Documenta Math. 241 A note on the p-adic Galois representations attached to

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 cor- respondence at these places, proving this for those forms not covered by the previous works of T. Saito and of D. Blasius and J. Rogawski.


Introduction
Let F be a totally real extension of Q of degree d.Let F be an algebraic closure of F and let G F := Gal(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 F v be an algebraic closure of F v and fix an F -embedding F ֒→ 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(F v /F v ).Let p be a rational prime and fix an algebraic closure Q p of Q p and an isomorphism ι : C ∼ → Q p .Via composition with ι the set I is identified with the embeddings of F into 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) ki |x| −w i 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 Here WD(σ) denotes the Weil-Deligne representation over Q p associated to a continuous representation σ : D v → GL n (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 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 | Dv is potentially semistable with Hodge-Tate type (k, w) and satisfies We recall that ρ v := ρ π | Dv is potentially semistable if is a free Q p ⊗ Qp F ur v,0 -module of rank 2, where here L is ranging over all finite extensions of F v , F ur v,0 is the union of all absolutely unramified subfields of F v , and B st is Fontaine's ring of semistable p-adic periods (the latter has a continuous action of D v = Gal(F v /F v ) with the property that B Gal(F v /L) st = L 0 , the maximal absolutely unramified subfield of L).We also recall that the module D HT (ρ v ) := (V ⊗ Qp B HT ) Dv is a graded Q p ⊗ Qp F v -module (recall that B HT := ⊕ n∈Z C Fv (n), C Fv := F v , with the obvious action of D v ).By ρ π | Dv having Hodge-Tate type (k, w), we mean that for j ∈ Hom Qp (F v , Q p ) the induced graded module D HT (ρ v ) ⊗ Q p ⊗Q p F,j 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 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 : 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).

Documenta Mathematica 14 (2009) 241-258
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| Dv 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 | Dv 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 Sym2 π and the results of [Mo] it follows that Sym 2 ρ v is crystalline1 and even that WD(Sym 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) ).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].
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.

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(F v /F v ), v|p, defined over a finite extension of Q p , we apply it to continuous representations over 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 (Q p ) a continuous representation.The subfields L of Q p that are finite over Q p form a countable set, and as each GL n (L) is closed in GL n (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 = ⊔ m i=1 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 Q p for v|p Let v|p be a place of F .Let B HT := ⊕ n∈ b Z C Fv (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 ur v,0 -algebras equipped with a continuous action of D v such that B Gal(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 ur v,0 -semilinear Frobenius morphism ϕ : B ? → B ? (that is, ϕ(ax) = frob p (a)ϕ(x) for all a ∈ F ur v,0 , where frob p ∈ Gal(F ur v,0 /Q p ) is the absolute arithmetic Frobenius).Additionally, B st is equipped with an F ur v,0 -linear and D v -equivariant monodromy operator N : For a finite-dimensional Q p -vector space V with a continuous Q p -linear action of D v we put where L/F v is a finite extension.Then D HT (V ) is a finite, graded L 0 -semilinear) Frobenius operator on D cris (V ) (resp.D L st (V )) that we also denote by ϕ.The action of the monodromy operator N on B st induces a Q p ⊗ Qp L 0 -linear nilpotent operator on D L st (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 . This also defines an action on D cris (V ).The action r is Q p ⊗ Qp F ur v,0 -linear, and we have 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 ur v,0 ֒→ Q p is any embedding, then it also follows that the induced action on is a Weil-Deligne representation over Q p (the subscript τ on the tensor sign means that we consider 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
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 , 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 Q p -endomorphism induced by ϕ ν(w) .We also recall that for a crystalline representation V there is

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 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 F ֒→ 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 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.

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].

Documenta Mathematica 14 (2009) 241-258
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 ).

The Shimura varieties
Let E 0 ⊆ 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 F ֒→ 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 τ 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 v for its conjugate (note that each place v|p of 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 i0 (α) < 0 and τ i (α) > 0 for i = i 0 : Φ(x, y) = t xJy.Then Φ has signature (2, 1) with respect to τ i0 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) , 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, , 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 h(z) = h(z, z).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 over2 E, K ⊂ G(A f ) being a neat open compact subgroup: in the notation of [Ko,§5] we take3 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.
and part of the level structure kv is a class modulo ).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,vmodule 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.

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 , the quadratic character of the extension E/F , and such that µ is unramified at each place over p.As explained5 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) E 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 ∞ ) : ǫ(σ ∞ )ǫ(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 Ev (V v ) 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 OE,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 n K , 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 n K is the n-fold self-product over the Shimura variety Sh K , and e is an idempotent in Z h (A n K × A n K )) such that for any prime ℓ and any isomorphism where the subscript ι ′ denotes that the objects on the right-hand side are the ℓ-adic Galois representations6 associated with the embedding ι ′ .Here ψ is the Hecke character 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 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 −1 ηψ )| Dv 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.

Essentially self-dual representations of GL
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 ′ : ) for all places v ∤ ℓ that are prime to the conductor of Π and the absolute discriminant of F .
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 m K , e), K small enough, such that the ℓ-adic realizations of M yield ρ Π | GE 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 ) 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 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 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 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 ′ | Dw ) ∼ = ι ′ Rec w (Π w ).

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.

An application of the symmetric square
F .Therefore, Π satisfies all the hypotheses in 2.3.3.In particular, there exist associated ℓ-adic representations w ) 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: , 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 Wintenberger7 -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 /±1that there is a crystalline representation ρ : 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 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].

The existence of a crystalline period
Recall that we are assuming that for each v|p, ), to prove (2) it suffices to show that WD(ρ v ) frobv =αv (̟v)q 1/2 v = 0, where frob v is a geometric frobenius at v, ̟ v is a uniformizer at v, and q v is the order of the residue field at v.This is equivalent to showing that at least one.To establish a lower bound on this rank, we make use of p-adic analytic families of cuspidal representations.
Let O denote the integer ring of F and let 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 For each v ∈ S p let and let U v ⊂ C c (GL 2 (O v )//I v ) be the abelian subalgebra generated by the characteristic functions Then there exists an f π ∈ π K S Ip that is an eigenvector for the (usual) action of the Hecke ring T S such that char(I v diag(̟ v , 1)I v ) acts with eigenvalue q 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 r0 -algebra R (so also an affinoid K-algebra) and a homomorphism φ : 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 → Q p gives the eigenvalues of the action of T S on an eigenvector f x ∈ π (ii) there exists x 0 ∈ X (K) with x 0 (1 + T ) = 1 such that φ x0 gives the eigenvalues of the action of T S on f π ; Documenta Mathematica 14 (2009) 241-258 (iv) there exists a continuous representation unramified away from S and such that for x as in (i) the representation ρ x : G F → GL 2 (Q p ) induced from ρ R by x is equivalent to ρ πx 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, 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 (̟ v )q 1/2 v .In particular, as Theorem 1 holds for ρ x ∼ = ρ πx we have that D cris (ρ x | Dv ) ϕ fv =x(φv) is a Q p ⊗ Qp 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 fv ).As the Hodge-Tate type of ρ x , x ∈ Σ, is (k x , w x ), each D HT (ρ x | Dv ) ⊗ Q p ⊗Q p Fv ,j Q p is non-zero in degrees 0 and n x + 1.It then follows easily from [Ki2,(5.15)] that 9 is also a Q p ⊗ Qp 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 ⊗ Z) be the subgroup of matrices with lower left entries in n ⊗ Z, and let 9 Proposition (5.14) and Corollary (5.15) of [Ki2] are only stated for representations of G Qp = Gal(Q p /Qp).But it is easily checked that the arguments extend to the case of the representations of Dv = Gal(F v /Fv) under consideration here; the necessary results with ϕ replaced by ϕ fv (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 ) ⊗ Qp Fv)[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 D κ (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 (Q p ) such that the induced map e b : A r → 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 ): ]).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 D κ (U ; m) that is stable under T S and such that for each a ∈ Z J >0 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 D κ (U ; m) (the latter is well-defined for r small enough), and furthermore is such that f 0 ∈ N ⊗ Ar,e0 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 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 Ar (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, Q p ) as in (i), let T x : G F → Q p be the continuous pseudo-representation associated with ρ πx (so T x = trace ρ πx ).Since for a place w ∤ np, T x (frob w ) = x • φ(char(GL 2 (O w )diag(̟ w , 1)GL 2 (O w )), ̟ 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, 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 2 A on which G F acts continuously and such that V x := M x ⊗ Ax,x Q p is isomorphic to the representation ρ πx , x ∈ Σ A or x any extension of φ 0 to A (here the Documenta Mathematica 14 (2009) 241-258 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) = aσ bσ cσ dσ , 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, 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 πx .As the latter representation is irreducible (this irreducibility is well-known, but see also the remark below) it follows that V x ∼ = ρ πx .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 × r0 .Then (i)-(iv) hold with R the quotient of A ⊗ Ar A r0 by any minimal prime (a finite A r0 -algebra and so an affinoid K-algebra) and with ρ R the representation of G F on the free R-module M ⊗ A R.
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, kv ) 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(x) ∨ • θ for all x ∈ O E ; (c) kv 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 overF v = E v to that in[Ko,  §5]  defining S K /Ev 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,].The p-divisible group A p of A decomposes under the action of O 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 := Q p ). Then O(B 1 ) = K < T >.Let A r := O(B r ); this is an affinoid K-algebra.