_{r}  P_{(X_{1},...,X_{r})|W_{1}⊗···⊗W_{r}}(\overline{f}(X)) = 0 ∈ End_{\overline{Q}_{ℓ}}(V),  (3.10.2)

where \overline{f} : \overline{𝔤}_{0} ⟶ \overline{𝔤} ↠ \overline{Q}_{ℓ} · 𝔤 ⊂ End_{\overline{Q}_{ℓ}}(V) is induced by f.

Step 3: For each i = 1, …, r we have W_{i} = ⊕_{u_{i}}W_{i,u_{i}}, where each W_{i,u_{i}} is a simple module for the \overline{Q}_{ℓ}- Lie algebra \overline{𝔤}_{i} = \overline{Q}_{ℓ} ⊗_{Q_{ℓ}} 𝔤_{i}. The Q_{ℓ}-Lie algebras 𝔤_{i,u_{i}} = Lie(Im(Γ ⟶ Aut_{\overline{Q}_{ℓ}}(W_{i,u_{i}}))) are reductive and 𝔤_{0} ⊂ ∏_{i=1}^{r} 𝔤_{i} ⊂ ∏_{i,u_{i}} 𝔤_{i,u_{i}} is a subalgebra projecting surjectively on each of the factors.

Applying Proposition 2.2 to \overline{𝔤}_{0} ⊂ ∏_{i,u_{i}} \overline{𝔤}_{i,u_{i}} and the simple \overline{𝔤}_{i,u_{i}}(=\overline{Q}_{ℓ} ⊗_{Q_{ℓ}} 𝔤_{i,u_{i}})-modules W_{i,u_{i}}, we obtain an isomorphism of \overline{Q}_{ℓ}-Lie algebras h : 𝔤^{(1)} × ⋯ × 𝔤^{(s)} ⟶^{∼} 𝒟\overline{𝔤}_{0}, where each 𝔤^{(j)} is a simple \overline{Q}_{ℓ}-Lie algebra and each 𝒟\overline{𝔤}_{i,u_{i}} is isomorphic to a product of several 𝔤^{(j)}’s (not necessarily distinct). Moreover, for each u = (u_{1}, …, u_{r}) the 𝒟\overline{𝔤}_{0}-module W_{⊗,u} = W_{1,u_{1}} ⊗ ⋯ ⊗ W_{r,u_{r}} satisfies h^{∗}(W_{⊗,u}) = M_{u}^{(1)} ⊠ ⋯ ⊠ M_{u}^{(s)}, where M_{u}^{(j)} is a tensor product (possibly empty) of minuscule representations of the simple Lie algebra 𝔤^{(j)}. In addition, each element of the centre 𝔷(\overline{𝔤}_{0}) ⊂ ∏_{i,u_{i}} 𝔷(\overline{𝔤}_{i,u_{i}}) acts on both W_{⊗,u} and V by a scalar.

These properties imply that the \overline{𝔤}_{0}-module W_{1} ⊗ ⋯ ⊗ W_{r} = ⊕_{u}W_{⊗,u} is a finite direct sum of tensor products of one-dimesional or minuscule representations of \overline{𝔤}_{0}.

Fix Cartan subalgebras 𝔥^{(j)} ⊂ 𝔤^{(j)} and 𝔥 ⊂ \overline{𝔤}_{0} such that 𝔥 ∩ 𝒟\overline{𝔤}_{0} = h(𝔥^{(1)} × ⋯ × 𝔥^{(s)}). The formula (3.10.2) applied to elements of 𝔥 implies that each weight of 𝔥 occurring in V must occur in W_{1} ⊗ ⋯ ⊗ W_{r}. Applying Proposition 1.6 we deduce that V is isomorphic (as a \overline{𝔤}_{0}-module) to a submodule of W_{1} ⊗ ⋯ ⊗ W_{r}, which is equivalent to the fact that there exists an open subgroup U ⊂ Γ such that ρ|_{U} is isomorphic to a subrepresentation of (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{U}, as claimed.

(2) Denote the representation ρ_{1}⊗⋯⊗ρ_{r} by ρ̃. According to (1) the restriction of ρ^{ss} to some open subgroup U ⊂ Γ is isomorphic to a subrepresentation of ρ̃|_{U}^{⊕m}, hence to ρ̃|_{U}^{⊕n} for some n ≤ m. It remains to show that ρ^{ss} and ρ^{⊕n} are isomorphic as representations of Γ.

As ρ^{ss} is semisimple, it is enough to consider any of its simple submodules, so we can assume that ρ^{ss} = ρ is irreducible. The statement (1) implies, by Frobenius reciprocity, that ρ is isomorphic to a simple \overline{Q}_{ℓ}[Γ]-submodule of Ind_{U}^{Γ}(ρ̃|_{U})^{⊕n} = (ρ̃⊗ Z[Γ/U])^{⊕n}, hence of ρ̃ ⊗ σ, for some irreducible representation σ : Γ/U ⟶ Aut_{\overline{Q}_{ℓ}}(V_{σ}) (after shrinking U we can assume that it is a normal subgroup of Γ). As

End_{\overline{Q}_{ℓ}[Γ]}(ρ̃⊗ σ) = End_{\overline{Q}_{ℓ}[U]}(ρ̃⊗ σ)^{Γ/U} = (End_{\overline{Q}_{ℓ}[U]}(ρ̃) ⊗_{\overline{Q}_{ℓ}} End_{\overline{Q}_{ℓ}}(σ))^{Γ/U} = End_{\overline{Q}_{ℓ}[Γ/U]}(σ) = \overline{Q}_{ℓ},

the (semisimple) representation ρ̃⊗ σ is irreducible, hence is isomorphic to ρ. We distinguish two cases. First case: σ(s)^{a} = id for all s ∈ Γ/U. If a = 1, then σ is the trivial representation and we are done. If a = 2, then σ(Γ/U) is an abelian group of exponent one or two, hence σ : Γ/U ⟶ {±1} is a trivial or a quadratic character. Second case: there exists s ∈ Γ such that σ(s)^{a} ≠ id, hence an eigenvalue c ≠ 1 of σ(s)^{a}. Relation (3.10.1) for g = us then gives

∀u ∈ U  P_{(_{ρ}(u)S)^{a}}(c(ρ̃(u)S)^{a}) = 0 ∈ End_{\overline{Q}_{ℓ}}(Ṽ),  (3.10.3)

where Ṽ = W_{1} ⊗ ⋯ ⊗ W_{r} and S = ρ̃(s) ∈ Aut_{\overline{Q}_{ℓ}}(Ṽ). Denote by G̃ the Zariski closure of ρ̃(U) in Aut_{\overline{Q}_{ℓ}}(Ṽ).

This is an affine algebraic group over \overline{Q}_{ℓ} whose connected component of identity G̃^{∘} acts irreducibly on Ṽ (thanks to the assumption on ρ̃ in (2)), hence G̃^{∘} is reductive. By continuity, we have

∀A ∈ G̃^{∘} P_{(AS)^{a}}(c(AS)^{a}) = 0 ∈ End_{\overline{Q}_{ℓ}}(Ṽ).

Lemma 3.11 below shows that S = 0, which is impossible. This contradiction implies that the second case never occurs.

(3) Combine (2) (for a = r = 1) with Theorem 3.7.

(3.11) Lemma. Let H be a connected reductive group over an algebraically closed field k ⊃ Q and r : H ⟶ GL(V) an irreducible rational representation of H. If a ≥ 1, c ∈ k^{×} ∖ {1} and S ∈ End_{k}(V) satisfy

∀h ∈ H(k)  P_{(r(h)S)^{a}}(c(r(h)S)^{a}) = 0 ∈ End_{k}(V),  (⋆)

then S = 0.

Proof. As in 1.3, a choice of a maximal torus T ⊂ H gives rise to weight decompositions V = ⨁_{μ} V (μ), End_{k}(V) = ⨁_{μ,μ′} Hom_{k}(V (μ), V (μ′)). Denote by λ the highest weight of V (for a fixed ordering of the roots of (H, T)).

If S ≠ 0, then there exists a non-zero irreducible rational subrepresentation M ⊂ End(V) of H containing S. Proposition 1.3(2) in the form 1.5 implies that there exists h_{0} ∈ H(k) such that the image of h_{0}Sh_{0}^{−1} in End_{k}(V (λ)) is non-zero. Condition (⋆) for S is invariant by conjugation by an element of H(k), which means that we can replace S by h_{0}Sh_{0}^{−1} and assume that the image S_{λ} of S in End_{k}(V (λ)) = k is non-zero.

Fix a cocharacter β : G_{m} ⟶ T such that β(λ) > 0 and β(λ) > β(μ) for all weights μ ≠ λ such that V (μ) ≠ 0. Denote by z the standard coordinate on G_{m} and by f(z) the image of P_{(r(β(z))S)^{a}}(c(r(β(z))S)^{a}) ∈ End_{k}(V)[z, z^{−1}] in End_{k}(V (λ))[z, z^{−1}] = k[z, z^{−1}]. The monomial of the highest degree occurring in f(z) is equal to c^{n−1}(c − 1)S_{λ}^{na} z^{β(λ)na}, where n = dim(V). As the coefficient c^{n−1}(c − 1)S_{λ}^{na} is non-zero, there exists u ∈ k^{×} such that f(u) ≠ 0, which implies that h = β(u) ∈ H(k) satisfies P_{(r(h)S)^{a}}(c(r(h)S)^{a}) ≠ 0. This contradiction with (⋆) shows that S = 0, as claimed.

(3.12) Theorem. Assume that the representations ρ and ρ_{i} from Theorem 3.3 satisfy the following conditions. (A”) Each ρ_{i} is strongly irreducible and W_{i} is a minuscule representation of the reductive \overline{Q}_{ℓ}-Lie algebra \overline{Q}_{ℓ} · Lie(ρ_{i}(Γ)). (C”) There exist an integer a ≥ 1, an open subgroup Γ′ ⊂ Γ and a dense subset Σ ⊂ Γ′ such that for each g ∈ Σ there are pairwise commuting elements u_{1}, …, u_{r} ∈ Aut_{\overline{Q}_{ℓ}}(V (g)), where V (g) ⊃ V is a finite dimensional vector space over \overline{Q}_{ℓ} depending on g, such that V is stable under u_{1} ⋯ u_{r}, ρ(g^{a}) = u_{1} ⋯ u_{r}|_{V} and P_{ρ_{i}(g^{a})}(u_{i}) = 0 (i = 1, …, r). Then: (1) Condition (C’) from Proposition 3.10 holds.

(2) The representation ρ is semisimple.

(3) The restriction of ρ to a suitable open subgroup U ⊂ Γ′ is isomorphic to a subrepresentation of (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{U}^{⊕m}, for some m ≥ 1.

(4) If ρ_{1} ⊗ ⋯ ⊗ ρ_{r} is strongly irreducible and if a = 1 or a = 2, then each irreducible constituent of ρ|_{Γ′} is isomorphic to (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{Γ′} ⊗ σ, for some character σ : Γ′ ⟶ {±1} satisfying σ^{a} = 1.

Proof. (1) This is a consequence of the following elementary fact: whenever u_{1}u_{2} = u_{2}u_{1} and P_{A_{i}}(u_{i}) = 0, then P_{A_{1}⊗A_{2}}(u_{1}u_{2}) = 0.

(2) In a minuscule representation each weight occurs with multiplicity one. As a result, Proposition 3.6 applies in the situation we consider, which means that we can assume – after replacing Γ′ by a open subgroup Γ_{0} ⊂ Γ′ and Σ by Σ ∩ U_{0}, for an open dense subset U_{0} ⊂ Γ_{0} – that for each g ∈ Σ all polynomials P_{ρ_{i}(g^{a})}(X) are without multiple roots, hence each u_{i} ∈ Aut_{\overline{Q}_{ℓ}}(V (g)) is semisimple, and so is the restriction to V of their product ρ(g^{a}). Therefore ρ(g) is semisimple and condition (C) in Theorem 3.3 is satisfied. Furthermore, condition (B) in Theorem 3.3 is satisfied if we replace Γ by a suitable open subgroup, thanks to (1) and Proposition 3.10(1). Applying Theorem 3.3 and taking into account 3.2 we deduce that ρ is semisimple. (3),(4) Combine (2) with Proposition 3.10(1) resp. 3.10(2).

4. Semisimplicity criteria for representations of profinite groups (bis)

(4.1) Our next goal is to prove a variant of Theorems 3.3 and 3.12 (Theorems 4.4 and 4.7 below) in which assumptions (A) and (A’) are not satisfied.

(4.2) Let Γ be a profinite group and Δ ◁ Γ an open normal subgroup. Denote by pr : Γ ⟶ Γ/Δ the projection map. There is a natural right action of Γ/Δ on the set of characters α : Δ ⟶ \overline{Q}_{ℓ}^{×}, namely

α^{pr(g)}(h) = α(ghg^{−1}),  g ∈ Γ, h ∈ Δ.  (4.2.1)

(4.3) From now on until 4.7 we assume that Γ/Δ is a cyclic group of order n > 1 and fix one of its generators σ. We change notation and write Γ_{n} instead of Δ. For characters α as in (4.2.1) the induced representations

I(α) = Ind_{Γ_{n}}^{Γ}(α)

are semisimple and have the following properties:

I(α)|_{Γ_{n}} ⟶^{∼} ⨁_{i=0}^{n−1} α^{σ^{i}},

I(α) ⟶^{∼} I(β) ⟺ ∃ i ∈ Z/nZ β = α^{σ^{i}},

I(α) ⊗ I(β) = Ind_{Γ_{n}}^{Γ}(α ⊗ I(β)|_{Γ_{n}}) ⟶^{∼} ⨁_{i=0}^{n−1} I(αβ^{σ^{i}}),

I(α_{1}) ⊗ ⋯ ⊗ I(α_{r}) ⟶^{∼} ⨁_{i_{2},...,i_{r}=0}^{n−1} I(α_{1}α_{2}^{σ^{i_{2}}} ⋯ α_{r}^{σ^{ir}}).

Fix a lift σ̃ ∈ pr^{−1}(σ) of σ and identify the representation space

I(α) = \{f : Γ ⟶ \overline{Q}_{ℓ} | f(hg) = α(h)f(g) ∀h ∈ Γ_{n}, ∀g ∈ Γ\}

with \overline{Q}_{ℓ}^{n} = ⊕_{i=1}^{n}\overline{Q}_{ℓ} · e_{i} via

f ↦ ∑_{i=1}^{n} f(σ^{i−1}) e_{i} = ( f(1) ; f(σ̃) ; ⋮ ; f(σ^{n−1}) ) .

The action (g ∗ f)(g_{1}) = f(g_{1}g) of Γ on I(α) then becomes

∀h ∈ Γ_{n} h(e_{i}) = α^{σ^{i−1}}(h) e_{i},  σ̃(e_{i}) = \{ \begin{matrix}e_{i−1}, & i ≠ 1 \\ α(σ^{n}) e_{n} & i = 1. \end{matrix}

In particular, σ^{n} ∈ Γ_{n} acts on I(α) by multiplication by the scalar α(σ^{n}). This scalar depends on σ̃, not just on σ, since

∀h ∈ Γ_{n} α ((hσ̃)^{n}) = α^{1+σ+···+σ^{n−1}}(h) α(σ^{n}).

The representation I(α) decomposes into a direct sum of n/d irreducible representations of dimension d

I(α) = ⨁ ; _{α} Ind_{Γ_{d}}^{Γ}(α̃),

where d = min{i ≥ 1 | α^{σ^{i}} = α} is a divisor of n, Γ_{d} = pr^{−1}(⟨σ^{d}⟩) is the inverse image of the cyclic group generated by σ^{d} and α̃ runs through all characters α̃ : Γ_{d} ⟶ \overline{Q}_{ℓ}^{×} extending α.

(4.4) Theorem. Let ρ_{i} = I(α_{i}) (i = 1, …, r), where α_{1}, …, α_{r} : Γ_{n} ⟶ \overline{Q}_{ℓ}^{×} are characters. Assume that a representation ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) satisfies the following conditions. (B) The semisimplification ρ^{ss} of ρ is isomorphic to a subrepresentation of (ρ_{1} ⊗⋯⊗ρ_{r})^{⊕m}, for some m ≥ 1. (S) There is a dense subset Σ ⊂ Γ such that, for each g ∈ Σ, ρ(g) is a semisimple element of Aut_{\overline{Q}_{ℓ}}(V). If n is a prime number, then the representation ρ is semisimple.

Proof. Note that the representations ρ_{i} and their tensor products are semisimple and that conditions (B) and (S) are satisfied by any subquotient of ρ. By induction, it is sufficient to consider only the case when ρ sits in an exact sequence

0 ⟶ Y ⟶ ρ ⟶ X ⟶ 0,

where X and Y are irreducible subrepresentations of ρ_{1} ⊗ ⋯ ⊗ ρ_{r}, hence X ⊕ X′ = I(α), Y ⊕ Y′ = I(β) for some α = α_{1}α_{2}^{σ^{i_{2}}} ⋯ α_{r}^{σ^{ir}} and β = α_{1}α_{2}^{σ^{j_{2}}} ⋯ α_{r}^{σ^{jr}}. After replacing ρ by ρ ⊕ X′ ⊕ Y′ we can assume that

0 ⟶ I(β) ⟶ ρ ⟶ I(α) ⟶ 0,

where β/α = φ^{σ−1} for a suitable character φ : Γ_{n} ⟶ \overline{Q}_{ℓ}^{×}. It is enough, therefore, to prove the following statement.

(4.5) Proposition. If n is a prime number and if α, β, φ : Γ_{n} ⟶ \overline{Q}_{ℓ}^{×} are characters such that β/α = φ^{σ−1}, then any representation ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) satisfying (S) which sits in an exact sequence

0 ⟶ I(β) ⟶ ρ ⟶ I(α) ⟶ 0

is isomorphic to I(β) ⊕ I(α).

Proof. Step 1: For each h ∈ Γ_{n} the element (hσ̃)^{n} = h_{n}σ^{n} (h_{n} = h(σ̃hσ^{−1}) ⋯ (σ^{n−1}hσ^{1−n})) acts on both I(α) and I(β) by the same scalar α(h_{n}σ^{n}) = β(h_{n}σ^{n}), since

(β/α)(h_{n}σ^{n}) = φ^{(σ−1)(1+σ+···+σ^{n−1})}(h) φ^{σ−1}(σ^{n}) = φ^{(σ^{n}−1)}(h) φ^{σ−1}(σ^{n}) = 1.

It follows that

\{h ∈ Γ_{n} | ρ(hσ̃) is semisimple\} = \{h ∈ Γ_{n} | ρ((hσ̃)^{n}) ∈ \overline{Q}_{ℓ}^{×} · id\}

is a closed subset of Γ_{n}, hence equal to Γ_{n}, thanks to (S). In particular, σ̃ acts semisimply on V. Step 2: According to Step 1 there exists a σ̃-equivariant \overline{Q}_{ℓ}-linear splitting of V ⟶ I(α), which we fix. Together with the identifications I(α) ⟶^{∼} \overline{Q}_{ℓ}^{n} ⟵^{∼} I(β) from 4.3 this allows us to identify V with \overline{Q}_{ℓ}^{n} ⊕ \overline{Q}_{ℓ}^{n} in such a way that g ∈ Γ acts on V by a matrix

ρ(g) = ( \begin{matrix}B & c(g)A \\ 0 & A \end{matrix} ) ,

where A ∈ GL_{n}(\overline{Q}_{ℓ}) (resp. B ∈ GL_{n}(\overline{Q}_{ℓ})) is given by the action of g on I(α) (resp. on I(β)) and

c ∈ Z^{1}(Γ, Hom(I(α), I(β))) = Z^{1}(Γ, M_{n}(\overline{Q}_{ℓ}))

is the 1-cocycle attached to the splitting. Continuity of ρ implies that the functions g ↦ A, B, c(g) are continuous, too. By construction, c(σ̃) = 0.

Note that there is an isomorphism of Γ-modules

Hom(I(α), I(β)) ⟶^{∼} I(α)^{∨} ⊗ I(β) ⟶^{∼} I(α^{−1}) ⊗ I(β) ⟶^{∼} ⨁_{i=0}^{n−1} I(β/α^{σ^{i}})

under which I(β/α^{σ^{i}}) corresponds to the set of matrices C ∈ M_{n}(\overline{Q}_{ℓ}) which have non-zero entries C_{ab} only on the shifted diagonal a + i ≡ b (mod n), since for each g ∈ Γ_{n} the matrix A is diagonal, with diagonal entries equal to α(g), α^{σ}(g), …, α^{σ^{n−1}}(g) (and similarly for B, when α is replaced by β). Denote by

p_{I(β/α^{σ^{j}})} : M_{n}(\overline{Q}_{ℓ}) ⟶^{∼} Hom(I(α), I(β)) ⟶^{∼} ⨁_{i=0}^{n−1} I(β/α^{σ^{i}}) ⟶ I(β/α^{σ^{j}})

the projection on the term I(β/α^{σ^{j}}) (corresponding to a shifted diagonal of M_{n}(\overline{Q}_{ℓ})).

Step 3: Assume first that β = α. For each h ∈ Γ_{n} ∩ Σ the semisimplicity of ρ(h) = ( \begin{matrix}A & c(h)A \\ 0 & A \end{matrix} ) implies

that all terms on the main diagonal of c(h) vanish, since A is diagonal. By continuity of c, it follows that

∀h ∈ Γ_{n}  p_{I(β/α)} (c(h)) = 0.  (4.5.1)

Step 4: Assume now that β ≠ α (hence φ^{σ} ≠ φ). Step 1 implies that

∀h ∈ Γ_{n}  0 = c((hσ̃)^{n}) = ∑_{j=0}^{n−1}(hσ̃)^{j} c(hσ̃) = ∑_{j=0}^{n−1}(hσ̃)^{j} c(h),  (4.5.2)

since c(hσ̃) = hc(σ̃) + c(h) = c(h).

Let h_{0}, h_{1} ∈ Γ_{n} be two elements such that h_{0} acts trivially on gr(V) = I(β) ⊕ I(α); set h_{2} = h_{0}h_{1}. The condition on h_{0} implies that c(h_{2}) = h_{0}c(h_{1}) + c(h_{0}) = c(h_{1}) + c(h_{0}) and that the action of h_{2}σ̃ on gr(V) coincides with that of h_{1}σ̃. Subtracting the relations (4.5.2) for h_{2} and h_{1}, we obtain

0 = ∑_{j=0}^{n−1} ((h_{2}σ̃)^{j} c(h_{2}) − (h_{1}σ̃)^{j} c(h_{1})) = ∑_{j=0}^{n−1}(h_{1}σ̃)^{j} c(h_{0}).  (4.5.3)

Write x := p_{I(β/α)}(c(h_{0})) as x = ∑_{i=1}^{n} x_{i}e_{i} (x_{i} ∈ \overline{Q}_{ℓ}) using the identification I(β/α) = I(φ^{σ−1}) ⟶^{∼} \overline{Q}_{ℓ}^{n} from 4.3. The formula

(h_{1}σ̃)^{j}(e_{i}) = (φ^{σ−1})^{σ^{i−1}+···+σ^{i−j}}(h_{1}) e_{i−j} = (φ^{σ^{i}−σ^{i−j}})(h_{1}) e_{i−j}

implies that (4.5.3) can be rewritten as

∀h_{1} ∈ Γ_{n} 0 = ∑_{j=0}^{n−1}(h_{1}σ̃)^{j} x = ∑_{i=1}^{n} ∑_{j=0}^{n−1} x_{i}(φ^{σ^{i}−σ^{i−j}})(h_{1}) e_{i−j} = ∑_{a=1}^{n} e_{a}/φ^{σ^{a}}(h_{1}) (∑_{j=0}^{n−1} φ^{σ^{a+j}}(h_{1}) x_{a+j}) ,

hence

∀h_{1} ∈ Γ_{n}  ∑_{i=1}^{n} φ^{σ^{i}}(h_{1}) x_{i} = 0.

Linear independence of the distinct characters φ^{σ}, …, φ^{σ^{n}} = φ (we are using our assumptions that φ^{σ} ≠ φ and n is a prime number) then implies that all x_{i} = 0. In other words,

p_{I(β/α)}(c(h_{0})) = x = 0.

Step 5: Applying Step 3 and Step 4 to the identifications I(α) ⟶^{∼} I(α^{σ^{i}}) ⟶^{∼} \overline{Q}_{ℓ}^{n} (when β/α^{σ^{i}} = φ_{i}^{σ−1} for φ_{i} = φ/α^{1+σ+···+σ^{i−1}}) we obtain

∀i = 0, …, n − 1 p_{I(β/α^{σ^{i}})}(c(h_{0})) = 0,

hence c(h_{0}) = 0 for every h_{0} ∈ Γ_{n} which acts trivially on gr(V). This implies that, for every h ∈ Γ_{n}, the value c(h) depends only on the image ρ_{gr(V)}(h) of ρ(h) in Aut(gr(V)). Step 6: According to Step 5, the restriction c_{n} of the cocycle c to Γ_{n}

c_{n} ∈ Z^{1}(Γ_{n}, Hom(I(α), I(β))) = ⨁_{i=0}^{n−1} Z^{1}(Γ_{n}, I(β/α^{σ^{i}}))

lies in the image of the inflation map

inf : Z^{1}(A, Hom(I(α), I(β))) = ⨁_{i=0}^{n−1} Z^{1}(A, I(β/α^{σ^{i}})) ⟶ ⨁_{i=0}^{n−1} Z^{1}(Γ_{n}, I(β/α^{σ^{i}})),

where A := ρ_{gr(V)}(Γ_{n}) ⊂ Aut(I(α) ⊕ I(β)) is an abelian profinite group.

If α^{σ^{i}} ≠ β, then H^{1}(A, β/α^{σ^{i}}) = 0, by Sah’s Lemma.

If α^{σ^{i}} = β, then (4.5.1) implies that p_{I(β/α^{σ^{i}})} c_{n} = 0.

In either case, the cohomology class of p_{I(β/α^{σ^{i}})} c_{n} vanishes in H^{1}(Γ_{n}, I(β/α^{σ^{i}})), hence the cohomology class of c lies in the group

Ker (res : H^{1}(Γ, Hom(I(α), I(β))) ⟶ H^{1}(Γ_{n}, Hom(I(α), I(β)))) ⟶^{∼} H^{1}(Γ/Γ_{n}, Hom(I(α), I(β))^{Γ_{n}}),

which is trivial, since Hom(I(α), I(β))^{Γ_{n}} is a Q-vector space. This finishes the proof of the fact that ρ ⟶^{∼} I(β) ⊕ I(α) (and of Theorem 4.4).

(4.6) Proposition. Let ρ_{i} = I(α_{i}) (i = 1, …, r) be as in Theorem 4.4. Let ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) be a representation satisfying the following condition. (S’) There exist an open subgroup Γ′ ⊂ Γ such that pr(Γ′) = Γ/Γ_{n} and a dense subset Σ ⊂ Γ′ such that

∀g ∈ Σ P_{(ρ_{1}⊗···⊗ρ_{r})(g)}(ρ(g)) = 0 ∈ End_{\overline{Q}_{ℓ}}(V).

Then there is an open subgroup U ⊂ Γ′ such that pr(U) = Γ/Γ_{n} and ρ^{ss}|_{U} = (ρ|_{U})^{ss} is isomorphic to a subrepresentation of (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{U}^{⊕m}, for some m ≥ 1.

Proof. Step 1: (S’) holds for all subquotients of ρ, which means that we can – and will – assume that ρ is irreducible. Note that conditions (A’) and (C’) from Proposition 3.10 are satisfied (with a = 1). Step 1 of the proof of Proposition 3.10(1) shows that

Ker(ρ) ⊇ Ker(ρ_{0}) = ⋂_{i=1}^{r} Ker(ρ_{i}) = ⋂_{i,j} Ker(α^{σ^{j}})

(where ρ_{0} = ρ_{1} ⊕ ⋯ ⊕ ρ_{r}), which means that we can replace Γ by Γ/Ker(ρ_{0}), hence assume that Γ_{n} is abelian. In this case ρ is necessarily of the form (cf. 4.3)

ρ = Ind_{Γ_{d}}^{Γ}(β̃) ⊂ I(β),  β̃: Γ_{d} ⟶ \overline{Q}_{ℓ}^{×},  β̃|_{Γ_{n}} = β,  d = min\{i ≥ 1 | β^{σ^{i}} = β\},  ρ|_{Γ_{n}} = ⨁_{i=0}^{d−1} β^{σ^{i}}.

Proof of Proposition 3.10(1) implies that there is an open subgroup A ⊂ Γ_{n} such that

β|_{A} ⊂ (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{A} = ⨁_{i_{1},...,i_{r}=0}^{n−1} α_{1}^{σ^{i_{1}}} ⋯ α_{r}^{σ^{ir}} |_{A},

hence there exist {i_{j}} for which the character χ = α_{1}^{σ^{i_{1}}} ⋯ α_{r}^{σ^{ir}} /β has finite order. We can replace each α_{j}^{σ^{i_{j}}} by α_{j} without changing I(α_{j}); then χ = α_{1} ⋯ α_{r}/β. Step 2: By continuity, the equality in (S’) holds for all g ∈ Γ′, in particular for all g ∈ Γ′ ∩ pr^{−1}(σ). Fix σ̃ ∈ Γ′ ∩ pr^{−1}(σ) and write g = σ̃h, where h ∈ Γ′_{n} = Γ′ ∩ Γ_{n}. If we denote by Nα the product αα^{σ} ⋯ α^{σ^{n−1}} (for any character α of Γ_{n}), then

P_{(ρ_{1}⊗···⊗ρ_{r})(_{σ}h)}(X) = (X^{n} − (N(α_{1} ⋯ α_{r}))(h) (α_{1} ⋯ α_{r})(σ^{n}))^{n^{r−1}}

and

P_{ρ(_{σ}h)}(X) = X^{d} − (β̃β̃^{σ} ⋯ β̃^{σ^{d−1}}) (h) β̃(σ^{d}), ρ(σ̃h)^{n} = (Nβ)(h) β(σ^{n}) · id_{V} .

Condition (S’) then yields

∀h ∈ Γ′_{n} (Nχ)(h) = (N(α_{1} ⋯ α_{r}/β)) (h) = (α_{1} ⋯ α_{r}/β)(σ^{n})^{−1} = χ(σ^{n})^{−1},

which is equivalent to χ(σ^{n}) = 1 and (Nχ)|_{Γ′_{n}} = 1, hence to

∀h ∈ Γ′_{n} χ((σ̃h)^{n}) = 1.

Step 3: The following open subgroup of Γ′_{n}

U_{n} = {h ∈ Γ′_{n} | ∀i ∈ Z χ^{σ^{i}}(h) = 1} = Γ′ ∩ ⋂_{i=0}^{n−1} Ker(χ^{σ^{i}})

is stable under the conjugation action by the cyclic group ⟨σ⟩. The extension class [Γ′] ∈ H^{2}(⟨σ⟩, Γ′_{n}) of

1 ⟶ Γ′_{n} ⟶ Γ′ ⟶ ⟨σ⟩ ⟶ 1

is equal to the image of σ^{n} ∈ (Γ′_{n})^{σ=1} under the periodicity isomorphism (depending on σ)

(Γ′_{n})^{σ=1}/(1 + σ + ⋯ + σ^{n−1})Γ′_{n} = Ĥ^{0}(⟨σ⟩, Γ′_{n}) ⟶^{∼} H^{2}(⟨σ⟩, Γ′_{n}).

The conclusion of Step 2 implies that σ^{n} ∈ U_{n}, hence

[Γ′] ∈ Im (H^{2}(⟨σ⟩, U_{n}) ⟶ H^{2}(⟨σ⟩, Γ′_{n})) ,

which means that there is a commutative diagram of group extensions

1 ⟶ U_{n} ⟶ U ⟶ ⟨σ⟩ ⟶ 1
∩ ↓  ∩ ↓  ∥
1 ⟶ Γ′_{n} ⟶ Γ′ ⟶ ⟨σ⟩ ⟶ 1
∩ ↓  ∩ ↓  ∥
1 ⟶ Γ_{n} ⟶ Γ ⟶ ⟨σ⟩ ⟶ 1

such that β|_{U_{n}} = (α_{1} ⋯ α_{r})|_{U_{n}}. We have ρ ⊂ I(β), hence

ρ|_{U} ⊂ I(β)|_{U} = Ind_{U_{n}}^{U}(β|_{U_{n}}) = Ind_{U_{n}}^{U}(α_{1} ⋯ α_{r}|_{U_{n}}) ⊂ ⨂_{i=1}^{r} Ind_{U_{n}}^{U}(α_{i}|_{U_{n}}) = (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{U},

as claimed.

(4.7) Theorem. Let ρ_{i} = I(α_{i}) (i = 1, …, r) be as in Theorem 4.4. Assume that n is a prime number and that α_{i}/α_{i}^{σ^{j}} is a character of infinite order, for each i = 1, …, r and j = 1, …, n−1. Let ρ : Γ ⟶ Aut_{\overline{Q}_{ℓ}}(V) be a representation satisfying the following condition. (S”) There exist an open subgroup Γ′ ⊂ Γ such that pr(Γ′) = Γ/Γ_{n} and a dense subset Σ ⊂ Γ′ such that for each g ∈ Σ there are pairwise commuting elements u_{1}, …, u_{r} ∈ Aut_{\overline{Q}_{ℓ}}(V (g)), where V (g) ⊃ V is a finite-dimensional vector space over \overline{Q}_{ℓ} depending on g, such that V is stable by u_{1} ⋯ u_{r}, ρ(g) = u_{1} ⋯ u_{r}|_{V} and P_{ρ_{i}(g)}(u_{i}) = 0 (i = 1, …, r). Then: (1) Condition (S’) from Proposition 4.6 holds.

(2) The representation ρ is semisimple.

(3) The restriction of ρ to a suitable open subgroup U ⊂ Γ′ satisfying pr(U) = Γ/Γ_{n} is isomorphic to a subrepresentation of (ρ_{1} ⊗ ⋯ ⊗ ρ_{r})|_{U}^{⊕m}, for some m ≥ 1.

Proof. (1) See the proof of Theorem 3.12(1).

(2) Thanks to (1), Proposition 4.6 applies to ρ, which yields an open subgroup U ⊂ Γ′ such that pr(U) = Γ/Γ_{n} for which ρ|_{U} satisfies condition (B) in Theorem 4.4 (where we replace (Γ, Γ_{n}, α_{i}) by (U, U_{n} = U ∩Γ_{n}, α_{i}|_{U_{n}})). Condition (S”) implies that, for each element g of Σ∩U ∩pr^{−1}(σ) (which is a dense subset of U ∩pr^{−1}(σ)), we have ρ(g) = u_{1} ⋯ u_{r}|_{V} with pairwise commuting u_{i} ∈ Aut_{\overline{Q}_{ℓ}}(V (g)) satisfying 0 = P_{ρ_{i}(g)}(u_{i}) = u_{i}^{n}−α_{i}(g^{n})id. It follows that each u_{i} is semisimple, and so is the restriction of their product ρ(g). After replacing σ by other generators of Γ/Γ_{n} = U/U_{n} we obtain the same statement for all g ∈ Σ∩(U ∖U_{n}). The assumption on α_{i}/α_{i}^{σ^{j}} implies that for all elements g of a suitable open dense subset U′_{n} ⊂ U_{n} the characteristic polynomials P_{ρ_{i}(g)}(X) (i = 1, …, r) have distinct roots. As above, this implies that ρ(g) is semisimple for each g ∈ Σ∩U′_{n}. This means that condition (S) in Theorem 4.4 is also satisfied by ρ|_{U} (with Σ replaced by (Σ ∩ (U ∖ U_{n})) ∪ (Σ ∩ U′_{n})). We deduce from Theorem 4.4 that ρ|_{U}, hence ρ as well, is semisimple.

(3) Combine (2) with Proposition 4.6 (which applies, by (1)).

5. Cohomology of quaternionic Shimura varieties

(5.1) In this section we apply the abstract results of §3 and §4 to Galois representations occurring in étale cohomology of quaternionic Shimura varieties. Historically, this was the first class of Shimura varieties of dimension dim > 1 to which the Langlands(-Kottwitz) method was applied [La], [VSL]. (5.2) Let F ⊂ \overline{Q} ⊂ C be a totally real number field of degree r = [F : Q]. Fix a quaternion algebra D over F which is not totally definite. Let H (resp. G = R_{F/Q}(H)) be the group D^{×}, viewed as an algebraic group over F (resp. over Q). Define G^{∗} to be the fibre product

\xymatrix{G^{∗} \ar[r] \ar[d] & G \ar[d]^{Nrd} \\ G_{m,Q} \ar[r] & R_{F/Q}(G_{m,F}),}

where Nrd is the reduced norm. (5.3) The set of infinite primes of F naturally decomposes as

{v : F ↪ R} = {v | D_{v} ≃ M_{2}(R)} ∪ {v | D_{v} ≃ H} = Ω ∪ Ω^{c},  |Ω| = t ≥ 1,  |Ω^{c}| = r − t.

We fix the corresponding isomorphisms

D ⊗ R ≃ M_{2}(R)^{Ω} × H^{Ω^{c}}, G(R) ≃ GL_{2}(R)^{Ω} × (H^{×})^{Ω^{c}}.

Let

h : S = R_{C/R}G_{m,C} → G_{R} ≃ GL(2)_{R}^{Ω} × (H^{×})^{Ω^{c}}  (5.3.1)

be the standard morphism x + iy ↦ ( \begin{matrix}x & y \\ −y & x \end{matrix} )^{Ω} × 1^{Ω^{c}}. Its G(R)-conjugacy class 𝒳 is naturally identified

with (C ∖ R)^{Ω} and h with the point {i}^{Ω}. (5.4) The Shimura variety Sh_{K}(G, 𝒳 ) has dimension t and its reflex field E = E(G, 𝒳 ) is equal to

E = Q(∑_{v∈Ω} v(a) | a ∈ F ) ⊂ F^{gal} ⊂ \overline{Q} ⊂ C.

Equivalently, Γ_{E} = Gal(\overline{Q}/E) = \{γ ∈ Γ_{Q} | γ(Ω) = Ω\}, if we consider Ω ⊂ Hom(F, C) = Hom(F, \overline{Q}). Define an intermediate field E ⊂ E′ ⊂ F^{gal} to be the fixed field of Γ_{E′} = Gal(\overline{Q}/E′) = \{γ ∈ Γ_{Q} | ∀v ∈ Ω γ(v) = v\}.

If t = 1 (the case of Shimura curves), then E = E′ = F. If t = r (the essentially PEL case), then E = Q and E′ = F^{gal}. (5.5) Our main objects of interest will be the cohomology groups H_{et}^{∗}(Sh(G, 𝒳 )⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) (in the notation of Introduction), where

ξ : G_{C} = GL(2)_{C}^{Hom(F,R)} ⟶ GL(N)_{C}

is an irreducible rational representation such that ξ|_{Z_{C}} factors through N_{F/Q}. Explicitly,

ξ = ⊗_{v|∞}ξ_{v},  ξ_{v} = Sym^{k_{v}−2}(Std^{∨}) ⊗ (det ∘ Std^{∨})^{(w−k_{v})/2} : GL(2)_{C} ⟶ GL(k_{v} − 1)_{C},  (5.5.1)

where Std^{∨} is the dual of the standard two-dimensional representation of GL(2), k : Hom(F, R) ⟶ Z_{≥2}, w ∈ Z and ∀v|∞ k_{v} ≡ w (mod 2). The reason why we consider Std^{∨} rather then Std is explained by the discussion in A5.6 below.

The corresponding ℓ-adic sheaf ℒ_{ξ,ℓ} is pure of weight t(w − 2). (5.6) If t = r (i.e., if the quaternion algebra D is totally indefinite), then the morphism (5.3.1) factors through G_{R}^{∗}. Its G^{∗}(R)-conjugacy class 𝒳 ^{∗} is naturally identified with (H^{+})^{Hom(F,R)} ∪ (H^{−})^{Hom(F,R)} ⊂ (C ∖ R)^{Hom(F,R)}, where H^{+} and H^{−} denote the upper and lower half planes in C, respectively.

The Shimura variety Sh(G^{∗}, 𝒳 ^{∗}) is of PEL type. It is defined over the common reflex field E = Q of (G^{∗}, 𝒳 ^{∗}) and (G, 𝒳 ).

(5.7) Proposition. If t = r, then the morphism i : Sh(G^{∗}, 𝒳 ^{∗}) ⟶ Sh(G, 𝒳 ) defined by the inclusion (G^{∗}, 𝒳 ^{∗}) ⊂ (G, 𝒳 ) induces an isomorphism of the connected components Sh(G^{∗}, 𝒳 ^{∗})^{an,+} ≃ Sh(G, 𝒳 )^{an,+} containing 𝒳 ^{+} × {1}, where (𝒳 ^{∗})^{+} = 𝒳 ^{+} = (H^{+})^{Hom(F,R)} ⊂ 𝒳 ^{∗} ⊂ 𝒳 . Moreover, the map i is an open immersion.

Proof. This is a well-known consequence of Chevalley’s theorem on units (cf. [De2, Cor. 2.0.12]). According to [De1, Prop. 1.15], the map i is an open immersion. It is enough, therefore, to show that for any pair of open compact subgroups K_{1}^{∗} ⊂ G^{∗}(Q̂), K_{1} ⊂ G(Q̂) such that K_{1}^{∗} ⊂ K_{1} there is a smaller pair (K_{2}^{∗}, K_{2}) ⊂ (K_{1}^{∗}, K_{1}) for which the following diagram can be completed by a diagonal morphism making the two triangles commutative.

\xymatrix{(K_{2}^{∗} ∩ G^{∗}(Q)_{+})\(𝒳 ^{∗})^{+} \ar[r] \ar[d] & (K_{2} ∩ G(Q)_{+})\𝒳 ^{+} \ar[d] \ar@{-->}[dl] \\ (K_{1}^{∗} ∩ G^{∗}(Q)_{+})\(𝒳 ^{∗})^{+} \ar[r] & (K_{1} ∩ G(Q)_{+})\𝒳 ^{+}}

Above, G(Q)_{+} = G(Q) ∩ G(R)_{+} (resp. G^{∗}(Q)_{+} = G^{∗}(Q) ∩ G^{∗}(R)_{+}), where G(R)_{+} (resp. G^{∗}(R)_{+}) is the subgroup of elements of G(R) (resp. of G^{∗}(R)) whose reduced norm lies in (R_{+}^{×})^{Hom(F,R)} (resp. in the diagonally embedded R_{+}^{×}). In concrete terms, it is enough to find (K_{2}^{∗}, K_{2}) such that

K_{2} ∩ G(Q)_{+} ⊂ (K_{1}^{∗} ∩ G^{∗}(Q)_{+})Z(R),

where Z ⊂ G is the centre. We can assume that

K_{1} ∩ G(Q) ⊂ (1 + MO_{B}) ∩ O_{B}^{×}, K_{1}^{∗} ∩ G^{∗}(Q) ⊃ (1 + M′O_{B})^{Nrd=1}

for some O_{F} -order O_{B} ⊂ B and integers 3 | M | M′. According to [C, Thm. 1] there exists an integer N ≥ 1 such that (1 + M′NO_{F}) ∩ O_{F}^{×} ⊂ (O_{F}^{×} ∩ (1 + M′O_{F}))^{2}. Taking K_{2} = (1 + M′NÔ_{B}) ∩ K_{1} and K_{2}^{∗} = K_{2} ∩ K_{1}^{∗}, the reduced norm of any a ∈ K_{2} ∩ G(Q) ⊂ 1 + M′NO_{B} is of the form Nrd(a) = u^{−2} for some u ∈ O_{F}^{×} ∩ (1 + M′O_{F}). It follows that

a = (au)u^{−1} ∈ (1 + M′NO_{B})^{Nrd=1}Z(R) ⊂ (K_{1}^{∗} ∩ G^{∗}(Q)_{+})Z(R),

as required.

In fact, a more sophisticated version of the above argument [TX, Lemma 2.5] shows that one can choose K_{2} and K_{2}^{∗} so that Sh_{K_{2}^{∗}}(G^{∗}, 𝒳 ^{∗})^{an,+} = Sh_{K_{2}}(G, 𝒳 )^{an,+}. However, we are not going to use this refined statement.

(5.8) Corollary. (1) There is a G(Q̂) -equivariant isomorphism Sh(G^{∗}, 𝒳 ^{∗})×^{U} G(Q̂) ≃ Sh(G, 𝒳 ), where U ⊂ G(Q̂) = D̂^{×} is the stabiliser of Sh(G^{∗}, 𝒳 ^{∗}).

(2) For any ξ as in 5.5 there is a Γ_{Q} × G(Q̂)-equivariant isomorphism

H_{et}^{j}(Sh(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) ≃ Ind_{U}^{G(_{Q})} H_{et}^{j}(Sh(G^{∗}, 𝒳 ^{∗}) ⊗_{E} \overline{Q}, i^{∗}ℒ_{ξ,ℓ}),

with smooth induction on the right hand side.

(5.9) If D ≄ M_{2}(F), then Sh_{K}(G, 𝒳 )^{an} is compact (for each open compact subgroup K ⊂ G(Q̂)) and the formula (0.4.1) applies (with m(π) = 1 in (0.3.2), by the multiplicity one theorem for automorphic representations of D_{A}^{×}):

H^{i} = H_{et}^{i}(Sh(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) = ⨁_{π^{∞}} V^{i}(π^{∞}) ⊗ π^{∞},  (5.9.1)

where π_{∞} ⊗ π^{∞} is an automorphic representation of G(A) = D_{A}^{×} such that

∀v ∈ Ω H^{∗}(𝔤l_{2}, O(2)R^{×}; π_{v} ⊗ ξ_{v}) ≠ 0  (5.9.2)

and

∀v ∈ Ω^{c} π_{v} ≃ ξ_{v}^{∨}.  (5.9.3)

If D ≃ M_{2}(F), then Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) and Sh_{K}(G, 𝒳 ) are Hilbert modular varieties and the formula (5.9.1) applies to the intersection cohomology of the Baily-Borel compactification j : Sh_{K}(G, 𝒳 ) ↪ Sh_{K}(G, 𝒳 )_{BB} = Sh_{K}(G, 𝒳 ) ∪ {cusps}:

H^{i} = \underrightarrow{lim}_{K} (H^{i})^{K} = \underrightarrow{lim}_{K} H_{et}^{i}(Sh_{K}(G, 𝒳 )_{BB} ⊗_{Q} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ}) = \underrightarrow{lim}_{K} H^{i}(Sh_{K,BB}),

with π = π_{∞} ⊗ π^{∞} being a discrete ( ⟺ cuspidal or one-dimensional) automorphic representation of GL(A_{F}) [BC]. As in the case D ≄ M_{2}(F), the Galois representation (H^{i})^{K} is pure of weight i + t(w − 2) at all unramified primes not dividing ℓ.

The canonical map (H^{i})^{K} = H^{i}(Sh_{K,BB}) ⟶ H^{i}(Sh_{K}) = H_{et}^{i}(Sh_{K}(G, 𝒳 ) ⊗_{Q} \overline{Q}, ℒ_{ξ,ℓ}) is almost always injective (see Proposition A6.17 below), the only exception being the case i = 2r, k_{v} = 2 for all v | ∞, when H^{2r}(Sh_{K,BB}) is dual to a Tate twist of H^{0}(Sh_{K,BB}) = H^{0}(Sh_{K}). (5.10) For fixed v ∈ Ω, the condition (5.9.2) can be made explicit as follows.

(5.10.1) If dim(π_{v}) < ∞, then necessarily dim(π) = 1 and π = χ ∘ Nrd for some χ : A_{F}^{×}/F^{×} ⟶ C^{×}. The central and infinitesimal characters of ξ_{v} ⊗ (χ_{v} ∘ Nrd) must be trivial [BW, Thm. I.5.3], hence k_{v} = 2 and χ_{v}^{2}(a)a^{2−w} = 1 for all a ∈ F_{v}^{×} = R^{×}, which implies that χ∥ · ∥_{F}^{1−w/2} is a character of finite order. In this case

dim H^{i}(𝔤l_{2}, O(2)R^{×}; π_{v} ⊗ ξ_{v}) = \{ \begin{matrix}1, & i = 0, 2 \\ 0, & otherwise. \end{matrix}

(5.10.2) If dim(π_{v}) = ∞, then the matching of central and infinitesimal characters of π_{v} and the dual of ξ_{v} [BW, Thm. I.5.3] implies that π_{v} is a discrete series representation of weight k_{v} (and appropriate central character). In this case

dim H^{i}(𝔤l_{2}, O(2)R^{×}; π_{v} ⊗ ξ_{v}) = \{ \begin{matrix}2, & i = 1 \\ 0, & otherwise. \end{matrix}

(5.11) Combining 5.10 with (5.9.3) and the strong multiplicity one theorem for automorphic representations of D_{A}^{×} (specifically, the fact that m(π) = 1 and π is determined by π^{∞}), we deduce that V^{i}(π^{∞}) ≠ 0 precisely in the following two mutually exclusive cases.

(A) π = χ∘Nrd, where χ : A_{F}^{×}/F^{×} ⟶ C^{×} is a character such that χ∥·∥_{F}^{1−w/2} is of finite order, dim(ξ) = 1 ( ⟺ ∀v|∞ k_{v} = 2) and

dim V^{i}(π^{∞}) = \{ \begin{matrix}\binom{t}{j}, & i = 2j (0 ≤ j ≤ t) \\ 0, & otherwise. \end{matrix}

This corresponds to the universal cohomology classes given by the cohomology of the dual compact symmetric space P^{1}(C)^{t}.

(B) π corresponds by the Jacquet-Langlands correspondence to a representation Π = JL(π) of GL_{2}(A_{F}) attached (up to a twist) to a holomorphic cuspidal Hilbert modular newform ϕ (this still holds in the case D ≃ M_{2}(F) when Π = π, since π was necessarily cuspidal). In this case

V (π^{∞}) = V^{t}(π^{∞}), dim V (π^{∞}) = 2^{t}.

In the case (A) we consider χ∥ · ∥_{F}^{1−w/2} as a Galois character of finite order Γ_{F} = Gal(\overline{Q}/F) ⟶ C^{×} ≃ \overline{Q}_{ℓ}^{×}, via the reciprocity map rec_{F} : A_{F}^{×}/F^{×} ⟶ Γ_{F}^{ab}. By abuse of language, we denote by χ : Γ_{F} ⟶ \overline{Q}_{ℓ}^{×} the tensor product of this Galois character of finite order by \overline{Q}_{ℓ}(w/2−1) (this is the ℓ-adic Galois representation attached to the algebraic Hecke character χ, [Sc, ch. 0, §5], [HT, p. 20]).

In the case (B) denote by ρ_{π} = ρ_{Π} : Γ_{F} ⟶ GL_{2}(\overline{Q}_{ℓ}) the Galois representation attached to Π. It is irreducible, unramified outside ℓ cond(Π) and satisfies

P_{ρ_{π}(Fr(P))}(X) = X^{2} − a_{P} X + ω(P)(NP)  (5.11.1)

for all finite primes P ∤ ℓ cond(Π) of F. Above, ω denotes the central character of Π (hence also of π) and a_{P} (resp. ω(P)) is the eigenvalue of the Hecke operator T_{P} (resp. S_{P}) acting on the spherical vector of π_{P} = Π_{P} (see A1.6 below). The character ω satisfies ω_{v}(a)a^{2−w} = 1 for all v | ∞, which implies that ω∥ · ∥_{F}^{2−w} has finite order. The Galois representation ρ_{π} is pure of weight 3 − w at all P ∤ ℓ cond(Π).

The Langlands-Kottwitz method yields the following information ([La], [VSL], [R], [BL]). (5.11.2) For π = χ ∘ Nrd as in (A) the dual representation is π^{∨} = χ^{−1} ∘ Nrd and

V^{2j}(π^{∞})^{ss} ≃ (⋀^{j} Ind_{Ω} \overline{Q}_{ℓ}(−1)) ⊗ det(Ind_{Ω}(χ^{−1})) = (⋀^{j} Ind_{Ω} \overline{Q}_{ℓ}(−1)) ⊗ Ind_{Ω}^{⊗}(χ^{−1}).

(5.11.3) If JL(π) = Π is attached to π as in (B), then

V^{t}(π^{∞})^{ss} ≃ Ind_{Ω}^{⊗}(ρ_{π^{∨}}),

where the dual representation π^{∨} satisfies JL(π^{∨}) = Π^{∨} ≃ Π ⊗ ω^{−1} and ρ_{π^{∨}} ≃ ρ_{π}^{∨}(−1) (cf. (A5.6.3); this Galois representation is pure of weight w − 1).

Had we used h^{−1} instead of h, then ρ_{π^{∨}} would have to be replaced by ρ_{π}. (5.12) Partial (tensor) induction. In (5.11.2-3) we have denoted by Ind_{Ω} and Ind_{Ω}^{⊗}, respectively, the partial induction and partial tensor induction functors which associate to a representation of Γ_{F} of dimension m a representation of Γ_{E} of dimension tm (resp. m^{t}). They are defined as follows.

The set X = Hom(F, \overline{Q}) = Hom(F, R) = Ω ∪ Ω^{c} is naturally identified with Γ_{Q}/Γ_{F}. A choice of a section s : X ⟶ Γ_{Q} of the canonical projection defines an injective group morphism

i_{s} : Γ_{Q} ↪ S_{X} ⋉ Γ_{F}^{X},  i_{s}(γ) = (σ, δ),  γ(s(x)) = s(σ(x))δ(x).

By definition,

Γ_{E} = i_{s}^{−1}((S_{Ω} ⋉ Γ_{F}^{Ω}) × (S_{Ω^{c}} ⋉ Γ_{F}^{Ω^{c}})).

Let M be any A[Γ_{F}]-module (where A is an arbitrary commutative ring). The wreath product S_{Ω} ⋉ Γ_{F}^{Ω} acts naturally on M^{⊕Ω} and M^{⊗Ω}. We let S_{Ω^{c}} ⋉ Γ_{F}^{Ω^{c}} act trivially and define

Ind_{Ω}(M) = i_{s}^{∗}(M^{⊕Ω}), Ind_{Ω}^{⊗}(M) = i_{s}^{∗}(M^{⊗Ω}).

The isomorphism classes of these two Γ_{E}-modules do not depend on s.

For γ ∈ Γ_{E′} the image of i_{s}(γ) in S_{Ω} ⋉ Γ_{F}^{Ω} lies in Γ_{F}^{Ω}; it is equal to (s(x)^{−1}γs(x))_{x∈Ω}. It follows that

Ind_{Ω}(M)|_{Γ_{E′}} ≃ ⨁_{x∈Ω}^{s(x)}M,  Ind_{Ω}^{⊗}(M)|_{Γ_{E′}} ≃ ⨂_{x∈Ω}^{s(x)}M,  (5.12.1)

where we have denoted by ^{s(x)}M the pull-back of M via the map

int(s(x))^{−1} : Γ_{E′} ⟶ s(x)^{−1}Γ_{E′}s(x) ⊂ Γ_{F} , γ ↦ s(x)^{−1}γs(x).

(5.13) It follows from (5.12.1) and (5.11.2-3) that

V^{2j}(π^{∞})^{ss}|_{Γ_{E′}} ≃ (\overline{Q}_{ℓ}(−j) ⊗ ⨂_{x∈Ω}^{s(x)}χ^{−1})^{⊕\binom{t}{j}}  (5.13.1)

in the case (A) and

V^{t}(π^{∞})^{ss}|_{Γ_{E′}} ≃ ⨂_{x∈Ω}^{s(x)}ρ_{π^{∨}}  (5.13.2)

in the case (B).

The finiteness of the class number implies that H_{g}^{1}(k, Q_{ℓ}) = 0 for any number field k. As a result, any representation σ : Γ_{k} ⟶ GL_{n}(\overline{Q}_{ℓ}) which is de Rham at all primes above ℓ and for which σ^{ss} ≃ α^{⊕n} for some character α : Γ_{k} ⟶ \overline{Q}_{ℓ}^{×} is necessarily semisimple, σ = σ^{ss}. In particular, (5.13.1) together with 3.2 imply that, in the case (A), each V^{2j}(π^{∞}) is semisimple. Consequently, we can replace V^{2j}(π^{∞})^{ss} by V^{2j}(π^{∞}) in (5.11.2). (5.14) In 5.11-13 we have summed up the information about the Galois representations V^{i}(π^{∞}) which can be obtained by the Langlands-Kottwitz method. In fact, this method yields (5.11.3) in a form which does not assume the existence of the Galois representation ρ_{π^{∨}}.

We are now going to revisit the representations V^{i}(π^{∞})|_{Γ_{E′}}, by applying Eichler-Shimura relations together with the abstract results of §3 and §4 (but, unlike in the Langlands-Kottwitz method, assuming the existence of ρ_{π^{∨}}).

(5.15) Fix π as in 5.11, i.e., assume that V^{i}(π^{∞}) ≠ 0. Fix a neat open compact subgroup K ⊂ G(Q̂) such that (π^{∞})^{K} ≠ 0. This implies that

0 ≠ V^{i}(π^{∞}) ⊗ (π^{∞})^{K} ⊂ H_{et}^{i}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}).  (5.15.1)

There exists a finite set S of primes of F such that K = K_{S}K^{S}, where K^{S} = ∏_{v∉S} K_{v} with (K_{v}, G(F_{v})) ≃ (GL_{2}(O_{F,v}), GL_{2}(F_{v})) for all v ∉ S. We can, and will, assume that S contains all infinite primes, all primes dividing 2ℓ and all primes at which F/Q and D are ramified. Denote by Q_{S}/Q the maximal subextension of \overline{Q}/Q which is unramified at all rational primes not lying below S. Note that Q_{S} ⊃ F^{gal}.

Fix an intermediate field E′ ⊂ Ẽ ⊂ F^{gal}. We are going to consider primes P_{S} of Q_{S} which are unramified in Q_{S}/Q and which satisfy

Fr_{Q_{S}/Q}(P_{S}) ∈ Gal(Q_{S}/Ẽ).  (5.15.2)

Denote by

p = P_{S} ∩ Z,  P = P_{S} ∩ O_{E},  P̃ = P_{S} ∩ O_{E}

the respective primes of Q, E and Ẽ below P_{S}. It follows from the condition (5.15.2) that

Q_{p} = E_{P} = Ẽ_{P}.

Moreover, for each x ∈ Ω

s(x)^{−1}Fr_{Q_{S}/Q}(P_{S})s(x) ∈ s(x)^{−1}Gal(Q_{S}/Ẽ)s(x) ⊂ s(x)^{−1}Gal(Q_{S}/E′)s(x) ⊂ Gal(Q_{S}/F),

which implies that the following primes of F

P_{x} = s(x)^{−1}P_{S} ∩ O_{F}  (x ∈ Ω)  (5.15.3)

are distinct and satisfy F_{P_{x}} = Q_{p}. (5.16) Eichler-Shimura relations. The following statements are discussed in A6 below.

In the situation of 5.15, the Shimura variety Sh_{K}(G, 𝒳 ) has (for sufficiently small K^{p}) a canonical model S_{K} over O_{E,P} = Z_{p}. Denote by S_{K}^{∘} = S_{K} ⊗_{Z_{p}} F_{p} its special fibre.

The Frobenius morphism φ : S_{K}^{∘} ⟶ S_{K}^{∘} has degree deg(φ) = p^{t} and the action of Fr(P_{S}) on H^{i}(Sh_{K}) = H_{et}^{i}(Sh_{K}(G, 𝒳 )⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) ≃ H_{et}^{i}(S_{K}^{∘} ⊗_{F_{p}} \overline{F}_{p}, ℒ_{ξ,ℓ}^{∘}) is given by the action (φ⊗id)^{∗} of φ⊗id : S_{K}^{∘} ⊗_{F_{p}} \overline{F}_{p} ⟶ S_{K}^{∘} ⊗_{F_{p}} \overline{F}_{p}.

A cohomological form of Eichler-Shimura relations in our situation asserts that (φ⊗id)^{∗} ∈ Aut_{\overline{Q}_{ℓ}}(H^{i}(Sh_{K})) naturally decomposes as a product of pairwise commuting cohomological partial Frobenius automorphisms

(φ ⊗ id)^{∗} = ∏_{x∈Ω} φ_{x}^{∗},  φ_{x}^{∗} ∈ Aut_{\overline{Q}_{ℓ}}(H^{i}(Sh_{K})),  φ_{x}^{∗}φ_{y}^{∗} = φ_{y}^{∗}φ_{x}^{∗}

satisfying the following quadratic equations, which generalise (0.8.1):

∀x ∈ Ω Q_{x}(φ_{x}^{∗}) = 0,  Q_{x}(Y) = Y^{2} − (T_{P_{x}}/S_{P_{x}})Y + p/S_{P_{x}}.  (5.16.1)

The relations (5.16.1) are proved in A6.4 and A6.19 (the case t = r) and A6.14 (the case t < r) below for Ẽ = F^{gal}.

In the totally indefinite case t = r (see A6.4) the automorphisms φ_{x}^{∗} are given by the action of φ_{x} ⊗ id on H^{i}(Sh_{K}), where φ_{x} : S_{K}^{∘} ⟶ S_{K}^{∘} are mutually commuting (φ_{x}φ_{y} = φ_{y}φ_{x}) geometric partial Frobenius morphisms of degree deg(φ_{x}) = NP_{x} = p. In the case t < r we construct geometric partial Frobenius morphisms on the special fibre of a closely related unitary Shimura variety, and then transfer the corresponding cohomological partial Frobenius automorphisms to the quaternionic Shimura variety (see A6.13).

The relations (5.16.1) imply that the action of Fr(P_{S}) = (φ ⊗ id)^{∗} = ∏_{x∈Ω} φ_{x}^{∗} on H^{i}(Sh_{K}) satisfies the following equation of degree 2^{t}:

Q(φ^{∗}) = 0,  Q = ⨂_{x∈Ω} Q_{x},  (5.16.2)

where ⨂ is the “Rankin-Selberg product”: writing formally Q_{x}(Y) = (Y − α_{x,1})(Y − α_{x,2}), then

Q(Y) = ∏_{i:Ω⟶{1,2}}(Y − ∏_{x∈Ω} α_{x,i(x)}).  (5.16.3)

In the essentially PEL case t = r (when E = Q and E′ = Ẽ = F^{gal}) the formula (5.16.2) follows from [Mo, Cor. 4.2.15] and [W, Theorem, p. 44] (taking into account Corollary 5.8(2)).

In the isotropic case D ≃ M_{2}(F) the relations (5.16.1-2) also hold for the action of the φ_{x}^{∗} on H^{i} = H^{i}(Sh_{K,BB}), in the notation of 5.9 (see A6.19). (5.17) For π = χ ∘ Nrd in the case (A), the formulas (5.16.1) and (5.16.2) imply, respectively (thanks to (5.15.1)) that

∀x ∈ Ω  (φ_{x}^{∗} − p χ^{−1}(P_{x}))(φ_{x}^{∗} − χ^{−1}(P_{x}))|_{V^{i}(π^{∞})⊗(π^{∞})^{K}} = 0  (5.17.1)

resp.

∏_{I⊂Ω}(Fr(P_{S}) − p^{|I|}χ^{−1}( ∏_{x∈Ω} P_{x}))|_{V^{i}(π^{∞})⊗(π^{∞})^{K}} = 0.  (5.17.2)

(5.18) Theorem. For π = χ∘Nrd in the case (A), the relation (5.16.2) (which holds for Ẽ = F^{gal}, thanks to [Mo] and [W] in the case t = r and (A6.14.3) in the case t < r) implies (5.13.1). In other words, the Eichler-Shimura relation for the usual Frobenius morphism (for P_{S} as in (5.15.2)) determines the isomorphism class of each V^{i}(π^{∞})|_{Γ_{E}} (thanks to the remarks at the end of 5.13).

Proof. As H^{2j} is pure of weight 2j + t(w − 2) and χ∥ · ∥_{F}^{1−w/2} is of finite order, it follows from (5.17.2) that, for each P_{S} satisfying (5.15.2), Fr(P_{S}) acts on V^{2j}(π^{∞})^{ss} by the scalar p^{j}χ^{−1}(∏_{x∈Ω} P_{x}). The set of elements Fr(P_{S}) is dense in Gal(Q_{S}/Ẽ), which implies that

V^{2j}(π^{∞})^{ss}|_{Γ_{E}} ≃ (\overline{Q}_{ℓ}(−j) ⊗ ⨂_{x∈Ω}^{s(x)}χ^{−1})^{⊕\binom{t}{j}},

as claimed. Note that one can rewrite the above formula in a more succint form as

V (π^{∞})^{ss}|_{Γ_{E}} ≃ (χ^{−1} ⊗ (\overline{Q}_{ℓ} ⊕ \overline{Q}_{ℓ}(−1)))^{⊗t}|_{Γ_{E}}.

As remarked in 5.13, the finiteness of the class number implies that each V^{2j}(π^{∞})^{ss} = V^{2j}(π^{∞}) is semisimple. (5.19) If we are in the case (B) and JL(π) = Π, write Ω = \{x_{1}, …, x_{t}\} and ρ_{i} = ^{s(x_{i})}ρ_{π^{∨}} : Γ_{E′} ⟶ GL_{2}(\overline{Q}_{ℓ}) (1 ≤ i ≤ t). Denote by ρ : Γ_{E′} ⟶ GL_{2^{t}}(\overline{Q}_{ℓ}) the action of Γ_{E′} on V^{t}(π^{∞}).

The relation (5.16.1) implies (again thanks to (5.15.1)) that, for each P_{S} satisfying (5.15.2),

ρ(Fr(P_{S})) = u_{1} ⋯ u_{t},  u_{i} = φ_{x_{i}}^{∗},  u_{i}u_{j} = u_{j}u_{i},  P_{ρ_{i}(Fr(P_{S}))}(u_{i}) = 0.  (5.19.1)

Similarly, the relation (5.16.2) implies that

P_{(ρ_{1}⊗···⊗ρ_{t})(Fr(P_{S}))} (ρ(Fr(P_{S}))) = 0,  (5.19.2)

hence

∀g ∈ Γ_{E}  P_{(ρ_{1}⊗···⊗ρ_{t})(g)} (ρ(g)) = 0,  (5.19.3)

by the Čebotarev density theorem.

(5.20) Theorem. Assume that we are in the case (B) and JL(π) = Π.

(1) The relation (5.16.2) implies that there exists a finite extension E′′/Ẽ and an integer m ≥ 1 such that

V^{t}(π^{∞})^{ss}|_{Γ_{E′′}} ⊂ (⨂_{x∈Ω}^{s(x)}ρ_{π^{∨}})|_{Γ_{E′′}}^{⊕m} .

If, in addition, ϕ has complex multiplication by a totally imaginary quadratic extension M of F, then E′′ satisfies M^{gal} ⊄ (M^{gal})^{+}E′′, where M^{gal} denotes the Galois closure of M in \overline{Q} and (M^{gal})^{+} the maximal totally real subfield of the CM field M^{gal}.

(2) If ϕ does not have complex multiplication and if the weights (k_{x})_{x∈Ω} are distinct, then the representation ⨂_{x∈Ω} ^{s(x)}ρ_{π^{∨}} of Γ_{E′} is strongly irreducible and the relation (5.16.2) implies that (5.13.2) holds after restricting to Γ_{E}.

(3) The relation (5.16.1) implies that the representation V^{t}(π^{∞}) is semisimple.

Proof. We must distinguish two cases.

If ϕ has complex multiplication, then ρ_{π^{∨}} = Ind_{Γ_{M}}^{Γ_{F}} (α) = I(α), where M ⊂ \overline{Q} is a totally imaginary

quadratic extension of F and α : Γ_{M} ⟶ \overline{Q}_{ℓ}^{×} a character. Let M′ be the Galois closure of M in \overline{Q}. It is also a CM field and its maximal totally real subfield F′ is a Galois extension of Q containing F^{gal} (hence Ẽ). Using the notation from 5.19, we have, for each i = 1, …, t,

ρ_{i}|_{Γ_{F′}} = Ind_{Γ_{M′}}^{Γ_{F′}} (α_{i}), α_{i} = ^{s(x_{i})}(α|_{Γ_{M′}} ).

Thanks to (5.19.3) (resp. (5.19.1)), Proposition 4.6 (resp. Theorem 4.7(2)) applies to the restrictions of ρ and ρ_{i} to Γ_{F′} (note that n = 2, σ̃ = c is the complex conjugation and the character α_{i}/α_{i}^{c} has infinite order, since the Hodge-Tate weights of α and α^{c} are distinct). The statement (1) (resp. (3)) of the theorem follows.

If ϕ has no complex multiplication, then \overline{Q}_{ℓ} · Lie(ρ_{π^{∨}}(Γ_{F})) = 𝔤l_{2}(\overline{Q}_{ℓ}) and ρ_{π^{∨}} is strongly irreducible. Thanks to (5.19.3) (resp. (5.19.1)), Proposition 3.10(1) (resp. Theorem 3.12(2)) applies to the representations ρ = V^{t}(π^{∞}) and ρ_{i} of Γ = Γ′ = Gal(Q_{S}/Ẽ) and to Σ = {Fr(P_{S})}. The statement (1) (resp. (3)) of the theorem follows. The statement (2) will follow from Proposition 3.10(2) once we show that ⨂_{x∈Ω} ^{s(x)}ρ_{π^{∨}} = ρ_{1} ⊗ ⋯ ⊗ ρ_{t} is strongly irreducible.

Fix 1 ≤ i ≠ j ≤ t. The \overline{Q}_{ℓ}-Lie algebra

𝔤 = \overline{Q}_{ℓ} · Lie((ρ_{i} ⊕ ρ_{j})(Γ)) ⊂ \overline{Q}_{ℓ} · Lie(ρ_{i}(Γ)) ⊕ \overline{Q}_{ℓ} · Lie(ρ_{j}(Γ)) = 𝔤l_{2}(\overline{Q}_{ℓ}) × 𝔤l_{2}(\overline{Q}_{ℓ})

satisfies p_{1}(𝔤) = p_{2}(𝔤) = 𝔤l_{2}(\overline{Q}_{ℓ}). If the desired strong irreducibility statement does not hold, the last part of Proposition 2.2 implies (together with Proposition 2.1) that 𝒟(𝔤) ⊂ 𝔰l_{2}(\overline{Q}_{ℓ})×𝔰l_{2}(\overline{Q}_{ℓ}) is the graph of a Lie algebra isomorphism 𝔰l_{2}(\overline{Q}_{ℓ}) ⟶^{∼} 𝔰l_{2}(\overline{Q}_{ℓ}). Every automorphism of 𝔰l_{2}(\overline{Q}_{ℓ}) is inner, which means that, after conjugating ρ_{j} by a suitable matrix A ∈ GL_{2}(\overline{Q}_{ℓ}), 𝒟(𝔤) will coincide with the diagonally embedded 𝔰l_{2}(\overline{Q}_{ℓ}). Moreover, the determinants of ρ_{i} and ρ_{j} differ by a character of finite order, which implies that 𝔤 itself coincides with the diagonally embedded 𝔤l_{2}(\overline{Q}_{ℓ}). As a result, the restrictions of ρ_{i} and ρ_{j} to an appropriate open subgroup U ⊂ Γ are isomorphic (this is equivalent, by Schur’s Lemma, to the existence of a character of finite order α : Γ ⟶ \overline{Q}_{ℓ}^{×} such that ρ_{i} ≃ ρ_{j} ⊗ α, but we do not need this fact). This is impossible, since the Hodge-Tate weights of ρ_{i} (with respect to E′ ⊂ \overline{Q} ⊂ C ≃ \overline{Q}_{ℓ}) are equal to (1 − (w + k_{x_{i}})/2, (k_{x_{i}} − w)/2), hence are different from those of ρ_{j}. This contradiction concludes the proof of (2).

(5.21) Remarks. (1) The proof of Theorem 5.20(2) shows that the conclusion holds more generally, namely, if we assume that ϕ has no complex multiplication and ^{s(x)}ρ_{π^{∨}} ≄ ^{s(y)}ρ_{π^{∨}} ⊗ α for any x ≠ y ∈ Ω and any character of finite order α of Γ_{E′}.

(2) Is it possible to deduce from Theorem 5.20(2) the full statement of the restriction of (5.13.2) to Γ_{E} (for ϕ without complex multiplication) by letting ϕ vary in a ℓ-adic family?

(5.22) Corollary. For every ξ as in 5.5, the action of Γ_{E} on H_{et}^{i}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) in the case D ≄ M_{2}(F) (resp. on H_{et}^{i}(Sh_{K}(G, 𝒳 )_{BB} ⊗_{Q} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ}) in the case D ≃ M_{2}(F)) is semisimple, and the same result holds for Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) (and i^{∗}ℒ_{ξ,ℓ})) if D ⊗ R ≃ M_{2}(R)^{r}.

6. Cohomology of quaternionic Shimura varieties (bis)

(6.1) In this section we investigate the cohomology of Sh(G^{∗}, 𝒳 ^{∗}) with coefficients in local systems that do not come from Sh(G, 𝒳 ). The notation is as in §5. We assume throughout §6 that D ⊗ R ≃ M_{2}(R)^{r} (i.e., that Ω = X, t = r).

(6.2) An irreducible algebraic representation ξ^{∗} of G_{C}^{∗} is a restriction to G_{C}^{∗} of a representation ⨂_{v∈X} ξ_{v} of G_{C}, where ξ_{v} = Sym^{k_{v}−2}(Std^{∨}) ⊗ (det ∘ Std^{∨})^{m_{v}} : GL(2)_{C} ⟶ GL(k_{v} − 1)_{C} (k_{v} ≥ 2, m_{v} ∈ Z). The corresponding ℓ-adic local system ℒ_{ξ^{∗},ℓ} = ℒ_{\underline{k},\underline{m}} on Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) (for small enough K^{∗} ⊂ G^{∗}(Q̂)) is pure of weight w^{∗} = ∑_{v∈X}(k_{v} − 2 + 2m_{v}) and satisfies ℒ_{\underline{k},\underline{m}} = ℒ_{\underline{k},0}(− ∑_{v∈X} m_{v}). This implies that ℒ_{\underline{k},\underline{m}} is a Tate twist of i^{∗}ℒ_{ξ,ℓ} for some ξ as in (5.5.1) iff

∀v, v′ ∈ X  k_{v} ≡ k_{v′} (mod 2)  (6.2.1)

(in other words, iff \underline{k} = (k_{v})_{v∈X} is a “motivic weight” in the language of [BR]). (6.3) We fix \underline{k} and \underline{m} = (m_{v})_{v∈X} and write H^{i} = H_{et}^{i}(Sh(G^{∗}, 𝒳 ^{∗}) ⊗_{Q} \overline{Q}, ℒ_{\underline{k},\underline{m}}) in the case D ≄ M_{2}(F) (resp. H^{i} = H_{et}^{i}(Sh(G^{∗}, 𝒳 ^{∗})_{BB} ⊗_{Q} \overline{Q}, j_{!∗}ℒ_{\underline{k},\underline{m}}) in the case D ≃ M_{2}(F)).

If the weight \underline{k} is motivic, Corollary 5.22 implies that H^{i} is a semisimple Γ_{Q}-module. If \underline{k} is not motivic, we are going to prove in Corollary 6.20 below an analogous result for the non CM part of H^{i}. The techniques of §5 do not apply in this case directly, only after a passage to an auxiliary totally imaginary quadratic extension F_{c} of F. For a good prime p that splits completely in F_{c}/F one cannot define the partial Frobenius morphism φ_{x} on the special fibre of Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) at p, only a certain twist of its square φ_{x}^{2}. Similarly, there is no Galois representation of Γ_{F} attached to a Hilbert modular form of non-motivic weight involved in the decomposition of H^{r}, but there is a Galois representation of Γ_{F_{c}} attached to a suitable twist of the base change of the Hilbert modular form to F_{c}, by [BR]. Working over F_{c} with such twisted objects leads to a proof of a variant of Theorem 5.20, but only in the non CM case (see Theorem 6.19 below). Instead of the Eichler-Shimura relation for the action of φ_{x} on the cohomology of Sh_{K}(G, 𝒳 ) given in A6.4 below we use the results of [Mo] and [W], which apply to the action of the twisted version of φ_{x}^{2}. This method also works for motivic weights; it reproves the non CM case of Theorem 5.20 for t = r. (6.4) As in §5, there is a Γ_{Q} × G^{∗}(Q̂)-equivariant decomposition

H^{i} = ⨁_{(π^{∗})^{∞}} V^{i}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞},

where π^{∗} = π_{∞}^{∗} ⊗ (π^{∗})^{∞} is an automorphic representation of G^{∗}(A) such that π_{∞}^{∗} is cohomological for ξ^{∗} in degree i and π^{∗} is one-dimensional or cuspidal (which is automatic if D ≄ M_{2}(F)). We are going to investigate the ℓ-adic representation V^{i}((π^{∗})^{∞}) of Γ_{Q}, which is pure of weight i + w^{∗} at all unramified primes. As in 5.11, V^{i}((π^{∗})^{∞}) ≠ 0 only in the following two cases. (6.5) Case (A): dim(π^{∗}) = 1. In this case π^{∗} = χ^{∗} ∘ Nrd, χ^{∗} : A^{×}/Q^{×} ⟶ C^{×}, k_{v} = 2 for all v ∈ X, ℒ_{ξ^{∗},ℓ} = \overline{Q}_{ℓ}(−m) = (i^{∗}ℒ_{ξ,ℓ})(−m), m = ∑_{v} m_{v}, ξ is the trivial representation of G_{C} and χ_{∞}^{∗}(a) = a^{2m}.

We can assume that m_{v} = 0 for all v ∈ X; then χ^{∗} can be identified (via the reciprocity map) with a Galois character of finite order χ^{∗} : Γ_{Q} ⟶ C^{×} ≃ \overline{Q}_{ℓ}^{×}. The arguments from 5.18 show that the Eichler-Shimura relation (proved in [Mo] and [W]) for the usual Frobenius acting on H^{i} (at good primes that split completely in F/Q) implies that V^{i}((π^{∗})^{∞}) = 0 if i ∉ {0, 2, …, 2r} and

V^{2j}((π^{∗})^{∞})|_{Γ_{F^{gal}}} ≃ \overline{Q}_{ℓ}(−j) ⊗ (χ^{∗})^{−1}|_{Γ_{F^{gal}}}^{⊕m_{j}}  (0 ≤ j ≤ r).  (6.5.1)

(6.6) Case (B): dim(π^{∗}) = ∞, π^{∗} cuspidal.

As the restriction of π_{∞}^{∗} to SL_{2}(F ⊗ R) is of infinite dimension and cohomological in degree i for ξ^{∗}, we have necessarily that i = r and π_{∞}^{∗}|_{SL_{2}(F ⊗R)} is a direct sum of tensor products ⨂_{v∈X} of (holomorphic or antiholomorphic) discrete series representations of weight k_{v} of SL_{2}(F_{v}) = SL_{2}(R).

The central character ω_{π^{∗}} : A^{×}/Q^{×} ⟶ C^{×} of π^{∗} satisfies (ω_{π^{∗}})_{∞} = ω_{ξ^{∗}}^{−1} : a ↦ a^{w^{∗}}, which implies that π^{∗}(−w^{∗}/2) = π^{∗} ⊗ (∥ · ∥_{Q} ∘ Nrd)^{−w^{∗}/2} has central character of finite order. As in [LSc, Prop. 3.5], every cuspidal automorphic form on G^{∗}(A) extends to a cuspidal automorphic form on G(A) = D_{A}^{×}. As a result, there exists a cuspidal automorphic representation π of D_{A}^{×} with central character ω_{π} of finite order such that π^{∗}(−w^{∗}/2) is isomorphic to a quotient of the restriction of π to G^{∗}(A). Fix such a π.

For each v ∈ X, π_{v} is a discrete series representation of weight k_{v} and central character ω_{π_{v}} = (sgn)^{k_{v}} of GL_{2}(R). The Jacquet-Langlands transfer JL(π) of π to GL_{2}(A_{F}) is cuspidal, since π is cuspidal in the case D ≃ M_{2}(F).

(6.7) The representation JL(π) corresponds, up to a twist, to a cuspidal holomorphic Hilbert modular newform ϕ of weight \underline{k} = (k_{v})_{v∈X}. Blasius and Rogawski [BR] attached compatible systems of Galois representations to suitable twists of JL(π) by Hecke characters. Their setup is the following.

Fix an auxiliary imaginary quadratic field E_{0} and let F_{c} = E_{0}F (if ϕ has complex multiplication by a totally imaginary quadratic extension M of F, assume that E_{0} ⊄ M^{gal}). Fix an embedding of E_{0} into C (as F ⊂ \overline{Q} ⊂ C by assumption, this defines a distinguished embedding F_{c} ↪ \overline{Q} ⊂ C) and denote by {σ_{x} : F_{c} ↪ C}_{x∈X} the induced CM type of F_{c}.

According to Proposition A6.15 there exists a character ψ : A_{F_{c}}^{×}/F_{c}^{×} ⟶ C^{×} satisfying

ψ|_{A_{F}^{×}} = ω_{π}^{−1}, ∀x ∈ X ψ_{x}(a) = (σ_{x}(a)/|σ_{x}(a)|)^{k_{x}};

fix such a ψ. The twisted base change

Π = BC_{F_{c}/F} (JL(π)) ⊗ ψ  (6.7.1)

is a cuspidal automorphic representation of GL_{2}(A_{F_{c}}) (since JL(π) does not have CM by F_{c}) such that

Π^{∨} ≃ Π^{c},  ω_{Π} = ψ/ψ^{c},  ∀x ∈ X (ω_{Π})_{x}(a) = (σ_{x}(a)/\overline{σ_{x}(a)})^{k_{x}}

(above, c denotes the non-trivial element of Gal(F_{c}/F) and (ψ^{c})(a) = ψ(c^{−1}(a))).

According to [BR, Thm. 2.6.1] there exists a semisimple Galois representation

ρ_{Π} = ρ_{Π,ℓ} : Γ_{F_{c}} ⟶ GL_{2}(\overline{Q}_{ℓ})

such that (note that our normalisations – including the values of (k_{x})_{x∈X} – differ from those of [BR])

L_{v}(ρ_{Π}, s) = L_{v}(Π, Std, s − 1/2)  (∀v ∤ ℓ cond(Π) cond(ψ)D_{F_{c}}).  (6.7.2)

(6.8) From now on, until 6.19, assume that V^{r}((π^{∗})^{∞}) ≠ 0. There exists an open compact subgroup K ⊂ G(Q̂) = D̂^{×} such that (π^{∞})^{K} ≠ 0 ≠ ((π^{∗})^{∞})^{K^{∗}}, where K^{∗} = K ∩ G^{∗}(Q̂); fix such a K.

Let S be a finite set of primes of F satisfying the properties listed in 5.15; we require, in addition, all primes ramified in F_{c}/F to be contained in S. Let p be a rational prime not lying below S. After shrinking K_{S} if necessary, there exists a smooth quasi-projective model S_{K^{∗}} (projective if D ≄ M_{2}(F)) of Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) over Z_{(p)} constructed in [Ko, §5] (cf. A6.3 below). As in 5.16, H^{i}(Sh_{K^{∗}}) is isomorphic to H_{et}^{i}(S_{K^{∗}}^{∘} ⊗_{F_{p}} \overline{F}_{p}, ℒ_{ξ^{∗},ℓ}^{∘}), where S_{K^{∗}}^{∘} denotes the special fibre of S_{K^{∗}} (this is also true in the case D ≃ M_{2}(F), as explained in A5.11.2). (6.9) Let P_{S} be a prime of Q_{S} unramified in Q_{S}/Q such that

Fr_{Q_{S}/Q}(P_{S}) ∈ Gal(Q_{S}/F_{c}^{gal}),  (6.9.1)

where F_{c}^{gal} denotes the Galois closure of F_{c} in \overline{Q}. The rational prime p = P_{S} ∩ Z is then as in 6.8; moreover, it splits completely in F_{c}/Q. Extend each element σ_{x} : F_{c} ↪ C (x ∈ X) of the CM type of F_{c} to an element s(x) ∈ Γ_{Q}. As in 5.15 and A5.9, we obtain primes above p in F and F_{c}, respectively, given by

P_{x} = s(x)^{−1}P_{S} ∩ O_{F} , P′_{x} = s(x)^{−1}P_{S} ∩ O_{F_{c}}

that depend only on σ_{x} and such that

pO_{F} = ∏_{x∈X} P_{x},  P_{x}O_{F_{c}} = P′_{x}P′′_{x},  F_{P_{x}} = (F_{c})_{P′_{x}} = Q_{p}.

(6.10) The pair (G(Q_{p}), K_{p}) is isomorphic to ∏_{x∈X}(GL_{2}(F_{P_{x}}), GL_{2}(O_{F,P_{x}})) = (GL_{2}(Q_{p}), GL_{2}(Z_{p}))^{X}; its subgroup (G^{∗}(Q_{p}), K_{p}^{∗}) corresponds to the elements whose determinant lies in the diagonally embedded sugroup (Q_{p}^{×}, Z_{p}^{×}) ⊂ (Q_{p}^{×}, Z_{p}^{×})^{X}.

As in A4.3, the cocharacter μ : G_{m,Q_{p}} ⟶ G_{Q_{p}} ≃ GL(2)_{Q_{p}}^{X} from A3.2 (attached to the Shimura datum

h) decomposes as μ = (μ_{P_{x}})_{x∈X}, where μ_{P_{x}}(a) = ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ). The corresponding parabolic P_{μ} ⊂ G_{Q_{p}} (see A1 below) is the product of upper Borel subgroups; its Levi subgroup M is the product of the diagonal maximal tori in GL(2)_{Q_{p}}. We identify M with its set of Q_{p}-points and we write M^{∗} = M ∩ G^{∗}(Q_{p}), L = M ∩ K_{p} and L^{∗} = M^{∗} ∩ K_{p}^{∗}.

The maps K_{p}^{∗}gK_{p}^{∗} ↦ K_{p}gK_{p} (resp. mL^{∗} ↦ mL) define embeddings of Hecke algebras

H(G^{∗}(Q_{p})//K_{p}^{∗}, Q) ↪ H(G(Q_{p})//K_{p}, Q) = ⨂_{P |p} H(H(F_{P})//K_{P} , Q) ≃ H(GL_{2}(Q_{p})//GL_{2}(Z_{p}), Q)^{X}

and

H(M^{∗}//L^{∗}, Q) ↪ H(M//L, Q) = ⨂_{P |p} H(M_{P} //L_{P} , Q) ≃ H((Q_{p}^{×} × Q_{p}^{×})//(Z_{p}^{×} × Z_{p}^{×}), Q)^{X}

that are compatible with the twisted Satake transforms

\overline{S}_{μ} : H(G^{∗}(Q_{p})//K_{p}^{∗}, Q) ⟶ H(M^{∗}//L^{∗}, Q), \overline{S}_{μ} : H(G(Q_{p})//K_{p}, Q) ⟶ H(M//L, Q)

from A1.4. These Hecke algebras contain the following important elements (with P = P_{x}, x ∈ X, NP = p):

• S_{P} , T_{P} , T_{P^{2}} ∈ H(H(F_{P})//K_{P} , Q), where S_{P} = K_{P} ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) K_{P}, T_{P} = K_{P} ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) K_{P} and T_{P^{2}} =

K_{P} ( \begin{matrix}p^{2} & 0 \\ 0 & 1 \end{matrix} ) K_{P} = T_{P}^{2} − (NP + 1)S_{P}.

• T_{P}^{2}/S_{P} , S_{p} = ∏_{P |p} S_{P} ∈ H(G^{∗}(Q_{p})//K_{p}^{∗}, Q).

• φ_{P} = μ_{P} (p)^{−1}L_{P} ∈ H(M_{P} //L_{P} , Q) (the partial Frobenius at P)

• φ = μ(p)^{−1}L^{∗} = ∏_{P |p} φ_{P} ∈ H(M^{∗}//L^{∗}, Q) (the total Frobenius)

• φ̃_{P} = φ_{P}^{2} ∏_{P′≠P} S_{P′}^{−1} = φ_{P}^{2} S_{P} S_{p}^{−1} ∈ H(M^{∗}//L^{∗}, Q) (twisted square of φ_{P}), ∏_{P |p} φ̃_{P} = φ^{2}S_{p}^{1−r}.

The important point is that the partial Frobenius φ_{P} does not lie in H(M^{∗}//L^{∗}, Q), but its twisted square φ̃_{P} does. (6.11) For a non-zero polynomial Q ∈ A[Y] with coefficients in a commutative ring A and a ∈ A^{×} let

(R_{a}Q)(Y) = a^{deg(Q)}Q(a^{−1}Y); then (R_{a_{1}}Q_{1}) ⊗ (R_{a_{2}}Q_{2}) = R_{a_{1}a_{2}}(Q_{1} ⊗ Q_{2}),  (6.11.1)

where the Rankin-Selberg product Q_{1} ⊗ Q_{2} is defined analogously as in (5.16.3). If Q is monic, factor it formally as Q(Y) = (Y − α_{1}) ⋯ (Y − α_{n}) and let

Q^{(2)}(Y) = (Y − α_{1}^{2}) ⋯ (Y − α_{n}^{2}) ∈ A[Y].

The Eichler-Shimura relations in our situation are consequences of the following abstract polynomial identity (see A1.6–A.1.8):

\overline{S}_{μ_{P}} (Q_{P})|_{Y =φ_{P}} = 0 ∈ H(M_{P} //L_{P} , Q),  (6.11.2)

where P = P_{x}, NP = p and

Q_{P} (Y) = Y^{2} − (T_{P} /S_{P})Y + (NP)/S_{P} ∈ H(H(F_{P})//K_{P} , Q).

It implies that

\overline{S}_{μ}( ⨂_{P |p} Q_{P} )|_{Y =φ} = 0 ∈ H(M^{∗}//L^{∗}, Q)  (6.11.3)

and

\overline{S}_{μ_{P}} (Q_{P}^{(2)})|_{Y =φ_{P}^{2}} = 0 ∈ H(M_{P} //L_{P} , Q),

hence

\overline{S}_{μ_{P}} (R_{S_{P} /S_{p}}Q_{P}^{(2)})|_{Y =_{φ_{P}}} = 0 ∈ H(M^{∗}//L^{∗}, Q), R_{S_{P} /S_{p}}Q_{P}^{(2)}(Y) = Y^{2}+(2NP −T_{P}^{2}/S_{P})Y/S_{p}+(NP/S_{p})^{2}.

(6.11.4)

(6.12) These polynomials are related to the Euler factors of the Galois representation ρ_{Π} as follows. If we write P = P_{x}, P′ = P′_{x} and P′′ = P′′_{x}, then

L_{P′}(BC_{F_{c}/F} (JL(π)), Std, s)^{−1} = L_{P} (π, Std, s)^{−1} = 1 − T_{P} p^{−1/2−s} + S_{P} p^{−2s}|_{π_{P}^{K_{P}}}

and the relations (6.7.1-2) imply that

P_{ρ_{Π}(Fr(P′))}(Y) = Y^{2} − T_{P} ψ(P′)Y + pS_{P} ψ(P′)^{2}|_{π_{P}^{K_{P}}} .

For x ∈ X let ρ_{x} be the representation ρ_{x} = ^{s(x)}ρ_{Π^{∨}} = ρ_{Π^{∨}} ∘ int(s(x))^{−1} : Γ_{F_{c}^{gal}} ⟶ GL_{2}(\overline{Q}_{ℓ}), as in 5.12. Its Euler factor at P_{S} is given by

P_{ρ_{x}(Fr(P_{S}))}(Y) = P_{ρ_{Π}^{∨}(P′_{x})}(Y) = P_{ρ_{Π}(P′′_{x})}(Y) = Y^{2}−T_{P_{x}}ψ(P′′_{x})Y +pS_{P_{x}}ψ(P′′_{x})^{2}|_{π_{Px}^{K_{Px}}} = (R_{S_{Px} ψ(P′′_{x})}Q_{P_{x}})|_{π_{Px}^{K_{Px}}} ,

(6.12.1)

since Π^{∨} ≃ Π^{c} ≃ Π ⊗ ω_{Π}^{−1} ≃ Π ⊗ (ψ^{c}/ψ).

(6.13) Proposition. There exists a character χ : Γ_{F_{c}^{gal}} ⟶ \overline{Q}_{ℓ}^{×} with the following property: for every P_{S} satisfying (6.9.1) we have

χ(Fr(P_{S})) = p^{w^{∗}/2} ∏_{x∈X} ψ(P′_{x}), ( ⨂_{x∈X} Q_{P_{x}})(Y)|_{(π_{p}^{∗})^{K^{∗}_{p}}} = P_{(χ⊗⨂_{x∈X} ρ_{x})(Fr(P_{S}))}(Y).

Proof. The character

χ̃ : ∏_{x∈X} (ψ∥ · ∥_{F_{c}}^{−w^{∗}/2r}) ∘ N_{F_{c}^{gal}/F_{c}} ∘ s(x)^{−1} : A_{F_{c}^{gal}}^{×}/(F_{c}^{gal})^{×} ⟶ C^{×}

sends a uniformiser at P_{S} ∩ O_{F_{c}^{gal}} to p^{w^{∗}/2} ∏_{x∈X} ψ(P′_{x}). The infinity type of χ̃ is algebraic: if τ : F_{c}^{gal} ↪ C extends E_{0} ↪ C, then

χ̃_{τ} (a) = |τ(a)|^{−w^{∗}}(τ(a)/|τ(a)|)^{k} = τ(a)^{r−m}\overline{τ(a)}^{r−k−m},

where k = ∑_{x∈X} k_{x} and w^{∗} = ∑_{x∈X}(k_{x} − 2 + 2m_{x}) = k − 2r + 2m. The ℓ-adic character χ : Γ_{F_{c}^{gal}} ⟶ \overline{Q}_{ℓ}^{×} attached to χ̃ ([Sc, ch. 0, §5], [HT, p. 20]) then satisfies χ(Fr(P_{S})) = p^{w^{∗}/2} ∏_{x∈X} ψ(P′_{x}), as claimed. The remaining statement follows from the fact that

R_{p^{−w^{∗}/2}}( ⨂_{x∈X} Q_{P_{x}})|_{(π_{p}^{∗})^{K^{∗}_{p}}} = ( ⨂_{x∈X} Q_{P_{x}})|_{π_{p}^{Kp}} = ⨂_{x∈X} (Q_{P_{x}} |_{π_{Px}^{K_{Px}}} ) = ⨂_{x∈X} (R_{a_{x}}P_{ρ_{x}(Fr(P_{S}))}) = R_{a}P_{(⨂ ρ_{x})(Fr(P_{S}))},

where

a_{x} = ψ(P′′_{x})^{−1}S_{P_{x}}|_{π_{Px}^{K_{Px}}} = ψ(P′′_{x})^{−1}(ω_{π})_{P_{x}}(P_{x}) = ψ(P′_{x}), a = ∏_{x∈X} a_{x} = p^{−w^{∗}/2}χ(Fr(P_{S})).

(6.14) Proposition. Denote by ρ the ℓ-adic representation of Γ_{Q} given by its action on V^{r}((π^{∗})^{∞}). The dense subset Σ = {Fr(P_{S}) | P_{S} as in (6.9.1)} ⊂ Γ_{F_{c}^{gal}} has the following properties.

(1) ∀g ∈ Σ P_{(χ⊗⨂_{x∈X} ρ_{x})}(ρ(g)) = 0.

(2) For every g ∈ Σ there exist mutually commuting endomorphisms u_{x} ∈ End_{\overline{Q}_{ℓ}}(V^{r}((π^{∗})^{∞}) (x ∈ X) and

non-zero scalars c, c_{x} ∈ \overline{Q}_{ℓ}^{×} such that

ρ(g^{2}) = c ∏_{x∈X} u_{x}, ∀x ∈ X P_{ρ_{x}(g^{2})}(c_{x}u_{x}) = 0.

Proof. If P_{S} | p is as in (6.9.1), then the group G^{∗} splits over Q_{p} and the main result of [W] (see also [Mo, Cor. 4.2.5]) applies. Explicitly, there is a stack (in fact, a scheme, after imposing a level structure outside p) of p-isogenies p − Isog_{K^{∗p}} ⟶ S_{K^{∗}} × S_{K^{∗}} (where S_{K^{∗}} is the Kottwitz model of Sh_{K^{∗}} over Z_{(p)}) and a commutative diagram (see A4 below for the notation and the sign conventions, which differ from those in [W])

\xymatrix{H(G^{∗}(Q_{p})_{−}//K_{p}^{∗}, Q) \ar[r]^{h} \ar[d]^{_{\overline{S}_{μ}}} & Q[p − Isog_{K^{∗p}} ⊗ Q_{p}] \ar[d]^{σ} \\ H(M_{−}^{∗}//L^{∗}, Q) \ar[r]^{\overline{h}} & Q[p − Isog_{K^{∗p}} ⊗ F_{p}]}

equipped with compatible G^{∗}(Q̂)-equivariant actions on H^{r}(Sh_{K^{∗}}) = H^{r}(Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) ⊗_{Q} \overline{Q}, ℒ_{ξ^{∗},ℓ}) (hence also on V^{r}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞} ⊂ (H^{r})^{K^{∗}}, since (H^{r})^{K^{∗}} = H^{r}(Sh_{K^{∗}}) if D ≄ M_{2}(F) and (H^{r})^{K^{∗}} injects into H^{r}(Sh_{K^{∗}}) if D ≃ M_{2}(F), by Proposition A6.17).

(1) The action of Fr(P_{S}) on H^{r}(Sh_{K^{∗}}) is given by the action of φ ∈ H(M_{−}^{∗}//L^{∗}, Q). Letting the relation (6.11.3) act on V^{r}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞} we obtain the Eichler-Shimura relation for the usual Frobenius

( ⨂_{x∈X} Q_{P_{x}})|_{(π_{p}^{∗})^{K^{∗}_{p}}} (Fr(P_{S})|_{V^{r}((π^{∗})^{∞})⊗(π^{∗})^{∞}}) = 0 ∈ End(V^{r}((π^{∗})^{∞}) ⊗ (π^{∗})^{∞}),

which is equivalent to

P_{(χ⊗⨂_{x∈X} ρ_{x})(Fr(P_{S}))}(ρ(Fr(P_{S}))) = 0,

thanks to Proposition 6.13.

(2) As explained in A5.5, the G^{∗}(Q̂)-equivariance implies that the action of each φ̃_{P_{x}} ∈ H(M_{−}^{∗}//L^{∗}, Q) on V^{r}((π^{∗})^{∞})⊗(π^{∗})^{∞} is of the form u_{x}⊗id, where u_{x} ∈ Aut_{\overline{Q}_{ℓ}}(V^{r}((π^{∗})^{∞})) (but we do not really need this fact) and u_{x}u_{y} = u_{y}u_{x}. The relation (6.11.4) combined with (6.12.1) implies that u_{x} is a root of the polynomial

(R_{S_{Px} /S_{p}}Q_{P_{x}}^{(2)})|_{(π_{p}^{∗})^{K^{∗}_{p}}} = R_{p^{w^{∗}}} (R_{S_{Px} /S_{p}}Q_{P_{x}}^{(2)})|_{(π_{Px})^{K_{Px}}} = R_{p^{w^{∗}}} R_{S_{Px} /S_{p}}R_{S_{Px}^{−2}ψ(P′′_{x})^{−2}}P_{ρ_{x}(Fr(P_{S}))^{2}} |_{(π_{Px})^{K_{Px}}} =

= R_{c_{x}^{−1}}P_{ρ_{x}(Fr(P_{S})^{2})}, c_{x} = p^{−w^{∗}}ψ(P′′_{x})^{2}S_{p}S_{P_{x}} |_{(π_{Px})^{K_{Px}}} ∈ \overline{Q}_{ℓ}^{×}.

Therefore P_{ρ_{x}(Fr(P_{S})^{2})}(c_{x}u_{x}) = 0. Finally

ρ(Fr(P_{S})^{2}) = φ^{2}|_{V^{r}((π^{∗})^{∞})} = S_{p}^{r−1} ∏_{x∈X} φ̃_{P_{x}}|_{V^{r}((π^{∗})^{∞})} = c ∏_{x∈X} u_{x}, c = S_{p}^{r−1}|_{π_{p}^{Kp}} .

(6.15) Corollary. If P_{S} is as in (6.9.1) and if, for each x ∈ X, the polynomial P_{ρ_{x}(Fr(P_{S})^{2})} has two distinct roots, then ρ(Fr(P_{S})) acts semisimply on V^{r}((π^{∗})^{∞}).

(6.16) Note that, if ϕ has complex multiplication by a (totally imaginary) quadratic extension M of F and

if Fr(P_{S})|_{M} ≠ id, then ρ_{x}(Fr(P_{S})) = ( \begin{matrix}0 & ∗ \\ ∗ & 0 \end{matrix} ) and P_{ρ_{x}(Fr(P_{S})^{2})}(Y) = (Y − a_{x})^{2} has a double root, for all

x ∈ X.

In general, the representations ρ_{Π} : Γ_{F_{c}} ⟶ GL_{2}(\overline{Q}_{ℓ}) and ρ_{x} = ^{s(x)}ρ_{Π^{∨}} = ^{s(x)}ρ_{Π^{c}} : Γ_{F_{c}^{gal}} ⟶ GL_{2}(\overline{Q}_{ℓ}) have the following properties.

(6.17) Proposition. (1) Let λ | ℓ be the prime of F_{c} above ℓ induced by the fixed embeddings F_{c} ⊂ \overline{Q} ⊂ C ≃ \overline{Q}_{ℓ}. For every x ∈ X the restriction of the representation ρ_{x} to Γ_{(F_{c})_{λ}} is Hodge-Tate. Its two Hodge-Tate weights are distinct; their difference is equal to k_{x} − 1.

(2) The representation ρ_{Π} is irreducible.

(3) If ϕ does not have complex multiplication, then the representation ρ_{Π} is strongly irreducible and, for each x ∈ X, \overline{Q}_{ℓ} · Lie(Im(ρ_{x})) ⊃ 𝔰l_{2}(\overline{Q}_{ℓ}).

Proof. (1) This is a special case of a general compatibility between the Hodge-Tate weights and automorphic weights for n-dimensional ℓ-adic Galois representations ρ_{Π} of Γ_{F_{c}} attached to self-dual (Π^{∨} ≃ Π^{c}) regular algebraic automorphic representations Π of GL_{n}(A_{F_{c}}), proved in [CH, Thm. 3.2.5].

(2) The argument from [T, Prop. 3.1] in the case of motivic weight applies.

(3) If ρ_{Π} is not strongly irreducible, then it is induced from a character Γ_{L} ⟶ \overline{Q}_{ℓ}^{×}, where [L : F_{c}] = 2. It follows that Π ⊗ α ≃ Π, where α : A_{F_{c}}^{×}/F_{c}^{×} ⟶ {±1} is the quadratic character attached to L/F_{c}. Therefore α^{c} = α, which implies that α = β ∘ N_{F_{c}/F} for some β : A_{F}^{×}/F^{×} ⟶ {±1} and BC_{F_{c}/F} (JL(π)) = BC_{F_{c}/F} (JL(π) ⊗ β), hence JL(π) ≃ JL(π) ⊗ β or JL(π) ≃ JL(π) ⊗ βη, where η is the quadratic character of A_{F}^{×}/F^{×} attached to F_{c}/F; thus JL(π) has complex multiplication by β ≠ 1 or by βη ≠ 1.

If ρ_{Π} is strongly irreducible, then the \overline{Q}_{ℓ}-Lie algebra \overline{Q}_{ℓ} · Lie(Im(ρ_{Π^{∨}})) ⊂ 𝔤l_{2}(\overline{Q}_{ℓ}) acts irreducibly on \overline{Q}_{ℓ}^{2}, which means that it must contain 𝔰l_{2}(\overline{Q}_{ℓ}).

(6.18) Proposition. If ϕ does not have complex multiplication, then there exists an open subgroup U ⊂ Γ_{F_{c}^{gal}} and a dense subset Σ_{U} ⊂ U such that each element of Σ_{U} acts semisimply on V^{r}((π^{∗})^{∞}).

Proof. Combine Proposition 6.17(3) with Proposition 3.6 (for ρ_{i} = ρ_{x} and a = 2) and Corollary 6.15.

(6.19) Theorem. Assume that we are in the case (B) and BC_{F_{c}/F} (JL(π)) ⊗ ψ = Π.

(1) There exists a finite extension E′′/F_{c}^{gal} and an integer m ≥ 1 such that

V^{r}((π^{∗})^{∞})^{ss}|_{Γ_{E′′}} ⊂ (⨂_{x∈X} ^{s(x)}ρ_{Π^{∨}}) ⊗ χ|_{Γ_{E′′}}^{⊕m} .

If, in addition, ϕ has complex multiplication by a totally imaginary quadratic extension M of F, then E′′ satisfies M^{gal} ⊄ (M^{gal})^{+}E′′.

(2) If ϕ does not have complex multiplication and if the weights (k_{x})_{x∈X} are distinct, then the representation ⨂_{x∈X} ^{s(x)}ρ_{π^{∨}} of Γ_{F_{c}^{gal}} is strongly irreducible and

V^{r}((π^{∗})^{∞})^{ss}|_{Γ_{F{gal \atop c}}} = (⨂_{x∈X} ^{s(x)}ρ_{Π^{∨}}) ⊗ χ|_{Γ_{F{gal \atop c}}}^{⊕m} .

(3) If ϕ does not have complex multiplication, then the representation V^{r}((π^{∗})^{∞}) of Γ_{Q} is semisimple.

Proof. The arguments used in the proof of Theorem 5.20 apply, with references to 5.19 to be replaced by those to Proposition 6.14 and Proposition 6.17. In concrete terms, (1) is a consequence of Proposition 6.14(1) and Proposition 3.10(1) (resp. and Proposition 4.6) if ϕ does not (resp. does) have complex multiplication. The statement (2) follows from Proposition 3.10(2) applied to ρ_{i} = ρ_{x} and a = 1; the assumptions (A’) and (C’) are consequences of Proposition 6.17(3) and Proposition 6.14(1), respectively, and the strong irreducibility is a consequence of the argument from 5.20 that uses the Hodge-Tate weights. The statement (3) follows from Theorem 3.12(2) applied to ρ_{i} = ρ_{x} and a = 2; the assumptions (A”) and (C”) are satisfied, respectively, thanks to Proposition 6.17(3) and Proposition 6.14(2).

(6.20) Corollary. For every ξ^{∗} as in 6.2, the action of Γ_{Q} on the non CM part of H_{et}^{i}(Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}) ⊗_{Q} \overline{Q}, ℒ_{ξ^{∗},ℓ}) in the case D ≄ M_{2}(F) (resp. on H_{et}^{i}(Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗})_{BB} ⊗_{Q} \overline{Q}, j_{!∗}ℒ_{ξ^{∗},ℓ}) in the case D ≃ M_{2}(F)) is semisimple.

Appendix: Eichler-Shimura relations

In this Appendix we indicate how the methods of [FC] and [W] for proving Eichler-Shimura relations in the split case also work for partial Frobenius morphisms. We try to keep consistent sign conventions, but it is conceivable that some of the formulas hold only up to a sign. This is not important, however, for the applications to semisimplicity in the main body of this article.

A1. The Satake transform (the split case)

(A1.1) In this section we recall various versions of the Satake transform in the simplest possible setting. Let K be a finite extension of Q_{p} with ring of integers O. Let | · | be the normalised valuation on K (so that the valuation of any uniformiser ϖ is equal to |ϖ| = q^{−1}, where q is the cardinality of the residue field O/ϖO). (A1.2) Let \underline{G} be a split connected reductive group over K; it is the general fibre of a group scheme over O (which will also be denoted by \underline{G}) with reductive special fibre.

Assume that we are given a cocharacter μ : G_{m,O} ⟶ \underline{G} (defined over O). Fix a maximal torus \underline{T} ⊂ \underline{G} such that μ factors through \underline{T}. This defines subgroup schemes \underline{T} ⊂ \underline{M} ⊂ \underline{P}_{μ} ⊂ \underline{G}, where \underline{M} is the centraliser of μ in \underline{G} and \underline{P}_{μ} is a parabolic subgroup of \underline{G} with Levi subgroup \underline{M} and unipotent radical \underline{N}_{μ} characterised by the fact that Lie(\underline{N}_{μ}) is the direct sum of those root spaces Lie(\underline{G})^{α} with respect to \underline{T} for which α∘μ > 0.

We obtain the corresponding groups of points T = \underline{T} (K) ⊂ M = \underline{M}(K) ⊂ P_{μ} = \underline{P}_{μ}(K) = M ⋉ N_{μ} ⊂ G = \underline{G}(K) ⊃ K = \underline{G}(O), where N_{μ} = \underline{N}_{μ}(K). The modulus morphism

δ_{μ} : M ⟶ q^{Z}, m ↦ | det(Ad(m) | Lie(N_{μ}))|

satisfies δ_{μ^{−1}} = δ_{μ}^{−1}. (A1.3) Let H(G//K, Q) be the Hecke algebra of locally constant functions with compact support f : G ⟶ Q satisfying f(kgk′) = f(g) for all g ∈ G and k, k′ ∈ K. The product is given by the convolution

(f_{1} ∗ f_{2})(h) = ∫_{G} f_{1}(g)f_{2}(g^{−1}h) dg,

where dg is the Haar measure on G giving K volume 1. The algebra H is commutative and the characteristic function char(K) of K is its unit. We define similarly H(M//M ∩ K, Q). (A1.4) Let du be the Haar measure on N_{μ} normalised by giving N_{μ} ∩ K volume 1. We define two twisted Satake transforms

\overline{S}_{μ}, S̃_{μ} : H(G//K, Q) ⟶ H(M//(M ∩ K), Q),

related by S̃_{μ} = δ_{μ} · \overline{S}_{μ}, by the formulas

(\overline{S}_{μ}f)(m) = ∫_{N_{μ}} f(mu) du, (S̃_{μ}f)(m) = ∫_{N_{μ}} f(um) du,

and the usual (normalised) Satake transform

S_{μ} = δ_{μ}^{1/2} · \overline{S}_{μ} = δ_{μ}^{−1/2} · S̃_{μ} : H(G//K, Q) ⟶ H(M//(M ∩ K), Q) ⊗ Z[q^{±1/2}].

(A1.5) The Hecke polynomial. In the special case when \underline{M} = \underline{T} the parabolic subgroup \underline{P}_{μ} = \underline{B} = \underline{T} ⋉\underline{U} is a Borel subgroup and the normalised Satake transform induces an isomorphism

S : H(G//K, Z) ⊗ Z[q^{±1/2}] ⟶^{∼} (H(T//(T ∩ K), Z) ⊗ Z[q^{±1/2}])^{W} .

The target group is canonically identified with R(Ĝ) ⊗ Z[q^{±1/2}], where R(Ĝ) is the Grothendieck ring of algebraic representations of the complex dual group Ĝ, via the bijections

Z[X_{∗}(\underline{T})] ⟶^{∼} H(T//(T ∩ K), Z), [λ] ↦ char(λ(ϖ)(T ∩ K)),

R(Ĝ) ⟶^{∼} Z[X^{∗}(T̂)]^{W} = Z[X_{∗}(\underline{T})]^{W} , [V] ↦ Trace(V |_{\underline{T}}).

Using this isomorphism, we define, for any algebraic representation V of Ĝ, the Hecke polynomial (“the characteristic polynomial”)

H_{V} (X) := ∑_{k=0}^{dim V} (−1)^{k}[Λ^{k}V] X^{k} ∈ (H(G//K, Z) ⊗ Z[q^{±1/2}])[X],  H̃_{V} (X) = X^{dim V} H_{V} (1/X).

For any cocharacter λ ∈ X_{∗}(\underline{T}) (considered as a character of the dual torus T̂ ⊂ Ĝ) there exists w ∈ W such that wλ will lie in the positive Weyl chamber with respect to \underline{B}. We denote by V_{λ} the irreducible representation of Ĝ with highest weight wλ and we let

H_{λ}(X) = H_{V_{λ}}(X),  H̃_{λ}(X) = H̃_{V_{λ}}(X).

For example, if \underline{G} = \underline{T} is a torus, then

H_{λ}(X) = 1 − char(λ(ϖ)K)X,  H̃_{λ}(X) = X − char(λ(ϖ)K).  (A1.5.1)

(A1.6) Consider the following toy model: \underline{G} = GL(2) and λ(t) = ( \begin{matrix}t & 0 \\ 0 & 1 \end{matrix} ). In this case \underline{T} is the diagonal

torus, V_{λ} is the standard two-dimensional representation of Ĝ = GL_{2}(C) and

H_{λ}(X) = (1 − X char(( \begin{matrix}ϖ & 0 \\ 0 & 1 \end{matrix} ) (T ∩ K))) (1 − X char(( \begin{matrix}1 & 0 \\ 0 & ϖ \end{matrix} ) (T ∩ K)))

= 1 − q^{−1/2}T_{ϖ}X + S_{ϖ}X^{2},

where

T_{ϖ} = char(K ( \begin{matrix}ϖ & 0 \\ 0 & 1 \end{matrix} ) K), S_{ϖ} = char(K ( \begin{matrix}ϖ & 0 \\ 0 & ϖ \end{matrix} ) K) ∈ H(G//K, Z).

Similarly, V_{λ^{−1}} is the dual of V_{λ}, hence V_{λ^{−1}} ⟶^{∼} V_{λ} ⊗ (Λ^{2}V_{λ})^{−1} and

H_{λ^{−1}}(X) = 1 − q^{−1/2}(T_{ϖ}/S_{ϖ})X + (1/S_{ϖ})X^{2},  H̃_{λ^{−1}}(X) = X^{2} − q^{−1/2}(T_{ϖ}/S_{ϖ})X + (1/S_{ϖ}),

qH̃_{λ^{−1}}(q^{−1/2}X) = X^{2} − (T_{ϖ}/S_{ϖ})X + (q/S_{ϖ}).  (A1.6.1)

(A1.7) Proposition ([Bu, Prop. 3.4], [W, Prop. 2.9]). If the cocharacter μ in A1.2 is minuscule, then (\overline{S}_{μ}(H̃_{μ^{−1}}))(q^{−⟨ρ,μ⟩}char(Mμ(ϖ)^{−1}M)) = 0 ∈ H(M//(M ∩ K), C), where 2ρ ∈ Z[X^{∗}(\underline{T})] denotes the sum of all positive roots of (\underline{G}, \underline{B}, \underline{T}).

(A1.8) In the situation of A1.6 the cocharacter μ = λ is minuscule, \underline{P}_{μ} = \underline{B} is the upper triangular Borel subgroup, ⟨ρ, μ⟩ = 1/2 and

(\overline{S}_{μ}(H̃_{μ^{−1}}))(X) = (X − q^{−1/2} char(( \begin{matrix}ϖ^{−1} & 0 \\ 0 & 1 \end{matrix} ) (T ∩ K))) (X − q^{1/2} char(( \begin{matrix}1 & 0 \\ 0 & ϖ^{−1} \end{matrix} ) (T ∩ K))) .

A2. Hecke correspondences and their action on cohomology

(A2.1) Let (G, 𝒳 ) be a pure Shimura datum, let K ⊂ G(Q̂) be an open compact subgroup. Throughout this Appendix (with the exception of A5.6, A5.11-12 and A6.16-20) we assume that G^{der} is anisotropic, hence the Shimura variety Sh_{K} = Sh_{K}(G, 𝒳 ) is projective (and smooth if K is small enough) over the reflex field E = E(G, 𝒳 ).

Any diagram (Sh_{K} ⟵^{q_{1}} Z ⟶^{q_{2}} Sh_{K}) with finite morphisms q_{i} defines a correspondence cl(Z) = (q_{1}, q_{2})_{∗}(Z) ∈ Corr(Sh_{K})_{Q} = CH^{dim Sh_{K}} (Sh_{K} × Sh_{K})_{Q}. The product of correspondences is given by A ∘ B = (p_{14})_{∗}((A × B) · Δ_{23}). For example, any finite morphism α : Sh_{K} ⟶ Sh_{K} has a graph Γ_{α} = cl(Sh_{K} ⟵^{id} Sh_{K} ⟶^{α} Sh_{K}) and its transpose ^{t}Γ_{α} = cl(Sh_{K} ⟵^{α} Sh_{K} ⟶^{id} Sh_{K}) satisfying Γ_{α} ∘ Γ_{β} = Γ_{β∘α} and ^{t}Γ_{α} ∘^{t}Γ_{β} = ^{t}Γ_{α∘β}.

For every reasonable cohomology theory H^{∗} with coefficients in a field of characteristic zero (such as H^{∗} = H_{et}^{∗}(− ⊗_{E} \overline{Q}, \overline{Q}_{ℓ})) the ring of correspondences Corr(Sh_{K})_{Q} naturally acts on the left on H^{∗}(Sh_{K}) via the formula

L(A) : x ↦ (p_{1})_{∗}([A] ∪ p_{2}^{∗}(x)),

where [A] ∈ H^{∗}(Sh_{K} × Sh_{K}) is the cohomology class of a correspondence A ∈ Corr(Sh_{K})_{Q}. In particular,

L(Sh_{K} ⟵^{q_{1}} Z ⟶^{q_{2}} Sh_{K}) = (q_{1})_{∗} ∘ q_{2}^{∗} : H^{∗}(Sh_{K}) ⟶ H^{∗}(Sh_{K}), L(Γ_{α}) = α^{∗}, L(^{t}Γ_{α}) = α_{∗}, L(A ∘ B) =

L(A) ∘ L(B).

(A2.2) For any g ∈ G(Q̂) the diagram

\xymatrix{Sh_{gKg^{−1}∩K} \ar[d]_{pr_{1}} \ar[r]^{[·g]} & Sh_{K∩g^{−1}Kg} \ar[d]^{pr_{2}} \\ Sh_{K} & Sh_{K}}

(where [·g] denotes the standard right action of g on the tower {Sh_{K}}) defines a Hecke correspondence

[KgK] = (Sh_{K} ⟵^{pr_{1}∘[·g^{−1}]} Sh_{K∩g^{−1}Kg} ⟶^{pr_{2}} Sh_{K}) ∈ Corr(Sh_{K})_{Q}

depending only on the double coset KgK ∈ K\G(Q̂)/K. Define the global Hecke algebra H(G(Q̂)//K, Q) as in A1.3, with K of volume 1. The Q-linear extension of the map char(KgK) ↦ [KgK] defines a ring homomorphism H(G(Q̂)//K, Q) ⟶ Corr(Sh_{K})_{Q}. The corresponding left action on cohomology

L([KgK]) = (pr_{1} ∘ [·g^{−1}])_{∗} ∘ pr_{2}^{∗} = (pr_{1})_{∗} ∘ [·g]^{∗} ∘ pr_{2}^{∗}

corresponds to the natural left action of G(Q̂) on H^{∗}(Sh_{K}) given by L(g) = [·g]^{∗}. (A2.3) As a multivalued map pr_{2} ∘ [·g] ∘ (pr_{1})^{−1}, the Hecke correspondence is given, using the standard notation [x, γ]_{K} for the class of (x, γ) ∈ 𝒳 × G(Q̂) in Sh_{K}, by

[x, γ]_{K} ↦ ∑[x, γg_{i}g]_{K},  K = ∐_{i} g_{i}(K ∩ gKg^{−1}),  KgK = ∐_{i} g_{i}gK.

See A5.5 below for the action of Hecke correspondences on cohomology with coefficients in a local system.

A3. The PEL data

(A3.1) Assume that we are given the following data: (B, ∗, V, ⟨ , ⟩_{F}), where B is a finite-dimensional simple Q-algebra, ∗ is a Q-linear positive involution on B, F = Z(B)^{∗=id} (a totally real number field), V is a non-zero left B-module of finite type and ⟨ , ⟩_{F} : V × V ⟶ F is a non-degenerate alternating F -bilinear form such that ⟨bv, v′⟩_{F} = ⟨v, b^{∗}v′⟩_{F} for all b ∈ B and v, v′ ∈ V.

The centre Z(B) = F_{c} of B is equal either to F, or to a totally imaginary quadratic extension of F. Set ⟨ , ⟩ = Tr_{F/Q} ∘ ⟨ , ⟩_{F} : V × V ⟶ Q; this is a non-degenerate alternating Q-bilinear form satisfying the same hermitian property as ⟨ , ⟩_{F}. The centraliser C = End_{B}(V) is a simple Q-algebra with centre F_{c} and an F -linear involution # given by the adjoint with respect to ⟨ , ⟩_{F}.

Let H = GSp_{B}(V, ⟨ , ⟩_{F}) be the algebraic group over F whose points with values in any F -algebra S are given by

H(S) = {h ∈ GL_{B}(V ⊗_{F} S) | ∃ν(h) ∈ S^{×} ∀v, v′ ∈ V ⟨hv, hv′⟩_{F} = ν(h)⟨v, v′⟩_{F} }
= {h ∈ (C ⊗_{F} S)^{×} | hh^{#} = ν(h) ∈ S^{×}}

and let G^{∗} be the algebraic group over Q such that

G^{∗}(R) = {g ∈ (C ⊗ R)^{×} | gg^{#} = ν(h) ∈ R^{×}}

for all Q-algebras R. As in 5.2, there is a cartesian diagram

\xymatrix{G^{∗} \ar[r] \ar[d]^{ν} & G \ar[d]^{ν} \\ G_{m,Q} \ar[r] & R_{F/Q}(G_{m,F}),}

where G = R_{F/Q}(H). We assume, from now on, that the group G^{der} is anisotropic.

(A3.2) Recall [Ko, §1] that (B ⊗_{F} \overline{F} , ∗⊗id) is isomorphic either to End(W)×End(W)^{op} with (a, b)^{∗} = (b, a) (type (A)), or to End(W) with ∗ being the adjoint map with respect to a symmetric (resp. alternating) bilinear form on W (type (C)) (resp. type (BD)).

From now on, assume that our datum is of type (A) (when F_{c} ≠ F) or (C) (when F_{c} = F). This implies that H, G and G^{∗} are connected reductive groups and that the derived group of H (hence of G) is simply connected ([Mi, Prop. 8.7]). Furthermore, there exists a morphism of R-algebras h : C ⟶ C ⊗ R such that h(\overline{z}) = h(z)^{#} for which the symmetric R-bilinear form ⟨v, h(i)v′⟩ : V_{R} × V_{R} ⟶ R is positive definite. The morphism h is unique up to conjugation by an element c ∈ (C ⊗ R)^{×} such that cc^{#} = 1 ([Ko, Lemma 4.3], [Mi, Prop. 8.12]).

It follows that h defines a Shimura datum (G^{∗}, 𝒳 ^{∗}) (resp. (G, 𝒳 )), where 𝒳 ^{∗} (resp. 𝒳 ) is the G^{∗}(R)- conjugacy class (resp. the G(R)-conjugacy class) of h. The real group G_{R} = ∏_{v|∞} H ⊗_{F,v} R is isomorphic to ∏_{v} GSp(2n)_{R} (resp. to ∏_{v} GU(a_{v}, b_{v}), a_{v} + b_{v} = n) if (B, ∗) is of type (C) (resp. of type (A)).

The action of h(i) defines a complex structure on V_{R}, hence a Hodge decomposition V_{C} = V^{−1,0} ⊕V^{0,−1} of weight -1, with h(z) ⊗ id acting as z (resp. \overline{z}) on V^{−1,0} (resp. on V^{0,−1}). The cocharacter μ = μ_{h} : G_{m,C} ⟶ G_{C} attached to h acts on V_{C} as follows: μ(z) acts as z ·id (resp. as id) on V^{−1,0} (resp. on V^{0,−1}).

The common reflex field E = E(G^{∗}, 𝒳 ^{∗}) = E(G, 𝒳 ) ⊂ \overline{Q} ⊂ C is the field generated over Q by the coefficients of the characteristic polynomial

det(X_{1}α_{1} + ⋯ + X_{t}α_{t} | V^{−1,0}),  (A3.2.1)

for any Q-basis {α_{j}} of B.

(A3.3) The arguments in [Ko, §7] show that the group G satisfies the Hasse principle. The key point is a description of the torus T = G/G^{der} in terms of tori _{k}T = R_{k/Q}(G_{m,k}). If (B, ∗) is of type (C), then ν induces an isomorphism ν : T ⟶^{∼} _{F} T. If it is of type (A) and n ≥ 1 is as in A3.2, then the map “determinant” together with ν induce an isomorphism T ⟶^{∼} {(a, b) ∈ _{F_{c}}T ×_{F} T | N_{F_{c}/F} (a) = b^{n}}. For n = 2k + 1 (resp. n = 2k), the map (a, b) ↦ ab^{−k} (resp. (a, b) ↦ (ab^{−k}, b)) defines an isomorphism β : T ⟶^{∼} _{F_{c}}T satisfying N_{F_{c}/F} ∘ β = ν (resp. β = (β_{1}, ν) : T ⟶^{∼} Ker(N_{F_{c}/F} : _{F_{c}}T ⟶ _{F} T) ×_{F} T). All tori _{k}T have trivial H^{1} and the Hasse principle holds for the norm N_{F_{c}/F}. It follows that T satisfies the Hasse principle, hence so does G (cf. [Mi, Lemma 8.20, 8.21]).

(A3.4) Unramified local data at p. Let p be a prime number; fix an embedding \overline{Q} ↪ \overline{Q}_{p}. Write Q̂^{(p)} = (∏_{ℓ≠p} Z_{ℓ}) ⊗ Q for the ring of finite adeles outside p. Assume that each term in the decomposition B_{Q_{p}} = ∏_{P |p} B ⊗_{F} F_{P} is a matrix algebra over an unramified extension of Q_{p} (in particular, p is unramified in F_{c}/Q). This assumtion will be further strengthened in A4.3 below.

Assume, furthermore, that we are given the following data: an open compact subgroup K^{p} ⊂ G(Q̂^{(p)}), a ∗-stable O_{F} ⊗ Z_{(p)}-order O_{B} ⊂ B such that O_{B} ⊗ Z_{p} is a maximal order in B_{Q_{p}} and an O_{F} ⊗ Z_{p}-lattice Λ ⊂ V_{Q_{p}} which is self-dual (up to a scalar in (F ⊗ Q_{p})^{×}) with respect to ⟨ , ⟩_{F}. Fix such a Λ and let K_{p} = {g ∈ G(Q_{p}) | g(Λ) = Λ}, K = K_{p}K^{p} ⊂ G(Q̂). As in [Ko, p.390], the characteristic polynomial det(b | V^{−1,0}) in (A3.2.1) will have coefficients in O_{E} ⊗ Z_{(p)}, if we choose {α_{j}} to be a Z_{(p)}-basis of O_{B}. (A3.5) A moduli problem. A p-integral model S_{K} of Sh_{K}(G, 𝒳 ) (for sufficiently small K^{p}) can be constructed as follows (see [R, 2.14] and [TX, 2.3-2.8, 4.6] for special cases).

For any α ∈ (F̂^{(p)})^{×} = (F ⊗ Q̂^{(p)})^{×} consider the following moduli stack ℳ_{α,K^{p}} over the category of locally noetherian schemes S over O_{E} ⊗ Z_{(p)}. Its objects over S are quadruples (A, ι, λ, \overline{(η, u)}), where

• A is an abelian scheme over S up to prime-to-p isogeny (notation: A is an object of (AV/S) ⊗ Z_{(p)});

• λ : A ⟶ Â is a Z_{(p)}-polarisation of degree prime to p;

• ι : O_{B} ⟶ End(A) is a ∗-morphism (with respect to ∗ on O_{B} and the Rosati involution coming from λ on End(A));

• for a fixed geometric point s of (every connected component of) S, \overline{(η, u)} is a π_{1}(S, s)-invariant K^{p}-level structure. By the latter we mean a K^{p}-orbit of pairs (η, u), where u ∈ (Ô_{F}^{(p)})^{×} and η : V ⊗Q̂^{(p)} ⟶^{∼} V^{(p)}(A_{s}) is a B ⊗ Q̂^{(p)}-linear isomorphism such that the Weil pairing ⟨ , ⟩_{λ} : V^{(p)}(A_{s}) × V^{(p)}(A_{s}) ⟶ Q̂^{(p)} on V^{(p)}(A_{s}) = Q ⊗ ∏_{ℓ≠p} T_{ℓ}(A_{s}) attached to λ satisfies

⟨η(x), η(y)⟩_{λ} = Tr_{F/Q}(αu⟨x, y⟩_{F}).

An element g ∈ K^{p} acts on (η, u) by (η, u)g = (η ∘ g, u ν(g)).

• the Kottwitz determinant condition should be satisfied: det(b | Lie(A)) = det(b | V^{−1,0}) as polynomial functions on O_{B}.

Morphisms between (A, ι, λ, \overline{(η, u)}) and (A′, ι′, λ′, \overline{(η′, u′)}) are given by O_{B}-linear isomorphisms f : A ⟶^{∼} A′ in (AV/S) ⊗ Z_{(p)} such that λ = f^{∗}(λ′) (= f̂∘ λ′ ∘ f) and \overline{(η′, u′)} = \overline{(f ∘ η, u)}.

Note that the degree of λ is determined by the above conditions. Moreover, (A, ι, λ, \overline{(η, u)}) has no non-trivial automorphism if K^{p} is small enough (cf. [R, 2.13]), which we assume, from now on.

This implies, as in [Ko, §5] and [R], that ℳ_{α,K^{p}} is represented by a smooth quasi-projective scheme over O_{E} ⊗ Z_{(p)}, which will be denoted by M_{α,K^{p}}.

There is an action of the group of totally positive units O_{F,+}^{×} on ℳ_{α,K^{p}} given by the formula

ε · (A, ι, λ, \overline{(η, u)}) = (A, ι, ι(ε)λ, \overline{(η, εu)}).

If ε = N_{F_{c}/F} (ε′) for some ε′ ∈ O_{F_{c}}^{×} ∩ K^{p} ⊂ F_{c}^{×} = Z_{G}(Q) ⊂ G(Q̂^{(p)}), then multiplication by ι(ε′) on A defines an isomorphism

[ι(ε′)] : (A, ι, λ, \overline{(η, u)}) ⟶^{∼} ε^{−1} · (A, ι, λ, \overline{(η, u)}).

It follows that the finite abelian group

Δ = O_{F,+}^{×}/N_{F_{c}/F} (O_{F_{c}}^{×} ∩ K^{p})

acts on M_{α,K^{p}}. It turns out that, after replacing K^{p} by a suitable open subgroup, the group Δ will act on the scheme M_{α,K^{p}} freely by permuting its connected components. This is proved in [Ki] in general and in A3.9-10 below in the cases (C) and (A even).

In particular, the quotient scheme M_{α,K^{p}}/Δ exists and is quasi-projective and smooth over O_{E} ⊗ Z_{(p)}. (A3.6) The moduli problem over C. Following [Ko, §8], we define a map

M_{α,K^{p}}(C) ⟶ Sh_{K}(G, 𝒳 )(C) = G(Q)\(𝒳 × G(Q_{p})/K_{p} × G(Q̂^{(p)})/K^{p})

(see also [TX, 2.4], [Mi, 6.3, 6.9]).

If (A, λ, ι, \overline{(η, u)}) is a quadruple over C, then H = H_{1}(A, Q) is a B-module via ι, equipped with a skew-Hermitian pairing ⟨ , ⟩_{H,λ} : H × H ⟶ F such that Tr_{F/Q} ∘ ⟨ , ⟩_{H,λ} is induced by λ.

As in [Ko, p. 338-339], one checks that H_{Q_{v}} and V_{Q_{v}} are isomorphic skew-Hermitian B_{Q_{v}}-modules, for all places v of Q. For v ≠ ∞, p this follows from the existence of η; for v = ∞ one uses the determinant condition and [Ko, Lemma 4.2]. For v = p, T_{p}(A) = H_{1}(A, Z_{p}) ⊂ H_{Q_{p}} is a self-dual O_{B} ⊗ Z_{p}-lattice and a variant of [Ko, Lemma 7.2] applies.

The validity of the Hasse principle for G implies that there is a B-linear isomorphism a : H ⟶^{∼} V, unique up to left multiplication by G(Q), which sends ⟨ , ⟩_{H,λ} to an F^{×}-multiple of ⟨ , ⟩_{F}. We fix such an isomorphism.

The natural complex structure h_{A} on H_{R} = Lie(A) defines a complex structure ah_{A} = (z ↦ a ∘ h_{A}(z) ∘ a^{−1}) on V which lies in 𝒳 (when interpreted as a pure real Hodge structure of weight -1 on V_{R}), thanks to [Ko, Lemma 4.2].

At p, a(T_{p}(A)) ⊂ V_{Q_{p}} is an O_{B} ⊗ Z_{p}-lattice, self-dual up to a scalar in (F ⊗ Q_{p})^{×}. A variant of [Ko, Lemma 7.3] shows that there exists g_{p} ∈ G(Q_{p}) (with g_{p}K_{p} depending only on a(T_{p}(A))) such that a(T_{p}(A)) = g_{p}Λ. Equivalently, η_{p} = a^{−1} ∘ g_{p} : V_{Q_{p}} ⟶^{∼} H_{Q_{p}} satisfies T_{p}(A) = η_{p}(Λ).

Finally, a ∘ η : V ⊗ Q̂^{(p)} ⟶^{∼} V ⊗ Q̂^{(p)} is an element of G(Q̂^{(p)}).

The map M_{α,K^{p}}(C) ⟶ Sh_{K}(G, 𝒳 )(C) given by sending (A, λ, ι, \overline{(η, u)}) to
[ah, g_{p}K_{p}, a ∘ η]_{K} ∈ G(Q)\(𝒳 × G(Q_{p})/K_{p} × G(Q̂^{(p)})/K^{p}) = Sh_{K}(G, 𝒳 )(C)  (A3.6.1)

is well-defined and factors through the quotient M_{α,K^{p}}(C)/Δ.

(A3.7) A p-integral model S_{K} of Sh_{K}(G, 𝒳 ). Choose a (finite) set Σ = {α} ⊂ (F̂^{(p)})^{×} of representatives of the double cosets

(F̂^{(p)})^{×} = ∐_{α∈Σ}(O_{F} ⊗ Z_{(p)})_{+}^{×} α (Ô_{F}^{(p)})^{×}.

The maps (A3.6.1) induce a bijection ([Fa, Prop. 3.6.3])

∐_{α∈Σ}(M_{α,K^{p}}(C)/Δ) ⟶^{∼} Sh_{K}(G, X)(C),  (A3.7.1)

which implies that the smooth quasi-projective O_{E} ⊗ Z_{(p)}-scheme

S_{K} := ∐_{α∈Σ}(M_{α,K^{p}}/Δ) = M_{K^{p}}/Δ, M_{K^{p}} = ∐_{α∈Σ} M_{α,K^{p}}

is a model of Sh_{K}(G, X). Moreover, (A3.7.1) identifies S_{K} ⊗ E with a canonical model of Sh_{K}(G, h), not of Sh_{K}(G, h^{−1}) – see the discussion in [Mi, p. 347]. (A3.8) Recall that G^{der} is anisotropic, by assumption, hence Sh_{K}(G, 𝒳 ) is compact. In this case each M_{α,K^{p}} (hence S_{K}, too) is projective over O_{E} ⊗ Z_{(p)}, by [Ko, p. 392] if C is a division algebra, and by [L, Thm. 4.6] in general. (A3.9) Deligne’s description of the set of geometric connected components π_{0}Sh_{K}(G, 𝒳 )(C) ([De1, Thm. 2.4], [Mi, Thm. 5.17]) in terms of the map G ⟶ G/G^{der} = T yields the following bijections (depending on a choice of a connected component of 𝒳 ):

π_{0}Sh_{K}(G, 𝒳 )(C) ⟶^{∼} (O_{F} ⊗ Z_{(p)})^{×}\F̂^{(p),×}/ν(K^{p})

in the case (C),

π_{0}Sh_{K}(G, 𝒳 )(C) ⟶^{∼} (O_{F_{c}} ⊗ Z_{(p)})^{×}\F̂_{c}^{(p),×}/β(K^{p})

in the case (A odd) (when n = 2k + 1) and

π_{0}Sh_{K}(G, 𝒳 )(C) ⟶^{∼} U_{1}(K^{p}) × U_{2}(K^{p}),

U_{1}(K^{p}) = Ker(N : (O_{F_{c}} ⊗ Z_{(p)})^{×} ⟶ (O_{F} ⊗ Z_{(p)})_{+}^{×})\Ker(N : F̂_{c}^{(p),×} ⟶ F̂^{(p),×})/β_{1}(K^{p})

U_{2}(K^{p}) = (O_{F} ⊗ Z_{(p)})_{+}^{×}\F̂^{(p),×}/ν(K^{p})

in the case (A even) (when n = 2k).

In the case (C),

π_{0}M_{α,K^{p}}(C) = (Ô_{F}^{(p)})^{×}/ν(K^{p}) = Ô_{F}^{×}/ν(K),

π_{0}(M_{α,K^{p}}/Δ)(C) = O_{F,+}^{×}\Ô_{F}^{×}/ν(K)

and the stabiliser in Δ of any connected component of M_{α,K^{p}}(C) is equal to

Δ_{0}(K) = (O_{F,+}^{×} ∩ ν(K)) /(O_{F}^{×} ∩ K)^{2}

(see [TX, 2.3, 2.4] in the case of Hilbert modular varieties, when H = GL(2)_{F}).

Similarly, in the case (A even),

π_{0}M_{α,K^{p}}(C) = U_{1}(K^{p}) × (Ô_{F}^{(p)})^{×}/ν(K^{p}) = U_{1}(K^{p}) × Ô_{F}^{×}/ν(K),

π_{0}(M_{α,K^{p}}/Δ)(C) = U_{1}(K^{p}) × O_{F,+}^{×}\Ô_{F}^{×}/ν(K)

and the stabiliser in Δ of any connected component of M_{α,K^{p}}(C) is equal to

Δ_{0}(K) = (O_{F,+}^{×} ∩ ν(K)) /N_{F_{c}/F} (O_{F_{c}}^{×} ∩ K).

(A3.10) Proposition ([TX, Lemma 2.5]). Assume that (B, ∗) is of type (C) or (A even). After replacing K^{p} by a suitable open subgroup if necessary one can achieve Δ_{0}(K) = 0, hence Δ will act on each M_{α,K^{p}}(C) freely by permuting certain connected components.

(A3.11) The Frobenius morphism. The absolute Frobenius morphism

φ : S_{K} ⊗ O_{E}/pO_{E} ⟶ S_{K} ⊗ O_{E}/pO_{E}  (A3.11.1)

is induced by the relative Frobenius morphism F_{A} : A ⟶ A^{(p)} on abelian schemes in characteristic p. More precisely, let S be a (locally noetherian) scheme over O_{E}/pO_{E} and (A, ι, λ, η, u) a 5-tuple representing an element of M_{α,K^{p}}(S). There is a canonical action ι^{(p)} : O_{B} ⟶ End(A^{(p)}) compatible with ι via F_{A} and a Z_{(p)}-polarisation λ^{(p)} : A^{(p)} ⟶ Â^{(p)} satisfying F_{A}^{∗}(λ^{(p)}) = pλ. The formula

φ(A, ι, λ, η, u) = (A^{(p)}, ι^{(p)}, λ^{(p)}, F_{A} ∘ η, pu)

gives an explicit description of

φ : M_{α,K^{p}} ⊗ O_{E}/pO_{E} ⟶ M_{α,K^{p}} ⊗ O_{E}/pO_{E}

and of the restriction of the map (A3.11.1) to (M_{α,K^{p}}/Δ) ⊗ O_{E}/pO_{E}. (A3.12) Partial Frobenius morphisms. One can write the map φ in (A3.11.1) as a product φ = ∏_{P |p} φ_{P} of mutually commuting partial Frobenius morphisms

φ_{P} : S_{K} ⊗ O_{E}/pO_{E} ⟶ S_{K} ⊗ O_{E}/pO_{E},

for primes P | p of F above p (see [TX, 4.6] in the case of Hilbert modular varieties).

Fix a totally positive element c ∈ F_{+}^{×} such that v_{P} (c) = 1 and v_{P′}(c) = 0 for all P′ | p, P′ ≠ P. Let S and (A, ι, λ, η, u) be as in A3.11. Consider A′ = A/Ker(F_{A})[P] and denote by f_{P} : A ⟶ A′ the quotient map. Again, there is a canonical morphism ι′ : O_{B} ⟶ End(A′) induced by ι and f_{P}. We define

φ_{P} (A, ι, λ, η, u) = (A′, ι′, λ′, η′, u′),  (A3.12.1)

where cλ = f_{P}^{∗} (λ′), η′ = f_{P} ∘ η and cαu = α′u′. The recipe (A3.12.1) is compatible with the right K^{p}-action on the pairs (η, u), with isomorphisms and with the action of Δ. However, it depends on the choice of c.

If we replace c by c̃, then c̃ = εc with ε ∈ O_{F,+}^{×} and the 5-tuple (A′, ι′, λ′, η′, u′) is replaced by (A′, ι′, ελ′, η′, εu′). This implies that the above formula gives rise to a well-defined partial Frobenius morphism

φ_{P} : (M_{α,K^{p}}/Δ) ⊗ O_{E}/pO_{E} ⟶ (M_{α′,K^{p}}/Δ) ⊗ O_{E}/pO_{E}.

hence to φ_{P} : S_{K} ⊗ O_{E}/pO_{E} ⟶ S_{K} ⊗ O_{E}/pO_{E}.

A4. The p-isogenies ([FC, VII.3-4], [W, §3-5])

We continue to assume that K^{p} is sufficiently small. (A4.1) Let v | p be the prime of E ⊂ \overline{Q} defined by the fixed embedding \overline{Q} ↪ \overline{Q}_{p}. As in [W, §3], a p-isogeny f : (A_{1}, ι_{1}, λ_{1}, \overline{(η_{1}, u_{1})}) ⟶ (A_{2}, ι_{2}, λ_{2}, \overline{(η_{2}, u_{2})}) between objects (A_{j}, ι_{j}, λ_{j}, \overline{(η_{j}, u_{j})}) of ℳ_{α_{j},K^{p}}(S) (for a scheme S over O_{E_{v}}) is an O_{B}-linear isogeny in (AV/S) ⊗ Z_{(p)} of p-power degree such that f ∘ \overline{η}_{1} = \overline{η}_{2} and f^{∗}(λ_{2}) = cλ_{1} for some c ∈ F_{+}^{×} such that v_{P} (c) ≥ 0 for all P | p in F.

A morphism f ⟶ f′ between two p-isogenies is given by a pair of O_{B}-linear isomorphisms g_{j} : A_{j} ⟶ A′_{j} (j = 1, 2) in (AV/S) ⊗ Z_{(p)} satisfying

g_{2} ∘ f = f′ ∘ g_{1},  g_{j}^{∗}(λ′_{j}) = λ_{j},  g_{j} ∘ \overline{η}_{j} = \overline{η}′_{j}  (j = 1, 2).

The p-isogenies (for all possible combinations of α_{1} and α_{2}) form a stack of groupoids p − Isog_{K^{p}} over (Sch/O_{E_{v}}), equipped with canonical projections p_{j} : p − Isog_{K^{p}} ⟶ ℳ_{K^{p}} = ∐_{α} ℳ_{α,K^{p}} (j = 1, 2) sending f to (A_{j}, ι_{j}, λ_{j}, \overline{(η_{j}, u_{j})}). As in [W], the restriction of each p_{j} to the substack p − Isog_{K^{p}}^{m} of p-isogenies with fixed value of m = (m_{P} = v_{P} (c))_{P |p} is represented by a proper surjective map (which is finite étale when restricted to (Sch/E_{v})). As a result, p − Isog_{K^{p}} is, in fact, a scheme equipped with a morphism (p_{1}, p_{2}) : p − Isog_{K^{p}} ⟶ M_{K^{p}} × M_{K^{p}}, where M_{K^{p}} = ∐_{α} M_{α,K^{p}}.

The composition of isogenies

(p − Isog_{K^{p}}) ×_{p_{2},M_{K^{p}} ,p_{1}} (p − Isog_{K^{p}}) ⟶ p − Isog_{K^{p}}, f_{1}, f_{2} ↦ f_{2} ∘ f_{1}

defines a ring structure on Q[p − Isog_{K^{p}}/S], the Q-vector space on the set of connected components of p − Isog_{K^{p}}(S) (see [FC, p. 252]). (A4.2) p-isogenies in characteristic zero. If f is a p-isogeny over S = Spec(C), then f identifies H_{1}(A_{1}, Q) = H_{1}(A_{2}, Q) = H and induces an injection T_{p}(f) : H_{1}(A_{1}, Z_{p}) = T_{p}(A_{1}) ↪ H_{1}(A_{2}, Z_{p}) = T_{p}(A_{2}) ⊂ H_{Q_{p}}. The corresponding elements g_{p,j} ∈ G(Q_{p}) from A3.6 satisfy g_{p,1}(Λ) = a(T_{p}(A_{1})) ⊂ a(T_{p}(A_{2})) = g_{p,2}(Λ), hence g_{p,2} = g_{p,1}g with g ∈ G(Q_{p})_{−} = {u ∈ G(Q_{p}) | g^{−1}(Λ) ⊂ Λ}. We define the type of f to be the double coset K_{p}gK_{p} ∈ K_{p}\G(Q_{p})_{−}/K_{p}; it depends only on f.

If C is replaced by an arbitrary algebraically closed field of characteristic zero, then there are isomorphisms φ_{j} : T_{p}(A_{j}) ⟶^{∼} Λ. They satisfy φ_{2} ∘ T_{p}(f) ∘ φ_{1}^{−1}(Λ) = g^{−1}Λ for some g ∈ G(Q_{p})_{−}; we define the type of f to be again K_{p}gK_{p}. The type of the geometric fibres of any p-isogeny over a base S over Spec(E) is locally constant on S.

Note a sign change compared to [W, 4.1]; this is forced on us by the formula (A3.6.1), which relates the moduli problem to the canonical model of the Shimura variety.

As in [FC, p. 253] and [W, 4.2], define a map

h : H(G(Q_{p})_{−}//K_{p}, Q) ⟶ Q[p − Isog_{K^{p}}/E]

by sending the characteristic function of any double coset K_{p}gK_{p} to the union of the connected components on which the p-isogeny has type K_{p}gK_{p}. This is a ring morphism (if we let K_{p} have volume 1) and its composition with

(p_{1}, p_{2}) : Q[p − Isog_{K^{p}}/E] ⟶ Corr(M_{K^{p}} ⊗ E)_{Q}

is given by char(K_{p}gK_{p}) ↦ (pr × pr)^{∗} ∘ [KgK], in the notation of A2.2 (where pr : M_{K^{p}} ⊗ E ⟶ S_{K} ⊗ E is the map (A3.6.1)).

In particular, the action of h(g) on étale cohomology of M_{K^{p}} ⊗ \overline{E} leaves stable the image under pr^{∗} of étale cohomology of S_{K} ⊗ \overline{E} = Sh_{K} ⊗_{E} \overline{E} and acts on the latter as the Hecke operator L([KgK]).

(A4.3) From now on, we impose the following additional assumption:

p splits completely in F_{c}/Q.  (A4.3.1)

This implies that E_{v} = Q_{p}, k(v) = F_{p}, F ⊗ Q_{p} = ∏_{P |p} F_{P}, F_{P} = Q_{p}, B_{Q_{p}} ≃ ∏_{i} M_{n_{i}}(Q_{p}), O_{B} ⊗ Z_{p} ≃ ∏_{i} M_{n_{i}}(Z_{p}), each group H ⊗_{F} F_{P} is split over F_{P} = Q_{p} (cf. the discussion in [Mi, 8.5-8.6]), G splits over Q_{p}. Of course, G(Q_{p}) = ∏_{P |p} H(F_{P}) and K_{p} = ∏_{P |p} K_{P}, where K_{P} is a maximal compact subgroup of H(F_{P}).

As in [W, 5.1], the conjugacy class [μ] of the cocharacter μ_{h} from A3.2 (considered over \overline{Q}_{p}, via the given embeddings \overline{Q}_{p} ↩ \overline{Q} ⊂ C) contains a cocharacter defined over Q_{p}, which extends to μ : G_{m,Z_{p}} ⟶ \underline{G}, where \underline{G} is a reductive model of G over Z_{p} defined by Λ. The decomposition Λ = ∏_{P |p} Λ_{P} defines, for each P | p in F, a cocharacter μ_{P} : G_{m,O_{F_{P}}} = G_{m,Z_{p}} ⟶ \underline{H}_{P}, where \underline{H}_{P} is a reductive model of H ⊗_{F} F_{P} over O_{F_{P}} = Z_{p}.

Fix μ as above; then Λ = Λ^{−1,0} ⊕ Λ^{0,−1}, where μ(z) = z · id (resp. μ(z) = id) on Λ^{−1,0} (resp. on Λ^{0,−1}). In [W], Λ^{−1,0} is denoted by Λ_{0} and Λ^{0,−1} by Λ_{1}. The centraliser \underline{M} = \{g ∈ \underline{G} | g(\underline{Λ}^{i,j}) = \underline{Λ}^{i,j}\} is a Levi factor of the parabolic \underline{P}_{μ} = \{g ∈ \underline{G} | (\underline{Λ}^{−1,0}) = \underline{Λ}^{−1,0}\} attached to μ as in A1.2. Let L = \underline{M}(Z_{p}) = K_{p} ∩ M, where M = \underline{M}(Q_{p}).

(A4.4) Ordinary p-isogenies ([FC, VII.4], [W, §5]). A p-isogeny over a field of characteristic p is ordinary if A_{1} (hence A_{2}, too) is an ordinary abelian variety. A general p-isogeny is ordinary if its fibres over points in characteristic p are ordinary. They form a subscheme p − Isog_{K^{p}}^{ord} of p − Isog_{K^{p}}.

Let (A, ι, λ, \overline{(η, u)}) ∈ ℳ_{α,K^{p}}(k), where k is an algebraically closed field of characteristic p. If A is ordinary, it is shown in [W, 5.2-3] that there are O_{B}-linear isomorphisms

T_{p}(A) ⟶^{∼} T_{p}(Â) ⟶^{∼} Hom_{Z_{p}}(Λ^{−1,0}, Z_{p}) ⟶^{∼} Λ^{0,−1},

with the first isomorphism induced by λ, the third by ⟨ , ⟩_{F} and with O_{B} acting on the third term by (b · u)(x) = u(b^{∗}x). Above, T_{p}(A) = \underleftarrow{lim}_{n} A(k)[p^{n}] denotes the physical Tate module of A.

This implies that a p-ordinary isogeny f over k gives rise to m_{1} ∈ End_{O_{B}⊗Z_{p}}(Λ^{0,−1}) and its dual f̂ to m_{0}^{∗} ∈ End_{O_{B}⊗Z_{p}}(Hom_{Z_{p}}(Λ^{−1,0}, Z_{p})), hence to m_{0} ∈ End_{O_{B}⊗Z_{p}}(Λ^{−1,0}). We define the type of f to be the double coset L(m_{0}, m_{1})^{−1}L ∈ L\M_{−}/L (note the change of sign with respect to [W, 5.3]), where M_{−} = {m ∈ M | m^{−1}(Λ^{i,j}) ⊂ Λ^{i,j}}.

For example, the Frobenius isogeny f = F_{A} has purely multiplicative kernel, which means that m_{1} = p and m_{0} = 1 ([W, 5.9]), so (m_{0}, m_{1}) = μ(p) and the type of f in our sense is equal to Lμ(p)^{−1}L.

More generally, if P | p is a prime of F above p, then the invariant (m_{0}, m_{1}) attached to the isogeny f_{P} : A ⟶ A/Ker(F_{A})[P] used in the definition of the partial Frobenius morphism φ_{P} : S_{K} ⊗ k(v) = S_{K} ⊗ F_{p} ⟶ S_{K} ⊗ k(v) is equal to (m_{0}, m_{1}) = μ_{P} (p); thus the type of f_{P} is Lμ_{P} (p)^{−1}L = L_{P} μ_{P} (p)^{−1}L_{P}.

The relation between the type of a p-isogeny in characteristic zero and the type of its reduction modulo p (assumed to be ordinary) is explained in [FC, p. 263]: let O be a complete DVR of mixed characteristic with residue field k = \overline{k} ⊃ F_{p}. Let f be a p-isogeny over O with ordinary special fibre \overline{f}. The Barsotti-Tate objects Λ′_{i} = T_{p}(A_{i/O}) sit in exact sequences

0 ⟶ (Λ′_{i})_{mult} ⟶ Λ′_{i} ⟶ (Λ′_{i})_{et} ⟶ 0

and f induces injections (Λ′_{1})_{?} ↪ (Λ′_{2})_{?} (? = ∅, mult, et). If we consider Λ′_{i} ⊂ H_{Q_{p}} = H_{1}(A_{i}, Q_{p}) and if we fix an isomorphism a : H_{Q_{p}} ⟶^{∼} V_{Q_{p}}, then we obtain lattices a(Λ′_{1}) = g_{1}(Λ) ⊂ a(Λ′_{2}) = g_{2}(Λ) ⊂ V_{Q_{p}}, where g = g_{1}^{−1}g_{2} ∈ \underline{P}_{μ}(Q_{p})_{−} = (P_{μ})_{−}. By definition, the type of f is equal to K_{p}gK_{p}. If we denote by m(g) ∈ M_{−} the projection of g onto the Levi part of P_{μ}, then the type of \overline{f} will be equal to L(m_{0}, m_{1})^{−1}L = Lm(g)L.

The arguments in [W, 5.5-7] show that the restrictions of the projections p_{1}, p_{2} : p − Isog_{K^{p}}^{ord} ⊗ F_{p} ⟶ M_{K^{p}}^{ord}⊗F_{p} to the subscheme of ordinary p-isogenies of a fixed type LmL are finite and flat and their geometric fibres have pure multiplicity given by explicit constants mult_{i}(LmL).

As in [W, 5.8], one defines a map

\overline{h} : H(M_{−}//L, Q) ⟶ Q[p − Isog_{K^{p}}^{ord} ⊗ k(v)] = Q[p − Isog_{K^{p}}^{ord} ⊗ F_{p}]

by sending the characteristic function of LmL to (1/mult_{1}(LmL)) times the union of all connected components corresponding to ordinary p-isogenies of type LmL (again, L is of volume 1).

In particular, the correspondence \overline{h}(char(Lμ(p)^{−1}L)) (resp. \overline{h}(char(Lμ_{P} (p)^{−1}L))) on M_{K^{p}} ⊗ k(v) = M_{K^{p}} ⊗F_{p} is equal to the pullback by pr×pr of the graph of the Frobenius morphism φ : S_{K} ⊗F_{p} ⟶ S_{K} ⊗F_{p} (resp. of the graph of the partial Frobenius φ_{P} : S_{K} ⊗ F_{p} ⟶ S_{K} ⊗ F_{p}). As a result, its action on étale cohomology of M_{K^{p}} ⊗ \overline{F}_{p} leaves stable the image under pr^{∗} of étale cohomology of S_{K} ⊗ \overline{F}_{p} and its action on the latter coincides with the action of φ ⊗ id (resp. of φ_{P} ⊗ id).

(A4.5) Proposition ([FC, p. 263], [W, Prop. 5.10]). The following diagram commutes (the map σ is given by specialisation of cycles).

\xymatrix{H(G(Q_{p})_{−}//K_{p}, Q) \ar[r]^{h} \ar[d]^{_{\overline{S}_{μ}}} & Q[p − Isog_{K^{p}}^{ord} ⊗ E_{v}] \ar[d]^{σ} \\ H(M_{−}//L, Q) \ar[r]^{\overline{h}} & Q[p − Isog_{K^{p}}^{ord} ⊗ F_{p}]}

Proof. As in [FC, p. 263], this follows from the discussion in A4.4 relating the types of f and \overline{f}. Note that [FC] work with classical objects, such as Γ\ℍ_{g}, defined as quotients by a left action, whereas Shimura varieties Sh_{K} = Sh/K are quotients by a right action. This accounts for a sign change in the formulas involving the action of the Hecke algebra. More precisely, one uses the Iwasawa decomposition G(Q_{p}) = \underline{P}_{μ}(Q_{p})K_{p} = P_{μ}K_{p} to determine the number of cosets K_{p}gK_{p}/K_{p} lying in the fibre of the map

G(Q_{p})/K_{p} = P_{μ}/(K_{p} ∩ P_{μ}) ⟶ M/L, g ↦ m(g)

above a fixed class mL. As P_{μ} = MU (with U ∩ K_{p} of volume 1), the above number is equal to

∫_{U}(char(K_{p}gK_{p}))(mu) du = (\overline{S}_{μ}char(K_{p}gK_{p}))(m),

as claimed.

A5. Eichler-Shimura relations for partial Frobenii

(A5.1) The principal geometric result of [W, §6] is the relative density theorem for the moduli problem giving rise to an integral model of Sh_{K^{∗}}(G^{∗}, 𝒳 ^{∗}). The theorem states that the ordinary p-isogenies in characteristic p are dense in all p-isogenies, provided the group G^{∗} is split over Q_{p}. Using Serre-Tate theory, this is deduced from a deformation statement about p-isogenies of principally quasi-polarised O_{B} ⊗ Z_{p}-modules [W, Prop. 6.10], which is then sufficient to prove only for O_{B} ⊗ Z_{p} = Z_{p} or Z_{p} × Z_{p}.

This reduction argument to a special case of [W, 6.10] is also valid in our situation (i.e., for the union of the Kottwitz models M_{K^{p}} = ∐_{α} M_{α,K^{p}}), which yields an equality Q[p − Isog_{K^{p}} ⊗ F_{p}] = Q[p − Isog_{K^{p}}^{ord} ⊗ F_{p}]. The commutative diagram in Proposition A4.5 then becomes

\xymatrix{H(G(Q_{p})_{−}//K_{p}, Q) \ar[r]^{h} \ar[d]^{_{\overline{S}_{μ}}} & Q[p − Isog_{K^{p}} ⊗ E_{v}] \ar[d]^{σ} \\ H(M_{−}//L, Q) \ar[r]^{\overline{h}} & Q[p − Isog_{K^{p}} ⊗ F_{p}]}  (A5.1.1)

(A5.2) Both Hecke algebras decompose into tensor products:

H(G(Q_{p})_{−}//K_{p}, Q) = ⨂_{P |p} H(H(F_{P})_{−}//K_{P} , Q)

H(M_{−}//L, Q) = ⨂_{P |p} H(M_{P −}//L_{P} , Q).

The discussion in A4.4 justifies the following definition: the partial Frobenius at P in Q[p − Isog_{K^{p}} ⊗ F_{p}] is defined as

φ_{P} = \overline{h}(L_{P} μ_{P} (p)^{−1}L_{P}).

Their product is equal to

φ = ∏_{P |p} φ_{P} = \overline{h}(Lμ(p)^{−1}L).

(A5.3) Theorem (the Eichler-Shimura relation for partial Frobenius morphisms). If p splits completely in F_{c}/Q and satisfies the assumptions from A3.4, then the following relation holds, for every prime P | p in F:

H̃_{μ_{P}^{−1}}(p^{−⟨ρ_{P} ,μ_{P} ⟩}φ_{P}) = 0 ∈ Q[p − Isog_{K^{p}} ⊗ F_{p}],

where ρ_{P} denotes the half sum of all positive roots of H ⊗_{F} F_{P}.

Proof. Combine (A5.1.1) with Proposition A1.7. (A5.4) Similarly, the full Frobenius φ satisfies the “Rankin-Selberg product” of the relations A5.3, in the sense of (5.16.2). This relation differs by a sign from the one stated in [W], but it is compatible with the reciprocity law giving the Galois action on H^{0}, which is dual to the action on π_{0}, hence is given by the reciprocity morphism attached to μ^{−1}, rather than to μ. (A5.5) The Eichler-Shimura relation for the action on cohomology. The centre Z_{G} of G contains the torus_{F} T and the quotient torus Z_{G}/_{F} T is anisotropic over R. Fix an irreducible algebraic representation ξ : G_{C} ⟶ GL(V_{ξ}) such that ξ|_{_{F} T} = N_{F/Q}^{m} for some m ∈ Z. This condition implies that ξ(Z_{G}(Q)∩K) = {1},

for all sufficiently small open compact subgroups K ⊂ G(Q̂), hence the local sections of G(Q)\(𝒳 × V_{ξ} × G(Q̂)/K) over Sh_{K}(C) define a localy constant sheaf of complex vector spaces ℒ_{ξ} on Sh_{K}(C).

As in 0.1, fix an isomorphism C ⟶^{∼} \overline{Q}_{ℓ}. It is explained in [HT, III.2] how to attach to ξ_{ℓ} a smooth ℓ-adic étale sheaf ℒ_{ξ,ℓ} on Sh_{K}. This sheaf is G(Q̂)-equivariant ([HT, III.2], [Ko, §6]), which means that there is a natural left action of G(Q̂) and H(G(Q̂)//K, Q) on H^{i}(Sh_{K}) = H_{et}^{i}(Sh_{K}(G, 𝒳 ) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}). Moreover, ξ_{ℓ} can be obtained by a suitable tensor construction from the representation V_{Q_{ℓ}} (which coincides with V_{ℓ}(A) for the abelian variety A appearing in the moduli problems ℳ_{α,K^{p}}). This implies that the ring Q[p−Isog_{K^{p}} ⊗E] also acts on H^{i}(M_{K^{p}}) = H^{i}(M_{K^{p}} ⊗\overline{Q}, pr^{∗}ℒ_{ξ,ℓ}) (cf. [FC, p. 253]). Moreover, ℒ_{ξ,ℓ} extends to the (proper) integral model S_{K^{p}} and H^{i}(M_{K^{p}}) is isomorphic to the étale cohomology of the special fibre of M_{K^{p}}, which is equipped with an action of Q[p − Isog_{K^{p}} ⊗ F_{p}]. The maps h and σ in (A5.1.1) are compatible with the actions on pr^{∗}H^{i}(Sh_{K}) ⊂ H^{i}(M_{K^{p}}) of the various rings appearing in the diagram, thanks to the discussion at the end of A4.2.

These compatibilities yield, together with Theorem A5.3, the Eichler-Shimura relation

∀P | p  H̃_{μ_{P}^{−1}}(p^{−⟨ρ_{P} ,μ_{P} ⟩}φ_{P} |_{H^{i}(Sh_{K})}) = 0 ∈ End(H^{i}(Sh_{K})).  (A5.5.1)

The action of φ_{P} on étale cohomology of the special fibre can be defined directly (thanks to the compatibility alluded to at the end of A4.4), by observing that ℒ_{ξ,ℓ} is obtained by a limit procedure from finite Galois étale covers S_{K′} ⟶ S_{K} (K′ = K_{p}K^{′p}, where K^{′p} is a suitable open normal subgroup of K^{p}) and that φ_{P} from A3.12 acts compatibly on both S_{K′} ⊗ F_{p} and S_{K} ⊗ F_{p}.

The decomposition (0.4.1) yields H^{i}(Sh_{K}) = ⨁ V^{i}(π^{∞}) ⊗ (π^{∞})^{K}, where π^{∞} is the non-archimedean component of an automorphic representation π of G(A) and V^{i}(π^{∞}) is a finite-dimensional ℓ-adic representation of Γ_{E}.

The action of each φ_{P} on H^{i}(Sh_{K}) commutes with the action of both G(Q̂^{(p)}) (by the functorial definition of φ_{P} in (A3.12.1)) and H(G(Q_{p})_{−}//K_{p}, Q) (since the Hecke algebra H(M_{−}//L, Q) is commutative). As a result, each term V^{i}(π^{∞}) ⊗ (π^{∞})^{K} ⊂ H^{i}(Sh_{K}) is φ_{P} -stable. Furthermore, dim End_{G(_{Q^{(p)}})}(π^{∞})^{K_{p}} = 1 by Schur’s Lemma, which implies that φ_{P} ∈ End V^{i}(π^{∞}).

Assume that (π^{∞})^{K} ≠ 0. If we write π^{∞} = ⊗′π_{v}, where π_{v} is a representation of H(F_{v}), then dim(π_{P}^{K_{P}}) = 1 and we obtain from (A5.5.1) the following relation (the notation means that we replace each element of the Hecke algebra H(G(Q_{p})_{−}//K_{p}, Q) by its eigenvalue on π_{P}^{K_{P}}):

(H̃_{μ_{P}^{−1}} |_{π_{P}^{K_{P}}} ) (p^{−⟨ρ_{P} ,μ_{P} ⟩}φ_{P} |_{V^{i}(π^{∞})⊗(π^{∞})^{K}} ) = 0 ∈ End(V^{i}(π^{∞}) ⊗ (π^{∞})^{K}).  (A5.5.2)

(A5.6) A toy model: GL(2). Let us discuss the relation (A5.5.2) in the simplest case F = B = Q and V = Q^{2}, when G = H = G^{∗} = GL(2)_{Q} and the Shimura varieties Sh_{K} are modular curves. They are not compact, but the relation still holds for V^{1}(π^{∞}) contributing to the cuspidal cohomology H_{!}^{1} discussed in 0.8.

The standard two-dimensional representation Std of GL(2) corresponds to H_{1}(A) of the universal abelian variety (= elliptic curve), hence is of weight -1. Its dual Std^{∨} corresponds to H^{1}(A) and is of weight 1.

A general irreducible algebraic representation of G is of the form ξ = Sym^{k−2}(Std^{∨})⊗(det ∘Std^{∨})^{(w−k)/2}, where k ≥ 2 and w ∈ Z, w ≡ k (mod 2). Its central character is ω_{ξ}(x) = x^{2−w}. The sheaf ℒ_{ξ,ℓ} is pure of weight w − 2, hence V^{1}(π^{∞}) ⊂ H_{!}^{1} is pure of weight w − 1. If π is a cuspidal automorphic representation of GL(2, A) such that π_{∞} is cohomological for ξ, then ω_{π_{∞}}(x) = x^{w−2}, which implies that the central character ω_{π} : A^{×}/Q^{×} ⟶ C^{×} of π is of the form ω_{π} = χ∥ · ∥_{Q}^{w−2}, where χ is a character of finite order, which will be identified with a Dirichlet character such that χ(−1) = (−1)^{w}.

If p ≠ ℓ is a prime such that π_{p} is unramified, then the Hecke operators T_{p} and S_{p} defined in A1.6 have the following eigenvalues on π_{p}^{K_{p}} (K_{p} = GL(2, Z_{p})):

S_{p}|_{π_{p}^{Kp}} = ω_{π_{p}}(p) = |p|_{p}^{2−w}χ(p) = p^{w−2}χ(p), T_{p}|_{π_{p}^{Kp}} = a_{p}.

The respective local L-factors at p of π and of the Galois representation ρ_{π} : Γ_{Q} ⟶ GL_{2}(\overline{Q}_{ℓ}) attached to π are given by

L_{p}(π, s)^{−1} = 1 − p^{−1/2}a_{p}p^{−s} + p^{w−2}χ(p)p^{−2s} = (1 − p^{−1/2}T_{p}p^{−s} + S_{p}p^{−2s})|_{π_{p}^{Kp}}

and

L_{p}(ρ_{π}, s)^{−1} = det(1 − p^{−s}ρ_{π}(Fr(p))) = 1 − a_{p}p^{−s} + p^{w−1}χ(p)p^{−2s} = L_{p}(π, s − 1/2)^{−1} = (1 − αp^{−s})(1 − βp^{−s})

(where Fr(p) is the geometric Frobenius). The formula (A1.6.1) together with the relation A5.3 imply that the action φ_{p}^{∗} ∈ Aut(H_{!}^{1}) of φ_{p} on H_{!}^{1} satisfies

Q_{p}(φ_{p}^{∗}) = 0,  Q_{p}(X) = X^{2} − (T_{p}/S_{p})X + p/S_{p},  (A5.6.1)

which means that Fr(p)|(V^{1}(π^{∞}) ⊗ (π^{∞})^{K}) = φ_{p}^{∗}|(V^{1}(π^{∞}) ⊗ (π^{∞})^{K}) is a root of the polynomial

Q_{p}(X)|_{π_{p}^{Kp}} = (X − pα^{−1})(X − pβ^{−1}) = det(X − ρ_{π}^{∨}(−1)(Fr(p)).

This being true for almost all p, the Čebotarev density theorem implies that

P_{ρ_{π}^{∨}(−1)}(V^{1}(π^{∞})) = 0.  (A5.6.2)

In fact, V^{1}(π^{∞}) is two-dimensional, isomorphic to ρ_{π}^{∨}(−1), which is, in turn, isomorphic to ρ_{π^{∨}}. Indeed, π^{∨} ≃ π ⊗ ω_{π}^{−1} and

S_{p}|_{(π_{p}^{∨})^{Kp}} = 1/(p^{w−2}χ(p)) = pα^{−1}β^{−1}, T_{p}|_{(π_{p}^{∨})^{Kp}} = a_{p}/(p^{w−2}χ(p)) = p(α^{−1} + β^{−1}),
L_{p}(π^{∨}, s)^{−1} = (1 − p^{1/2}α^{−1}p^{−s})(1 − p^{1/2}β^{−1}p^{−s}),
L_{p}(ρ_{π^{∨}}, s)^{−1} = (1 − pα^{−1}p^{−s})(1 − pβ^{−1}p^{−s}) = L_{p}(ρ_{π}^{∨}(−1), s)^{−1}.

The relation (A5.6.2) thus reads as follows:

P_{ρ_{π^{∨}}} (V^{1}(π^{∞})) = 0.  (A5.6.3)

Note that π_{∞}^{∨} is cohomological for ξ^{∨} = Sym^{k−2}(Std) ⊗ (det ∘Std)^{(w−k)/2} ≃ ξ ⊗ (det ∘Std)^{w−2}, which means that ℒ_{ξ^{∨},ℓ} is pure of weight 2 − w and V^{1}((π^{∨})^{∞}) is pure of weight 3 − w. We deduce from (A5.6.3) that

P_{ρ_{π}}(V^{1}((π^{∨})^{∞})) = 0.  (A5.6.4)

In fact, V^{1}((π^{∨})^{∞}) = ρ_{π}. (A5.7) Shimura varieties of type (A). Assume that (B, ∗) is of type (A). In this case F_{c} is a CM field and [F_{c} : F] = 2. For each prime v | ∞ of F fix an embedding σ_{v} : F_{c} ↪ \overline{Q} ⊂ C extending v : F ↪ R; then Φ = {σ_{v}} is a CM type of F_{c}. This induces an isomorphism

B ⊗ R = ∏_{v|∞} B ⊗_{F,v} R = ∏_{v|∞} B ⊗_{F_{c},σ_{v}} C ⟶^{∼} ∏_{v|∞} M_{N}(C)

under which V^{−1,0} ⟶^{∼} ⨁_{v|∞} ((C^{N})^{a_{v}} ⊕ (\overline{C}^{N})^{b_{v}}), where a_{v} + b_{v} = n and C ⊗ R ⟶^{∼} ∏_{v|∞} M_{n}(C).

As explained in [HT, I.6], there is a canonical isomorphism H ⊗_{F} F_{c} ⟶^{∼} C^{×} ×G_{m,F_{c}}, where we consider C^{×} = GL_{B}(V) as an algebraic group over F_{c}. This induces an isomorphism

H ⊗_{F} k ⟶^{∼} (C^{×})_{k} × G_{m,k},  (A5.7.1)

for any field embedding F_{c} ↪ k. In particular, the choice of Φ yields an isomorphism

G_{C} = ∏_{v|∞}(H ⊗_{F} F_{c}) ⊗_{F_{c},σ_{v}} C ⟶^{∼} ∏_{v|∞}(GL(n)_{C} × G_{m,C})

under which μ = μ_{h} is given by μ(z) = ((zI_{a_{v}}, I_{b_{v}})_{v|∞}, z). (A5.8) Let ξ, ℒ_{ξ} and ℒ_{ξ,ℓ} be as in A5.5. Assume that π = π_{∞} ⊗ π^{∞} is an automorphic representation of G(A) = H(A_{F}) such that π_{∞} is cohomological for ξ. We further assume that π admits a transfer to a cuspidal automorphic representation (Π, ψ) of GL(n, A_{F_{c}}) × A_{F_{c}}^{×} (cf. [HT, Thm. VI.2.1]). All we need to know is that: (a) Π^{∨} ≃ Π^{c}; (b) Π is cohomological for a suitable algebraic representation ξ′ of R_{F_{c}/Q}(GL(n))_{C}; (c) ψ = ω_{π}^{c} is an algebraic Hecke character of F_{c}; (d) let u ∤ ∞ be a prime of F that splits in F_{c} as u = ww^{c}. The inclusion F_{c} ↪ (F_{c})_{w} = F_{u} defines, by (A5.7.1), an isomorphism H(F_{u}) ⟶^{∼} GL(n, F_{u})×F_{u}^{×}. If the representation π_{u} of the left hand side is unramified, then it is isomorphic to the representation (Π_{u}, ψ_{u}) of the right hand side.

The cuspidality of Π together with (a) and (b) imply [CH] that there is a Galois representation

ρ_{Π} : Γ_{F_{c}} ⟶ GL_{n}(\overline{Q}_{ℓ})

such that

L_{w}(Π, Std_{n}, s) = L_{w}(ρ_{Π}, s + (n − 1)/2),

for all primes w ∤ ℓ∞ at which Π is unramified. Similarly, one can attach to ψ a one-dimensional Galois representation ρ_{ψ} : Γ_{F_{c}} ⟶ \overline{Q}_{ℓ}^{×} such that L(ψ, s) = L(ρ_{ψ}, s). (A5.9) We are going to make the relation (A5.5.2) explicit in terms of ρ_{Π} and ρ_{ψ}. Fix a finite set of primes S ⊃ {ℓ, ∞} of Q such that F_{c}/Q, H and π are ramified only at places above S.

Let Q_{S} ⊃ F_{c}^{gal} be as in 5.15 and let P_{S} be a prime of Q_{S} not above S such that

Fr_{Q_{S}/Q}(P_{S}) ∈ Gal(Q_{S}/F_{c}^{gal}).  (A5.9.1)

Then P_{S} ∩ Z = (p), where p ∉ S is a prime that splits completely in F_{c}/Q. Fix an embedding \overline{Q} ↪ \overline{Q}_{p} inducing P_{S}; then all the assumptions on p imposed in A3 and A4 are satisfied.

Fix F_{c} ⊂ \overline{Q}. For each prime v | ∞ in F extend σ_{v} : F_{c} ↪ \overline{Q} to an automorphism σ̃_{v} : \overline{Q} ⟶^{∼} \overline{Q}. The prime P′_{v} = σ_{v}^{−1}P_{S} ∩ O_{F_{c}} of F_{c} above p depends only on σ_{v}, and P_{v} = P′_{v} ∩ O_{F} splits in F_{c} as P_{v} = P′_{v}(P′_{v})^{c}.

For a representation ρ of Gal(Q_{S}/F_{c}^{gal}) denote by _{σ_{v}}ρ the representation (_{σ_{v}}ρ)(g) = ρ(σ_{v}^{−1}gσ̃_{v}) of the same group; then

(_{σ_{v}}ρ)(Fr(P_{S})) = ρ(σ_{v}^{−1}Fr(P_{S})σ̃_{v}) = ρ(Fr(P′_{v})).

(A5.10) Fix v | ∞ in F and consider the relation (A5.5.2) for P = P_{v}. Firstly, the embedding F_{c} ↪ (F_{c})_{P′_{v}} = F_{P} = Q_{p} gives a canonical isomorphism H(F_{P}) ⟶^{∼} GL(n, F_{P}) × F_{P}^{×} under which μ_{P} can be chosen as μ_{P} (z) = ((zI_{a}, I_{b}), z), where a = a_{v} and b = b_{v} = n − a. Secondly, the number ⟨2ρ_{P} , μ_{P} ⟩ is equal to the dimension of the symmetric space for SU(a, b), namely, to ab = a(n − a). Thirdly, V_{μ_{P}^{−1}} is the

representation (⋀^{a} Std_{n}^{∨}) ⊗ Std_{1}^{∨} of the dual group { \atop GL(n)} × Ĝ_{m} = GL(n, C) × C^{×}.

Write

L_{P′_{v}}(ρ_{Π}, s) = ∏_{i=1}^{n}(1 − α_{i}p^{−s})^{−1};

then

L_{P′_{v}}(Π, Std_{n}, s) = ∏_{i=1}^{n}(1 − t_{i}p^{−s})^{−1}, t_{i} = p^{(1−n)/2}α_{i}.

The Satake parameters of Π_{P′_{v}} are given, therefore, by the semisimple element t = diag(t_{1}, …, t_{n}) of the dual GL(n). Similarly, the Satake parameter of ψ_{P′_{v}} is equal to u = ρ_{ψ}(Fr(P′_{v})). By definition (and by the fact that π_{P} ≃ (Π, ψ)_{P′_{v}}),

H̃_{μ_{P}^{−1}}(X)|_{π_{P}^{K_{P}}} = det (X − (t, u)|(⋀^{a} Std_{n}^{∨}) ⊗ Std_{1}^{∨})) = ∏_{|I|=a}(X − t_{I}^{−1}u^{−1}) = ∏_{|I|=a}(X − p^{(n−1)a/2}α_{I}^{−1}u^{−1}),

where I ⊂ {1, …, n}, t_{I} = ∏_{i∈I} t_{i} and α_{I} = ∏_{i∈I} α_{i}. This implies that

p^{C} H̃_{μ_{P}^{−1}}(p^{−a(n−a)/2}X)|_{π_{P}^{K_{P}}} = det (X − Fr(P′_{v})|(⋀^{a}ρ_{Π}^{∨}) ⊗ ρ_{ψ}^{∨}(a(a + 1)/2 − an)) ,  (A5.10.1)

for some C ∈ Z. Consider the representation

ρ_{v} = _{σ_{v}} ((⋀^{a_{v}}ρ_{Π}^{∨}) ⊗ ρ_{ψ}^{∨}(a_{v}(a_{v} + 1)/2 − a_{v}n)) : Gal(Q_{S}/F_{c}^{gal}) ⟶ GL_{n_{v}}(\overline{Q}_{ℓ}).

The right hand side of (A5.10.1) is then equal to det(X − ρ_{v}(Fr(P_{S}))) = P_{ρ_{v}(Fr(P_{S}))}(X) and the relation (A5.5.2) reads as follows:

P_{ρ_{v}(Fr(P_{S}))}(φ_{P_{v}} |_{V^{i}(π^{∞})⊗(π^{∞})^{K}}) = 0.  (A5.10.2)

Of course,

∏_{v|∞}(φ_{P_{v}} |_{V^{i}(π^{∞})⊗(π^{∞})^{K}}) = Fr(P_{S})|_{V^{i}(π^{∞})⊗(π^{∞})^{K}} .

(A5.11) The isotropic case. What happens in the general PEL situation of A3 if we drop the assumption made in A3.1 that the group G^{der} is anisotropic (but if we keep the assumptions from A3.2 and A3.4)? If G^{der} is isotropic, then Sh_{K}(G, 𝒳 ) is no longer proper over E and the discussion in A5.5 needs to be modified as follows (as explained to the author by B. Stroh).

The (pull-back of) the sheaf ℒ_{ξ,ℓ} extends to the union of the Kottwitz models M_{K^{p}} = ∐_{α∈Σ} M_{α,K^{p}} from A3.5 and there is a canonical Γ_{E} × G(Q̂)-equivariant isomorphism

H_{et}^{i}(M_{K^{p}} ⊗ \overline{Q}, pr^{∗}(ℒ_{ξ,ℓ})) ≃ H_{et}^{i}(M_{K^{p}} ⊗ \overline{k(v)}, pr^{∗}(ℒ_{ξ,ℓ}))  (A5.11.1)

([FC, Thm. VI.6.1] in the case of Siegel modular varieties, [LS, Thm. 6.1] in general). The point is that the cohomology of pr^{∗}(ℒ_{ξ,ℓ}) is, up to a Tate twist, a direct summand of the cohomology of the constant sheaf \overline{Q}_{ℓ} on a suitable Kuga-Sato variety. The integral model of the Kuga-Sato variety over M_{K^{p}} admits a smooth toroidal compactification whose boundary is a relative normal crossing divisor, which means that the general result of [SGA 4\frac{1}{2}, Th. de finitude, App. 1.3.3(i)] applies.

Passing to Δ-invariants, one obtains from (A5.11.1) an isomorphism

H_{et}^{i}(Sh_{K} ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) ≃ H_{et}^{i}(S_{K} ⊗ \overline{k(v)}, ℒ_{ξ,ℓ}).  (A5.11.2)

Under the assumption (A4.3.1) the arguments of [W] establish the existence of the diagram (A5.1.1) even in the isotropic case. Hecke correspondences on Sh_{K} (resp. the p-isogenies on M_{K^{p}}) are proper correspondences in the sense that their projections on each of the factors in Sh_{K} × Sh_{K} (resp. M_{K^{p}} × M_{K^{p}}) are proper (and generically finite). As explained in [FC, VII.2], one can generalise the definitions recalled in A2 to this situation and define

• an action of H(G(Q_{p})_{−}//K_{p}, Q) on H_{et}^{i}(Sh_{K} ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ});

• an action of Q[p − Isog_{K^{p}} ⊗ E_{v}] on H_{et}^{i}(M_{K^{p}} ⊗ \overline{Q}, pr^{∗}(ℒ_{ξ,ℓ}));

• an action of Q[p − Isog_{K^{p}} ⊗ F_{p}] on H_{et}^{i}(M_{K^{p}} ⊗ \overline{k(v)}, pr^{∗}(ℒ_{ξ,ℓ}));

as in A5.5, these actions are compatible with the isomorphisms (A5.11.1-2). As a result, the Eichler-Shimura relation (A5.5.1) holds (if p splits completely in F_{c}/Q) for the action of φ_{P} on H^{i}(Sh_{K}) = H_{et}^{i}(Sh_{K} ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}).

(A5.12) The decompositions (0.3.2) and (0.4.1) no longer hold in the isotropic case. For any irreducible smooth representation π^{∞} of G(Q̂) one can consider the π^{∞}-eigenspace in H^{i}(Sh) = \underrightarrow{lim}_{K} H^{i}(Sh_{K}), namely,

H^{i}(Sh)[π^{∞}] = Im(V^{i}(π^{∞}) ⊗ π^{∞} ↪ H^{i}(Sh)),  V^{i}(π^{∞}) = Hom_{G(_{Q})}(π^{∞}, H^{i}(Sh)).  (A5.12.1)

In general, the action of G(Q̂) on H^{i}(Sh) is not semisimple (as pointed out by the referee, this happens already for G = GL(2)_{Q}, i = 1 and ξ = 1, when some π^{∞} can occur as a subquotient but not as a submodule). This means that, a priori, H^{i}(Sh)[π^{∞}] could be smaller than the corresponding generalised eigenspace (in other words, the space Hom_{G(_{Q})}(π^{∞}, H^{i}(Sh)/H^{i}(Sh)[π^{∞}]) could be non-zero).

As in the isotropic case, V^{i}(π^{∞}) is of finite dimension over \overline{Q}_{ℓ} and the action of Γ_{E} on H^{i}(Sh) gives rise to a representation

Γ_{E} ⟶ Aut_{G(_{Q})}(H^{i}(Sh)[π^{∞}]) = Aut_{\overline{Q}_{ℓ}}(V^{i}(π^{∞})).

If K = K_{p}K^{p} is as in A5.11 (with p split in F_{c}/Q) and (π^{∞})^{K} ≠ 0, then the G(Q̂)-equivariance of the φ_{P} ’s implies again that the subspace V^{i}(π^{∞}) ⊗ (π^{∞})^{K} = H^{i}(Sh)[π^{∞}]^{K} ⊂ H^{i}(Sh)^{K} = H^{i}(Sh_{K}) is φ_{P} -stable and that each φ_{P} acts on it through an action φ_{P} ∈ Aut_{\overline{Q}_{ℓ}}(V^{i}(π^{∞})) on V^{i}(π^{∞}).

As in A5.5, restricting (A5.5.1) to H^{i}(Sh)[π^{∞}]^{K} yields the formula (A5.5.2) for the space V^{i}(π^{∞}) defined in (A5.12.1).

An important subspace of V^{i}(π^{∞}) arises as follows. The analytic intersection cohomology of the Baily-Borel compactification j : Sh_{K}(G, 𝒳 ) ↪ Sh_{K}(G, 𝒳 )_{BB} is isomorphic to the L^{2}-cohomology of Sh_{K}(G, 𝒳 )^{an} ([Lo], [SS]) and the latter space admits a decomposition analogous to (0.3.2), by [BC, Th. A] combined with [BG, Prop. 5.6]. As a result, there is a Γ_{E} × G(Q̂)-equivariant decomposition of the intersection étale cohomology

H^{i}(Sh_{BB}) = \underrightarrow{lim}_{K} H_{et}^{i}(Sh_{K}(G, 𝒳 )_{BB} ⊗_{E} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ})

of the form

H^{i}(Sh_{BB}) = ⨁_{π=π_{∞}⊗π^{∞}} m_{disc}(π) H^{i}(𝔤, K_{∞}; π_{∞} ⊗ ξ) ⊗ π^{∞} = ⨁_{π^{∞}} V_{disc}^{i}(π^{∞}) ⊗ π^{∞},  (A5.12.2)

where π runs through discrete automorphic representations of G(A) and m_{disc}(π) denotes the multiplicity of π′ = (ω′_{π})^{−1}π in the discrete part L_{disc}^{2}(G, ω_{π}) ⊂ L^{2}(G, ω_{π}).

In general, the canonical Γ_{E}-equivariant map V_{disc}^{i}(π^{∞}) ⟶ V^{i}(π^{∞}) induced by H^{i}(Sh_{BB}) ⟶ H^{i}(Sh) is not injective, nor surjective (cf. Proposition A6.17). It would be of interest to define compatible actions of various rings in the diagram (A5.1.1) on the intersection étale cohomology of Sh_{K}(G, 𝒳 )_{BB} (for which, again, isomorphisms analogous to (A5.11.1-2) hold, thanks to [LS, Thm. 6.1]) and deduce the formula (A5.5.2) on V_{disc}^{i}(π^{∞}). All that we can say at the moment is that \overline{V} _{disc}^{i}(π^{∞}) = Im(V_{disc}^{i}(π^{∞}) ⟶ V^{i}(π^{∞})) ⊂ V^{i}(π^{∞}) is Γ_{E}-stable and that, for P_{S} as in (A5.9.1), the action of Fr(P_{S}) on \overline{V} _{disc}^{i}(π^{∞}) is given by the restriction of the action of ∏_{P |p} φ_{P} ∈ Aut(V^{i}(π^{∞})), with each φ_{P} satisfying (A5.5.2).

This issue does not arise for the contribution of cuspidal representations π to (A5.12.2), since cuspidal cohomology injects into H^{i}(Sh).

A6. Quaternionic Shimura varieties

(A6.1) Throughout A6 we assume that F ⊂ \overline{Q} ⊂ C, r = [F : Q], D, Ω, t = |Ω|, E and ξ are as in 5.1-5.5. In A6.1–A6.14 we assume that D ≠ M_{2}(F). The Shimura varieties involved and their integral models will then all be proper.

Denote by d ↦ \overline{d} the main involution on D. Let v | p be the prime of E induced by a fixed embedding \overline{Q} ↪ \overline{Q}_{p}. Let K = K_{S}K^{S} ⊂ (D ⊗ Q̂)^{×} be an open compact subgroup as in 5.15 (with S containing all primes of F dividing 2ℓ∞ and all primes at which F/Q and D ramify).

The action of Γ_{E} on H_{ξ,K}^{i} = H_{et}^{i}(Sh_{K}(D^{×}) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}) then factors through Gal(Q_{S}/E). The goal of A6 is to verify the Eichler-Shimura relation (5.16.1) for Ẽ = F^{gal}, namely, that for every prime P_{S} of Q_{S} satisfying Fr_{Q_{S}/Q}(P_{S}) ∈ Gal(Q_{S}/F^{gal}) ( ⟺ p = P_{S} ∩ Z does not lie below S and p splits completely in F/Q) the action of Fr(P_{S}) on H_{ξ,K}^{i} can be written as

Fr(P_{S})|_{H_{ξ,K}^{i}} = ∏_{x∈Ω} φ_{x}^{∗},  φ_{x}^{∗}φ_{y}^{∗} = φ_{y}^{∗}φ_{x}^{∗},  (φ_{x}^{∗})^{2} − (T_{P_{x}}/S_{P_{x}})φ_{x}^{∗} + p/S_{P_{x}} = 0.  (A6.1.1)

These relations follow directly from Theorem A5.3 if t = r (see A6.4), but require an auxiliary unitary Shimura variety if t < r (see A6.14). (A6.2) The PEL data in the case t = r. In the totally indefinite case D ⊗ R ≃ M_{2}(R)^{r} one only needs to use the fact that the main involution A ↦ \overline{A} = Tr(A) · I − A on M_{2}(R) is not positive, but is conjugate

to the positive involution A ↦ A^{t} by the matrix ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ).

Consider the following PEL data of type (C), in the sense of A3: B = D, ∗ = a positive involution on D ( ⟺ d^{∗} = u\overline{d}u^{−1} for some u ∈ D^{×}, unique up to F^{×}, such that \overline{u} = −u and Nrd(u) = −u^{2} ∈ F_{+}^{×}), V = D with a left action of D given by d·x = xd^{∗}. The F -bilinear form ⟨x, y⟩_{F} = Trd(xu\overline{y}) on V is skew-symmetric and satisfies ⟨d · x, y⟩_{F} = ⟨x, d^{∗} · y⟩_{F}.

The centraliser C = End_{B}(V) is isomorphic to D, with d ∈ D acting by left multiplication x ↦ dx. As ⟨dx, y⟩_{F} = ⟨x, \overline{d}y⟩_{F}, the involution # on C coincides with the main involution on D and

GSp_{B}(V, ⟨ , ⟩_{F}) = D^{×}, ν = Nrd.

A morphism h : C ⟶ C ⊗ R = D ⊗ R as in A3.2 is given, for example, by h(i) = u/\sqrt{−u^{2}} (with a totally positive square root), hence is conjugate to the one in (5.3.1). This identifies Sh(D^{×}) with the Shimura variety attached to the above PEL data (B, ∗, V, ⟨ , ⟩_{F}) of type (C). (A6.3) Given a rational prime p which does not lie below S, we can choose u ∈ D^{×} in A6.2 in such a way that u^{2} ∈ (O_{F} ⊗ Z_{p})^{×}, which implies that unramified local data O_{B} and Λ as in A3.4 exist. We can assume that K_{S} is sufficiently small; then the construction from A3.7 yields a smooth projective model S_{K} of Sh_{K}(D^{×}) over O_{E_{v}}, to which the local system ℒ_{ξ,ℓ} naturally extends. Let S_{K}^{∘} = S_{K} ⊗ k(v) be the special fibre of S_{K}. (A6.4) The Eichler-Shimura relation in the case t = r. In the situation of A6.3, assume, in addition, that p = P_{S} ∩ Z, where P_{S} is a prime of Q_{S} such that Fr_{Q_{S}/Q}(P_{S}) ∈ Gal(Q_{S}/F^{gal}). This is equivalent to requiring p to split completely in F/Q; in fact, pO_{F} = ∏_{x∈X} P_{x}, where P_{x} are as in (5.15.3) (and F_{P_{x}} = Q_{p}).

The absolute Frobenius morphism φ : S_{K}^{∘} = S_{K} ⊗ F_{p} ⟶ S_{K}^{∘} decomposes as a product of mutually commuting partial Frobenius morphisms φ = ∏_{x∈X} φ_{P_{x}} defined in A3.12. The action of Fr(P_{S}) on H^{i} = H_{et}^{i}(S_{K}^{∘} ⊗ \overline{F}_{p}, ℒ_{ξ,ℓ}) is given by the action (φ ⊗ id)^{∗} of φ ⊗ id : S_{K}^{∘} ⊗ \overline{F}_{p} ⟶ S_{K}^{∘} ⊗ \overline{F}_{p}. It follows that

Fr(P_{S})|_{H^{i}} = ∏_{x∈X} φ_{x}^{∗},  φ_{x}^{∗} = (φ_{P_{x}} ⊗ id)|_{H^{i}},  φ_{x}^{∗}φ_{y}^{∗} = φ_{y}^{∗}φ_{x}^{∗}.

Theorem A5.3 applies to each φ_{P_{x}} and yields, thanks to (A1.6.1), the sought for Eichler-Shimura relation (φ_{x}^{∗})^{2} − (T_{P_{x}}/S_{P_{x}})φ_{x}^{∗} + p/S_{P_{x}} = 0 ∈ End(H^{i}). (A6.5) The auxiliary PEL data in the case 1 ≤ t < r. In this case the quaternionic Shimura variety Sh(D^{×}) is not of the form considered in A3-A5, but it can be related to other Shimura varieties defined in terms of suitable PEL data of type (A) (and signature (1, 1)^{t} × (2, 0)^{r−t}). The following construction, which differs from the standard one ([De1, §6], [R, p. 11]), was communicated to the author by C. Cornut.

Fix injections F ↪ F_{c} ↪ D, where F_{c} is a CM field and [F_{c} : F] = 2. Fix elements η ∈ F_{c}^{×} and j ∈ D^{×} such that η^{∗} = −η, j^{∗} = −j, jη = η^{∗}j = −ηj, where ∗ denotes the main involution of D (whose restriction to F_{c} is positive). For every infinite prime v | ∞ of F we have j_{v}^{2} ∈ F_{v}^{×} = R^{×} and sgn(j_{v}^{2}) = +1 (resp. = −1) if v ∈ Ω (resp. if v ∈ Ω^{c}).

Consider the following PEL data of type (A) in the sense of A3: (B, ∗) = (F_{c}, ∗), V = D as a left F_{c}-module, ⟨x, y⟩_{F} = Trd(ηxy^{∗}). Explicitly,

∀x_{1}, x_{2}, y_{1}, y_{2} ∈ F_{c} ⟨x_{1} + x_{2}j, y_{1} + y_{2}j⟩_{F} = Tr_{F_{c}/F} (η(x_{1}y_{1}^{∗} − j^{2}x_{2}y_{2}^{∗})).

The morphism

τ : F_{c}^{×} × D^{×} ⟶ GL_{F_{c}}(V), (λ, d) ↦ (x ↦ λxd^{∗})

gives rise to an exact sequence of algebraic groups over F

1 ⟶ G_{m,F} ⟶^{Δ} R_{F_{c}/F} (G_{m,F_{c}}) × D^{×} ⟶^{τ} H = GSp_{B}(V, ⟨ , ⟩_{F}) ⟶ 1,  (A6.5.1)

where Δ(a) = (a, a^{−1}). Note that G_{m,F} has trivial H^{1}, which implies that the map τ in (A6.5.1) is surjective on adelic and F -rational points. (A6.6) The previous data define a CM type Φ = {σ_{v} : F_{c} ↪ Q}_{v|∞} of F_{c} characterised by σ_{v}(η) = −sgn(j_{v}^{2})|η_{v}|i. Denote by σ_{v}^{∨} : C ≃ F_{c} ⊗_{F} F_{v}(↪ D ⊗_{F} F_{v}) the maps induced by the inverse of σ_{v}. Let

h = (h_{F_{c}}, h_{D}) : C ⟶ (F_{c} ⊗ R) × (D ⊗ R)  (A6.6.1)

be the morphism whose components h_{v} : C ⟶ (F_{c} ⊗_{F} F_{v}) × (D ⊗_{F} F_{v}) are given as follows: h_{v} = (1, σ_{v}^{∨}) (resp. h_{v} = (σ_{v}^{∨}, 1)) if v ∈ Ω (resp. if v ∈ Ω^{c}).

Explicitly,

h_{v}(i)(x_{v} + y_{v}j_{v}) = \frac{η_{v}}{|η_{v}|} (x_{v} − y_{v} sgn(j_{v}^{2})j_{v}),

∀x_{1}, x_{2}, y_{1}, y_{2} ∈ F_{c} ⊗ R ⟨x_{1} + x_{2}j, h(i)(y_{1} + y_{2}j)⟩_{F} = ∑_{v|∞} |η_{v}|Tr_{F_{c}⊗_{F} F_{v}/F_{v}}(x_{1,v}y_{1,v}^{∗} + |j_{v}^{2}|x_{2,v}y_{2,v}^{∗}),

which means that h satisfies the positivity property from A3.2. Moreover, h_{D} is conjugate to the morphism (5.3.1) and

V_{v}^{−1,0} = \{ \begin{matrix}\overline{F_{c} ⊗_{F} F_{v}} + (F_{c} ⊗_{F} F_{v})j_{v} & v ∈ Ω, \\ (F_{c} ⊗_{F} F_{v}) + (F_{c} ⊗_{F} F_{v})j_{v} & v ∈ Ω^{c}, \end{matrix}

which implies that the signatures of H in the sense of A5.7 are equal to (a_{v}, b_{v}) = (1, 1) (resp. (2, 0)) if v ∈ Ω (resp. if v ∈ Ω^{c}). (A6.7) The base change of the exact sequence (A6.5.1) to F_{c} becomes isomorphic, via (A5.7.1), to

1 ⟶ G_{m,F_{c}} ⟶^{Δ_{c}} G_{m,F_{c}} × G_{m,F_{c}} × GL_{F_{c}}(V) ⟶^{τ_{c}} GL_{F_{c}}(V) × G_{m,F_{c}} ⟶ 1,  (A6.7.1)

where Δ_{c}(a) = (a, a, a^{−1}) and τ_{c}(a_{1}, a_{2}, g) = (a_{1}g, a_{1}a_{2} det(g)). Above, we have identified G_{m,F_{c}} ⊗_{F} F_{c} with G_{m,F_{c}} × G_{m,F_{c}} in the usual way: for every F_{c}-algebra R, the corresponding isomorphism (F_{c} ⊗_{F} R)^{×} ⟶^{∼} R^{×} × R^{×} sends a ⊗ r to (ar, a^{∗}r).

The cocharacters

μ = (μ_{x} : G_{m,C} ⟶ H ⊗_{F,σ_{x}} C)_{x∈X} : G_{m,C} ⟶ G_{C}

attached to the morphism h from (A6.6.1) are given, up to conjugation, by

∀x ∈ X  μ_{x}(z) = \{ \begin{matrix}τ_{c}(1, 1, ( \begin{matrix}z \\ & 1 \end{matrix} )) & x ∈ Ω, \\ τ_{c}(z, 1, ( \begin{matrix}1 \\ & 1 \end{matrix} )) & x ∈ Ω^{c}. \end{matrix}  (A6.7.2)

(A6.8) Assume that v is a finite prime of F at which D splits and which splits in F_{c}/F as vO_{F_{c}} = v′v′′. Choose one of the factors (say, v′) and identify F_{v} as an F_{c}-algebra via F_{c} ↪ (F_{c})_{v′} = F_{v}. The sequence (A6.7.1) then gives

1 ⟶ G_{m,F_{v}} ⟶ G_{m,(F_{c})_{v′}} × G_{m,(F_{c})_{v′′}} × GL(2)_{F_{v}} ⟶^{τ_{c}} H ⊗_{F} F_{v} ⟶ 1,  (A6.8.1)

with H ⊗_{F} F_{v} split over F_{v}.

Consider a cocharacter μ : G_{m,F_{v}} ⟶ H ⊗_{F} F_{v} given by one of the formulas in (A6.7.2). The corresponding Hecke polynomial of μ^{−1} is as follows.

(Case Ω): If μ(z) = τ_{c}(1, 1, ( \begin{matrix}z \\ & 1 \end{matrix} )), then (A1.6.1) implies that

(Nv) H̃_{μ^{−1}}((Nv)^{−1/2}Y) = Y^{2} − (T_{v}/S_{v})Y + (Nv)/S_{v}.  (A6.8.2)

(Case Ω^{c}): If μ(z) = τ_{c}(z, 1, I_{2}), then (A1.5.1) implies that

H̃_{μ^{−1}}(Y) = Y − 1/S_{v′},  S_{v′} = char(ϖ_{v′}O_{(F_{c})_{v′}}^{×}) ∈ H((F_{c})_{v′}^{×}//O_{(F_{c})_{v′}}^{×} , Z).  (A6.8.3)

(A6.9) Quaternionic and unitary Shimura data [R, §1]. Consider the following algebraic groups over F: F_{c}^{×} = R_{F_{c}/F} G_{m,F_{c}}, D^{×} and τ : (F_{c}^{×} × D^{×})/Δ(F^{×}) ⟶^{∼} H = GSp_{F_{c}}(V). Their respective restrictions of scalars to Q (notably G = R_{F/Q}(H)) are equipped with the Shimura data h_{F_{c}}, h_{D} (conjugate to the one from (5.3.1)) and h_{G} = τ_{R} ∘ h, where h = h_{F_{c}} × h_{D} was defined in (A6.6.1). Their reflex fields are equal to

E(G, h_{G}) = E(F_{c}^{×}, h_{F_{c}}) = E(Φ|Ω^{c}) = {γ ∈ Γ_{Q} | ∀v ∈ Ω^{c} γσ_{v} = σ_{v}} ⊃ E(D^{×}, h_{D}) = E.

The morphism τ induces a map

Sh(F_{c}^{×}) × (Sh(D^{×}) ⊗_{E} E(Φ|Ω^{c})) ⟶ Sh(G),

which is F̂_{c}^{×} × D̂-equivariant (in particular, Δ(F̂^{×}) acts along its fibres).

If K ⊂ D̂^{×} is an open compact subgroup which is small enough in the sense that K = K_{q}K^{q} for a prime q such that K_{q} ∩ (F ⊗ Q_{q})^{×} ⊂ 1 + q^{e}(O_{F} ⊗ Z_{q}), where e = 1 if q > 2 (resp. e = 2 if q = 2), then there exist open compact subgroups K(F_{c}) ⊂ F̂_{c}^{×} (defined in [R, (1.4)]) and K(G) = τ(K(F_{c}) × K) ⊂ G(Q̂) such that

Sh_{K(F_{c})}(F_{c}^{×}) × (Sh_{K}(D^{×}) ⊗_{E} E(Φ|Ω^{c})) ⟶ Sh_{K(G)}(G)

is a Galois covering with Galois group Δ(F^{×}\F̂^{×}/(K ∩ F̂^{×})). In particular, its fibres are the Δ(F̂^{×})-orbits. (A6.10) The Künneth formula. Fix an algebraic representation ξ′ of (_{F_{c}}T)_{C} whose restriction to (_{F} T)_{C} coincides with ω_{ξ} (= N_{F/Q}^{2−w}); then ξ′ ⊗ ξ = ξ_{G} ∘ τ for an algebraic representation ξ_{G} of G_{C}. The Künneth formula combined with the discussion in the previous paragraph implies that the cohomology groups

H_{ξ}^{∗} = H_{et}^{∗}(Sh(D^{×}) ⊗_{E} \overline{Q}, ℒ_{ξ,ℓ}), H_{ξ′}^{∗} = H_{et}^{∗}(Sh(F_{c}^{×}) ⊗_{E(Φ|Ω^{c})} \overline{Q}, ℒ_{ξ′,ℓ}),
H_{ξ_{G}}^{∗} = H_{et}^{∗}(Sh(G) ⊗_{E(Φ|Ω^{c})} \overline{Q}, ℒ_{ξ_{G},ℓ})

are related as follows (note that Sh(F_{c}^{×}) has dimension zero):

H_{ξ_{G}}^{i} ⟶^{∼} (H_{ξ′}^{0} ⊗ H_{ξ}^{i})^{Δ(_{F^{×}})} .  (A6.10.1)

Assume that π is as in 5.11 and 5.15: it is an automorphic representation of D_{A}^{×} such that π_{∞} is cohomological (in degree i) with respect to ξ. Fix an open compact subgroup K ⊂ D̂^{×} such that (π^{∞})^{K} ≠ 0. As in 5.15,

0 ≠ V^{i}(π^{∞}) ⊗ (π^{∞})^{K} ⊂ H_{ξ,K}^{i} = (H_{ξ}^{i})^{K},

with F̂^{×} acting on this subspace by ω_{π^{∞}}.

Assume, furthermore, that χ : A_{F_{c}}^{×}/F_{c}^{×} ⟶ C^{×} is a character such that

χ_{∞} = (ξ′)^{−1},  χ|_{A_{F}^{×}} = ω_{π}  (A6.10.2)

(these two conditions are compatible, since ω_{π_{∞}} = ω_{ξ}^{−1}). Denote by χ^{∞} : F̂_{c}^{×} ⟶ C^{×} ≃ \overline{Q}_{ℓ}^{×} its finite part. If K(F_{c}) ⊂ F̂_{c}^{×} is sufficiently small in the sense that

F_{c}^{×} ∩ K(F_{c}) ⊂ O_{F,+}^{×},  K(F_{c}) ⊂ Ker(χ^{∞}),  (A6.10.3)

then the χ^{∞}-eigenspace

V (χ^{∞}) = V^{0}(χ^{∞}) = {f ∈ H_{ξ′}^{0} | ∀a ∈ F̂_{c}^{×} a · f = χ^{∞}(a)f}

satisfies dim(V (χ^{∞}) ⊗ (χ^{∞})^{K(F_{c})}) = 1.

The representation χ ⊗ π of A_{F_{c}}^{×} × D_{A}^{×} is of the form χ ⊗ π = π_{G} ∘ τ, where π_{G} is an automorphic representation of G(A) = H(A_{F}), with (π_{G})_{∞} cohomological (in degree i) with respect to ξ_{G}. The previous discussion combined with (A6.10.1) implies that, for K sufficiently small,

V^{i}(π_{G}^{∞}) ⊗ (π_{G}^{∞})^{K(G)} = V (χ^{∞}) ⊗ V^{i}(π^{∞}) ⊗ (χ^{∞} ⊗ (π^{∞})^{K})^{Δ(_{F^{×}})} = V (χ^{∞}) ⊗ V^{i}(π^{∞}) ⊗ (χ^{∞} ⊗ (π^{∞})^{K}) ≠ 0,

(A6.10.4)

hence

V^{i}(π_{G}^{∞}) = V (χ^{∞}) ⊗ V^{i}(π^{∞}) ≠ 0.

In particular, there are canonical identifications

End(V^{i}(π_{G}^{∞})) = End(V^{i}(π^{∞})),  End(V^{i}(π_{G}^{∞}) ⊗ (π_{G}^{∞})^{K(G)}) = End(V^{i}(π^{∞}) ⊗ (π^{∞})^{K}).  (A6.10.5)

(A6.11) The partial Frobenii. Assume that K = K_{S}K^{S} ⊂ D̂^{×} is as in 5.15 and that K_{S} is sufficiently small. Let p be a rational prime which does not lie below S and which satisfies the following two conditions: η^{2}, j^{2} ∈ (O_{F} ⊗ Z_{p})^{×} and every prime P | p above p in F splits in F_{c}/F: P O_{F_{c}} = P′P′′.

These conditions imply that there exist unramified data O_{B} and Λ at p (as in A3.4) for the PEL data from A6.5. As before, this gives a smooth projective model S_{K(G)} of Sh_{K(G)}(G) over the ring of integers of E(Φ|Ω^{c})_{v}, where v | p (note that K(G)_{p} ≃ τ((O_{F_{c}} ⊗ Z_{p})^{×}, GL_{2}(O_{F} ⊗ Z_{p})) in this case; cf. [R, (1.4)]).

Moreover, the construction in A3.12 defines, for primes P of F above p, partial Frobenius morphisms φ_{P} : S_{K(G)}^{∘} = S_{K(G)} ⊗ k(v) ⟶ S_{K(G)}^{∘} satisfying φ = ∏_{P |p} φ_{P}.

(A6.12) The Eichler-Shimura relation for the unitary Shimura variety. Denote by S_{c} the following finite set of primes of F: S_{c} = S ∪ {v | v ramified in F_{c}/F or ord_{v}(η^{2}) ≠ 0 or ord_{v}(j^{2}) ≠ 0} and assume that P_{S_{c}} is a finite prime of Q_{S_{c}} ⊂ \overline{Q} such that

Fr_{Q_{Sc} /Q}(P_{S_{c}}) ∈ Gal(Q_{S_{c}}/F_{c}^{gal}).  (A6.12.1)

Extend each σ_{x} : F_{c} ↪ \overline{Q} (x ∈ X) to an element σ̃_{x} ∈ Γ_{Q}. As in A5.9 (and 5.15) we obtain primes P′_{x} = σ_{x}^{−1}P_{S_{c}} ∩ O_{F_{c}} and P_{x} = σ_{x}^{−1}P_{S_{c}} ∩ O_{F} of F_{c} and F, respectively, which depend only on σ_{x} and lie above a rational prime p satisfying the conditions from A6.11. Moreover, (A6.12.1) implies that p splits completely in F_{c}/Q:

pO_{F} = ∏_{x∈X} P_{x},  P_{x}O_{F_{c}} = P′_{x}P′′_{x},  Q_{p} = F_{P_{x}} = (F_{c})_{P′_{x}} = E(Φ|Ω^{c})_{v}.

The discussion from A6.8 applies to each v = P_{x} and v′ = P′_{x}. After identifying H ⊗_{F} F_{P_{x}} with GL(2)_{Q_{p}} × GL(1)_{Q_{p}} as in (A6.8.1), the cocharacters μ_{P_{x}} : G_{m,F_{Px}} ⟶ H ⊗_{F} F_{P_{x}} from A4.3 can be chosen as in A6.8, with the case Ω (resp. the case Ω^{c}) occurring if x ∈ Ω (resp. if x ∈ Ω^{c}).

Theorem A5.3 applies to the action of each partial Frobenius φ_{P_{x}} : S_{K(G)}^{∘} = S_{K(G)} ⊗ F_{p} ⟶ S_{K(G)}^{∘} on H_{ξ_{G},K(G)}^{i} = (H_{ξ_{G}}^{i})^{K(G)} = H_{et}^{i}(S_{K(G)}^{∘} ⊗ \overline{F}_{p}, ℒ_{ξ_{G},ℓ}) and yields, thanks to (A6.8.2) and (A6.8.3),

∀x ∈ Ω  Q_{x}(φ_{P_{x}} ⊗ id)|_{H_{ξ_{G},K(G)}^{i}} = 0,  Q_{x}(Y) = Y^{2} − (T_{P_{x}}/S_{P_{x}})Y + p/S_{P_{x}}. (A6.12.2)
∀x ∈ Ω^{c}  (φ_{P_{x}} ⊗ id − 1/S_{P′_{x}})|_{H_{ξ_{G},K(G)}^{i}} = 0.  (A6.12.3)

The action of Fr_{Q_{Sc} /Q}(P_{S_{c}}) on H_{ξ_{G},K(G)}^{i} is given by the action of φ ⊗ id = ∏_{x∈X}(φ_{P_{x}} ⊗ id). We have H_{ξ_{G},K(G)}^{i} = ⨁ V^{i}(π_{G}^{∞}) ⊗ (π_{G}^{∞})^{K(G)}. If

V^{i}(π_{G}^{∞}) ⊗ (π_{G}^{∞})^{K(G)} ≠ 0,  (A6.12.4)

then π_{G} ∘ τ = χ ⊗ π, where χ and π are as in (A6.10.4). For all x ∈ X, the action of φ_{P_{x}} ⊗ id on V^{i}(π_{G}^{∞}) ⊗ (π_{G}^{∞})^{K(G)} ⊂ H_{ξ_{G},K(G)}^{i} defines, via the identification (A6.10.5), elements

φ_{x}^{∗} ∈ End(V^{i}(π^{∞}) ⊗ (π^{∞})^{K}), φ_{x}^{∗}φ_{y}^{∗} = φ_{y}^{∗}φ_{x}^{∗}

satisfying

∀x ∈ Ω Q_{x}(φ_{x}^{∗}) = 0,  ∀x ∈ Ω^{c} φ_{x}^{∗} = χ^{∞}(P′_{x})^{−1}.  (A6.12.5)

On the other hand, the action of g = Fr(P_{S_{c}}) satisfies

∏_{x∈X}(φ_{P_{x}} ⊗ id)|_{V^{i}(π_{G}^{∞})⊗(π_{G}^{∞})^{K(G)}} = g|_{V^{i}(π_{G}^{∞})⊗(π_{G}^{∞})^{K(G)}} = g|_{V (χ^{∞})} ⊗ g|_{V^{i}(π^{∞})⊗(π^{∞})^{K}} ,

g|_{V (χ^{∞})} = ∏_{x∈Ω^{c}} χ^{∞}(P′_{x})^{−1}

(the last equality holds thanks to the reciprocity map for the Shimura variety Sh(F_{c}^{×}) [R, p. 10]), which implies, thanks to (A6.12.5), that

∏_{x∈Ω} φ_{x}^{∗} = Fr(P_{S_{c}})|_{V^{i}(π^{∞})⊗(π^{∞})^{K}} .  (A6.12.6)

One can show that, under the above assumptions, each φ_{x}^{∗} (x ∈ Ω) is induced by a geometric morphism φ_{x} : S_{K}^{∘} ⊗ \overline{F}_{p} ⟶ S_{K}^{∘} ⊗ \overline{F}_{p} (using the discussion in [R, p. 48-49]), but we are not going to use this fact. (A6.13) It is convenient to carry out the constructions in A6.12 for K = K_{S} = K^{S} = K_{p}K^{p} (where p ∉ S_{c} is fixed) and then pass to the limit K^{p} ⟶ {1} (in other words, replace K^{p} by an arbitrarily small open subgroup K^{′p} ⊂ K^{p} and work with all groups of the form K′ = K_{p}K^{′p} ⊂ K).

This yields, for all x ∈ Ω, G(Q̂^{(p)})-equivariant automorphisms φ_{x}^{∗} of the space

(H_{ξ_{G}}^{i})^{K_{p}(F_{c})×K_{p}} = (H_{ξ′}^{0} ⊗H_{ξ}^{i})^{Δ(_{F^{×}})·(K_{p}(F_{c})×K_{p})} = ⨁_{χ,π}(V (χ^{∞})⊗(χ^{∞})^{K_{p}(F_{c})})⊗V^{i}(π^{∞})⊗(π^{∞})^{K_{p}}, (A6.13.1)

where χ : A_{F_{c}}^{×}/F_{c}^{×} ⟶ C^{×} is unramified at all primes above p, χ_{∞} = (ξ′)^{−1}, π is an automorphic representation of D_{A}^{×} such that π_{∞} is cohomological in degree i with respect to ξ (⟹ ω_{π_{∞}} = ω_{ξ}^{−1}), π_{p} = ⊗_{x∈X}π_{P_{x}} is unramified and χ_{A_{F}^{×}} = ω_{π}.

As

dim V (χ^{∞}) ⊗ (χ^{∞})^{K_{p}(F_{c})} = dim V (χ^{∞}) ⊗ χ^{∞} = 1,

G(Q̂^{(p)})-equivariance of φ_{x}^{∗} together with irreducibility of π^{∞} and Schur’s Lemma imply (as in A5.5) that

φ_{x}^{∗} ∈ End_{G(_{Q^{(p)}})}((V (χ^{∞}) ⊗ (χ^{∞})^{K_{p}(F_{c})}) ⊗ V^{i}(π^{∞}) ⊗ (π^{∞})^{K_{p}}) =

= End_{(D⊗_{Q^{(p)}})^{×}}(V^{i}(π^{∞}) ⊗ (π^{∞})^{K_{p}}) = End V^{i}(π^{∞}).

We have not shown that φ_{x}^{∗} ∈ End V^{i}(π^{∞}) is independent of χ. Such an independence follows from the geometric description of φ_{x}^{∗} that was alluded to at the end of A6.12, but we do not need it for the applications in 5.18–5.22.

We deduce from (A6.12.5) and (A6.12.6) that

∀x ∈ Ω Q_{x}(φ_{x}^{∗}) = 0 ∈ End V^{i}(π^{∞}),  ∏_{x∈Ω} φ_{x}^{∗} = Fr(P_{S_{c}}) ∈ End V^{i}(π^{∞}).  (A6.13.2)

(A6.14) The Eichler-Shimura relation for the quaternionic Shimura variety. The relations (A6.13.2) give what we need for (5.16.1) in the case t < r, provided we can reverse the above arguments and find F_{c} ↪ D and χ for which (A6.12.4) holds.

In order to do that, assume that we are given K = K_{S}K^{S} as in 5.15 (with K_{S} sufficiently small) and a prime P_{S} of Q_{S} satisfying (5.15.2) with Ẽ = F^{gal} (which implies that the rational prime p below P_{S} splits completely in F/Q). There are infinitely many totally imaginary quadratic extensions F_{c}/F such that all primes of F above p (resp. all primes at which D is ramified) are split (resp. inert or ramified) in F_{c}/F. Fix such an extension F_{c} and embeddings F ↪ F_{c} ↪ D (they exist, by construction). There also exist elements η, j ∈ D as in A6.5 satisfying η^{2}, j^{2} ∈ (O_{F} ⊗ Z_{p})^{×}. By construction, p splits completely in F_{c}/Q, which implies that the prime P_{S} extends to a prime P_{S_{c}} satisfying (A6.12.1). Fix ξ′ as in A6.10.

Finally, for every automorphic representation π = π_{∞} ⊗ π^{∞} of D_{A}^{×} such that

0 ≠ V^{i}(π^{∞}) ⊗ (π^{∞})^{K} ⊂ H_{ξ,K}^{i} = (H_{ξ}^{i})^{K},  (A6.14.1)

Proposition A6.15 below implies that there exists a character χ : A_{F_{c}}^{×}/F_{c}^{×} ⟶ C^{×} which is unramified at all primes above p and which satisfies χ|_{A_{F}^{×}} = ω_{π} and χ_{∞} = (ξ′)^{−1}. The pair (χ, π) then contributes to the sum (A6.13.1) and the arguments in A6.13 give mutually commuting elements (for x ∈ Ω)

φ_{x}^{∗} ∈ End V^{i}(π^{∞}),  ∀x ∈ Ω Q_{x}(φ_{x}^{∗}) = 0,  ∏_{x∈Ω} φ_{x}^{∗} = Fr(P_{S})|_{V^{i}(π^{∞})}.  (A6.14.2)

As a result, (5.6.1) holds (for Ẽ = F^{gal}). The relations (A6.14.2) imply the Eichler-Shimura relation (5.16.2) for the full Frobenius, namely, that

Q(Fr(P_{S}))|_{H_{ξ,K}^{i}} = 0,  (A6.14.3)

where Q = ⨂_{x∈Ω} Q_{x} is the Rankin-Selberg polynomial defined in (5.16.3).

(A6.15) Proposition. (1) For every pair of characters α : A_{F}^{×}/F^{×} ⟶ C^{×} and β : (F_{c} ⊗ R)^{×} ⟶ C^{×} satisfying β|_{(F ⊗R)^{×}} = α_{∞} there exists a character χ : A_{F_{c}}^{×}/F_{c}^{×} ⟶ C^{×} such that χ|_{A_{F}^{×}} = α and χ_{∞} = β.

(2) For every finite set T of finite primes of F there exists χ as in (1) satisfying ord_{w}(cond(χ)) = ord_{w}(cond(α)) for all primes w of F_{c} above T. In particular, if α is unramified at v ∈ T, then χ is unramified at all w | v.

Proof. (1) Denote by C_{k} = A_{k}^{×}/k^{×} the idele class group of a number field k and by C_{k}^{1} = Ker(∥ · ∥_{k} : C_{k} ⟶ R_{+}^{×}) its compact subgroup of unit norm ideles. We have

C_{F_{c}} ⊃ C_{F} · (F_{c} ⊗ R)^{×} = C_{F} · U(1)^{r},  C_{F_{c}}^{1} ⊃ C_{F}^{1} · U(1)^{r},  C_{F} ∩ U(1)^{r} = C_{F}^{1} ∩ U(1)^{r} = {±1}^{r}.

The compactness of the groups involved implies that the product morphism m : C_{F}^{1} × U(1)^{r} ⟶ C_{F_{c}}^{1} (m(x, y) = xy) is strict: it induces a topological isomorphism between the compact groups Coim(m) = (C_{F}^{1} × U(1)^{r})/Ker(m) with quotient topology and Im(m) = C_{F}^{1} · U(1)^{r} with topology induced from C_{F_{c}}^{1}.

The compatibility between α and β implies that the restriction of α ⊗ β : C_{F}^{1} × U(1)^{r} ⟶ U(1) to Ker(m) = {(a, −a) | a ∈ {±1}^{r}} is trivial. It follows that there exists a unique character C_{F}^{1} · U(1)^{r} ≃ Coim(m) ⟶ U(1) whose restriction to the first (resp. to the second) factor is given by α (resp. by β). Such a character extends to a character ψ : C_{F_{c}}^{1} ⟶ U(1).

Fix an infinite prime v_{∞} of F and define continuous sections

s : R_{+}^{×} ⟶ (F ⊗ R)_{+}^{×} ⟶ A_{F}^{×}, s_{c} : R_{+}^{×} ⟶ (F ⊗ R)_{+}^{×} ⟶ A_{F_{c}}^{×}

of ∥·∥_{F} and ∥·∥_{F_{c}}, respectively, by s(t)_{v} = 1 for all v ≠ v_{∞} and s(t)_{v_{∞}} = t, and s_{c}(t) = s(t^{1/2}). The formula

χ(x) = β(s_{c}(∥x∥_{F_{c}})) ψ(x/s_{c}(∥x∥_{F_{c}})), x ∈ A_{F_{c}}^{×}

then defines a character χ : C_{F_{c}} ⟶ C^{×} with the required properties.

(2) Let χ be as in (1). For each v ∈ T, the restriction χ_{v} of χ to (F_{c} ⊗_{F} F_{v})^{×} satisfies χ_{v}|_{F_{v}^{×}} = α_{v}. Recall the following elementary fact: given finite abelian groups H ⊂ G ⊃ G_{1} and a character λ : G ⟶ U(1) such that λ_{H∩G_{1}} = 1, then there exists a character λ′ : G ⟶ U(1) such that λ′|_{H} = λ|_{H} and λ′|_{G_{1}} = 1. Applying this statement to G = O_{F_{c}⊗_{F} F_{v}}^{×} ⊃ H = O_{F_{v}}^{×} (rather, to their quotients by a suitable piece of the canonical filtration on G) and λ = χ_{v} we deduce that there exists a character of finite order γ^{(v)} of (F_{c} ⊗_{F} F_{v})^{×}/F_{v}^{×} such that ord_{w}(cond(γ^{(v)}χ_{v})) = ord_{w}(cond(α)), for all w | v in F_{c}. The value γ^{(v)}(x) depends only on x^{1−c} = x/x^{c} ∈ ^{1−c}(F_{c} ⊗_{F} F_{v})^{×} ⊂ (F_{c} ⊗_{F} F_{v})^{×}, where c is the non-trivial element of Gal(F_{c}/F). The character x^{1−c} ↦ γ^{(v)}(x) of ^{1−c}(F_{c} ⊗_{F} F_{v})^{×} extends to a character of finite order δ^{(v)} of (F_{c} ⊗_{F} F_{v})^{×}, which means that γ^{(v)} = ^{1−c}δ^{(v)}. There exists a global character of finite order δ : C_{F_{c}} ⟶ C^{×} such that δ_{v} = δ^{(v)} for all v ∈ T; the character χ′ = (^{1−c}δ)χ then has the required properties. (A6.16) The Eichler-Shimura relation in the case D ≃ M_{2}(F). In this case H = GL(2)_{F} and Sh_{K} is a Hilbert modular variety of dimension t = r and reflex field E = Q. It is defined by the simplest possible PEL datum of type (C): B = F, ∗ = id, V = F^{2}, ⟨ , ⟩_{F} = the standard symplectic form.

The Baily-Borel compactification j : Sh_{K} ↪ Sh_{K,BB} = Sh_{K} ∪ {cusps} has as a boundary a reduced zero-dimensional scheme of cusps. For any ξ as in (5.5.1), the canonical G(Q̂)-equivariant map

can_{i} : H^{i}(Sh_{K,BB}^{an}) = H^{i}(Sh_{K,BB}^{an}, j_{!∗}ℒ_{ξ}) = H^{i}(Sh_{K,BB}^{an}, τ_{≤r−1}Rj_{∗}ℒ_{ξ}) ⟶ H^{i}(Sh_{K}^{an}, ℒ_{ξ}) = H^{i}(Sh_{K}^{an})

has the following property.

(A6.17) Proposition. The map can_{i} is injective in all cases except when i = 2r and k_{v} = 2 for all v | ∞.

Proof. The exact sequence

⋯ ⟶ H^{i}(Sh_{K,BB}^{an}) ⟶ H^{i}(Sh_{K}^{an}) ⟶ H^{0}({cusps}^{an}, (τ_{≥r}Rj_{∗}ℒ_{ξ})[i]) ⟶ H^{i+1}(Sh_{K,BB}^{an}) ⟶ ⋯

implies that can_{i} is an isomorphism for i < r and injective for i = r. In the decomposition (A5.12.2) of \underrightarrow{lim}_{K} H^{i}(Sh_{K,BB}^{an}), only cuspidal or one-dimensional automorphic representations π = π_{∞} ⊗ π^{∞} of GL_{2}(A_{F})

appear, with π_{∞} cohomological in degree i for ξ. For cuspidal π we have i = r, when the injectivity of can_{i} has been proved. It remains to investigate the (non-)injectivity of can_{i} (for i > r) when restricted to the cohomology classes corresponding to one-dimensional π, i.e., to the universal cohomology classes in the case when k_{v} = 2 for all v | ∞. This is done, for example, in [Fr, Lemma III.5.6] (in the classical language) or in [Ha, Prop. 3.2.4]. (A6.18) In the exceptional case i = 2r and k_{v} = 2 for all v | ∞ the corresponding étale sheaf ℒ_{ξ,ℓ} is a Tate twist of the constant sheaf \overline{Q}_{ℓ} and the étale intersection cohomology group H_{et}^{2r}(Sh_{K,BB} ⊗_{Q} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ}) is dual to a Tate twist of H_{et}^{0}(Sh_{K,BB} ⊗_{Q} \overline{Q}, j_{!∗}\overline{Q}_{ℓ}) = H_{et}^{0}(Sh_{K} ⊗_{Q} \overline{Q}, \overline{Q}_{ℓ}). (A6.19) If p is a rational prime that splits completely in F/Q and such that K = K_{p}K^{p} with K_{p} ≃ GL_{2}(O_{F} ⊗ Z_{p}), then the discussion in A5.11 implies that the Eichler-Shimura relation A6.4 holds for the action on étale cohomology of the open Hilbert modular variety H_{et}^{i}(Sh_{K} ⊗_{Q} \overline{Q}, ℒ_{ξ,ℓ}). It follows from Proposition A6.17 that these relations also hold for the action on H_{et}^{i}(Sh_{K,BB} ⊗_{Q} \overline{Q}, j_{!∗}ℒ_{ξ,ℓ}), unless we are in the exceptional case. However, A6.18 implies that they also hold in the exceptional case. (A6.20) In the case D ≃ M_{2}(F), both Proposition A6.17 and A6.18 remain valid for the Hilbert modular variety attached to (G^{∗}, 𝒳 ^{∗}) and for any ξ^{∗} from 6.2. As a result, the Eichler-Shimura relation used in the proof of Proposition 6.14 holds, thanks to the discussion in A6.19.

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