On some consequences of a theorem of J. Ludwig

We prove some qualitative results about the $p$-adic Jacquet--Langlands correspondence defined by Scholze, in the $GL(2,Q_p)$, residually reducible case, by using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration the $p$-adic Jacquet--Langlands correspondence can also deal with principal series representations in a non-trivial way, unlike its classical counterpart.


Introduction
Let F be a finite extension of Q p , and let L be a further sufficiently large finite extension of F , which will serve as the field of coefficients. Let O be the ring of integers in L, ̟ a uniformizer in O and let O/̟ = k be the residue field.
To a smooth admissible representation π of GL n (F ) on an O-torsion module Scholze in [45] attaches a sheaf F π on the adic space P n−1 Cp and shows that the cohomology groups H í et (P n−1 Cp , F π ) are admissible representations of D × p , the group of units in a central division over F with invariant 1/n, and carry a continuous commuting action of G F , the absolute Galois group of F . His construction is expected to realize both p-adic Langlands and p-adic Jacquet-Langlands correspondences. However, these groups seem to be very hard to compute, even to decide whether they are zero or not is highly non-trivial.
In order to extend his construction to admissible unitary Banach space representations Π of GL n (F ), the following seems like a sensible thing to do: choose an open bounded GL n (F )-invariant lattice Θ in Π, then Θ/̟ m is an admissible smooth representation of GL n (F ) on an O-torsion module and we may consider the limit lim ← −m H í et (P n−1 Cp , F Θ/̟ m ) equipped with the p-adic topology. We would like to invert p and obtain a Banach space, but the O-module might not be O-torsion free, and once we quotient out by the torsion, it might not be Hausdorff. If F = Q p and n = 2 and π is a principal series representation of GL 2 (Q p ) then Judith Ludwig shows in [31] the vanishing of H í et (P 1 Cp , F π ), when i = 2. All the other groups for i > 2 are known to vanish due to Scholze in this case. Moreover, it follows from his results that H 0 (P 1 Cp , F π ) also vanishes, if π SL2(Qp) is equal to zero. Thus if Date: May 15, 2018. 1 we restrict our attention only to those representations π, which have all irreducible subquotients isomorphic to irreducible principal series representations, then the functor π → H 1 et (P 1 Cp , F π ) is exact. Such subcategories of smooth representations of GL 2 (Q p ) on O-torsion modules have been studied in [36] and shown to be related to the reducible 2-dimensional mod p representations of G Qp . The exactness of the functor π → H 1 et (P 1 Cp , F π ) allows us to use arguments of Kisin in [27], who used the exactness of Colmez's functor to make a connection between the deformation theory of GL 2 (Q p )-representations and deformation theory of 2-dimensional G Qprepresentations. For simplicity we assume that p > 2 in this paper. Theorem 1.1. Let r : G Qp → GL 2 (L) be a continuous representation withr ss = χ 1 ⊕ χ 2 , where χ 1 , χ 2 : G Qp → k × are characters, such that χ 1 χ −1 2 = ω ±1 , where ω is the mod p cyclotomic character. Let Π be an admissible unitary L-Banach space representation of GL 2 (Q p ) corresponding to r via the p-adic Langlands correspondence for GL 2 (Q p ). ThenŠ 1 (Π) = 0.
Previously such a result was known only in the case, when r is a (twist of a) potentially semi-stable, non-crystabelline representation, which lies on an automorphic component of a potentially semistable deformation ring, proved by Chojecki and Knight in [12]. They prove it by patching and showing that locally algebraic vectors inŠ 1 (Π) are non-zero. Their argument relies on the theorem of Emerton [20] which allows to interpret classical automorphic forms as locally algebraic vectors in completed cohomology, and enables them to handle only the representations which are "discrete series at p". For example, their argument does not work for representations which become crystalline after restriction to a Galois group of an abelian extension of Q p (prinicipal series at p), or representations, which do not become potentially semistable after twisting by a character (non-classical).
Our argument works as follows: by the mod p local Langlands correspondence tō r ss one may associate two principal series representations π 1 and π 2 . We know from [15], [36] that the semi-simplification of Π 0 /̟ is isomorphic to π 1 ⊕ π 2 , where Π 0 is a unit ball in Π. Using the exactness results described above it is enough to show that at least one of H 1 et (P 1 Cp , F π1 ), H 1 et (P 1 Cp , F π2 ) does not vanish. To show that, it is enough to find some π with all irreducible subquotients isomorphic to either π 1 or π 2 , such that H 1 et (P 1 Cp , F π ) does not vanish. As we have already mentioned, it seems impossible to compute H 1 et (P 1 Cp , F π ) in general. However, Scholze manages to do so for certain representations coming from geometry. If π = S(U p , O/̟ n ) is a 0-th completed cohomology group of a tower of zero dimensional Shimura varieties associated to a quaternion algebra D 0 over 1 Q which is split at p and ramified at ∞, Scholze shows in [45] that H 1 et (P 1 Cp , F π ) is isomorphic as G Qp × D × p -representation to the 1-st completed cohomology group H 1 (U p , O/̟ n ) of a tower of Shimura curves associated to a quaternion algebra D, which is ramified at p, split at ∞, and has the same ramification as D 0 at all the other places. Here U p denotes some fixed tame level. Scholze also shows that this isomorphism respects the action by Hecke operators on both sides. We show that a localization of S(U p , O/̟ n ) at a maximal ideal m of the Hecke algebra corresponding to an absolutely irreducible Galois representationρ : G Q,S → GL 2 (k), such that the semi-simplification of ρ| G Qp is equal χ 1 ⊕ χ 2 as above, has all irreducible subquotients isomorphic to π 1 or π 2 as above. By applying Scholze's functor to this representation we obtain H 1 (U p , O/̟ n ) m , which we can show is non-zero by using the classical Jacquet-Langlands correspondence.
Let The Theorem is proved by using the observation of Kisin in [27] that exact functors take flat modules to flat modules and the results of Gee-Newton [22] on miracle flatness in non-commutative setting.
One can show that a Banach space representation of D × p has δ-dimension zero if and only if it is finite dimensional as an L-vector space. Moreover, the zero dimensional representations build a Serre subcategory, and thus one may pass to a quotient category. Informally this means that two 1-dimensional Banach space representations are isomorphic in the quotient category if they differ by a 0-dimensional Banach space representation. As Kohlhaase has pointed out to me, the quotient category of Banach space representation of δ-dimension at most 1 by the zero dimensional Banach space representations is noetherian and has an involution, hence it is also artinian, and hence every object in the quotient category has finite length. From this it is easy to deduce the following. We got quite excited about this Corollary at first, since we hoped that it might imply thatŠ 1 (Π) is of finite length as a Banach space representation of D × p by some formal representation theoretic arguments. However, here is an example suggesting that one should be cautious.
have an action of the same Hecke algebra T(U p ) m . Moreover, there is a surjective homomorphism of local rings Rρ ։ T(U p ) m , where Rρ is the universal deformation ring ofρ.
In this case, there is an isomorphism of admissible unitary κ(x)-Banach space where Π is the absolutely irreducible κ(x)-Banach space representations corresponding to ρ x | G Qp via the p-adic Langlands correspondence for GL 2 (Q p ). In particular, One should think of the first part of the theorem as a global Jacquet-Langlands correspondence between the p-adic automorphic forms on D × 0 and D × . In particular, since D 0 is split at p, one may always find a classical automorphic form, which is principal series at p, such that ( where m x is the corresponding maximal ideal. The theorem tells us that there is a p-adic automorphic form on D × corresponding to it. Such a form cannot be classical, since the classical Jacquet-Langlands correspondence cannot cope with principal series. It was pointed out to us by Sean Howe that in his 2017 University of Chicago PhD thesis, [24], he proves an analogue of the first part of the Theorem above in the setting when D 0 is a quaternion algebra over Q ramified at p and ∞ and D = GL 2 , for the maximal ideals corresponding to overconvergent Galois representations. He conjectures that the analogue of (1) holds in his setting. of elements with reduced norm equal to 1. LetŠ 1 (Π) 1−alg be the subspace of locally algebraic vectors for the action of D ×,1 p onŠ 1 (Π). ThenŠ 1 (Π) 1−alg is a finite dimensional L-vector space. If it is non-zero then ρ x | G Qp is a twist of a potentially semistable representation, which does not become crystalline after restriction to the Galois group of any abelian extension of Q p . The quotientŠ 1 (Π)/Š 1 (Π) 1−alg contains an irreducible closed subrepresentation of δ-dimension 1.
To the best of our knowledge the existence of irreducible admissible unitary Banach space representations of D × p of dimension 1 has not been known before. The locally analytic representations of D × p constructed by Kisin-Strauch in [30] should have the dimension equal to the dimension of the D × p -orbit in P 1 (Q p 2 ), which is equal to dim D × p − dim Q × p 2 = 4 − 2 = 2. If F = Q, as we assume in this introduction, then, after twisting, Theorem 1.5 follows readily from Theorem 1.4 by using Emerton's results on locally algebraic vectors in completed cohomology, [20]. However, when F is a totally real field, then one has to deal with p-adic automorphic representations, which have locally algebraic vectors at all places above p, except for one, where they have locally algebraic vectors after a twist by a character, which is not locally algebraic. Let  Cp , F π2 ) must be an infinite dimensional k-vector space and admissible as smooth representations of D × p . This means that they are built together out of finite dimensional pieces in some non-semisimple way. This non-semisimplicity makes the study of the mod-p Jacquet-Langlands correspondence complicated and passing to semi-simplification seems not to carry much information, since if the central character is fixed only finitely many isomorphism classes of irreducible subquotients can appear.
If we work in characteristic zero, then this problem need not appear. One may show that any irreducible unitary Banach space representation of D × p on a finite dimensional vector space is of the form Sym b L 2 ⊗ det a ⊗τ ⊗ η • Nrd, where τ is a smooth irreducible representation and η : Q × p → L × is a unitary character. Such representations appear in the classical Jacquet-Langlands correspondence (up to a twist) and we speculate that if the Galois representation corresponding to Π does not become potentially semi-stable after twisting by a character then such representations should not appear as subquotients ofŠ 1 (Π). It seems very reasonable to us in view of Corollary 1.3 and Theorem 1.5 to expect thatŠ 1 (Π) is of finite length as a Banach space representation of D × p . This raises a natural question, whether one can construct the irreducible representations in Theorem 1.5 directly, and prove local-global compatibility for them. We hope to pursue these questions in future work.
1.2. Patching. We have decided not to use patching in this paper, since this would add another level of technicalities. However, let us indicate, which parts of the paper can be improved upon if one chooses to use patching.
In the miracle flatness theorem of Gee-Newton, see Proposition 4.1 for the form we use it, one needs the commutative ring to be regular, so we cannot apply it to T(U p ) m and S(U p , O) m directly, however if one patches under favourable assumptions one may arrange that the patched ring R ∞ is regular, and then show that the patched module M ∞ is flat over R ∞ by the same argument. This would imply that S(U p , O) m is flat over T(U p ) m . This is how Gee-Newton prove their big R equals to big T theorem. As a consequence one would know that the multiplicity n in (1) is independent of x, and one would not need to assume that ρ x | G Qp is irreducible. Instead we use Cohen's structure for complete local rings to prove flatness of S(U p , O) m over some (random) formally smooth subring of T(U p ) m of the same dimension. As a consequence we are forced to assume that ρ x | G Qp is irreducible, since we cannot exclude that at points corresponding to reducible Galois representations, the fibre contains only one of the two irreducible Banach space that should appear there. Since we can not exclude that one of H A further improvement could be made in Theorem 1.5. Right now we use in an essential way that Π corresponds to the restriction to G Qp of a global Galois representation. One could spread this result out by first showing that the patched module M ∞ is projective as a GL 2 (Q p )-representation in a suitable category, as we do in [11], so that if r : G Qp → GL 2 (L) is an irreducible representation satisfyinḡ r ss = χ 1 ⊕ χ 2 then the corresponding GL 2 (Q p )-representation Π can be obtained by specializing M ∞ at some closed point y of Arguing as in [11,Lem. 4.18] and using Theorem 1.5 one sees that the Galois representations corresponding to the closed points in m-Spec R ∞ [1/p] in the support M ′ ∞ (σ • ) have the same p-adic Hodge theoretic properties, so for example if η is trivial, then all of them are potentially semistable, have the same Hodge-Tate weights, and the same inertial type, moreover the restriction of the associated Weil-Deligne representation to the Weil group of Q p is of "discrete series type". As a part of patching we obtain an ideal a ∞ of R ∞ -generated by an M ∞ -regular sequence y 1 , . . . , is non-zero and we may convert this into a representation theoretic information using [38,Prop. 2.22], to deduce that there is some Then as in the proof of Theorem 6.13 the p-adic Hodge theoretic data of ρ x | G Qp (and hence of r) will determine σ. This allows us to conclude that if Hom D × p (σ,Š 1 (Π)) is non-zero thenŠ 1 (Π) 1−alg is isomorphic to a finite direct sum of copies of σ and thus is a finite dimensional L-vector space. Arguing as in the proof of Theorem 6.13 we conclude thatŠ 1 (Π)/Š 1 (Π) 1−alg is non-zero and any irreducible Banach space subrepresentation has δ-dimension equal to 1. The details of this argument will appear in a subsequent paper.
1.3. Outline. We end the introduction with a brief outline how the paper is structured. Section 3 contains the local part of the paper. We first review the results of Ludwig and Scholze, we then recall some of results about the representation theory of GL 2 (Q p ) proved in [36]. In subsection 3.3 in order to make things transparent for the reader we dualize everything at least twice. In section 4 we recall the notion of dimension for finitely generated modules over Auslander regular rings and the results of Gee-Newton on miracle flatness in non-commutative setting. In section 5 we prove the local-global compatibility results. The main objective of this section is to be able to apply the results of [36] to the completed cohomology. These results were known to the authors of [11] at the time of writing that paper. However, the results proved in section 5 form a technical backbone of the paper, so it seems like a good idea to write down the details. The main ingredients are the theory of capture [15, §2.4], [39, §2.1], which is based on the ideas of Emerton in [18], the results of Berger-Breuil [5] on the universal unitary completions of locally algebraic principal series representations and the results of Colmez in [13] on the compatibility of the p-adic and classical local Langlands correspondence. In section 6 we prove the theorems stated in the introduction. I thank Yongquan Hu for the invitation and stimulating discussions, I thank the Morning Side Center for Mathematics for providing excellent working conditions. The comments of Yiwen Ding made during my talk at Peking University helped to improve the exposition of the paper. I thank Jan Kohlhaase for sharing his insights into dimension theory of Auslander regular rings with me and Shu Sasaki for discussions about Shimura curves. I thank Konstantin Ardakov, Matthew Emerton and Toby Gee for helpful correspondence. I thank Florian Herzig, Judith Ludwig and James Newton for their comments on an earlier draft. The author is partially supported by the DFG, SFB/TR45.

Notation
Our conventions on local class field theory, classical local Langlands correspondence and p-adic Hodge theory agree with those of [11]. In particular, uniformisers correspond to geometric Frobenii and the p-adic cyclotomic character ε : G Qp → Z × p has Hodge-Tate weight −1. We will denote its reduction modulo p by ω. We will denote by χ cyc : G Q → Z × p the global p-adic cyclotomic character. If G is a p-adic analytic group then we employ the notation scheme introduced in [19] for the categories of its representations. In particular, Mod , where we view ζ as a character of G Qp via local class field theory. This normalisation differs from [36] by a twist of cyclotomic character and coincides with the normalisation of [11]. In particular, if π is a principal series representation Ind G B ωχ 1 ⊗ χ 2 thenV(π ∨ ) = χ 1 viewed as a character of G Qp via the local class field theory.
If Π is an admissible unitary L-Banach space representation of G with central character ζ then may choose an open bounded G-invariant lattice Θ in Π. Its Schikhof dual Θ d is defined as Hom cont O (Θ, O) equipped with weak topology, see [43]. It is easy to see that Θ d is an object of C ζ (O), see [36,Lem. 4.11]. Thus we may apply the functorV to Θ d to obtain a continuous G Qp -representation on a compact O-module. We defineV(Π) : does not depend on the choice of Θ, since any two are commensurable. The functor Π →V(Π) is contravariant. If Π is absolutely irreducible and occurs as a subquotient of a unitary parabolic induction of a unitary character, then we say that Π is ordinary. Otherwise, we say that Π is non-ordinary. In this case it is shown in [36], [15] thatV(Π) is a 2-dimensional representation of G Qp and (taking into account our normalisations) detV(Π) = ζε −1 . A deep theorem of Colmez proved in [13] relates the existence of locally algebraic vectors in Π to the property ofV(Π) being potentially semi-stable with distinct Hodge-Tate weights. With our conventions Colmez's result says that If A is a commutative ring then we denote by m-Spec A the set of its maximal ideals. If x ∈ m-Spec A then κ(x) will denote its residue field. We will typically be considering m-Spec A when A = R[1/p], where R is a complete local noetherian O-algebra with residue field k. In this case κ(x) is a finite extension of L.

Local part
Let F be a finite extension of Q p . We fix an algebraic closure F of F and let G F = Gal(F /F ). Let C p be the completion of F and letF be the completion of the maximal unramified extension of F in F . Let G = GL n (F ) and let D/F be a central division algebra over F with invariant 1/n.

3.1.
Results of Ludwig and Scholze. We continue to denote by L a (sufficiently large) finite extension of F with the ring of integers O, uniformiser ̟ and residue field k. To a smooth representation π of G on an O-torsion module, Scholze associates a Weil-equivariant sheaf F π on theétale site of the adic space P n−1 F , see [45,Prop. 3.1]. If π is admissible then he shows that for any i ≥ 0 theétale cohomology groups H í et (P n−1 Cp , F π ) carry a continuous D × × G F -action, which make them into smooth admissible representations of D × . Moreover, they vanish for i > 2(n − 1), see [45,Thm. 3.2,4.4]. His construction is expected to realize both p-adic Jacquet-Langlands and the p-adic Langlands correspondences. The trouble is that these groups seem to be impossible to calculate in most cases. It is known that the natural map is an isomorphism, see [45,Prop. 4.7]. In particular, if π SLn(F ) = 0 then H 0 vanishes. We want to extend the results of Scholze to the category of locally admissible representations. Recall that a smooth representation of G or D × on an O-torsion module is locally admissible if it is equal to the union of its admissible subrepresentions. In this case we may write π = lim − → π ′ , where the limit is taken over all admissible subrepresentations of π. The proof of [45,Prop.3.1] shows that the natural map is an isomorphism, since it induces an isomorphism on stalks at geometric points. In our setting cohomology commutes with direct limits. For all i ≥ 0 we have isomorphisms is a locally admissible representation of D × , which vanishes for i > 2(n − 1).
If n = 2 then the cohomology vanishes for i > 2. If we additionally assume that F = Q p and π is a principal series representation Ludwig has shown in [31,Thm.4.6] that H 2 et (P 1 Cp , F π ) vanishes. Thus we get the following: If π is a locally admissible representation of GL 2 (Q p ), such that all its irreducible subquotients are principal series, then

3.2.
Representation theory of GL 2 (Q p ). From now on we assume that n = 2 and F = Q p so that G = GL 2 (Q p ). Let us recall some representation theory of G. The category Mod l.adm G (O) of smooth locally admissible G-representations on O-torsion modules decomposes into a direct sum of indecomposable subcategories called blocks. Blocks containing an absolutely irreducible k-representation of G correspond to semi-simple representationsr ss : G Qp → GL 2 (k), such thatr ss is either absolutely irreducible or a direct sum of 1-dimensional representations.
Let us describe these blocks explicitly. We denote by [π] the isomorphism class of a representation π of G. Ifr ss is absolutely irreducible then let Brss = {[π]}, where π is the supersingular representation of G corresponding tor ss under the semisimple mod p Langlands correspondence, [6]. Ifr ss = χ 1 ⊕ χ 2 then we consider χ 1 and χ 2 as characters of Q × p via the local class field theory, and let which maps x to x|x|, and let ω : Q × p → k × be its reduction modulo ̟. We then let Brss be the set of isomorphism classes of irreducible subquotients of π 1 and π 2 . It can have from one up to four elements depending on χ 1 χ where πrss is the maximal G-invariant subspace of π, such that the isomorphism classes of all its irreducible subquotients lie in Brss, and πr ss is the maximal Ginvariant subspace of π such that none of its irreducible subquotients lie in Brss.
In this paper we are especially interested in the case, whenr ss = χ 1 ⊕ χ 2 and In this case both representations π 1 and π 2 in (4) are irreducible principle series representations and so every π ∈ Mod l.adm . is exact and covariant.
We introduce these dual categories, because it is much more convenient to work with compact torsion-free O-modules than with discrete divisible O-modules.
We denote the category of unitary admissible L-Banach space representations of G by Ban adm G (L). If Π is in Ban adm G (L) and Θ is an open bounded G-invariant lattice in Π then it follows from [36,Lemma 4.4] that the Schikhof dual We refer the reader to [43, §1] for the properties of Schikhof dual. We thus may apply the functorsŠ i to it to obtain a compact O-moduleŠ i (Θ d ) with a continuous G Qp × D × -action. We denote the Schikhof dual of this module byŠ i (Θ d ) d and equip it with the p-adic topology.
Proof. Since where the transition maps in the projective system are induced by multiplication by ̟. The short The exact sequence is obtained by passing to the limit by noting that the system satisfies the Mittag-Leffler condition.
If we let M : We The definition does not depend on the choice of Θ, since any two are commensurable. To motivate this definition we observe that (6) implies that we have natural isomorphisms we obtain a long exact sequence: The terms for i ≥ 3 vanish, because of the results of Scholze explained in the previous section. Moreover, since Π is admissible Banach space representation we know that Θ/̟ is an admissible smooth G-representation. Scholze [43]. Now let us note, that even if we could show thatŠ 1 (Θ d /̟) is non-zero, we cannot rule out using (7)  Proposition 3.5. Assume thatr ss = χ 1 ⊕ χ 2 with χ 1 χ −1 2 = ω ±1 and let π 1 and π 2 be the principal series representation defined in (4), so that Brss consists of the isomorphism classes of π 1 and π 2 . Then the following assertions are equivalent: rss is absolutely irreducible, non-ordinary thenŠ 1 (Π) = 0. Proof. Any locally admissible representation of GL 2 (Q p ) is equal to the union of its subrepresentations of finite length, see [19,Thm.2.3.8]. If (i) holds then (ii) follows from (3) and Corollary 3.2. Part (ii) trivially implies (iii), which trivially implies (iv). For the proof of (iv) implies (i) recall that Π is non-ordinary if and only if it does not occur as a subquotient of a unitary parabolic induction of any unitary character of the maximal torus in G. If Θ is an open bounded G-invariant lattice in Π then the semi-simplification of Θ/̟ is isomorphic to π 1 ⊕ π 2 by [36, Thm.11.1]. Ludwig's theorem implies that H 2 (P 1 Cp , F Θ/̟ ) = 0 and the isomorphism (2) implies that the same holds for H 0 . Topological Nakayama's lemma together with (7) implies thatŠ i (Θ d ) = 0 for i = 0 and i = 2. We deduce that we have a short exact sequence: is isomorphic to a product of copies of O. In particular, if S 1 (Θ) = 0 thenŠ 1 (Π) = 0. Thus (iv) implies thatŠ 1 (Θ d ) = 0 and (8) implies that H 1 et (P 1 Cp , F Θ/̟ ) = 0. Since the semisimplification of Θ/̟ is isomorphic to π 1 ⊕ π 2 , Corollary 3.2 implies (i).
Remark 3.6. We will show later on that part (i) of the Proposition does not hold by showing that completed cohomology gives a counterexample to (ii). However, we can not rule out that one of the groups can vanish in (i) (unless of course π 1 ∼ = π 2 ). Most likely both groups are non-zero since there is no natural way to distinguish one of the principal series in the block.
In particular, if M is a pro-flat A-module, in the sense that the functor m → m ⊗ A M is exact, then so isŠ 1 (M ).
Proof. We refer the reader to [10] for the basics on pseudocompact rings and completed tensor products. Since the functor S 1 : Mod l.adm commutes with direct sums and is exact by Corollary 3.2, the functorŠ 1 : C G (O)rss → C G Qp ×D × (O) commutes with products and is also exact. The proposition is a formal consequence of these two properties, see the proof Proposition 2.4 in [38], which is based on ideas of Kisin in [27].
If M is A-flat then Proof. The first assertion follows from Lemma A.15 of [22], which says that if we mod out one relation then the dimension either goes down by one or stays the same. Other assertions follow from Proposition A.30 in [22] by observing that Since k ⊗ A M ∼ = (k ⊗ A R) ⊗ R M , it admits k ⊗ R M as a quotient, and since k ⊗ A R is an R-module of finite length, it has a filtration of finite length with graded pieces isomorphic to subquotients of k ⊗ R M . Lemma A.8 in [22] implies The corollary now follows from the proposition.

Local-global compatibility
Let p be a prime and let F be a totally real number field with a fixed place p above p, such that F p = Q p , and a fixed infinite place ∞ F . Let D 0 be a quaternion algebra with centre F , ramified at all the infinite places of F and split at p. Let Σ be a set of finite ramification places of D 0 . We fix a maximal order O D0 of D 0 , and for each finite place v ∈ Σ an isomorphism (O D0 ) v ∼ = M 2 (O Fv ). For each finite place v of F we will denote by N(v) the order of the residue field at v, and by ̟ v ∈ F v a uniformizer.
Denote by A f F ⊂ A F the finite adeles and the adeles respectively.
We assume that U p = GL 2 (Z p ) = K and that U v is a pro-p group at other places above p. We may write and a finite index set I. In the arguments that follow we are free to replace U by a smaller open subgroup by shrinking U v at any place v different from p. In particular, we may assume that (13) ( [39]. We will use standard notation for subgroups of U , so for example U p The group (D 0 ⊗ Q p ) × acts continuously on S(U p , A) by right translations. It follows from (13) where C denotes the space of continuous functions. Let ψ : (A f F ) × /F × → O × be a continuous character such that ψ is trivial of (A f F ) × ∩ U p . We may consider ψ as an A-valued character, via (14) induces an isomorphism of U p -representations: where C ψ denotes the continuous functions on which the centre acts by the character ψ. One may think of S ψ (U p , A) as the space of algebraic automorphic forms on D × 0 with tame level U p and no restrictions on the level or weight at places dividing p. We want to introduce a variant by fixing the level and weight at places dividing p, different from p. Let λ be a continuous representation U p p on a free O-module of finite rank, such that (A f F ) × ∩ U p p acts on λ by the restriction of ψ to this group. We let S ψ,λ (U p , A) := Hom U p p (λ, S ψ (U p , A)).

We will omit λ as an index if it is the trivial representation. Let us note that a presentation O[[U p p ]] ⊕n → O[[U p
p ]] ⊕m ։ λ gives us an exact sequence: If A is an O/̟ n -module then there is an open subgroup V p p of U p p , which acts trivially on λ/̟ n . If we let V p := U p V p p then by taking V p p -invariants of (16) we have an exact sequence (17) 0 If the topology on A is discrete, for example if A = L/O or A = O/̟ n then we have The action of (D 0 ⊗ Q p ) × on S ψ (U p , A) by right translations induces a continuous action of (D 0 ⊗ F F p ) × ∼ = GL 2 (Q p ) = G on S ψ (U p , A) and S ψ,λ (U p , A). Let ζ : Q × p → O × be the character obtained by restricting ψ to F × p .
Proof. The first assertion follows from (15), which also implies that we have an isomorphism of K-representations   (15) and (19).
Let S be a finite set of places of F containing Σ, all the places above p, all the infinite places and all the places v, where U v is not maximal and all the ramification places of ψ. Let T univ Proof. If U ′ p is an open subgroup of U p then S ψ (U p , O/̟ n ) U ′ p is a finitely generated O/̟ n -module and Chinese remainder theorem implies that ( p . By passing to a direct limit over all such U ′ p and n ≥ 1 we obtain the assertion for S ψ (U p , L/O). The argument for S ψ,λ (U p , L/O) is the same. The last assertion follows from Lemma 5.2.
We let equipped with the p-adic topology. It follows from (15) that for all n ≥ 1 the map Letr denote the restriction ofρ to the decomposition group at p, which we identify with the absolute Galois group of Q p . It is enough to prove the statement for S ψ (U p , k) m with U p p arbitrary small, since then (17) and (18) imply the assertion in general.
Let π be an irreducible subrepresentation of S ψ (U p , k) m . After enlarging L we may assume that π is absolutely irreducible. Let σ be an irreducible Ksubrepresentation of π. Thenσ is isomorphic to Sym b k 2 ⊗ det a , for uniquely determined integers 0 ≤ b ≤ p − 1, 0 ≤ a ≤ p − 2. It follows from [4] that where the Hecke operators T and S correspond to the double cosets K p 0 0 1 K and K p 0 0 p K, respectively. Moreover, there are λ, µ ∈ k such that π is a quotient of . We claim that one may read off λ, µ and the possible values of (a, b) from the restriction ofρ to the decomposition group at p. It then follows from [4] and [6], which describe the irreducible quotients of c-Ind G K σ/(T − λ, S − µ), that π lies in Brss. These arguments are by now fairly standard and appear in the weight part of Serre's conjecture, so we only give a sketch.
We first modify the setting slightly: if ψ ′ : (A f F ) × /F × → O × is a character congruent to ψ modulo ̟ and λ ′ is a representation of U p p on a finite O-module with central character ψ ′ and U p p is a pro-p group, then the U p p -invariants of λ ′ /̟ are non-zero, and thus S ψ (U p , k) m is a G-invariant subspace of S ψ ′ ,λ ′ (U p , k) m . We may choose ψ ′ = χ b+2a cyc α, where a and b are as above and α is the Teichmüller lift of the character ψχ −b−2a cyc (mod ̟) and χ cyc is the p-adic cyclotomic character and where the tensor product is taken over all embeddings ι : F ֒→ L, which do not factor through F p . Note that since U p p is assumed to be pro-p the character ψχ −b−2a and thus λ ′ has central character ψ ′ . Note that since ζ is also the central character ofσ, we have ζ(x) ≡ x b+2a (mod ̟), for all x ∈ Z × p . To ease the notation we drop the superscript ′ from ψ ′ and λ ′ .
Since S ψ,λ (U p , k) m is an admissible representation by Lemma 5.1, the k-vector space is finite dimensional. Thus we may assume that π is contained in S ψ,λ (U p , k)[m]. Let Since S ψ,λ (U p , L/O) m is admissible and injective in Mod sm K,ζ (O) by Lemmas 5.1, 5.2, using (20) we see that Hom K (σ • , S ψ,λ (U p , O) m ) is a free O-module of finite rank, which is congruent to Hom K (σ, S ψ,λ (U p , k) m ) modulo ̟. It follows from [16,Lemme 6.11] that after replacing L with a finite extension contains an eigenvector φ for all the Hecke operators in T univ S with eigenvalues lifting those given by m. By evaluating φ we obtain an automorphic form f on D × 0 such that the associated Galois representation ρ f liftsρ and its restriction to the decomposition group at p is crystalline with Hodge-Tate weights (1 − a, −a − b), where we adopt the conventions of [11], so that the cyclotomic character has Hodge-Tate weight equal to −1. We note that the difference of the two Hodge-Tate weights is equal to b + 1, which is between 1 and p. In particular,σ is a Serre weight for r. The possibilities for the pair (a, b) are listed in the proof of [11,Lemma 2.15]. The compatibility of local and global Langlands correspondence implies that the Gsubrepresentation of S ψ,λ (U p , O) m ⊗ O L generated by the image of φ is of the form Ψ ⊗ σ, where Ψ is a smooth unramified principal series representation. Moreover, one may read off the Satake parameters of Ψ from the Weil-Deligne representation associated to ρ f | GF p , see [11,Proposition 2.9]. It follows from [7] that λ and µ are reductions modulo ̟ of the Satake parameters of Ψ, rescaled by a suitable power of p, see the proof of [11,Lemma 2.15]. It then follows from [4] and [6], which describe the irreducible quotients of c-Ind G K σ/(T − λ, S − µ), that π lies in Brss. One could sum up the last part of the argument as the compatibility of p-adic and mod-p Langlands correspondences.
Let R ps trr be the universal deformation ring parameterizing 2-dimensional pseudocharacters of G Qp lifting trr. We have two actions of R ps trr on S ψ (U p , L/O) m . The first action is given by the composition where R univ ρ is the universal global deformation ring ofρ, and the first arrow is obtained by considering the restriction of the trace of the universal deformation ofρ to G Fp . The second arrow is obtained by associating Galois representations to automorphic forms, see [45,Proposition 5.7]. The third arrow is given by [45,Corollary 7.3]. The second action , is given by interpreting R ps trr as the centre of the category Mod l.adm G (O)rss using [36]. In order to apply the results of [36], ifr ss = χ⊕χω for some character χ : G Qp → k × then we assume that p ≥ 5. In [36] we work with a fixed central character, but this can be unfixed by using the ideas of [11,§6.5,Cor. 6.23]. Since the centre of the category acts naturally on every object in the category, Proposition 5.4 implies that R ps trr acts naturally on S ψ (U p , L/O) m and using (20) we obtain a natural action of R ps trr on S ψ (U p , O) m , which we denote by θ 2 . Proposition 5.5. Assume that ψ is of finite, prime to p order and ψ is trivial on Then the two actions θ 1 and θ 2 of R ps trr on S ψ (U p , L/O) m coincide. Proof. The result follows from three ingredients: the theory of capture [15, §2.4], [39, §2.1], which is based on the ideas of Emerton in [18], the results of Berger-Breuil [5] on the universal unitary completions of locally algebraic principal series representations and the results of Colmez in [13] on the compatibility of the p-adic and classical local Langlands correspondence, as we will now explain.
It follows from (20) that it is enough to show that the two actions coincide on Let τ := Ind K J χ, and let V a := τ ⊗ L Sym 2a L 2 ⊗ det −a , a ≥ 0. Note that the central character of V a is equal to ζ. It follows from the proof of Proposition 2.7 in [39] that if ϕ is a continuous K-equivariant endomorphism of Π(U p ), which kills for all a ≥ 0, then ϕ is zero. Thus it is enough to show that the two actions of R ps trr on the above modules coincide, so that θ 1 (r) − θ 2 (r) annihilates them for all r ∈ R ps trr and all a ≥ 0. In the following let V = V sm ⊗ V alg , where V alg = Sym 2a L 2 ⊗ det −a , and V sm = τ or τ ⊗ η • det. Since V is a locally algebraic representation of K, we have Hom K (V, Π(U p )) = Hom K (V, Π(U p ) l.alg ), where Π(U p ) l.alg is the subspace of locally algebraic vectors in Π(U p ). This subspace can be identified with a subspace of classical automorphic forms on D × 0 , see [48,Lemma 1.3], [28, §3.1.14], [20, §3]. In particular, the action of T(U p ) m [1/p] on Π(U p ) l.alg , and hence on Hom K (V, Π(U p )), is semi-simple. Since S ψ (U p , O) d m is finitely generated as an O[[K]]-module, the vector space Hom K (V, Π(U p )) is finite dimensional. Thus for a fixed V , after replacing L by a finite extension, we may assume that Hom K (V, Π(U p )) has a basis of eigenvectors for the action of T(U p ) m [1/p].
Let φ ∈ Hom K (V, Π(U p )) be such an eigenvector. Then it is enough to show that φ is an eigenvector for the action R ps trr via θ 2 and that the annihilators of φ for the two actions coincide, since then θ 1 (r) − θ 2 (r) will kill φ for all r ∈ R ps trr . Let x ∈ m-Spec R ps trr [1/p] be the kernel for the action R ps trr [1/p] on φ via θ 1 . Then by unravelling the definition of θ 1 we get that x corresponds to the pseudo-character tr(ρ f | GF p ), where ρ f is the Galois representation attached to the automorphic form f , corresponding to the Hecke eigenvalues given by the action of T(U p ) m [1/p] on φ. Moreover, by the same argument as in the proof of Proposition 5.4, ρ f | GF p , is potentially semistable with Hodge-Tate weights (1 + a, −a). As in the proof of Proposition 5.4 the G-subrepresentation of Π(U p ) generated by the image of φ is of the form Ψ ⊗ V alg , where Ψ is an irreducible smooth representation with Hom K (V sm , Ψ) = 0. The theory of types, see [23], implies that Ψ ∼ = ( Hence, ρ f | GF p is potentially crystalline. Moreover, Ψ determines the Weil-Deligne representation associated to ρ f | GF p via the classical local Langlands correspondence. The universal unitary completion of Ψ ⊗ V alg is absolutely irreducible, by [5, 5.3.4], [8, 2.2.1], and will coincide with the closure of Ψ ⊗ V alg in Π(U p ), which we denote by Π. The action of R ps trr [1/p] on Π(U p ) via θ 2 preserves Π, since it acts as the center of the category. Schur's lemma implies that the annihilator of Π is a maximal ideal of R ps trr [1/p], which we denote by y. Note that we have shown that the action of R ps trr [1/p] via θ 2 preserves φ and the annihilator is equal to y. So we have to show that x = y.
If Π is non-ordinary then r :=V(Π) is an absolutely irreducible two dimensional potentially crystalline representation of G Qp liftingr and it follows from [36,Proposition 11.3] that y corresponds to tr r. The compatibility of p-adic and classical Langlands correspondences, proved by Colmez in [13], implies that r and ρ f | GF p have the same Weil-Deligne representations. Moreover, the Hodge-Tate weights of r are determined by V alg and are equal to those of ρ f . Since the Hodge-Tate weights of r and ρ f | GF p are equal, it follows from [14, §4.5] that r ∼ = ρ f | GF p and and so x = y.
Corollary 5.6. The two actions of R ps trr on S ψ (U p , L/O) m and S ψ,λ (U p , L/O) m via θ 1 and θ 2 coincide.
This induces an isomorphism between the G-endomorphism rings Moreover, for i = 1, 2 we have φ • θ i = θ i • tw χ , where tw χ is the automorphism of R ps trr obtained by sending a deformation to its twist by χ. Thus if we prove the assertion for ψ then we may deduce the assertion for ψχ 2 for all χ as above.
Letψ be the reduction of ψ modulo ̟ and let [ψ] be the Teichmüller lift ofψ, then the character ψ −1 [ψ] takes values in 1 + p and thus we may take its square root by the usual binomial formula. If χ := ψ −1 [ψ] then ψχ 2 = [ψ] has order prime to p. Corollary 5.7. Let x ∈ m-Spec T(U p ) m [1/p] be such that the restriction of the corresponding Galois representation ρ x : G F,S → GL 2 (κ(x)) to G Fp is irreducible. Then there is an isomorphism of L-Banach space representations of G: for some integer n ≥ 0, where Π ∈ Ban adm G (L) corresponds to ρ x | GF p via the p-adic Langlands correspondence for GL 2 (Q p ).
Proof. Let y : R ps trr → L be the homomorphism corresponding to tr(ρ x | GF p ). It follows from Corollaries 6.8, 8.14, 9.36 and Proposition 10.107 in [36] that (21) holds for any Π ′ ∈ Ban adm G (L)rss of finite length, on which R ps trr acts as the center of the category via y. Corollary 5.6 and Lemma 5.8 below imply that we may apply this observation to Proof. There are only finitely many isomorphism classes of irreducible objects in Brss. Let π 1 , . . . , π k be a set of representatives. For each i let π i ֒→ J i be an injective envelope of π i in Mod l.adm G (O). Dually we let P i := J ∨ i so that P i ։ π ∨ i is a projective envelope of π ∨ i in C(O). Since the category Mod l.adm G (O)rss is locally finite we may embed τ into an injective envelope of its G-socle, which is isomorphic to a direct sum of finitely many copies of J i . This implies that dually there is a surjection ⊕ k i=1 P ⊕mi i ։ τ ∨ , where m i are finite multiplicities. Thus it is enough to prove the statement for τ = J i , 1 ≤ i ≤ k. It follows from [36,Lemma 3.3] that k ⊗ R ps trr P i is of finite length if and only if Hom C(O) (P j , k ⊗ R ps trr P i ) is a finite dimensional k-vector space for 1 ≤ j ≤ k. Since P j is projective the natural map k ⊗ R ps trr Hom C(O) (P j , P i ) → Hom C(O) (P j , k ⊗ R ps trr P i ) is an isomorphism. Thus it is enough to show that Hom C(O) (P j , P i ) is a finitely generated R ps trr -module for 1 ≤ i, j ≤ k. This assertion follows from Proposition 6.3, Corollary 8.11 in conjuction with Proposition B.26, Corollaries 9.25 and 9.27, Lemma 10.90 in [36].
Lemma 5.9. Let τ ∈ Mod l.adm G (O)rss and suppose that we are given a faithful action on τ of a local R ps trr -algebra R with residue field k via R ֒→ End G (τ ). Assume that the composition R ps trr → R → End G (τ ) coincides with the action of R ps trr on τ as the centre of category Mod l.adm G (O)rss . If the G-socle of τ is of finite length then R is finite over R ps trr . Proof. Let P i be as in the proof of Lemma 5.8 and let P = ⊕ n i=1 P i . Then P is a projective generator for the category C(O)rss and thus R acts faithfully on m := Hom C(O) (P, τ ∨ ). The proof of Lemma 5.8 shows that m ⊗ R ps trr k is a finite dimensional k-vector space. Since m is a compact R ps trr -module we deduce that m is finitely generated over R ps trr and hence also over R. If v 1 , . . . , v n are generators of m as an R-module then the map r → (rv 1 , . . . , rv n ) induces an embedding of R-modules R ֒→ m ⊕n . Since R ps trr is noetherian we deduce that R is a finitely generated R ps trr -module.
Proposition 5.10. Let τ and R ֒→ End G (τ ) be as in Lemma 5.9. Assume that τ is killed by ̟ and has a central character. If τ is admissible then the Krull dimension of R is less or equal to 2.
Proof. Since τ is admissible, its G-socle is of finite length. Lemma 5.9 implies that R is a finite R ps trr -module. Since we are interested only in the Krull dimension we may assume that R is the image of R ps trr in End G (τ ). Since τ has a central character the map R ps trr → End G (τ ) factors through the quotient R ps trr ։ R ps,ψ trr , which parameterises pseudocharacters with a fixed determinant. Since ̟ kills τ we get a surjection R ps,ψ trr /̟ ։ R. Now R ps,ψ trr /̟ is an integral domain of Krull dimension 3, see [36, §A], so we have to show that R ps,ψ trr /̟ does not act faithfully on τ . This assertion follows from [25,Thm. 3.23]. We will sketch the proof of this result along the lines of the proof of [25,Thm. 3.23] for the convenience of the reader. We first translate the faithfulness of the action into a commutative algebra statement.
Let π 1 , . . . π k be as in the proof of Lemma 5.8, but let π i ֒→ J i now denote an injective envelope of π i in Mod l.adm G,ζ (O), so that in the rest of the proof we work with a fixed central character. Let P := ⊕ k i=1 P i , where P i := J ∨ i and let E = End C(O) (P ). Then P is a projective generator of C ζ (O)rss and if we let m := Hom C(O) (P, τ ∨ ) then evaluation induces a natural isomorphism m ⊗ E P ∼ = −→ τ ∨ . Thus it is enough to show that R ps,ψ trr /̟ does not act faithfully on m. Since m is finitely generated over R ps,ψ trr , it is enough to produce a finitely generated module M of R ps,ψ trr /̟ of dimension at least 1, such that m⊗ R ps,ψ trr M has dimension 0. Since this would imply that the intersection of the supports of m and M in Spec R ps,ψ trr /̟ is equal to the closed point and thus m cannot be supported on the whole of Spec R ps,ψ trr /̟. Let σ be a smooth irreducible k-representation of K, such that Hom K (σ, τ ) is non-zero. Since R ps,ψ trr is noetherian and E is a finitely generated R ps,ψ trr -module, we may choose a presentation E ⊕n → E ⊕m ։ m. By applying ⊗ E P we obtain a presentation P ⊕n → P ⊕m ։ τ ∨ . By applying the functor Hom cont O[[K]] ( * , σ ∨ ) ∨ we obtain an isomorphism of R ps,ψ trr -modules (22) Hom cont The left-hand-side of (22) is non-zero by the choice of σ, and is a finite dimensional k-vector space since τ is admissible. Since E is a finitely generated R ps,ψ trr -module, E⊗ R ps,ψ trr E is a finitely generated E-module and we have an exact sequence of E-modules E ⊕a → E ⊗ R ps,ψ trr E ։ E, for some a ≥ 0. This induces an exact sequence of E-modules M (σ) ⊕a → E ⊗ R ps,ψ trr M (σ) ։ M (σ). This induces an exact sequence of k-vector spaces Since m is a finitely generated R ps,ψ trr -module and the ring is noetherian, the completed tensor product coincides with the usual tensor product and (23) implies that m ⊗ R ps,ψ trr M (σ) is a non-zero finite dimensional k-vector space.
We claim that M (σ) is a finitely generated R ps,ψ trr -module of dimension 1. As explained above the claim implies the proposition. We have an isomorphism of R ps,ψ trr -modules M (σ) There is some i such that M i (σ) is non-zero, since otherwise it would follow from (22), (23) that Hom K (σ, τ ) is zero. If M i (σ) is non-zero then it is a finitely generated R ps,ψ trr -module of dimension 1. In most cases this follows from [38,Thm. 5.2], since it implies that the element denoted by x in loc. cit. is a parameter for M i (σ). The rest of the cases are handled in [25,Cor. 3.11] by a similar argument.
Corollary 5.11. Let τ and R ֒→ End G (τ ) be as in Lemma 5.9. Assume that τ has a central character and is ̟-divisible (equivalently τ ∨ is ̟-torsion free). Then the Krull dimension of R is at most 3.
Proof. Since τ ∨ is ̟-torsion free we have an exact sequence 0 → τ ∨ ̟ → τ ∨ → τ ∨ /̟ → 0. Let P be the projective generator of C ζ (O)rss as in the proof of Proposition 5.10. We have an exact sequence of R-modules As explained in the proof of Lemma 5.9, Hom C(O) (P, τ ∨ ) is a finitely generated faithful R-module. If we let R be the quotient of R which acts faithfully on Hom C(O) (P, τ ∨ /̟) then its Krull dimension is equal to the dimension of the support of Hom C(O) (P, τ ∨ /̟) in Spec R, which is one less than the dimension of the support of Hom C(O) (P, τ ∨ ) in Spec R, as ̟ is regular on the module, and thus is equal to dim R − 1, as the action of R on Hom C(O) (P, τ ∨ ) is faithful.
It follows from Proposition above applied to R and τ [̟] that the Krull dimension of R is at most 2, thus the Krull dimension of R is at most 3. Proof. This follows from Lemma 5.9, Corollary 5.11 applied to τ = S ψ,λ (U p , L/O) m .
Remark 5.13. The proof of Corollary 5.12 goes back to the workshop on Galois representations and Automorphic forms in Princeton in 2011, and was motivated by discussions with Matthew Emerton there. It remained unpublished, since after we communicated the result to Frank Calegari, see [40], together with Patrick Allen they proved much more general results concerning finiteness of global deformation rings over local deformation rings, see [1]. However, one advantage of our argument is that we don't have to assume anything about the image of the global Galois representationρ. Proof. This follows from the proof of Corollary 7.5 in [41]. Alternatively, one can use [29,Prop. 5.4,5.7,Thm. 5.13]. However, the proof dearest to the author's heart is to show that dim π Kn grows as Cp n , for some constant C, as K n runs over principal congruence subgroups of K. If π is principal series then this can be done by hand, the result for special series can be deduced from this. The most interesting case, when π is supersingular, can be deduced from the exact sequence in Theorem 6.3 in [35]. Let us sketch the argument using the notation of loc. cit.. This gives an upper bound on the growth of K n -invariants, which has to be of the right order, since both M σ and Mσ are infinite dimensional k-vector spaces.
Yet another alternative is to use results of Morra [32], who actually computes the dimensions dim π In , where I n is a certain filtration of Iwahori subgroup by open normal subgroups.
Corollary 5.6 and Lemma 5.8 imply that F is of finite length as a G-representation. Lemma 5.14 implies that the fibre has dimension less than or equal to one. If it is zero then all irreducible subquotients of F would be characters and by looking at the graded pieces of the m-adic filtration on where m is the maximal ideal of R, we would deduce that all the irreducible subquotients of S ψ (U p , O) d m are characters. Since the central character is fixed and p > 2 there are no non-trivial extensions between 1-dimensional G-representations over k. This would imply that SL 2 (Q p ) acts trivially on S ψ,λ (U p , O) d m , which is impossible, since it would imply that SL 2 (Z p ) acts trivially on a projective object in Mod pro K,ζ (O). Hence, the dimension of the fibre is 1. It follows from Corollary 5.12 that the Krull dimension of R is at most 3. The assertion follows from Corollary 4.2.

Main result
We keep the notation of the previous section. Let D be the quaternion algebra over F , which is ramified at p, split at ∞ F and has the same ramification behaviour as D 0 at all the other places. We fix an isomorphism This allows us to view the subgroup U p of (D 0 ⊗ F A f F ) × , considered in the previous section, as a subgroup of (D ⊗ F A f F ) × . Let D p := D ⊗ F F p . Then D p is the nonsplit quaternion algebra over F p = Q p . We let U p = O × Dp and U = U p U p . If K is an open subgroup of U := U p U p then we let X(K) be the corresponding Shimura curve for D/F defined over F . We let Theorem 6.2 in [45] gives an isomorphism of T univ where S 1 is the functor π → H 1 et (P 1 , F π ). Let ψ : (A f F ) × /F × → O × be a continuous character such that ψ is trivial of (A f F ) × ∩ U p . To ease the notation we will use the same symbol to denote the restriction of ψ to the intersection of (A f F ) × with various subgroups of (D ⊗ F A f F ) × with the exception of ζ := ψ| F × p . We will also view ψ as a character of G F,S via the class field theory and denote by the same letter its restriction to various decomposition groups.
Let automorphic forms on D × and D × 0 , respectively, see Theorem 5.3 in [34] and [20, §3]. The assumption that S ψ (U p , L/O) m = 0 implies that S ψ (U p , O) m ⊗ O L is non-zero, and using Lemma 5.3 we may find a locally algebraic representation γ of U p p × GL 2 (Z p ) of the form λ ⊗ (σ ⊗ Sym b L 2 ⊗ det a ), such that σ is a supercuspidal type and Hom U p p GL2(Zp) (γ, S ψ (U p , O) m ⊗ O L) = 0. An eigenvector for the Hecke operator on this finite dimensional vector space will give us an automorphic form on D × 0 , which we may transfer to an automorphic form on D × by the classical Jacquet-Langlands correspondence, which will turn give us a non-zero vector in the locally algebraic vectors of H 1 ψ (U p , O) m ⊗ O L.
From now on we assume thatr ss = χ 1 ⊕ χ 2 with χ 1 χ −1 2 = ω ±1 . Let π 1 , π 2 be the principal series representations defined in (4). Proposition 3.5 implies that at least one ofŠ 1 (π ∨ 1 ) andŠ 1 (π ∨ 2 ) is non-zero. Proposition 6.5. The maximum of δ(Š 1 (π ∨ 1 )) and δ(Š 1 (π ∨ 2 )) is equal to 1. Proof. Let A be the ring in Proposition 5.15 and let K be an open uniform pro-p subgroup of D × p . It follows from Propositions 3.7 and 6.1 that H 1 ψ,λ (U p , O) d m is A-flat. Since it is non-zero, the fibre F := k ⊗ A H 1 ψ,λ (U p , O) d m is also non-zero and (10) implies that its δ-dimension is equal to δ( H 1 ψ,λ (U p , O) d m ) − 3. It follows from Proposition 6.2 that the δ-dimension of H 1 ψ,λ (U p , O) d m is equal to 4, see the argument in the proof of the analogous statement for S ψ,λ (U p , O) d m in Proposition 5.15. Hence the δ-dimension of the fibre is equal to 1.
Proof. After twisting by a character χ : (A f F ) × /F × → 1 + p, which is trivial on U p ∩ (A f F ) × we may assume that the restriction of ψ to (A f F ) × ∩ U p is locally alegbraic.
Let σ := σ alg ⊗σ sm be an irreducible locally algebraic representation of U p = O × Dp with central character ζ, where σ alg = Sym b L 2 ⊗ det a and σ sm is smooth. To σ sm one may attach an inertial type τ : I Qp → GL 2 (L), such that τ extends to a representation of the Weil group W Qp , the kernel of τ is an open subgroup of It follows from Proposition 6.9 that Hom Up (σ ⊗ η • Nrd, ( H 1 ψ,λ (U p , O) ⊗ O L)[m x ]) = 0, where σ is as above and η is either trivial or pr(χ cyc ). If η is trivial then the claim implies that y lies in Spec R r (w, τ, ψ) ds . We note that our assumption on r implies that any reducible potentially semistable lift r with Hodge-Tate weights w is crystabelline and its WD(r)| W Qp is a direct sum of distinct characters. Such representations cannot correspond to points on irreducible components of discrete series type. Hence ρ x | GF p is irreducible and thus is determined by its trace.
If η is not trivial then by twisting by its inverse and using the claim again, we deduce that tr(ρ x | GF p ⊗ η −1 ) gives a point in m-Spec R r (w, τ, ψη −2 ) ds [1/p]. Note that the character ψη −2 = ψpr(χ cyc ) −1 is locally algebraic, but the representation λ⊗η −1 •Nrd is not. That is why we cannot appeal directly to the results of Emerton in this case. Thus ρ x | GF p determines the integers a and b, the representation σ sm up to its conjugate by the uniformizer ̟ D of D × p , and whether η is trivial or not, by comparing whether the Hodge-Tate weight of det ρ x | GF p has the same parity as the Hodge-Tate weight of ψ| GF p . This implies thatŠ 1 (Π) 1−alg is isomorphic as a representation of U p to a finite direct sum of copies of σ ⊗ η • Nrd or its conjugate by ̟ D . In particular,Š 1 (Π) is finite dimensional. Remark 6.12. If F = Q then the proof of Proposition 6.11 can be simplified, since after twisting we may directly appeal to the results of Emerton on locally algebraic vectors in completed cohomology. Theorem 6.13. Assume the set up of Theorem 6.7 and the notation of Lemma 6.11. The quotientŠ 1 (Π)/Š 1 (Π) 1−alg contains an irreducible closed subrepresentation of δ-dimension 1.