A note on the p-adic Galois representations

attached to Hilbert modular forms

Christopher Skinner

Received: October 21, 2008

Communicated by Don Blasius

Abstract. We show that the p-adic Galois representations attached to Hilbert modular forms of motivic weight are potentially semistable at all places above p and are compatible with the local Langlands correspondence at these places, proving this for those forms not covered by the previous works of T. Saito and of D. Blasius and J. Rogawski.

2000 Mathematics Subject Classification: 11F80, 11F41

1 Introduction

Let F be a totally real extension of Q of degree d. Let \overline{F} be an algebraic closure of F and let G_{F} := Gal(\overline{F} /F). Let I := Hom_{Q}(F, C) be the set of embeddings of F into C. The set I indexes the archimedean places of F. For each finite place v of F let \overline{F}_{v} be an algebraic closure of F_{v} and fix an F -embedding \overline{F} ↪ \overline{F}_{v}. These determine a choice of a decomposition group D_{v} ⊂ G_{F} for each v and an identification of D_{v} with Gal(\overline{F}_{v}/F_{v}). Let p be a rational prime and fix an algebraic closure \overline{Q}_{p} of Q_{p} and an isomorphism ι : C →^{∼} \overline{Q}_{p}. Via composition with ι the set I is identified with the embeddings of F into \overline{Q}_{p}.

Let π be a cuspidal automorphic representation of GL_{2}(A_{F}). Then π is a restricted tensor product π = ⊗′π_{v} with v running over all places of F. Assume that each π_{i}, i ∈ I, is a discrete series representation with Blattner parameter k_{i} ≥ 2 and central character x ↦ sgn(x)^{k_{i}}|x|_{i}^{−w} with w an integer independent of i. We say that π has infinity type (k, w), k := (k_{i})_{i∈I}. Assume also that each k_{i} ≡ w ( mod 2). In this case, π is an automorphic representation associated with a Hilbert modular eigenform of weight k. We recall that attached to π (and ι) is a two-dimensional semisimple Galois representation

ρ_{π} : G_{F} → GL_{2}(\overline{Q}_{p})

such that

WD(ρ_{π}|_{D_{v}})^{Fr-ss} ≅ ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}) ∀v ∤ p∞.  (1)

Here WD(σ) denotes the Weil-Deligne representation over \overline{Q}_{p} associated to a continuous representation σ : D_{v} → GL_{n}(\overline{Q}_{p}) for a place v ∤ p∞ (see [Ta, (4.2.1)]), and the superscript ‘Fr-ss’ denotes its Frobenius semi-simplification. Also, Rec_{v}(τ) denotes the Frobenius semi-simple Weil-Deligne representation over C associated with an irreducible admissible representation τ of GL_{n}(F_{v}) by the local Langlands correspondence, and ιRec_{v}(τ) is the Weil-Deligne representation over \overline{Q}_{p} obtained from Rec_{v}(τ) by change of scalars via the isomorphism ι. We choose Rec_{v} so that when n = 1, Rec_{v} is the inverse of the Artin map of local class field theory normalized so that uniformizers correspond to geometric frobenius elements. The existence of a ρ_{π} satisfying (1) was established by Carayol [Ca2], Wiles [W], Blasius and Rogawski [BR], and Taylor [Tay1], following the work of Eichler, Shimura, Deligne, Langlands, and others on the Galois representations associated with elliptic modular eigenforms.

The purpose of this note is to complete the proof of the analog of (1) at places v | p:

Theorem 1 Let v | p be a place of F. The representation ρ_{π}|_{D_{v}} is potentially semistable with Hodge-Tate type (k, w) and satisfies

WD(ρ_{π}|_{D_{v}})^{Fr-ss} ≅ ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}).  (2)

We recall that ρ_{v} := ρ_{π}|_{D_{v}} is potentially semistable if

D_{pst}(ρ_{v}) := ⋃_{L/F_{v}}(ρ_{v} ⊗_{Q_{p}} B_{st})^{Gal(\overline{F}_{v}/L)}

is a free \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}^{ur}-module of rank 2, where here L is ranging over all finite extensions of F_{v}, F_{v,0}^{ur} is the union of all absolutely unramified subfields of \overline{F}_{v}, and B_{st} is Fontaine’s ring of semistable p-adic periods (the latter has a continuous action of D_{v} = Gal(\overline{F}_{v}/F_{v}) with the property that B_{st}^{Gal(\overline{F}_{v}/L)} = L_{0}, the maximal absolutely unramified subfield of L). We also recall that the module D_{HT} (ρ_{v}) := (V ⊗_{Q_{p}} B_{HT})^{D_{v}} is a graded \overline{Q}_{p} ⊗_{Q_{p}} F_{v}-module (recall

that B_{HT} := ⊕_{n∈Z}C_{F_{v}}(n), C_{F_{v}} := ˆ \overline{F}_{v}, with the obvious action of D_{v}). By ρ_{π}|_{D_{v}} having Hodge-Tate type (k, w), we mean that for j ∈ Hom_{Q_{p}}(F_{v}, \overline{Q}_{p}) the induced graded module D_{HT} (ρ_{v}) ⊗_{\overline{Q}_{p}⊗_{Qp} F,j} \overline{Q}_{p} is non-zero in degrees (w − k_{i(j)})/2 and (w + k_{i(j)} − 2)/2, where i(j) ∈ I is the induced embedding of F into \overline{Q}_{p}. To make sense of the left-hand side of (2) we recall that Fontaine has defined an action of the Weil-Deligne group on D_{pst}(ρ_{v}). Given an embedding τ : F_{v,0}^{ur} ↪ \overline{Q}_{p} we obtain a Weil-Deligne representation over \overline{Q}_{p} on WD(ρ_{v})_{τ} := D_{pst}(ρ_{v}) ⊗_{\overline{Q}_{p}⊗_{Qp} F_{v,0}^{ur},τ} \overline{Q}_{p}. This representation is independent of τ up to equivalence, and we have denoted an element of its equivalence class by WD(ρ_{v}). The right-hand side of (2) has the same meaning as in (1).

p-adic Galois representations 243

Saito proved that Theorem 1 holds when either d is odd or there exists a finite place w such that π_{w} is square-integrable [Sa1, Sa2]; this builds on the aforementioned work of Carayol. Under the same hypotheses or when d is even and some k_{i} is strictly larger than 2, Blasius and Rogawski proved that ρ|_{D_{v}} is potentially semistable of Hodge-Tate type (k, w), and when additionally π_{p} = ⊗_{v|p}π_{v} is unramified they essentially showed that the full conclusion of the theorem holds [BR] (some additional, albeit minor, observations are required to extend their arguments to all such cases). The theorem is of course also known for those π that are the automorphic induction of a (necessarily) algebraic Hecke character of an imaginary quadratic extension of F (such representations are often called CM representations). In this case, Theorem 1 follows from the results in [Se]. These results account for the cases where ρ_{π} is known to arise from a motive; the conclusion of the theorem then follows from various deep comparison theorems between suitable cohomology theories.

It remains to deal with the cases where ρ_{π} is not known to arise from a motive, namely those cases where each k_{i} = 2, each π_{v} is a principal series representation, and π is not a CM representation. In [Tay2] it is shown that if ρ_{π} is residually irreducible and π_{v}, v|p, is unramified, then ρ_{π}|_{D_{v}} is crystalline with the predicted Hodge-Tate weights. For p > 2 unramified in F, the same result is proved in [Br] without the hypothesis that ρ_{π} be residually irreducible. For those ρ_{π} that are residually irreducible, Kisin [Ki1] deduced Theorem 1 from his results on potentially semistable deformation rings, Taylor’s construction of the representations ρ_{π}, and Saito’s results. In this paper, we prove Theorem 1 by a different approach. A simple base change argument reduces the theorem, in the cases not covered by Saito’s results, to that where d is even and each π_{v}, v|p, is unramified. From the automorphy of the symmetric square Sym^{2}π and the results of [Mo] it follows that Sym^{2}ρ_{v} is crystalline^{1} and even that WD(Sym^{2}ρ_{v}) ≅ ιRec_{v}(Sym^{2}π_{v} ⊗ | · |_{v}^{−1}). From results of Wintenberger [Win1, Win2] we then deduce that ρ_{v} is crystalline up to a (possibly trivial) quadratic twist and hence that WD(ρ_{v}) is isomorphic to a (possibly trivial) quadratic twist of ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}). There exists a suitable p-adic analytic family of eigensystems of cuspidal representations of GL_{2}(A_{F}) (essentially due to Buzzard [Bu1] in the cases needed) that contains an eigensystem attached to ρ_{π}. For members of this family with sufficiently regular weights Theorem 1 is known by the work of Blasius and Rogawski. An appeal to a result of Kisin then shows that WD(ρ_{v}) has at least one D_{v}-eigenspace predicted by (2), from which we then conclude that (2) holds.

After completing the first draft of this paper, the author learned that Tong Liu [L] has also proven Theorem 1, at least for p > 2, by an argument that is a generalization of that of Kisin [Ki1].

Acknowledgements. The author’s work on this paper was inspired by a question about what was known regarding Theorem 1 asked by Henri Darmon at the summer school on the stable trace formula, automorphic forms, and Galois representations held at BIRS in August of 2008. The referee prodded the author to write a note with more details. The author’s research is supported by grants DMS-0701231 and DMS-0803223 from the National Science Foundation and by a fellowship from the David and Lucile Packard Foundation.

1As remarked at the end of 2.4.1, a similar use of the symmetric square yields a proof of the Ramanujan conjecture for π. This conjecture has previously been established in [B2].

Documenta Mathematica 14 (2009) 241–258

2 The proof of Theorem 1

We keep to the notation from the introduction. We assume some familiarity on the part of the reader with p-adic Hodge theory, particularly the theory of Hodge-Tate weights and the notions of crystalline and semistable representations. A good reference is [Fo]. While p-adic Hodge theory is usually applied to continuous representations of Gal(\overline{F}_{v}/F_{v}), v|p, defined over a finite extension of Q_{p}, we apply it to continuous representations over \overline{Q}_{p}. This should cause no confusion as the latter are always defined over a finite extension of Q_{p}. While this is well-known, references seem rare, so we provide a quick proof.

Let Γ be a compact group and ρ : Γ → GL_{n}(\overline{Q}_{p}) a continuous representation. The subfields L of \overline{Q}_{p} that are finite over Q_{p} form a countable set, and as each GL_{n}(L) is closed in GL_{n}(\overline{Q}_{p}), the subgroups Γ_{L} := ρ^{−1}(GL_{n}(L)) form a countable set of closed subgroups of Γ whose union is Γ. Since Γ is compact, it carries a Haar measure with total measure finite and non-zero. As the countable union of measurable sets each having measure zero also has measure zero, it follows that some Γ_{L} must have non-zero measure and hence have finite index in Γ. Write Γ = ⊔_{i=1}^{m}g_{i}Γ_{L}. Then ρ takes values in GL_{n}(L′) where L′ is the finite extension of Q_{p} generated by L and the entries of the ρ(g_{i}).

2.1 Weil-Deligne representations over \overline{Q}_{p} for v|p

Let v|p be a place of F. Let B_{HT} := ⊕_{n∈Ẑ}C_{F_{v}}(n) with the obvious action of D_{v}. Let B_{cris} ⊂ B_{st} be Fontaine’s rings of crystalline and semistable padic periods, respectively. Recall that the latter are naturally F_{v,0}^{ur}-algebras

equipped with a continuous action of D_{v} such that B_{?}^{Gal(\overline{F}_{v}/L)} = L_{0} for any finite extension L/F_{v}, ? = cris, st, and that furthermore they are both equipped with a compatible F_{v,0}^{ur}-semilinear Frobenius morphism φ : B_{?} → B_{?} (that is, φ(ax) = frob_{p}(a)φ(x) for all a ∈ F_{v,0}^{ur}, where frob_{p} ∈ Gal(F_{v,0}^{ur}/Q_{p}) is the absolute arithmetic Frobenius). Additionally, B_{st} is equipped with an F_{v,0}^{ur}-linear and D_{v}-equivariant monodromy operator N : B_{st} → B_{st} such that B_{cris} = B_{st}^{N=0}.

For a finite-dimensional \overline{Q}_{p}-vector space V with a continuous \overline{Q}_{p}-linear action of D_{v} we put

D_{HT} (V) := (V ⊗_{Q_{p}} B_{HT})^{D_{v}}, D_{cris}(V) := (V ⊗_{Q_{p}} B_{cris})^{D_{v}},

and

D_{st}^{L}(V) := (V ⊗_{Q_{p}} B_{st})^{Gal(\overline{F}_{v}/L)}, D_{pst}(V) := ⋃_{L/F_{v}} D_{st}^{L}(V),

where L/F_{v} is a finite extension. Then D_{HT} (V) is a finite, graded \overline{Q}_{p} ⊗_{Q_{p}} F_{v}- module. Also, D_{cris}(V) is a finite \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}-module, D_{st}^{L}(V) is a finite \overline{Q}_{p} ⊗_{Q_{p}} L_{0}-module, and D_{pst}(V) is a finite \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}^{ur}-module, each of rank at most dim_{\overline{Q}_{p}}(V). The action of φ induces a \overline{Q}_{p}-linear, F_{v,0}-semilinear (resp. L_{0}-semilinear) Frobenius operator on D_{cris}(V) (resp. D_{st}^{L}(V)) that we also denote by φ. The action of the monodromy operator N on B_{st} induces a \overline{Q}_{p} ⊗_{Q_{p}} L_{0}-linear nilpotent operator on D_{st}^{L}(V) that we also denote by N and which satisfies N ∘ φ = pφ ∘ N. These are compatible with varying L, so φ and N are defined on D_{pst}(V) as well. Note that D_{cris}(V) = D_{st}^{F_{v}}(V)^{N=0}.

Let W_{v} ⊂ D_{v} be the Weil group of F_{v}. The action of D_{v} on V and B_{st} induces a \overline{Q}_{p}-linear, F_{v,0}^{ur}-semilinear action r_{sl} of W_{v} on D_{pst}(V). We define another action r of W_{v} on D_{pst}(V): for w ∈ W_{K} we let r(w) = r_{sl}(w) ∘ φ^{ν(w)} with ν(w) ∈ Z such that w acts on F_{v,0}^{ur} as frob_{p}^{−ν(w)}. This also defines an action on D_{cris}(V). The action r is \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}^{ur}-linear, and we have

N ∘ r(w) = N ∘ r_{sl}(w) ∘ φ^{ν(w)} ∘ N = r_{sl}(w) ∘ N ∘ φ^{ν(w)} = p^{ν(w)}r(w) ∘ N.

It follows that the pair (r, N) defines an action of the Weil-Deligne group W′_{v} of F_{v} on D_{pst}(V). Moreover, if τ : F_{v,0}^{ur} ↪ \overline{Q}_{p} is any embedding, then it also follows that the induced action on

WD(V)_{τ} := D_{pst}(V) ⊗_{\overline{Q}_{p}⊗_{Qp} F_{v,0}^{ur},τ} \overline{Q}_{p}

is a Weil-Deligne representation over \overline{Q}_{p} (the subscript τ on the tensor sign means that we consider \overline{Q}_{p} as a \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}^{ur}-algebra via the homomorphism id ⊗ τ). Furthermore, d ⊗ x ↦ φ(d) ⊗ x defines an isomorphism WD(V)_{τ∘frob_{p}} →^{∼} WD(V)_{τ} of Weil-Deligne representations over \overline{Q}_{p}, hence the equivalence class of WD(V)_{τ} is independent of the choice of τ. We let WD(V) be any member of this equivalence class.

We recall that V is potentially semistable if D_{pst}(V) is a free \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}^{ur}- module of rank equal to dim_{\overline{Q}_{p}} V or, equivalently, dim_{\overline{Q}_{p}} WD(V) = dim_{\overline{Q}_{p}} V.

Similarly, V is crystalline if D_{cris}(V) is a free \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}-module of rank dim_{\overline{Q}_{p}} V. This is equivalent to (V ⊗_{Q_{p}} B_{cris})^{I_{v}} being a free \overline{Q}_{p} ⊗_{Q_{p}} F_{0,v}^{ur}- module of rank equal to dim_{\overline{Q}_{p}} V, where I_{v} ⊂ D_{v} is the inertia subgroup. Thus, V is crystalline if and only if V is potentially semistable and both N and I_{v} act trivially on D_{pst}(V). In particular, V is crystalline if and only if dim_{\overline{Q}_{p}} WD(V) = dim_{\overline{Q}_{p}}(V), WD(V) is unramified (i.e., N = 0 and the inertia group I_{v} acts trivially). Consequently, for V crystalline the eigenvalues of w ∈ W_{v} on WD(V)^{Fr-ss} are just the roots of the characteristic polynomial of the \overline{Q}_{p}-endomorphism induced by φ^{ν(w)}. We also recall that for a crystalline representation V there is \overline{Q}_{p} ⊗_{Q_{p}} F_{v}-filtration on D_{cris}(V) ⊗_{F_{v,0}} F_{v} whose associated graded module is just D_{HT} (V).

Suppose now that π_{v} is unramified. From the preceding paragraph it follows that (2) holds if ρ_{v} = ρ_{π}|_{D_{v}} is crystalline and if for all w ∈ W_{v}

det(1 − T φ^{ν(w)}|D_{cris}(V) ⊗_{\overline{Q}_{p}⊗_{Qp} F_{v,0},τ} \overline{Q}_{p}) = det(1 − T w|ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}))

(3)

for some (equivalently, each) embedding τ : F_{v,0} ↪ \overline{Q}_{p}.

2.2 Reduction to d even and π_{v} unramified

As mentioned in the introduction, Saito has proven Theorem 1 when the degree d of F is odd or some π_{v} is square-integrable [Sa1],[Sa2]. We may therefore assume that d is even and that π_{v} is a principal series representation for finite places v. Theorem 1 then asserts that each ρ_{v} is potentially crystalline with predicted Hodge-Tate weights. Clearly, this is true for ρ_{v} = ρ_{π}|_{D_{v}} if and only if there is a finite extension F′/F such that it is true for ρ_{π}|_{D_{v′}}, v′|v the place of F′ determined by the fixed embedding \overline{F} ↪ \overline{F}_{v}. Additionally, if ρ_{v} is potentially crystalline with the predicted Hodge-Tate weights, then to establish (2) it is enough to show that

trace(w|WD(ρ_{v})) = trace(w|ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}))  (4)

for all w ∈ W_{v} with ν(w) > 0.

Let v|p. For a given w ∈ W_{v} such that ν(w) > 0 there exists an abelian extension F′/F such that (a) the base change π′ of π to GL_{2}(A_{F′}) is cuspidal and unramified at each place over p and (b) w ∈ W_{v′} ⊆ W_{v} for v′|v the place of F′ determined by the fixed embedding \overline{F} ↪ \overline{F}_{v}. That (a) can be satisfied is a consequence of each local constituent of π being a principal series representation (we are, of course, using that base change is known for GL_{2} for abelian extensions). That (b) can be simultaneously satisfied with (a) is a simple consequence of ν(w) > 0. Note that the extension F′/F may depend on w. As ρ_{π′} ≅ ρ_{π}|_{G_{F′}}, it follows that WD(ρ_{π′}|_{D_{v′}}) ≅ WD(ρ_{π}|_{D_{v}})|_{W′_{v′}}. Similarly,

Rec_{v′}(π_{v′} ⊗ | · |_{v′}^{−1/2}) ≅ Rec_{v}(π_{v} ⊗ | · |_{v}^{−1/2})|_{W′_{v′}}. Therefore if Theorem 1 holds for π′, then ρ_{v} is potentially crystalline with the predicted Hodge-Tate weights and (4) holds for the given w. This shows that if Theorem 1 holds whenever the representation is unramifed at all primes above p then it also holds for π. Consequently, it suffices to prove Theorem 1 under the assumption that each π_{v}, v|p, is unramified.

2.3 Galois representations in the cohomology of certain Shimura varieties

As mentioned in the introduction, Blasius and Rogawski have essentially proved Theorem 1 in the case where some k_{i} > 2 and each π_{v}, v|p, is unramified [BR].

We explain this here, giving the necessary modifications required to make their argument cover all such cases. We also record some additional consequences for Galois representations associated with essentially self-dual representations of GL_{3}(A_{F}).

2.3.1 The Shimura varieties

Let E_{0} ⊆ \overline{F} be an imaginary quadratic extension of Q in which p splits and set E = F E_{0}. Fix a place v_{0} of E_{0} above p. For convenience we assume that for each place v|p of F the fixed embedding \overline{F} ↪ \overline{F}_{v} induces the valuation v_{0} on E_{0}. Fix an embedding E_{0} ↪ C such that - again for convenience - composition with ι also induces the valuation v_{0}. Let ϕ be the CM type of E consisting of those embeddings E ↪ C extending the fixed embedding of E_{0}. For τ ∈ ϕ we write τ̄ for the composition of τ with complex conjugation. Restriction to F determines a bijection between ϕ and I, and we write τ_{i} for the element of ϕ extending i ∈ I. Via composition with ι, ϕ determines a place of E above each place v|p of F; the fixed decomposition group D_{v} is also a decomposition group for the place of E above p so determined, hence we also denote this place by v, writing v̄ for its conjugate (note that each place v|p of F splits in E). If M is an O_{E}-module, then M_{∞} := M ⊗C decomposes as M_{∞} ≅ ∏_{τ∈ϕ} M_{τ} ⊕Mτ̄ with M_{σ} := M ⊗_{O_{E},σ} C for any embedding σ : E ↪ C. Similarly, M_{p} := M ⊗ Z_{p} decomposes as M_{p} ≅ ∏_{v} M_{v} ⊕ Mv̄ with M_{w} := M ⊗_{O_{E}} O_{E,w} for a place w|p of E.

Fix i_{0} ∈ I. Let Φ be the Hermitian E-pairing on V := E^{3} (viewed as column vectors) defined by the diagonal matrix J := diag(α, 1, 1) with α ∈ F^{×} such that τ_{i_{0}}(α) < 0 and τ_{i}(α) > 0 for i ≠ i_{0}: Φ(x, y) = ^{t}x̄Jy. Then Φ has signature (2, 1) with respect to τ_{i_{0}} and signature (3, 0) with respect to all other τ_{i}. Let U(Φ)_{/Q} be the unitary group of Φ and G := GU(Φ)_{/Q} its similitude group. We note that G(C) ≅ C^{×} × ∏_{τ∈ϕ} GL_{C}(V_{τ}), where the projection to the C^{×}-factor is the similitude character, and the projection to the second factor is via the corresponding projection of GL_{E⊗C}(V_{∞}). Similarly, G(Q_{p}) ≅ Q_{p}^{×} × ∏_{v} GL_{E_{v}}(V_{v}), where v runs over the place of F dividing p (or the fixed places of E over these). Let ψ := trace_{E/Q}βΦ with β a totally imaginary element of E_{0}. Then there exists an O_{E}-lattice Λ ⊂ V such that ψ identifies Λ_{p} with its Z_{p}-dual.

Let S := Res_{C/R}G_{m}, so S(R) = (C ⊗_{R} R)^{×} for any R-algebra R. We identify S(C) = (C ⊗ C)^{×} with C^{×} × C^{×} via z ⊗ w ↦ (zw, z̄w). Let h : S → G_{/R} be the homomorphism such that for (z, w) ∈ S(C)

h(z, w) = (zw) × ∏_{τ∈ϕ} \begin{cases}diag(z, w, w) & τ = τ_{i_{0}} \\ diag(w, w, w) & τ ≠ τ_{i_{0}}. \end{cases}

Let h(z) = h(z, z̄). We assume that β is such that ψ(x, h(i)x) is positive definite for x ∈ V ⊗ R. As explained in [Ko], associated with E, V, ψ, and h is a family of PEL moduli spaces S_{K} over^{2} E, K ⊂ G(A_{f}) being a neat open compact subgroup: in the notation of [Ko, §5] we take^{3} B = E with ∗ the nontrivial automorphism fixing F and (V, (−, −)) = (V, ψ); then C = End_{E}(V) and the G of loc. cit. is the group G defined above, and we take for the ∗- homomorphism C → C ⊗ R the R-linear extension of z ↦ h(z). The varieties S_{K} are smooth over E and, being solutions to PEL moduli problems, are equipped with ‘universal’ abelian varieties A_{K}/S_{K}. As explained in [Ko, §8], S_{K} is naturally identified with a disjoint union of a finite number of copies of the canonical model Sh_{K} over E of the Shimura variety associated with G, h^{−1}, and K, indexed by the isomorphism classes of Hermitian E-spaces (V′, ψ′) that are everywhere locally isomorphic to (V, ψ). We identify Sh_{K} with the copy corresponding to the class of (V, ψ) and let A_{K}/Sh_{K} be the restriction of the universal abelian variety.

Suppose K = K_{p}K^{p} with K^{p} ⊆ G(A_{f}^{p}) and K_{p} ⊂ G(Q_{p}) identified with a subgroup Z_{p}^{×} × ∏_{v|p} K_{v} ⊆ Z_{p}^{×} × ∏_{v} GL_{O_{E,v̄}}(Λv̄). Let v|p be a fixed place. If K_{v} = GL_{O_{E,v}}(Λ_{v}), then an argument of Carayol [Ca1, §5] shows that A_{K} and S_{K} have good reduction at v. A model of S_{K} over O_{F,v} = O_{E,v} is obtained by considering a moduli problem as in [Ca1, 5.2.2]. To be be precise, one considers the functor from the category of locally Noetherian O_{F,v}-schemes to the category of sets that sends an O_{F,v}-scheme R to the set of isomorphism classes of quadruples (A, i, θ, k̄^{v}) where (a) A is an abelian scheme over R of relative dimension 3d and i : O_{E} ↪ End_{R}(A) is an embedding such that Lie(A)_{v} is a locally free O_{R}-module of rank one on which O_{F,v} = O_{E,v} acts via the structure map O_{F,v} → O_{R} and such that Lie(A)_{v′} = 0 for all v′|p, v′ ≠ v; (b) θ is a prime-to-p polarization of A satisfying θ ∘ i(x) = i(x̄)^{∨} ∘ θ for all x ∈ O_{E}; (c) k̄^{v} is a K-level structure as^{4} in [Ca1, 5.2.2(c)] but with V_{Z} in the definition of W there replaced by Λ. That this functor is isomorphic over F_{v} = E_{v} to that in [Ko, §5] defining S_{K}_{/E_{v}} follows from the arguments in [Ca1, 2.4-2.6,5.2.2]. That it is representable by a smooth, projective scheme S_{K} over O_{F,v} follows from the arguments in [Ca1, 5.3-5.5]. The p-divisible group A_{p} of A decomposes under the action of O_{E,p} = O_{E} ⊗ Z_{p} as A_{p} = ∏_{v′|p} A_{v′} × A_{}_{v̄′}. The condition on Lie(A)_{v′} in (a) then implies that A_{v′} is ind-étale if v′ ≠ v, and part of the level structure k̄^{v} is a class modulo ∏_{v′≠v} K_{v′} of O_{E,p}-linear R-isomorphisms k_{p}^{v} : ∏_{v′≠v} A[p^{n}]_{v′} →^{∼} ∏_{v′≠v}(p^{−n}Λ/Λ)_{v′} with n any integer so large that K_{v′} contains the kernel of the reduction map GL_{O_{E,v′}} (Λ_{v′}) → GL_{O_{E,v′}} (Λ_{v′}/p^{n}Λ_{v′}) (see [Ca1, 5.2.3(ii)]). The condition that Λ_{p} is self-dual ensures that over F_{v} this moduli problem is equivalent to one with a usual K-level structure. The representability of this moduli problem by a scheme S_{K} over O_{F,v} follows from the arguments in [Ca1, 5.3] and the properness from those in [Ca1, 5.5]. The smoothness of this scheme follows exactly as in [Ca1, 5.4]. The key point is that for R a local artinian O_{F,v}-module, the conditions on the dimension of A and on Lie(A)_{v} in (a) imply that A_{v} is a divisible O_{F,v}- module of height 3 whose formal (or connected) part has height 1 (we are keeping to the terminology in the Appendix of [Ca1]). The smoothness then follows by the deformation argument given in loc. cit. Over E_{v}, S_{K} is just S_{K}, and A_{K} is the base change of the universal abelian scheme A_{K}/S_{K}. Hence S_{K}, Sh_{K}, and A_{K} have good reduction at v.

2The reflex field in this case is τ_{i_{0}}(E) ⊂ C which we identify with E via τ_{i_{0}}.

3As we are only defining the moduli spaces over E at this point, the conditions at p in [Ko, §5] are superfluous.

4When adapting the arguments of [Ca1] to the setting of this paper, the roles of the superscripts 1 and 2 in loc. cit. are switched. This is a result of our choice of the homomorphism h and the identification of E with the reflex field. A homomorphism S → G_{/R} more naturally generalizing that in loc. cit. would be (z, w) ↦ h(w, z). We have chosen h here so that Sh_{K} is the Shimura variety in [BR].

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2.3.2 Theorem 1 when some k_{i} > 2 and each π_{v} unramified

We can now explain how the arguments in [BR] yield Theorem 1 when d > 1, some k_{i} > 2, and each π_{v}, v|p, is unramified. Without loss of generality we may assume that w = max_{i∈I} k_{i}; choosing a different w amounts to replacing ρ_{π} by a Tate-twist. We may assume that E_{0} has been chosen so that the base change π_{E} of π to GL_{2}(A_{E}) is cuspidal (equivalently, π is not a CM representation associated to a Hecke character of E). Fix an algebraic Hecke character μ of A_{E}^{×} satisfying μ|_{A_{F}^{×}} = ω_{E/F}, the quadratic character of the extension E/F, and such that μ is unramified at each place over p. As explained^{5} in [BR, Prop. 4.1.2], there exists a global L-packet τ on the quasi-split unitary group U(2)_{/F} such that its non-standard base change to GL_{2}(A_{E}) (with respect to μ) is π_{E} ⊗ η| · |_{E}^{1/2} with η an algebraic Hecke character of A_{E}^{×} that is unramified at each place above p. It follows from [BR, Lem. 4.2.1] that there exists a global character θ of U(1)_{/F} unramified at all places above p for which the L-packet ρ = τ ⊗ θ of U(2) × U(1) is such that the endoscopic L-packet Π(ρ_{f}) for U(Φ) contains an element σ_{f} with d(σ_{f}) := #{σ_{∞} ∈ Π(ρ_{∞}) : ϵ(σ_{∞})ϵ(σ_{f}) = 1} = 2. Let χ be an algebraic character of the center of G extending the central character of Π(ρ) and unramified at all places above p (cf. [BR, §1.2]). The pair (σ_{f}, χ) defines an admissible representation π(σ_{f}, χ) of G(A_{Q}^{∞}). From the definition of σ_{f} it follows that σ_{p} is an unramified representation of U(Q_{p}) ≅ ∏_{v} GL_{Ev̄}(Vv̄) in the sense that it is a tensor product of unramified principal series representations of each factor. In particular, as χ is unramified at each place above p, π(σ_{f}, χ)^{K} ≠ 0 for K = K_{p}K^{p} with K_{p} identified with Z_{p}^{×} × ∏_{v|p} GL_{O_{E,v}}(Λ_{v}) and K^{p} sufficiently small.

As explained in the proof of [BR, Thm. 3.3.1], associated with π(σ_{f}, χ) is a motive M = (A_{K}^{n}, e) with coefficients in a number field T ⊂ C (this motive is denoted M_{0} in loc. cit.; n is some integer depending on the weights of π, μ, θ, and χ, A_{K}^{n} is the n-fold self-product over the Shimura variety Sh_{K}, and e is an idempotent in Z_{h}(A_{K}^{n} × A_{K}^{n})) such that for any prime ℓ and any isomorphism ι′ : C →^{∼} \overline{Q}_{ℓ}, the ℓ-adic realization M_{ℓ} of M satisfies

5The representations of GL_{2}(A_{F}) in [BR] are normalized so that what is denoted by π there equals π ⊗ | · |_{F}^{1/2} in this paper.

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M_{ℓ,ι′} := M_{ℓ} ⊗_{T ⊗Q_{ℓ},ι′} \overline{Q}_{ℓ} ≅ ρ_{π,ι′}|_{G_{E}} ⊗ ρ_{ηψ,ι′},  (5)

where the subscript ι′ denotes that the objects on the right-hand side are the ℓ-adic Galois representations^{6} associated with the embedding ι′. Here ψ is the Hecke character z ↦ χ(N_{E/E_{0}}(z̄)) of A_{E}^{×}. Equivalently,

WD(M_{ℓ,ι′}|_{D_{w}}) ≅ ι′Rec_{w}(π_{E,w} ⊗ η_{w}ψ_{w}| · |_{w}^{−1/2})  (6)

for all places w ∤ ℓ of E, D_{w} being any decomposition group for w. More precisely, (6) is only shown in [BR, Thm. 3.3.1] for those w ∤ ℓ coprime to the conductor of π and the absolute discriminant of E. But this together with the existence of the ℓ-adic representations associated with π, ηψ, and ι′ implies (5), from which (6) follows for all places w ∤ ℓ. This relies on more than is proved in loc. cit.; it also requires the work of Carayol and Taylor on the existence of the ℓ-adic representations.

As A_{K} has good reduction at v|p, it follows - from the theorems of Faltings and of Katz and Messing cited in [BR, §5] together with (5) and (6) - that for a place v|p of F the representation M_{p,ι} is crystalline at v and for all w ∈ W_{v}

det(1 − Xφ^{ν(w)}|D_{cris}(M_{p,ι}|_{D_{v}}) ⊗_{\overline{Q}_{p}⊗_{Qp} F_{v,0},τ} \overline{Q}_{p})
= det(1 − Xw|ιRec_{v}(π_{v} ⊗ η_{v}ψ_{v}| · |_{v}^{−1/2}).

(7)

As η and ψ are both unramified at all places above p, ρ_{ηψ|·|_{E}} is crystalline at v. It then follows that ρ_{v} ≅ (M_{p,ι} ⊗ ρ_{ηψ}^{−1})|_{D_{v}} is crystalline, and so (3) follows from (7). That ρ_{v} has Hodge-Tate type (k, w) is immediate from [BR, Thm. 2.5.1(ii)] and Faltings’ proof of the deRham conjecture.

2.3.3 Essentially self-dual representations of GL_{3}(A_{F})

Let Π = ⊗′Π_{v} be a cuspidal automorphic representation of GL_{3}(A_{F}) for which each Π_{i}, i ∈ I, is such that its corresponding representation Rec_{i}(Π_{i}) of the Weil group of F_{i} satisfies

Rec_{i}(Π_{i})|_{C^{×}} ≅ z^{a_{i}}z̄^{b_{i}} ⊕ z^{b_{i}}z̄^{a_{i}} ⊕ (zz̄)^{(a_{i}+b_{i})/2}, a_{i} ≠ b_{i} ∈ Z, a_{i} + b_{i} ∈ 2Z. (8)

Suppose also that Π^{∨} ≅ Π⊗ψ for some Hecke character ψ (then ψ is necessarily algebraic). As explained in [B1, 4.1-4.6], it is a consequence of the results in [Mo] that for each prime ℓ and each isomorphism ι′ : C →^{∼} \overline{Q}_{ℓ} there is an ℓ-adic Galois representation ρ_{Π,ι′} : G_{F} → GL_{3}(\overline{Q}_{ℓ}) satisfying WD(ρ_{Π,ι′}|_{D_{v}}) ≅ ι′Rec_{v}(Π_{v}) for all places v ∤ ℓ that are prime to the conductor of Π and the absolute discriminant of F.

6For an algebraic Hecke character ψ of a number field, we denote by ρ_{ψ,ι′} the ℓ-adic Galois representation associated with ψ and ι′, normalized so that the restriction of the Galois character to the decomposition group at a place w ∤ ℓ is just the image of the local character ψ_{w} under the inverse of the Artin map, composed with ι′.

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The proof of the existence of ρ_{Π,ι′} follows the arguments in [BR]. In particular, letting E be as in 2.3.2, if the base change of Π to E is still cuspidal then, as explained in the proof of [B1, Thm. 4.2], there is a motive M = (A_{K}^{m}, e), K small enough, such that the ℓ-adic realizations of M yield ρ_{Π}|_{G_{E}} twisted by a representation associated with an algebraic Hecke character of A_{E}^{×}. If Π is unramified at each v|p then one can take K = K_{p}K^{p} with K_{p} identified with Z_{p}^{×} × ∏_{v|p} GL_{O_{E,v}}(Λ_{v}) and the Hecke character can be taken unramified at each v|p. Then arguing as in 2.3.2 shows that ρ_{Π} := ρ_{Π,ι} is crystalline at each v|p and such that D_{HT} (ρ_{Π}|_{D_{v}}) ⊗_{\overline{Q}_{p}⊗_{Qp} F_{v},j} \overline{Q}_{p}, j ∈ Hom_{Q_{p}}(F_{v}, \overline{Q}_{p}), is nonzero in degrees −a_{i(j)}, −b_{i(j)}, and −(a_{i(j)} + b_{i(j)})/2, i(j) ∈ I being the induced embedding of F. Furthermore, if WD(ρ_{Π,ι′}|_{D_{v}}) ≅ ι′Rec_{v}(Π_{v}) for some ℓ ≠ p (only an additional condition if p is not prime to the absolute discriminant of F), then these arguments also show that WD(ρ_{Π}|_{D_{v}}) ≅ ιRec_{v}(Π_{v}).

Remark. Suppose Π_{v} is unramified at each v|p. From the good reduction of the Shimura variety Sh_{K} with K_{p} as in 2.3.2 or 2.3.3, it follows easily from the Weil conjectures that the Frobenius-at-v eigenvalues of any ℓ-adic representation ρ_{Π,ι′}, ℓ ≠ p, have absolute value as predicted by the Ramanujan conjecture for Π_{v} when considered as elements of C via ι′. Therefore, if WD(ρ_{Π,ι′}|_{D_{v}}) ≅ ι′Rec_{v}(Π_{v}), then the Ramanujan conjecture is true for Π_{v}. This argument shows (at least) that if q is a prime such that Π_{w} is unramified for all w|q, then the Ramanujan conjecture is true for Π_{w}, w|q, provided there is some prime ℓ ≠ q such that the ℓ-adic representation ρ_{Π,ι′} satisfies WD(ρ_{Π,ι′}|_{D_{w}}) ≅ ι′Rec_{w}(Π_{w}).

2.4 Theorem 1 for the remaining cases

As a consequence of the work of Saito [Sa1, Sa2], the remarks in 2.2, and the results of [BR] as described in 2.3.2, to complete the proof of Theorem 1 it remains to consider the case where d is even, each k_{i} = 2, each π_{v}, v|p, is unramified, and π is not a CM representation. Replacing π by a twist by an integral power of | · |_{F} if necessary (which corresponds to twisting ρ_{π} by a power of the cyclotomic character), we may also assume that w = 2. Hereon we assume we are in this case.

2.4.1 An application of the symmetric square

Let Π := Sym^{2}π ⊗ | · |_{F}^{−1}, with Sym^{2}π the symmetric square lift of π to GL_{3}(A_{F}) (cf. [GJ]). As π is not a CM representation, Π is cuspidal. Since Rec_{i}(π_{i})|_{C^{×}} ≅ (z̄/z)^{1/2} ¯⊕(z/z̄)^{1/2}, Rec_{i}(Π_{i})|_{C^{×}} ≅ Sym^{2}Rec_{i}(π_{i}|·|_{i}^{−1/2})|_{C^{×}} satisfies (8) with a_{i} = −2 and b_{i} = 0. Furthermore, as π^{∨} ≅ π⊗ω^{−1}, ω the central character of π, it follows that Π^{∨} ≅ Π ⊗ ω^{−2}| · |_{F}^{2} . Therefore, Π satisfies all the hypotheses in 2.3.3. In particular, there exist associated ℓ-adic representations ρ_{Π,ι′}. Clearly ρ_{Π,ι′} ≅ Sym^{2}ρ_{π,ι′}, so WD(ρ_{Π,ι′}|_{D_{w}}) ≅ ι′Rec_{w}(Sym^{2}π_{w} ⊗ | · |_{w}^{−1}) for all w ∤ ℓ. Since π_{v}, and therefore Π_{v}, is unramified at each v|p, as explained in 2.3.3 we can conclude from this that for each v|p: (i) Sym^{2}ρ_{π}|_{D_{v}} is crystalline for v|p, (ii) WD(Sym^{2}ρ_{π}|_{D_{v}}) ≅ ιRec_{v}(Sym^{2}π_{v} ⊗ | · |_{v}^{−1}), and (iii) D_{HT} (Sym^{2}ρ_{π}|_{D_{v}}) ⊗_{\overline{Q}_{p}⊗_{Qp} F_{v},j} \overline{Q}_{p}, j ∈ Hom_{Q_{p}}(F_{v}, \overline{Q}_{p}), is non-zero in degrees 2, 1, and 0.

Let v|p. By conclusion (iii) of the preceding paragraph, the graded module D_{HT} (Sym^{2}ρ_{v}) is the symmetric square of the expected graded module for ρ_{v}. It then follows from results of Wintenberger^{7} - Thm. 1.1.3, Prop. 1.2, and Remarks 1.1.4 of [Win1] or Thm. 2.2.2 of [Win2], applied to the isogeny GL_{2} → GL_{2}/±1 - that there is a crystalline representation ρ : D_{v} → GL_{2}(\overline{Q}_{p}) such that Sym^{2}ρ = Sym^{2}ρ_{v}. From this it follows that ρ_{v} is isomorphic to a (possibly trivial) quadratic twist of ρ. In particular, ρ_{v} is potentially crystalline. Therefore WD(Sym^{2}ρ_{v}) ≅ Sym^{2}WD(ρ_{v}), and it then follows from conclusion (ii) of the preceding paragraph that Sym^{2}WD(ρ_{v}) ≅ Sym^{2}ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}). From this it follows that WD(ρ_{v}) is isomorphic to a (possibly trivial) quadratic twist of ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}). It also follows that ρ_{v} is of Hodge-Tate type (k, w) (= ((2)_{i∈I}, 2) in this case).

Remark. We can also use Sym^{2}π to show that the Ramanujan conjecture holds for π. We may assume that π is not a CM representation. Let q be a prime. It then follows from the remark at the end of 2.3.3 that if π_{w} is unramified at each w|q, then the Ramanujan conjecture holds for each Sym^{2}π_{w} and hence for π_{w}. A simple base change argument like that in 2.2 then shows that the Ramanujan conjecture holds at all places where π is a principal series. In particular, this establishes the Ramanujan conjecture for those π for which there is no finite place v with π_{v} square-integrable. That the Ramanujan conjecture is known when such a v exists follows from Carayol’s work [Ca2]. The Ramanujan conjecture has already been established for π by Blasius [B2].

2.4.2 The existence of a crystalline period

Recall that we are assuming that for each v|p, π_{v} ≅ π(α_{v}, β_{v}) is an unramified principal series^{8}. As WD(ρ_{v}) is isomorphic to a (possibly trivial) quadratic twist of ιRec_{v}(π_{v} ⊗ | · |_{v}^{−1/2}), to prove (2) it suffices to show that WD(ρ_{v})^{frob_{v}=α_{v}(ϖ_{v})q_{v}^{1/2}} ≠ 0, where frob_{v} is a geometric frobenius at v, ϖ_{v} is a uniformizer at v, and q_{v} is the order of the residue field at v. This is equivalent to showing that D_{cris}(ρ_{v}|_{D_{v}})^{φ^{fv} =q_{v}^{1/2}α_{v}(ϖ_{v})} is a \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}-module of rank at least one. To establish a lower bound on this rank, we make use of p-adic analytic families of cuspidal representations.

7Note that ‘weakly admissible = admissible’ has been proved by Colmez and Fontaine, and ‘de Rham = potentially semistable’ has been proved (independently in some cases) by André, Berger, Kedlaya, and Mebkhout, and so the hypotheses on which these results depend are known to hold.

8By π(α, β) we mean the usual principal series representation that is the induction to GL_{2}(F_{v}) of the character ( \begin{matrix}a & ∗ \\ 0 & d \end{matrix}) ↦ α(a)β(d)|a/d|_{v}^{1/2} of the upper-triangular Borel.

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Let O denote the integer ring of F and let O_{p} := O ⊗ Z_{p} →^{∼} ∏_{v|p} O_{v}. Let S_{p} := {v|p} be the set of places of F over p and let S_{π} be the set of finite places of F at which π is ramified. Let S := S_{π} ∪ S_{p} and K^{S} := ∏_{v∉S,v∤∞} GL_{2}(O_{v}). Let H^{S} be the abelian Hecke algebra

H^{S} := C_{c}(GL_{2}(A_{F,f}^{S})//K^{S}).

For each v ∈ S_{p} let

I_{v} := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(O_{v}) : ϖ_{v}|c\}, I_{p} := ∏_{v|p} I_{v},

and let U_{v} ⊂ C_{c}(GL_{2}(O_{v})//I_{v}) be the abelian subalgebra generated by the characteristic functions

U_{v} := char(I_{v}diag(ϖ_{v}, 1)I_{v}).

Put

U_{p} := ⊗_{v|p}U_{p} and T^{S} := U_{p} ⊗ H^{S}.

Then there exists an f_{π} ∈ π^{K^{S}I_{p}} that is an eigenvector for the (usual) action of the Hecke ring T^{S} such that char(I_{v}diag(ϖ_{v}, 1)I_{v}) acts with eigenvalue q_{v}^{1/2}α_{v}(ϖ_{v}).

Let K ⊂ \overline{Q}_{p} be a finite extension of Q_{p} containing each i(F), i ∈ I, and the eigenvalues for the action of T^{S} on f_{π}. Let |K^{×}| = {|x|_{p} : x ∈ K^{×}}. For r ∈ |K^{×}|, we denote by B_{r} the usual closed rigid ball over K of radius r (so B_{r}(C_{p}) = \{x ∈ C_{p} : |x|_{p} ≤ r\}, where C_{p} := ˆ \overline{Q}_{p}). Then O(B_{1}) = K < T >. Let A_{r} := O(B_{r}); this is an affinoid K-algebra. From the work of Buzzard [Bu1, Bu2] one can deduce that if r_{0} ∈ |K^{×}| is sufficiently small, then there exists a reduced finite torsion-free A_{r_{0}}-algebra R (so also an affinoid K-algebra) and a homomorphism ϕ : T^{S} → R satisfying (i)-(iii) below. For x ∈ Hom_{K}(R, \overline{Q}_{p}) put ϕ_{x} := x ∘ ϕ. Then:

(i) if x is such that x(1 + T) = (1 + q)^{n_{x}}, n_{x} ∈ p(p − 1)Z_{>0} (q = p if p odd and q = 4 if p = 2), then there exists a cuspidal representation π_{x} of GL_{2}(A_{F}) with infinity type (k_{x}, w_{x}) = ((n_{x} + 2)_{i∈I}, n_{x} + 2)) and which is unramified at all v|p and such that ϕ_{x} : T^{S} → \overline{Q}_{p} gives the eigenvalues

of the action of T^{S} on an eigenvector f_{x} ∈ π_{x}^{K^{S}I_{p}};

(ii) there exists x_{0} ∈ X (K) with x_{0}(1 + T) = 1 such that ϕ_{x_{0}} gives the eigenvalues of the action of T^{S} on f_{π};

(iii) if ϕ_{v} := ϕ(U_{v}) ∈ R^{×}, then |x(ϕ_{v})|_{p} is constant for all x; (iv) there exists a continuous representation

ρ_{R} : G_{F} → GL_{2}(R)

unramified away from S and such that for x as in (i) the representation ρ_{x} : G_{F} → GL_{2}(\overline{Q}_{p}) induced from ρ_{R} by x is equivalent to ρ_{π_{x}} and that induced by x_{0} is equivalent to ρ_{π}.

Assuming the existence of R and ϕ, we can complete the proof of Theorem 1. Let Σ ⊂ Hom_{K}(R, \overline{Q}_{p}) be the set of x as in (i). Then Σ is Zariski dense by the finiteness of R over A_{r}. As explained in 2.3.2 we know that Theorem 1 holds for each π_{x}, x ∈ Σ. Let v|p and x ∈ Σ. Then π_{x,v} ≅ π(μ_{x}, λ_{x}), an unramified principal series with x(ϕ_{v}) = μ_{x}(ϖ_{v})q_{v}^{1/2}. In particular, as Theorem 1 holds for ρ_{x} ≅ ρ_{π_{x}} we have that D_{cris}(ρ_{x}|_{D_{v}})^{φ^{fv} =x(ϕ_{v})} is a \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}-module of rank at least one for all x ∈ Σ, where f_{v} is the residue class degree of F_{v} (so q_{v} = p^{f_{v}}). As the Hodge-Tate type of ρ_{x}, x ∈ Σ, is (k_{x}, w_{x}), each D_{HT} (ρ_{x}|_{D_{v}})⊗_{\overline{Q}_{p}⊗_{Qp} F_{v},j} \overline{Q}_{p} is non-zero in degrees 0 and n_{x} +1. It then follows easily from [Ki2, (5.15)] that^{9}

D_{cris}(ρ_{π}|_{D_{v}})^{φ^{fv} =q_{v}^{1/2}α_{v}(ϖ_{v})} = D_{cris}(ρ_{x_{0}}|_{D_{v}})^{φ^{fv} =x_{0}(ϕ_{v})}

is also a \overline{Q}_{p} ⊗_{Q_{p}} F_{v,0}-module of rank at least one.

While the existence of R and ϕ is essentially proved in the work of Buzzard, there is no convenient reference in [Bu1]. So we conclude by explaining how their existence follows from this work. Let D be the quaternion algebra over F that is split at all finite places and compact modulo the center at all archimedean places. Fix a maximal order O_{D} of D, and for each finite place v of F fix an isomorphism O_{D,v} ≅ M_{2}(O_{v}). This identifies GL_{2}(A_{F,f}) with (D ⊗_{F} A_{F,f})^{×}. Let 𝔫 be the conductor of π and let U_{0} ⊆ GL_{2}(O ⊗ Ẑ) be the subgroup of matrices with lower left entries in 𝔫 ⊗ Ẑ, and let U = U_{0} ∩ I_{p}. Let J := {v|p}. For a ∈ Z_{>0}^{J} let U_{a} := ∏_{v∈J} U_{v}^{a_{v}}. For v ∈ J let σ_{v} := ord_{v}(α_{v}(ϖ_{v})q_{v}^{1/2}), and let σ_{a} := ∑_{v∈J} a_{v}σ_{v}.

For r ∈ |K^{×}| with r ≤ 1 we define a homomorphism κ : O_{p}^{×} × O_{p}^{×} → A_{r}^{×} by

κ((x_{v}), (y_{v})) = ∏_{v∈S_{p}} ∏_{j∈Hom_{Qp} (F_{v},\overline{Q}_{p})} j(y_{v})(1 + T)^{log_{p} Nm_{Fv /Qp} (x_{v})}.

9Proposition (5.14) and Corollary (5.15) of [Ki2] are only stated for representations of G_{Q_{p}} = Gal(\overline{Q}_{p}/Q_{p}). But it is easily checked that the arguments extend to the case of the representations of D_{v} = Gal(\overline{F}_{v}/F_{v}) under consideration here; the necessary results with φ replaced by φ^{f_{v}} (e.g., Corollary (3.7)) are easily deduced from those for φ. A key point is that our hypotheses on the weights in the family X ensure that the polynomial P (X) ∈ (O(X) ⊗_{Q_{p}} F_{v})[X] provided by Sen’s theory as in [Ki2, (2.2)] is of the form P (X) = XQ(X)

with the constant coefficient of Q not a zero-divisor.

Documenta Mathematica 14 (2009) 241–258

Let W be the rigid analytic weight space over K defined in §8 of [Bu1]. Then B_{r} is identified with a reduced affinoid subspace of W such that κ is the induced weight in the sense of loc. cit. Let m ∈ |K^{×}| be so small that the A_{r}-Banach module S_{κ}^{D}(U; m) of overconvergent automorphic forms is defined (notation as in [Bu1, §9]). This is equipped with an A_{r}-linear action of T^{S} such that each U_{v} is a completely continuous operator. For b ∈ B_{r}(\overline{Q}_{p}) such that the induced map e_{b} : A_{r} → \overline{Q}_{p} sends 1 + T to (1 + q)^{n_{b}} with n_{b} ∈ p(p − 1)Z_{≥0} we have a T^{S}-equivariant inclusion of the classical forms of weight (k_{b}, w_{b}): S_{k_{b},w_{b}}^{D}(U) ⊆ S_{κ}^{D}(U; m) ⊗_{O(B_{r}),e_{b}} \overline{Q}_{p}, w_{b} := (n_{b} + 2)_{i∈I} ∈ Z^{I} (see [Bu1, §11]). By the Jacquet-Langlands correspondence, there exists f_{0} ∈ S_{2,2}(U) having the same T^{S}-eigenvalues as f_{π}. Recall that by the theory of Fredholm series and orthonormalizable Banach modules as developed by Coleman, Ash and Stevens, and Buzzard, if r is small enough then there is a finite A_{r}-direct summand N ⊂ S_{κ}^{D}(U; m) that is stable under T^{S} and such that for each a ∈ Z_{>0}^{J} the Fredholm series for U_{a} on N is a factor of the slope σ_{a} part of the Fredholm series P_{a}(X) ∈ A_{r}{X} associated to the completely continuous operator U_{a} on S_{κ}^{D}(U; m) (the latter is well-defined for r small enough), and furthermore is such that f_{0} ∈ N ⊗_{A_{r},e_{0}} K. If r is sufficiently small then for any b ∈ A_{r} with n_{b} ∈ p(p − 1)Z_{>0} it follows from the arguments in [Bu2, §7] (see also the comment at the end of §11 of [Bu1]) that N_{b} := N ⊗_{A_{r},e_{b}} \overline{Q}_{p} is comprised of classical forms in S_{n_{b}+2,n_{b}+2}^{D}(U) (n_{b} is divisible by a high power of p; the smaller r is, the larger the power of p). By the definition of N, any T^{S}-eigenform in N_{b} is such that the eigenvalue of U_{v} has slope σ_{v}, and so if r is small enough relative to σ_{v} then it is easily seen that the v-constituent of the irreducible representation of GL_{2}(A_{F,f}) generated by f is not special and therefore must be an unramified principal series.

Let R be the A_{r}-algebra generated by the image of T^{S} in End_{A_{r}}(N); this is a finite torsion-free A_{r}-algebra and so an affinoid K-algebra. Note that there exists a K-homomorphism ϕ_{0} : R → K giving the eigenvalues of the T^{S}-action on f_{0}. Let A be the normalization of the quotient of R by a minimal prime containing the kernel of ϕ_{0}. This is a reduced finite torsion-free A_{r}-algebra and so also an affinoid K-algebra. Let ϕ : T^{S} → A be the canonical homomorphism. It follows from the definitions that (i), (ii), and (iii) hold with R replaced by A. For each x ∈ Hom_{K}(A, \overline{Q}_{p}) as in (i), let T_{x} : G_{F} → \overline{Q}_{p} be the continuous pseudo-representation associated with ρ_{π_{x}} (so T_{x} = trace ρ_{π_{x}}). Since for a place w ∤ 𝔫p, T_{x}(frob_{w}) = x∘ϕ(char(GL_{2}(O_{w})diag(ϖ_{w}, 1)GL_{2}(O_{w})), ϖ_{w} ∈ O_{w} a uniformizer, it follows easily from the Cebotarev density theorem and the Zariski density of the set Σ_{A} of x ∈ Hom_{K}(A, \overline{Q}_{p}) as in (i) that there is a continuous pseudo-representation T : G_{F} → A such that T_{x} = x ∘ T. From the general theory of pseudo-representations (cf. [Tay3]) there is a semisimple Galois representation ρ_{A} : G_{F} → GL_{2}(F_{A}), F_{A} the field of fractions of A, such that T = trace ρ_{A}. It is easy to see that there is a finite A-module M ⊂ F_{A}^{2} on which G_{F} acts continuously and such that V_{x} := M_{x} ⊗_{A_{x},x} \overline{Q}_{p} is isomorphic to the representation ρ_{π_{x}}, x ∈ Σ_{A} or x any extension of ϕ_{0} to A (here the subscript x on M and A denotes the localization at the kernel of x). Such a module M is given explicitly as follows. Fix a basis of ρ_{A} such that for some i ∈ I the corresponding complex conjugation in G_{F} is diagonalized (with eigenvalues 1 and −1). Writing ρ_{A}(σ) = ( \begin{matrix}a_{σ} & b_{σ} \\ c_{σ} & d_{σ} \end{matrix} ), we have a_{σ}, d_{σ}, b_{σ}c_{σ′} ∈ A for all σ, σ′ ∈ G_{F} and that these define continuous functions of σ and σ′. It follows that the A-submodules B and C of F_{A} generated by {b_{σ} : σ ∈ G_{F} } and {c_{σ} : σ ∈ G_{F} }, respectively, are fractional ideals of A satisfying CB ⊆ A (note that by the semisimplicity of ρ_{A} and the diagonalization of the chosen complex conjugation, B = 0 if and only of C = 0). We can then take M = A ⊕ A if C = 0 and M = A ⊕ C otherwise. Being a finite A-module, M is a Banach A-module and the continuity of the action of G_{F} on M is clear from the continuity of the functions a_{σ}, d_{σ}, and b_{σ}c_{σ′}. As A is normal, for any x ∈ Hom_{K}(A, \overline{Q}_{p}) the localization A_{x} is a DVR, and so M_{x} is a free A_{x}-module of rank two. The representation V_{x} is then two-dimensional and its associated pseudo-representation is x ∘ T. Therefore if x ∈ Σ_{A} or x any extension of ϕ_{0} to A, the pseudo-representation associated with V_{x} equals that associated with ρ_{π_{x}}. As the latter representation is irreducible (this irreducibility is well-known, but see also the remark below) it follows that V_{x} ≅ ρ_{π_{x}}. As A is normal and finite over A_{r}, there is an f ∉ T A_{r} (in fact one can pick f not to be zero on any given finite set of points of B_{r}) such that M_{f} is free over A_{f}. Let r_{0} ≤ r be so small that f ∈ A_{r_{0}}^{×}. Then (i)-(iv) hold with R the quotient of A ⊗_{A_{r}} A_{r_{0}} by any minimal prime (a finite A_{r_{0}}-algebra and so an affinoid K-algebra) and with ρ_{R} the representation of G_{F} on the free R-module M ⊗_{A} R.

Remark. We recall that there is a quick proof of the irreducibility of ρ_{π} using that it is potentially semistable (really only that it is Hodge-Tate), which was established in 2.4.1. If ρ_{π} ≅ χ_{1} ⊕ χ_{2}, then each χ_{i} is potentially semistable and hence is the Galois representation associated to an algebraic Hecke character ψ_{i} of F (cf. [Se], esp. III,2.3-2.4). It then follows that L(π ⊗ ψ_{2}^{−1}, s − 1/2) = L(ψ_{1}/ψ_{2}, s)ζ_{F} (s). As ψ_{i} = ψ′_{i}| · |_{F}^{a_{i}} with a_{i} ∈ Z and ψ′_{i} finite and since we may assume a_{1} ≥ a_{2}, L(ψ_{1}/ψ_{2}, 1) = L(ψ′_{1}/ψ′_{2}, a_{1} − a_{2} + 1) ≠ 0. But this implies that L(π ⊗ ψ_{2}^{−1}, s) has a pole at s = 1, contradicting the cuspidality of π.

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Christopher Skinner

Department of Mathematics Princeton University Fine Hall, Washington Road Princeton, NJ 08544-1000 USA