Patching and the p-adic Langlands program for GL_{2}(ℚ_{p})

Ana Caraiani, Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paškūnas and Sug Woo Shin

Abstract

We present a new construction of the p-adic local Langlands correspondence for GL_{2}(ℚ_{p}) via the patching method of Taylor–Wiles and Kisin. This construction sheds light on the relationship between the various other approaches to both the local and global aspects of the p-adic Langlands program; in particular, it gives a new proof of many cases of the second author’s local-global compatibility theorem, and relaxes a hypothesis on the local mod p representation in that theorem.

1. Introduction

The primary goal of this paper is to explain how (under mild technical hypotheses) the patching construction of [CEG^{+}16], when applied to the group GL_{2}(ℚ_{p}), gives rise to the p-adic local Langlands correspondence for GL_{2}(ℚ_{p}), as constructed in [Col10b], and as further analyzed in [Paš13] and [CDP14]. As a by-product, we obtain a new proof of many cases of the local-global compatibility theorem of [Eme11] (and of some cases not treated there).

1.1 Background We start by recalling the main results of [CEG^{+}16] and the role we expect them to play in the (hypothetical) p-adic local Langlands correspondence. Let F be a finite extension of ℚ_{p}, and let G_{F} be its absolute Galois group. One would like to have an analogue of the local Langlands correspondence for all finite-dimensional, continuous, p-adic representations of G_{F}. Let E be another finite extension of ℚ_{p}, which will be our field of coefficients, assumed large enough, with ring of integers O, uniformizer ϖ and residue field 𝔽. To a continuous Galois representation r : G_{F} → GL_{n}(E) one would like to attach an admissible unitary E-Banach space representation Π(r) of G := GL_{n}(F) (or possibly a family of such Banach space representations). Ideally, such a construction should be compatible with deformations, should encode the classical local Langlands correspondence and should be compatible with a global p-adic correspondence, realized in the completed cohomology of locally symmetric spaces.

It is expected that the Banach spaces Π(r) should encode the classical local Langlands

2010 Mathematics Subject Classification 11S37, 22E50 Keywords: p-adic Langlands, local–global compatibility, Taylor–Wiles patching

A.C. was partially supported by the NSF Postdoctoral Fellowship DMS-1204465 and NSF Grant DMS-1501064. M.E. was partially supported by NSF grants DMS-1003339, DMS-1249548, and DMS-1303450. T.G. was partially supported by a Leverhulme Prize, EPSRC grant EP/L025485/1, Marie Curie Career Integration Grant 303605, and by ERC Starting Grant 306326. D.G. was partially supported by NSF grants DMS-1200304 and DMS-1128155. V.P. was partially supported by the DFG, SFB/TR45. S.W.S. was partially supported by NSF grant DMS-1162250 and a Sloan Fellowship.

A. Caraiani et al.

correspondence in the following way: if r is potentially semi-stable with regular Hodge–Tate weights, then the subspace of locally algebraic vectors Π(r)^{l.alg} in Π(r) should be isomorphic to π_{sm}(r) ⊗ π_{alg}(r) as a G-representation, where π_{sm}(r) is the smooth representation of G corresponding via classical local Langlands to the Weil–Deligne representation obtained from r by Fontaine’s recipe, and π_{alg}(r) is an algebraic representation of G, whose highest weight vector is determined by the Hodge–Tate weights of r.

1.2 Example. If F = ℚ_{p}, n = 2 and r is crystalline with Hodge–Tate weights a < b, then π_{sm}(r) is a smooth unramified principal series representation, whose Satake parameters can be calculated in terms of the trace and determinant of Frobenius on D_{cris}(r), and π_{alg}(r) = Sym^{b−a−1}E^{2} ⊗det^{1−a}. (We note that in the literature different normalisations lead to different twists by a power of det.)

Such a correspondence has been established in the case of n = 2 and F = ℚ_{p} by the works of Breuil, Colmez and others, see [Bre08], [Col10a] as well as the introduction to [Col10b]. Moreover, when n = 2 and F = ℚ_{p}, this correspondence has been proved (in most cases) to satisfy localglobal compatibility with the p-adically completed cohomology of modular curves, see [Eme11]. However, not much is known beyond this case. In [CEG^{+}16] we have constructed a candidate for such a correspondence using the Taylor–Wiles–Kisin patching method, which has been traditionally employed to prove modularity lifting theorems for Galois representations. We now describe the end product of the paper [CEG^{+}16].

Let r̄ : G_{F} → GL_{n}(𝔽) be a continuous representation and let R_{p}^{□} be its universal framed deformation ring. Under the assumption that p does not divide 2n we construct an R_{∞}[G]-module M_{∞}, which is finitely generated as a module over the completed group algebra R_{∞}[[GL_{n}(O_{F})]], where R_{∞} is a complete local noetherian R_{p}^{□}-algebra with residue field 𝔽. If y ∈ Spec R_{∞} is an E-valued point then

Π_{y} := Hom_{O}^{cont}(M_{∞} ⊗_{R_{∞},y} O, E)

is an admissible unitary E-Banach space representation of G. The composition R_{p}^{□} → R_{∞} →^{y} E defines an E-valued point x ∈ Spec R_{p}^{□} and thus a continuous Galois representation r_{x} : G_{F} → GL_{n}(E). We expect that the Banach space representation Π_{y} depends only on x and that it should be related to r_{x} by the hypothetical p-adic Langlands correspondence; see §6 of [CEG^{+}16] for a detailed discussion. We show in [CEG^{+}16, Theorem 4.35] that if π_{sm}(r_{x}) is generic and x lies on an automorphic component of a potentially crystalline deformation ring of r̄ then Π_{y}^{l.alg} ≅ π_{sm}(r_{x}) ⊗ π_{alg}(r_{x}) as expected; moreover, the points x such that π_{sm}(r_{x}) is generic are Zariski dense in every irreducible component of a potentially crystalline deformation ring. (It is expected that every irreducible component of a potentially crystalline deformation ring is automorphic; this expectation is motivated by the Fontaine–Mazur and Breuil–Mézard conjectures. However, it is intrinsic to our method that we would not be able to access these non-automorphic components even if they existed.)

However, there are many natural questions regarding our construction for GL_{n}(F) that we cannot answer at the moment and that appear to be genuinely deep, as they are intertwined with questions about local-global compatibility for p-adically completed cohomology, with the Breuil–Mézard conjecture on the geometry of local deformation rings and with the Fontaine– Mazur conjecture for global Galois representations. For example, it is not clear that Π_{y} depends only on x, it is not clear that Π_{y} is non-zero for an arbitrary y, and that furthermore Π_{y}^{l.alg} is non-zero if r_{x} is potentially semistable of regular weight, and it is not at all clear that M_{∞} does not depend on the different choices made during the patching process.

Patching and the p-adic Langlands program for GL_{2}(ℚ_{p})

1.3 The present paper In this paper, we specialize the construction of [CEG^{+}16] to the case F = ℚ_{p} and n = 2 (so that G := GL_{2}(ℚ_{p}) and K := GL_{2}(ℤ_{p}) from now on) to confirm our expectation that, firstly, M_{∞} does not depend on any of the choices made during the patching process and, secondly, that it does recover the p-adic local Langlands correspondence as constructed by Colmez.

We achieve the first part without appealing to Colmez’s construction (which relies on the theory of (φ, Γ)-modules). The proof that M_{∞} is uniquely determined highlights some key features of the GL_{2}(ℚ_{p}) setting beyond the use of (φ, Γ)-modules: the classification of irreducible mod p representations of GL_{2}(ℚ_{p}) in terms of Serre weights and Hecke operators, and the fact that the Weil–Deligne representation and the Hodge–Tate weights determine a (irreducible) 2-dimensional crystalline representation of G_{ℚ_{p}} uniquely (up to isomorphism).

When combined with the results of [Paš13] (which do rely on Colmez’s functor V̌), we obtain that M_{∞} realizes the p-adic Langlands correspondence as constructed by Colmez.

We also obtain a new proof of local-global compatibility, which helps clarify the relationship between different perspectives and approaches to p-adic local Langlands.

1.4 Arithmetic actions In the body of the paper we restrict the representations r̄ we consider by assuming that they have only scalar endomorphisms, so that End_{G_{ℚp}} (r̄) = 𝔽, and that r̄ ≅ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) ⊗ χ for any character χ : G_{ℚ_{p}} → 𝔽^{×}. For simplicity, let us assume in this introduction that r̄ is irreducible and let R_{p} be its universal deformation ring. Then R_{p}^{□} is formally smooth over R_{p}. Moreover, (as F = ℚ_{p} and n = 2) we may also assume that R_{∞} is formally smooth over R_{p}^{□}, and thus over R_{p}.

The following definition is meant to axiomatize the key properties of the patched module M_{∞}.

1.5 Definition. Let d be a non-negative integer, let R_{∞} := R_{p}[[x_{1}, …, x_{d}]] and let M be a nonzero R_{∞}[G]-module. We say that the action of R_{∞} on M is arithmetic if the following conditions hold:

(AA1) M is a finitely generated module over the completed group algebra R_{∞}[[K]];

(AA2) M is projective in the category of pseudo-compact O[[K]]-modules;

(AA3) for each pair of integers a < b, the action of R_{∞} on

M(σ^{∘}) := Hom_{O[[K]]}^{cont}(M, (σ^{∘})^{d})^{d}

factors through the action of R_{∞}(σ) := R_{p}(σ)[[x_{1}, …, x_{d}]]. Here R_{p}(σ) is the quotient of R_{p} constructed by Kisin, which parameterizes crystalline representations with Hodge–Tate weights (a, b), σ^{∘} is a K-invariant O-lattice in σ := Sym^{b−a−1}E^{2} ⊗ det^{1−a} and (∗)^{d} := Hom_{O}^{cont}(∗, O) denotes the Schikhof dual. Moreover, M(σ^{∘}) is maximal Cohen–Macaulay over R_{∞}(σ) and the R_{∞}(σ)[1/p]-module M(σ^{∘})[1/p] is locally free of rank 1 over its support.

(AA4) for each σ as above and each maximal ideal y of R_{∞}[1/p] in the support of M(σ^{∘}), there is a non-zero G-equivariant map

π_{sm}(r_{x}) ⊗ π_{alg}(r_{x}) → Π_{y}^{l.alg}

where x is the image of y in Spec R_{p}.

The last condition says that M encodes the classical local Langlands correspondence. This is what motivated us to call such actions arithmetic. (In fact in the main body of the paper we use a reformulation of condition (AA4), see §3.1 and Remark 3.3.) To motivate (AA3), we note for the sake of the reader familiar with Kisin’s proof of the Fontaine–Mazur conjecture [Kis09] that the modules M(σ^{∘}) are analogues of the patched modules denoted by M_{∞} in [Kis09], except that Kisin patches algebraic automorphic forms for definite quaternion algebras and in this paper we will ultimately be making use of patching arguments for algebraic automorphic forms on forms of U(2).

1.6 Uniqueness of M_{∞} As already mentioned, the patched module M_{∞} of [CEG^{+}16] carries an arithmetic action of R_{∞} for some d. In order to prove that M_{∞} is uniquely determined, it is enough to show that for any given d, any R_{∞}[G]-module M with an arithmetic action of R_{∞} is uniquely determined. The following is our main result, which for simplicity we state under the assumption that r̄ is irreducible.

1.7 Theorem. Let M be an R_{∞}[G]-module with an arithmetic action of R_{∞}.

(i) If π is any irreducible G-subrepresentation of the Pontryagin dual M^{∨} of M then π is isomorphic to the representation of G associated to r̄ by the mod p local Langlands correspondence for GL_{2}(ℚ_{p}).

(ii) Let π ↪ J be an injective envelope of the above π in the category of smooth locally admissible representations of G on O-torsion modules. Let P̃ be the Pontryagin dual of J. Then P̃ carries a unique arithmetic action of R_{p} and, moreover,

M ≅ P̃ ˆ⊗_{R_{p}} R_{∞}

as R_{∞}[G]-modules.

The theorem completely characterizes modules with an arithmetic action and shows that M_{∞} does not depend on the choices made in the patching process. A further consequence is that the Banach space Π_{y} depends only on the image of y in Spec R_{p}, as expected.

Let us sketch the proof of Theorem 1.7 assuming for simplicity that d = 0. The first step is to show that M^{∨} is an injective object in the category of smooth locally admissible representations of G on O-torsion modules and that its G-socle is isomorphic to π. This is done by computing Hom_{G}(π′, M^{∨}) and showing that Ext_{G}^{1}(π′, M_{∞}^{∨}) vanishes for all irreducible 𝔽-representations π′ of G; see Proposition 4.2 and Theorem 4.15. The arguments here use the foundational results of Barthel–Livné [BL94] and Breuil [Bre03a] on the classification of irreducible mod p representations of G, arguments related to the weight part of Serre’s conjecture, and the fact that the rings R_{p}(σ) are formally smooth over O, whenever σ is of the form Sym^{b−a−1}E^{2} ⊗ det^{1−a} with 1 ⩽ b − a ⩽ p.

This first step allows us to conclude that M^{∨} is an injective envelope of π, which depends only on r̄. Since injective envelopes are unique up to isomorphism, we conclude that any two modules with an arithmetic action of R_{p} are isomorphic as G-representations. Therefore, it remains to show that any two arithmetic actions of R_{p} on P̃ coincide. As R_{p} is O-torsion free, it is enough to show that two such actions induce the same action on the unitary E-Banach space Π := Hom_{O}^{cont}(M, E). Since M is a projective O[[K]]-module by (AA2) one may show using the “capture” arguments that appear in [CDP14, §2.4], [Eme11, Prop. 5.4.1] that the subspace of Kalgebraic vectors in Π is dense. Since the actions of R_{p} on Π are continuous it is enough to show that they agree on this dense subspace. Since the subspace of K-algebraic vectors is semi-simple as a K-representation, it is enough to show that the two actions agree on σ-isotypic subspaces in Π for all irreducible algebraic K-representations σ. These are precisely the representations σ in axiom (AA3). Taking duals one more time, we are left with showing that any two arithmetic actions induce the same action of R_{p} on M(σ^{∘})[1/p] for all σ as above.

At this point we use another special feature of 2-dimensional crystalline representations of G_{ℚ_{p}}: the associated Weil–Deligne representation together with Hodge–Tate weights determine a 2-dimensional crystalline representation of G_{ℚ_{p}} up to isomorphism. Using this fact and axioms (AA3) and (AA4) for the arithmetic action we show that the action of the Hecke algebra H(σ) := End_{G}(c-Ind_{K}^{G} σ) on M(σ^{∘})[1/p] completely determines the action of R_{p}(σ) on M(σ^{∘})[1/p]; see the proof of Theorem 4.30 as well as the key Proposition 2.13. Since the action of H(σ) on M(σ^{∘})[1/p] depends only on the G-module structure of M, we are able to conclude that the two arithmetic actions are the same. The reduction from the case when d is arbitrary to the case when d = 0 is carried out in §4.16.

1.8 Remark. As we have already remarked, the arguments up to this point make no use of (φ, Γ)-modules. Indeed the proof of Theorem 1.7 does not use them. One of the objectives of this project was to find out how much of the p-adic Langlands for GL_{2}(ℚ_{p}) correspondence can one recover from the patched module M_{∞} without using Colmez’s functors, as these constructions are not available for groups other than GL_{2}(ℚ_{p}), while our patched module M_{∞} is. Along the same lines, in section §5 we show that to a large extent we can recover a fundamental theorem of Berger–Breuil [BB10] on the uniqueness of unitary completions of locally algebraic principal series without making use of (φ, Γ)-modules, see Theorem 5.1 and Remark 5.3.

1.9 Remark. As already explained, Theorem 1.7 implies that Π_{y} depends only on the image of y in Spec R_{p}. However, we are still not able to deduce using only our methods that Π_{y} is non-zero for an arbitrary y ∈ m-Spec R_{∞}[1/p]. Since M_{∞} is not a finitely generated module over R_{∞} theoretically it could happen that Π_{y} ≠ 0 for a dense subset of m-Spec R_{∞}[1/p], but Π_{y} = 0 at all other maximal ideals. We can only prove that this pathological situation does not occur after combining Theorem 1.7 with the results of [Paš13].

In §6 we relate the arithmetic action of R_{p} on P̃ to the results of [Paš13], where an action of R_{p} on an injective envelope of π in the subcategory of representations with a fixed central character is constructed using Colmez’s functor; see Theorem 6.18. Then by appealing to the results of [Paš13] we show that Π_{y} and r_{x} correspond to each other under the p-adic Langlands correspondence as defined by Colmez [Col10b], for all y ∈ m-Spec R_{∞}[1/p], where x denotes the image of y in m-Spec R_{p}[1/p].

It follows from the construction of M_{∞} that after quotienting out by a certain ideal of R_{∞} we obtain a dual of completed cohomology, see [CEG^{+}16, Corollary 2.11]. This property combined with Theorem 1.7 and with the results in §6 enables us to obtain a new proof of local-global compatibility as in [Eme11] as well as obtaining a genuinely new result, when \overline{ρ}|_{G_{ℚp}} is isomorphic to ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} ) ⊗ χ, where ω is the cyclotomic character modulo p. (See Remark 7.7.)

1.10 Prospects for generalization Since our primary goal in this paper is to build some new connections between various existing ideas related to the p-adic Langlands program, we have not striven for maximal generality, and we expect that some of our hypotheses on r̄ : G_{ℚ_{p}} → GL_{2}(\overline{𝔽}_{p}) could be relaxed. In particular, it should be possible to prove results when p = 2 by using results of Thorne [Tho] to redo the patching in [CEG^{+}16]. It may also be possible to extend our results to cover more general r̄ (recall that we assume that r̄ has only scalar endomorphisms, and that it is not a twist of an extension of the trivial character by the mod p cyclotomic character). In Section 6.27 we discuss the particular case where r̄ has scalar semisimplification; as this discussion (and the arguments of [Paš13]) show, while it may well be possible to generalise our arguments, they will necessarily be considerably more involved in cases where r̄ does not satisfy the hypotheses that we have imposed.

Since the patching construction in [CEG^{+}16] applies equally well to the case of GL_{2}(F) for any finite extension F/ℚ_{p}, or indeed to GL_{n}(F), it is natural to ask whether any of our arguments can be extended to such cases (where there is at present no construction of a padic local Langlands correspondence). As explained in Remark 3.2, the natural analogues of our axioms (AA1)-(AA4) hold, even in the generality of GL_{n}(F). Unfortunately, the prospects for proving analogues of our main theorems are less rosy, as it seems that none of the main inputs to our arguments will hold. Indeed, already for the case of GL_{2}(ℚ_{p^{2}}) there is no analogue available of the classification in [Bre03a] of the irreducible 𝔽-representations of GL_{2}(ℚ_{p}), and it is clear from the results of [BP12] that any such classification would be much more complicated.

Furthermore, beyond the case of GL_{2}(ℚ_{p}) it is no longer the case that crystalline representations are (essentially) determined by their underlying Weil–Deligne representations, so there is no possibility of deducing that a p-adic correspondence is uniquely determined by the classical correspondence in the way that we do here, and no hope that an analogue of the results of [BB10] could hold. Finally, it is possible to use the constructions of [Paš04] to show that for GL_{2}(ℚ_{p^{2}}) the patched module M_{∞} is not a projective GL_{2}(ℚ_{p^{2}})-module.

1.11 Outline of the paper

In Section 2 we recall some well-known results about Hecke algebras and crystalline deformation rings for GL_{2}(ℚ_{p}). The main result in this section is Proposition 2.15, which describes the crystalline deformation rings corresponding to Serre weights as completions of the corresponding Hecke algebras. In Section 3 we explain our axioms for a module with an arithmetic action, and show how the results of [CEG^{+}16] produce patched modules M_{∞} satisfying these axioms.

Section 4 proves that the axioms determine M_{∞} (essentially) uniquely, giving a new construction of the p-adic local Langlands correspondence for GL_{2}(ℚ_{p}). It begins by showing that M_{∞} is a projective GL_{2}(ℚ_{p})-module (Theorem 4.15), before making a category-theoretic argument that allows us to “factor out” the patching variables (Proposition 4.22). We then use the “capture” machinery to complete the proof.

In Section 5 we explain how our results can be used to give a new proof (not making use of (φ, Γ)-modules) that certain locally algebraic principal series representations admit at most one unitary completion. Section 6 combines our results with those of [Paš13] to show that our construction is compatible with Colmez’s correspondence, and as a byproduct extends some results of [Paš13] to a situation where the central character is not fixed.

Finally, in Section 7 we explain how our results give a new proof of the second author’s localglobal compatibility theorem, and briefly explain how such results can be extended to quaternion algebras over totally real fields (Remark 7.8).

1.12 Notation

We fix an odd prime p, an algebraic closure \overline{ℚ}_{p} of ℚ_{p}, and a finite extension E/ℚ_{p} in \overline{ℚ}_{p}, which will be our coefficient field. We write O = O_{E} for the ring of integers in E, ϖ = ϖ_{E} for a uniformiser, and 𝔽 := O/ϖ for the residue field. We will assume without comment that E and 𝔽 are sufficiently large, and in particular that if we are working with representations of the absolute Galois group of a p-adic field K, then the images of all embeddings K ↪ \overline{ℚ}_{p} are contained in E.

1.12.1 Galois-theoretic notation If K is a field, we let G_{K} denote its absolute Galois group. Let ε denote the p-adic cyclotomic character, and \overline{ε} = ω the mod p cyclotomic character. If K is a finite extension of ℚ_{p} for some p, we write I_{K} for the inertia subgroup of G_{K}. If R is a local ring we write 𝔪_{R} for the maximal ideal of R. If F is a number field and v is a finite place of F then we let Frob_{v} denote a geometric Frobenius element of G_{F_{v}}.

If K/ℚ_{p} is a finite extension, we write Art_{K} : K^{×} ⟶^{∼} W_{K}^{ab} for the Artin map normalized to send uniformizers to geometric Frobenius elements. To avoid cluttering up the notation, we will use Art_{ℚ_{p}} to regard characters of ℚ_{p}^{×}, ℤ_{p}^{×} as characters of G_{ℚ_{p}}, I_{ℚ_{p}} respectively, without explicitly mentioning Art_{ℚ_{p}} when we do so.

If K is a p-adic field and ρ a de Rham representation of G_{K} over E and if τ : K ↪ E then we will write HT_{τ} (ρ) for the multiset of Hodge–Tate numbers of ρ with respect to τ. By definition, the multiset HT_{τ} (ρ) contains i with multiplicity dim_{E}(ρ ⊗_{τ,K} ˆ \overline{K}(i))^{G_{K}}. Thus for example HT_{τ} (ε) = {−1}. If ρ is moreover crystalline then we have the associated filtered φmodule D_{cris}(ρ) := (ρ ⊗_{ℚ_{p}} B_{cris})^{G_{K}}, where B_{cris} is Fontaine’s crystalline period ring.

1.12.2 Local Langlands correspondence Let n ∈ ℤ_{⩾1}, let K be a finite extension of ℚ_{p}, and let rec denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations of GL_{n}(K) over ℂ to isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of W_{K} defined in [HT01]. Fix an isomorphism ı : \overline{ℚ}_{p} → ℂ. We define the local Langlands correspondence rec_{p} over \overline{ℚ}_{p} by ı ∘ rec_{p} = rec ∘ ı. Then r_{p}(π) := rec_{p}(π ⊗ | det |^{(1−n)/2}) is independent of the choice of ı. In this paper we are mostly concerned with the case that n = 2 and K = ℚ_{p}.

1.12.3 Notation for duals If A is a topological O-module, we write A^{∨} := Hom_{O}^{cont}(A, E/O) for the Pontryagin dual of A. We apply this to O-modules that are either discrete or profinite, so that the usual formalism of Pontryagin duality applies.

If A is a pseudocompact O-torsion free O-module, we write A^{d} := Hom_{O}^{cont}(A, O) for its Schikhof dual.

If F is a free module of finite rank over a ring R, then we write F^{∗} := Hom_{R}(F, R) to denote its R-linear dual, which is again a free R-module of the same rank over R as F.

If R is a commutative O-algebra, and if A is an R-module that is pseudocompact and Otorsion free as an O-module, then we may form its Schikhof dual A^{d}, which has a natural R-module structure via the transpose action, extending its O-module structure. If F is a finite rank free R-module, then A ⊗_{R} F is again an R-module that is pseudocompact as an O-module (if F has rank n then it is non-canonically isomorphic to a direct sum of n copies of A), and there is a canonical isomorphism of R-modules (A ⊗_{R} F)^{d} ⟶^{∼} A^{d} ⊗_{R} F^{∗}.

1.12.4 Group-theoretic notation Throughout the paper we write G = GL_{2}(ℚ_{p}) and K = GL_{2}(ℤ_{p}), and let Z = Z(G) denote the centre of G. We also let B denote the Borel subgroup of G consisting of upper triangular matrices, and T denote the diagonal torus contained in B.

If χ : T → E^{×} is a continuous character, then we define the continuous induction (Ind_{B}^{G} χ)_{cont} to be the E-vector space of continuous functions f : G → E satisfying the condition f(bg) = χ(b)f(g) for all b ∈ B and g ∈ G; it forms a G-representation with respect to the right regular action. If χ is in fact a smooth character, then we may also form the smooth induction (Ind_{B}^{G} χ)_{sm}; this is the E-subspace of (Ind_{B}^{G} χ)_{cont} consisting of smooth functions, and is a G-subrepresentation of the continuous induction.

If χ_{1} and χ_{2} are continuous characters of ℚ_{p}^{×}, then the character χ_{1} ⊗ χ_{2} : T → E^{×} is defined via ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) ↦ χ_{1}(a)χ_{2}(d). Any continuous E-valued character χ of T is of this form, and χ is smooth if and only if χ_{1} and χ_{2} are.

2. Galois deformation rings and Hecke algebras

2.1 Galois deformation rings Recall that we assume throughout the paper that p is an odd prime. Fix a continuous representation r̄: G_{ℚ_{p}} → GL_{2}(𝔽), where as before 𝔽/𝔽_{p} is a finite extension. Possibly enlarging 𝔽, we fix a sufficiently large extension E/ℚ_{p} with ring of integers O and residue field 𝔽.

We will make the following assumption from now on.

2.2 Assumption. Assume that End_{G_{ℚp}} (r̄) = 𝔽, and that r̄ ≅ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) ⊗ χ, for any character χ : G_{ℚ_{p}} → 𝔽^{×}.

In particular this assumption implies that r̄ has a universal deformation O-algebra R_{p}, and that either r̄ is (absolutely) irreducible, or r̄ is a non-split extension of characters.

We begin by recalling the relationship between crystalline deformation rings of r̄, and the representation theory of G := GL_{2}(ℚ_{p}) and K := GL_{2}(ℤ_{p}). Given a pair of integers a ∈ ℤ and b ∈ ℤ_{⩾0}, we let σ_{a,b} be the absolutely irreducible E-representation det^{a} ⊗Sym^{b}E^{2} of K. Note that this is just the algebraic representation of highest weight (a + b, a) with respect to the Borel subgroup given by the upper-triangular matrices in G.

We say that a representation r : G_{ℚ_{p}} → GL_{2}(\overline{ℚ}_{p}) is crystalline of Hodge type σ = σ_{a,b} if it is crystalline with Hodge–Tate weights (1 − a, −a − b),^{1} we write R_{p}(σ), for the reduced, p-torsion free quotient of R_{p} corresponding to crystalline deformations of Hodge type σ.

2.3 The morphism from the Hecke algebra to the deformation ring We briefly recall some results from [CEG^{+}16, §4], specialised to the case of crystalline representations of GL_{2}(ℚ_{p}).

Set σ = σ_{a,b}, and let H(σ) := End_{G}(c-Ind_{K}^{G} σ). The action of K on σ extends to the action of G. This gives rise to the isomorphism of G-representations:

(c-Ind_{K}^{G} 1) ⊗ σ ≅ c-Ind_{K}^{G} σ, f ⊗ v ↦ [g ↦ f(g)σ(g)v].

The map

H(𝟙) → H(σ), ϕ ↦ ϕ ⊗ id_{σ}  (2.4)

1Note that this convention agrees with those of [CEG^{+}16] and [Eme11].

is an isomorphism of E-algebras by Lemma 1.4 of [ST06]. Using the above isomorphism we will identify elements of H(σ) with E-valued K-biinvariant functions on G, supported on finitely many double cosets.

2.5 Proposition. Let S ∈ H(σ) be the function supported on the double coset of ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ), with

value p^{2a+b} at ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ), and let T ∈ H(σ) be the function supported on the double coset^{2} of ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ),

with value p^{a+b} at ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ). Then H(σ) = E[S^{±1}, T] as an E-algebra.

Proof. This is immediate from (2.4) and the Satake isomorphism.

Let r, s be integers with r < s, and let t, d ∈ E with d ∈ E^{×}. We let D := D(r, s, t, d) be the two-dimensional filtered φ-module that has e_{1}, e_{2} as a basis of its underlying E-vector space, has its E-linear Frobenius endomorphism φ being given by

φ(e_{1}) = e_{2}, φ(e_{2}) = −de_{1} + te_{2},

and has its Hodge filtration given by

Fil^{i} D = D if i ⩽ r, Fil^{i} D = Ee_{1} if r + 1 ⩽ i ⩽ s, and Fil^{i} D = 0 if i > s.

We note that t is the trace and d is the determinant of φ on D, and both are therefore determined uniquely by D. The same construction works if E is replaced with an E-algebra A. We will still write D(r, s, t, d) for the resulting φ-module with A-coefficients if the coefficient algebra is clear from the context.

2.6 Lemma. If V is an indecomposable 2-dimensional crystalline representation of G_{ℚ_{p}} over E with distinct Hodge–Tate weights (s, r) then there exists a unique pair (t, d) ∈ E × E^{×} such that D_{cris}(V) ≅ D(r, s, t, d). Moreover, v_{p}(d) = r + s and v_{p}(t) ⩾ r.

Proof. This is well known, and is a straightforward computation using the fact that D_{cris}(V) is weakly admissible. For the sake of completeness, we sketch the proof; the key fact one employs is that V ↦ D_{cris}(V) is a fully faithful embedding of the category of crystalline representations of G_{ℚ_{p}} into the category of weakly admissible filtered φ-modules. (Indeed, it induces an equivalence between these two categories, but that more difficult fact isn’t needed for this computation.) We choose e_{1} to be a basis for Fil^{s} D_{cris}(V); the assumption that V is indecomposable implies that Fil^{s} D_{cris}(V) is not stable under φ, and so if we write e_{2} := φ(e_{1}) then e_{1}, e_{2} is a basis for D_{cris}(V), and φ has a matrix of the required form for a uniquely determined t and d. The asserted relations between v_{p}(t), v_{p}(d), r, and s follow from the weak admissibility of D_{cris}(V).

In fact, it will be helpful to state a generalisation of the previous result to the context of finite dimensional E-algebras. (Note that the definition of D(r, s, t, d) extends naturally to the case when t and d are taken to lie in such a finite-dimensional algebra.)

2.7 Lemma. If A is an Artinian local E-algebra with residue field E′, and if V_{A} is a crystalline representation of rank two over A whose associated residual representation V_{E′} := E′ ⊗_{A} V_{A} is indecomposable with distinct Hodge–Tate weights (s, r), and if D_{cris}(V_{A}) denotes the filtered φ-module associated to V_{A}, then there exists a unique pair (t, d) ∈ A×A^{×} such that D_{cris}(V_{A}) ≅ D(r, s, t, d).

2The function supported on the double coset KgK with value 1 at g, viewed as an element of H(1), acts on v ∈ V^{K} by the formula [KgK]v = ∑_{hK⊂KgK} hv.

Proof. Choose a basis \overline{e}_{1} for Fil^{s} D_{cris}(V_{E′}), and choose e_{1} ∈ Fil^{s} D_{cris}(V_{A}) lifting \overline{e}_{1}. By Nakayama’s lemma, e_{1} generates Fil^{s} D_{cris}(V_{A}), and by considering the length of Fil^{s} D_{cris}(V_{A}) as an E-vector space, we see that Fil^{s} D_{cris}(V_{A}) is a free A-module of rank one.

Let e_{2} = ϕ(e_{1}), and write \overline{e}_{2} for the image of e_{2} in D_{cris}(V_{E′}). As in the proof of Lemma 2.6, \overline{e}_{1}, \overline{e}_{2} is a basis of D_{cris}(V_{E′}), and thus by another application of Nakayama’s lemma, e_{1}, e_{2} are an A-basis of D_{cris}(V_{A}). The matrix of ϕ in this basis is evidently of the required form.

2.8 Corollary. If V is an indecomposable 2-dimensional crystalline representation of G_{ℚ_{p}} over E with distinct Hodge–Tate weights (s, r), for which D_{cris}(V) ≅ D(r, s, t_{0}, d_{0}), then the formal crystalline deformation ring of V is naturally isomorphic to E[[t − t_{0}, d − d_{0}]].

Proof. This is immediate from Lemma 2.7, and the fact that V ↦ D_{cris}(V) is an equivalence of categories.

Suppose that r̄ has a crystalline lift of Hodge type σ. By [CEG^{+}16, Thm. 4.1] there is a natural E-algebra homomorphism η : H(σ) → R_{p}(σ)[1/p] interpolating a normalized local Langlands correspondence r_{p} (introduced in Section 1.12). In order to characterise this map, one considers the composite

H(σ) ⟶^{η} R_{p}(σ)[1/p] ↪ (R_{p}(σ))^{an},

where (R_{p}(σ))^{an} denotes the ring of rigid analytic functions on the rigid analytic generic fibre of Spf R_{p}(σ). Over (R_{p}(σ))^{an}, we may consider the universal filtered φ-module, and the underlying universal Weil group representation (given by forgetting the filtration). The trace and determinant of Frobenius on this representation are certain elements of (R_{p}(σ))^{an} (which in fact lie in R_{p}(σ)[1/p]), and η is characterised by the fact that it identifies appropriately chosen generators of H(σ) with this universal trace and determinant.

It is straightforward to give explicit formulas for these generators of H(σ), but we have found it interesting (in part with an eye to making arguments in more general contexts) to also derive the facts that we need without using such explicit formulas.

Regarding explicit formulas, we have the following result.

2.9 Proposition. The elements η(S), η(T) ∈ R_{p}(σ)[1/p] are characterised by the following property: if x : R_{p}(σ)[1/p] → \overline{ℚ}_{p} is an E-algebra morphism, and V_{x} is the corresponding twodimensional \overline{ℚ}_{p}-representation of G_{ℚ_{p}}, then x(η(T)) = p^{a+b}t, and x(η(S)) = p^{2a+b−1}d, where t, d are respectively the trace and the determinant of φ on D_{cris}(V_{x}).

Proof. Lemma 2.7 implies that there are uniquely determined t, d ∈ \overline{ℚ}_{p}, such that D_{cris}(V_{x}) ≅ D(r, s, t, d), where r = −a − b and s = 1 − a. The Weil–Deligne representation associated to D(r, s, t, d) is an unramified 2-dimensional representation of W_{ℚ_{p}}, on which the geometric Frobenius Frob_{p} acts by the matrix of crystalline Frobenius on D(r, s, t, d), which is ( \begin{matrix}0 & −d \\ 1 & t \end{matrix} ). Thus

WD(D_{cris}(V_{x})) = rec_{p}(χ_{1}) ⊕ rec_{p}(χ_{2}),

where χ_{1}, χ_{2} : ℚ_{p}^{×} → \overline{ℚ}_{p}^{×} are unramified characters, such that χ_{1}(p)+χ_{2}(p) = t and χ_{1}(p)χ_{2}(p) = d.

If π = (Ind_{B}^{G} | ▪ |χ_{1} ⊗ χ_{2})_{sm} then π ⊗ | det |^{−1/2} ≅ ι_{B}^{G}(χ_{1} ⊗ χ_{2}), where ι_{B}^{G} denotes smooth normalized parabolic induction. Then

r_{p}(π) = rec_{p}(ι_{B}^{G}(χ_{1} ⊗ χ_{2})) = rec_{p}(χ_{1}) ⊕ rec_{p}(χ_{2}).

The action of H(1) on π^{K} is given by sending [K( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} )K] to p|p|χ_{1}(p)+χ_{2}(p) = t and [K( \begin{matrix}p & 0 \\ 0 & p \end{matrix} )K] to |p|χ_{1}(p)χ_{2}(p) = p^{−1}d. By [CEG^{+}16, Thm. 4.1] and the fact that the evident isomorphism between π^{K} = Hom_{K}(1, 1 ⊗ π) and Hom_{K}(σ, σ ⊗ π) is equivariant with respect to the actions by H(1) and H(σ) via the isomorphism (2.4), we see that

x(η(T)) = p^{−r}t = p^{a+b}t, x(η(S)) = p^{−r−s}d = p^{2a+b−1}d.  (2.10)

Since R_{p}(σ)[1/p] is a reduced Jacobson ring, the formulas determine η(T) and η(S) uniquely.

2.11 Corollary. η(S) and η(T) are contained in the normalisation of R_{p}(σ) in R_{p}(σ)[1/p].

Proof. It follows from (2.10) and Lemma 2.6 that for all closed points x : R_{p}(σ)[1/p] → \overline{ℚ}_{p}, we have x(η(S)), x(η(T)) ∈ \overline{ℤ}_{p}. The result follows from [dJ95, Prop. 7.3.6].

A key fact that we will use, which is special to our context of 2-dimensional crystalline representations of G_{ℚ_{p}}, is that the morphism

(Spf R(σ))^{an} → (Spec H(σ))^{an}

induced by η is an open immersion of rigid analytic spaces, where the superscript an signifies the associated rigid analytic space. We prove this statement (in its infinitesimal form) in the following result.

2.12 Lemma. Let σ = σ_{a,b} with a ∈ ℤ and b ∈ ℤ_{⩾0}. Then

dim_{κ(y)} κ(y) ⊗_{H(σ)} R_{p}(σ)[1/p] ⩽ 1, ∀y ∈ m-Spec H(σ).

Proof. Let us assume that A := κ(y)⊗_{H(σ)}R_{p}(σ)[1/p] is non-zero. If x, x′ ∈ m-Spec A then Frobenius on D_{cris}(V_{x}) and D_{cris}(V_{x′}) will have the same trace and determinant (since, by Proposition 2.9, these are determined by the images of T and S in κ(y)); denote them by t and d. It follows from Lemma 2.6 that D_{cris}(V_{x}) ≅ D_{cris}(V_{x′}) and hence x = x′. Since D(r, s, t, d) can be constructed over κ(y) (as t and d lie in κ(y)), so can V_{x} and thus κ(x) = κ(y). To complete the proof of the lemma it is enough to show that the map 𝔪_{y} → 𝔪_{x}/𝔪_{x}^{2} is surjective. Since we know that R_{p}(σ)[1/p] is a regular ring of dimension 2 by [Kis08, Thm. 3.3.8], it is enough to construct a 2-dimensional family of deformations of D_{cris}(V_{x}) to the ring of dual numbers κ(y)[ϵ], which induces a non-trivial deformation of the images of S, T. That this is possible is immediate from Corollary 2.8 and Proposition 2.9.

2.13 Proposition. Let σ = σ_{a,b} with a ∈ ℤ and b ∈ ℤ_{⩾0}. Let y ∈ m-Spec H(σ) be the image of x ∈ m-Spec R_{p}(σ)[1/p] under the morphism induced by η : H(σ) → R_{p}(σ)[1/p]. Then η induces an isomorphism of completions:

{ \atop H(σ)}_{𝔪_{y}} ⟶^{≅} { \atop R_{p}(σ)[1/p]}_{𝔪_{x}}.

Proof. This can be proved by explicit computation, taking into account Corollary 2.8 and Proposition 2.9.

We can also deduce it in more pure thought manner as follows: Since H(σ) ≅ E[T, S^{±1}] by Proposition 2.5 and R_{p}(σ)[1/p] is a regular ring of dimension 2 as in the preceding proof, both completions are regular rings of dimension 2. It follows from Lemma 2.12 that κ(y) = κ(x) and the map induces a surjection on tangent spaces. Hence the map is an isomorphism.

If 0 ⩽ b ⩽ p − 1, then σ_{a,b} has a unique (up to homothety) K-invariant lattice σ_{a,b}^{∘}, which is isomorphic to det^{a} ⊗Sym^{b}O^{2} as a K-representation. We let \overline{σ}_{a,b} be its reduction modulo ϖ.

Then \overline{σ}_{a,b} is the absolutely irreducible 𝔽-representation det^{a} ⊗Sym^{b}𝔽^{2} of GL_{2}(𝔽_{p}); note that every (absolutely) irreducible 𝔽-representation of GL_{2}(𝔽_{p}) is of this form for some uniquely determined a, b with 0 ⩽ a < p − 1. We refer to such representations as Serre weights.

If \overline{σ} = \overline{σ}_{a,b} is a Serre weight with the property that r̄ has a lift r : G_{ℚ_{p}} → GL_{2}(\overline{ℤ}_{p}) that is crystalline of Hodge type σ = σ_{a,b}, then we say that \overline{σ} is a Serre weight of r̄.

Again we consider σ = σ_{a,b} with any a ∈ ℤ and b ∈ ℤ_{⩾0}. Let

σ^{∘} := det^{a} ⊗ Sym^{b}O^{2}, \overline{σ} := det^{a} ⊗ Sym^{b}𝔽^{2}

so that σ^{∘}/ϖ = \overline{σ}. We let

H(σ^{∘}) := End_{G}(c-Ind_{K}^{G} σ^{∘}), H(\overline{σ}) := End_{G}(c-Ind_{K}^{G} \overline{σ}).

Note that H(σ^{∘}) is p-torsion free, since c-Ind_{K}^{G} σ^{∘} is.

2.14 Lemma. (1) For any σ, there is a natural isomorphism H(σ) ≅ H(σ^{∘})[1/p], and a natural inclusion H(σ^{∘})/ϖ ↪ H(\overline{σ}). Furthermore, the O-subalgebra O[S^{±1}, T] of H(σ) is contained in H(σ^{∘}).

(2) If, in addition σ = σ_{a,b} with 0 ⩽ b ⩽ p−1, then O[S^{±1}, T] = H(σ^{∘}), and there is a natural isomorphism H(σ^{∘})/ϖ ≅ H(\overline{σ}).

Proof. The isomorphism of (1) follows immediately from the fact that c-Ind_{K}^{G} σ^{∘} is a finitely generated O[G]-module. To see the claimed inclusion, apply Hom_{G}(c-Ind_{K}^{G} σ^{∘}, −) to the exact sequence

0 → c-Ind_{K}^{G} σ^{∘} →^{ϖ} c-Ind_{K}^{G} σ^{∘} → c-Ind_{K}^{G} \overline{σ} → 0

so as to obtain an injective map

H(σ^{∘})/ϖ ↪ Hom_{G}(c-Ind_{K}^{G} σ^{∘}, c-Ind_{K}^{G} \overline{σ}) ≅ H(\overline{σ}).

To see the final claim of (1), we recall that from (2.4) and Frobenius reciprocity we have natural isomorphisms

H(𝟙) ≃ H(σ) ≃ Hom_{K}(σ, c-Ind_{K}^{G} σ);

the image of ϕ ∈ H(𝟙) under the composite map sends v ∈ σ to the function g ↦ ϕ(g^{−1})σ(g)v. A direct computation of the actions of S, T on the standard basis of σ^{∘} then verifies that S^{±1} and T lie in the O-submodule Hom_{K}(σ^{∘}, c-Ind_{K}^{G} σ^{∘}) of H(σ).

To prove (2), we note that it follows from [Bre03b, §2] and [BL94] that the composite 𝔽[S^{±1}, T] → H(σ^{∘})/ϖ → H(\overline{σ}) is an isomorphism. Since the second of these maps is injective, by (1), we conclude that each of these maps is in fact an isomorphism, confirming the second claim of (2). Furthermore, this shows that the inclusion O[S^{±1}, T] ↪ H(σ^{∘}) of (1) becomes an isomorphism both after reducing modulo ϖ as well as after inverting ϖ (because H(σ) is generated by S^{±1} and T by Proposition 2.5). Thus it is an isomorphism, completing the proof of (2).

The following lemma is well known, but for lack of a convenient reference we sketch a proof.

2.15 Lemma. Assume that r̄ satisfies Assumption 2.2. Then r̄ has at most two Serre weights. Furthermore, if we let \overline{σ} = \overline{σ}_{a,b} be a Serre weight of r̄ then the following hold:

(i) The deformation ring R_{p}(σ) is formally smooth of relative dimension 2 over O. (ii) The morphism of E-algebras η : H(σ) → R_{p}(σ)[1/p] induces a morphism of O-algebras

H(σ^{∘}) → R_{p}(σ).

(iii) The character ω^{1−2a−b} det r̄ is unramified, and if we let – μ = (ω^{1−2a−b} det r̄)(Frob_{p}), and

– if r̄ is irreducible, then λ = 0, and

– if r̄ is reducible, then we can write r̄ ≅ ω^{a+b} ⊗(\begin{matrix}\overline{χ}_{1} & ∗ \\ 0 & \overline{χ}_{2}ω^{−b−1} \end{matrix}) for unramified characters

\overline{χ}_{1}, \overline{χ}_{2}, and let λ = \overline{χ}_{1}(Frob_{p}), then the composition

α : H(σ^{∘}) → R_{p}(σ) → 𝔽

maps T ↦ λ and S ↦ μ.

(iv) Let { \atop H(σ^{∘})} be the completion of H(σ^{∘}) with respect to the kernel of α. Then the map H(σ^{∘}) → R_{p}(σ) induces an isomorphism of local O-algebras { \atop H(σ^{∘})} →^{≅} R_{p}(σ). In coordinates, we have R_{p}(σ) = O[[S − μ̃, T − λ̃]], where the tilde denotes the Teichmüller lift.

(v) If we set π := (c-Ind_{K}^{G} \overline{σ}) ⊗_{H(\overline{σ}),α} 𝔽, then π is an absolutely irreducible representation of G, and is independent of the choice of Serre weight \overline{σ} of r̄.

Proof. The claim that r̄ has at most two Serre weights is immediate from the proof of [BDJ10, Thm. 3.17], which explicitly describes the Serre weights of r̄. Concretely, in the case at hand these weights are as follows (see also the discussion of [Eme11, §3.5], which uses the same conventions as this paper). If r̄ is irreducible, then we may write

r̄|_{I_{ℚp}} ≅ ω^{m−1} ⊗ (\begin{matrix}ω_{2}^{n+1} & 0 \\ 0 & ω_{2}^{p(n+1)} \end{matrix}) ,

where ω_{2} is a fundamental character of niveau 2, and 0 ⩽ m < p − 1, 0 ⩽ n ⩽ p − 2. Then the Serre weights of r̄ are \overline{σ}_{m,n} and \overline{σ}_{m+n,p−1−n} (with m + n taken modulo p − 1). If r̄ is reducible, then we may write

r̄|_{I_{ℚp}} ≅ ω^{m+n} ⊗ (\begin{matrix}1 & ∗ \\ 0 & ω^{−n−1} \end{matrix}) ,

where 0 ⩽ m < p − 1, 0 ⩽ n < p − 1. Then (under Assumption 2.2) if n ≠ 0, the unique Serre weight of r̄ is \overline{σ}_{m,n}, while if n = 0, then \overline{σ}_{m,0} and \overline{σ}_{m,p−1} are the two Serre weights of r̄.

Part (2) follows from (1) by Lemma 2.14 (2) and Corollary 2.11. We prove parts (1), (3) and (4) simultaneously. If \overline{σ} is not of the form \overline{σ}_{a,p−1}, the claims about R_{p}(σ) are a standard consequence of (unipotent) Fontaine–Laffaille theory; for example, the irreducible case with O = ℤ_{p} is [FM95, Thm. B2], and the reducible case follows in the same way. The key point is that the corresponding weakly admissible modules are either reducible, or are uniquely determined by the trace and determinant of φ, by Lemma 2.6. Concretely, if r̄ is irreducible, then the crystalline lifts of r̄ of Hodge type σ_{a,b} correspond exactly to the weakly admissible modules D(−(a + b), 1 − a, t, d) where v_{p}(t) > −a − b and \overline{p^{2a+b−1}d} = μ. The claimed description of the deformation ring then follows.

Similarly, if r̄ is reducible, then it follows from Fontaine–Laffaille theory (and Assumption 2.2) that any crystalline lift of Hodge type \overline{σ}_{a,b} is necessarily reducible and indecomposable, and one finds that these crystalline lifts correspond precisely to those weakly admissible modules with D(−(a + b), 1 − a, t, d) where v_{p}(t) = −a − b, \overline{p^{a+b}t} = λ, and \overline{p^{2a+b−1}d} = μ.

This leaves only the case that \overline{σ} is of the form \overline{σ}_{a,p−1}. In this case the result is immediate from the main result of [BLZ04], which shows that the above description of the weakly admissible modules continues to hold.

Finally, (5) is immediate from the main results of [BL94, Bre03a], together with the explicit description of \overline{σ}, λ, established above. More precisely, in the case that r̄ is irreducible, the absolute irreducibility of π is [Bre03a, Thm. 1.1], and its independence of the choice of \overline{σ} is [Bre03a, Thm. 1.3]. If r̄ is reducible and has only a single Serre weight, then the absolute irreducibility of π is [BL94, Thm. 33(2)]. In the remaining case that r̄ is reducible and has two Serre weights, then Assumption 2.2 together with the explicit description of λ, μ above implies that λ^{2} ≠ μ, and the absolute irreducibility of π is again [BL94, Thm. 33(2)]. The independence of π of the choice of \overline{σ} is [BL94, Cor. 36(2)(b)].

2.16 Remark. It follows from the explicit description of π that it is either a principal series representation or supersingular, and neither one-dimensional nor an element of the special series. (This would no longer be the case if we allowed r̄ to be a twist of an extension of the trivial character by the mod p cyclotomic character, when in fact π would be an extension of a onedimensional representation and a special representation, which would also depend on the Serre weight if r̄ is peu ramifié.)

2.17 Remark. If π has central character ψ, then det r̄ = ψω^{−1}.

3. Patched modules and arithmetic actions

We now introduce the notion of an arithmetic action of (a power series ring over) R_{p} on an O[G]-module. It is not obvious from the definition that any examples exist, but we will explain later in this section how to deduce the existence of an example from the results of [CEG^{+}16] (that is, from the Taylor–Wiles patching method). The rest of the paper is devoted to showing a uniqueness result for such actions, and thus deducing that they encode the p-adic local Langlands correspondence for GL_{2}(ℚ_{p}). We anticipate that the axiomatic approach taken here will be useful in other contexts (for example, for proving local-global compatibility in the p-adic Langlands correspondence for GL_{2}(ℚ_{p}) in global settings other than those considered in [Eme11] or [CEG^{+}16]).

3.1 Axioms Fix an integer d ⩾ 0, and set R_{∞} := R_{p} ˆ⊗_{O}O[[x_{1}, …, x_{d}]]. Then an O[G]-module with an arithmetic action of R_{∞} is by definition a non-zero R_{∞}[G]-module M_{∞} satisfying the following axioms.

(AA1) M_{∞} is a finitely generated R_{∞}[[K]]-module. (AA2) M_{∞} is projective in the category of pseudocompact O[[K]]-modules.

Let σ^{∘} be a K-stable O-lattice in σ = σ_{a,b}. Set

M_{∞}(σ^{∘}) := (Hom_{O[[K]]}^{cont}(M_{∞}, (σ^{∘})^{d}))^{d} ,

where we are considering continuous homomorphisms for the profinite topology on M_{∞} and the p-adic topology on (σ^{∘})^{d}. This is a finitely generated R_{∞}-module by (AA1) and [Paš15, Cor. 2.5].

(AA3) For any σ, the action of R_{∞} on M_{∞}(σ^{∘}) factors through R_{∞}(σ) := R_{p}(σ)[[x_{1}, …, x_{d}]]. Furthermore, M_{∞}(σ^{∘}) is maximal Cohen–Macaulay over R_{∞}(σ), and the R_{∞}(σ)[1/p]-module M_{∞}(σ^{∘})[1/p] is locally free of rank one over its support.

For each σ^{∘}, we have a natural action of H(σ^{∘}) on M_{∞}(σ^{∘}), and thus of H(σ) on M_{∞}(σ^{∘})[1/p].

(AA4) For any σ, the action of H(σ) on M_{∞}(σ^{∘})[1/p] is given by the composite

H(σ) →^{η} R_{p}(σ)[1/p] → R_{p}(σ)[[x_{1}, …, x_{d}]][1/p],

where H(σ) →^{η} R_{p}(σ)[1/p] is defined in [CEG^{+}16, Thm. 4.1].

3.2 Remark. While these axioms may appear somewhat mysterious, as we will see in the next subsection they arise very naturally in the constructions of [CEG^{+}16]. (Indeed, those constructions give modules M_{∞} satisfying obvious analogues of the above conditions for GL_{n}(K) for any finite extension K/ℚ_{p}; however, our arguments in the rest of the paper will only apply to the case of GL_{2}(ℚ_{p}).)

In these examples, axioms (AA1) and (AA2) essentially follow from the facts that spaces of automorphic forms are finite-dimensional, and that the cohomology of zero-dimensional Shimura varieties is concentrated in a single degree (degree zero). Axioms (AA3) and (AA4) come from the existence of Galois representations attached to automorphic forms on unitary groups, and from local-global compatibility at p for automorphic forms of level prime to p.

The following remark explains how axiom (AA4) is related to Breuil’s original formulation of the p-adic Langlands correspondence in terms of unitary completions of locally algebraic vectors; see also the proof of Proposition 6.17 below.

3.3 Remark. Axiom (AA4) is, in the presence of axioms (AA1)-(AA3), equivalent to an alternative axiom (AA4’), which expresses a pointwise compatibility with the classical local Langlands correspondence, as we now explain. Write R_{∞}(σ) := R_{p}(σ)[[x_{1}, …, x_{d}]]. If y is a maximal ideal of R_{∞}(σ)[1/p] which lies in the support of M_{∞}(σ^{∘})[1/p], then we write

Π_{y} := Hom_{O}^{cont}(M_{∞} ⊗_{R_{∞},y} O_{κ(y)}, E).

We write x for the corresponding maximal ideal of R_{p}(σ)[1/p], r_{x} for the deformation of r̄ corresponding to x, and set π_{sm}(r_{x}) := r_{p}^{−1}(WD(r_{x})^{F −ss}), which is the smooth representation of G corresponding to the Weil–Deligne representation associated to r_{x} by the classical Langlands correspondence r_{p} (normalised as in Section 1.12). We write π_{alg}(r_{x}) for the algebraic representation of G whose restriction to K is equal to σ.

(AA4’) For any σ and for any y and x as above, there is a non-zero G-equivariant map

π_{sm}(r_{x}) ⊗ π_{alg}(r_{x}) → Π_{y}^{l.alg}.

That (AA1)-(AA4) imply (AA4’) is a straightforward consequence of the defining property of the map η. Conversely, assume (AA1)-(AA3) and (AA4’), and write

\overline{R}_{∞}(σ) := R_{∞}(σ)/Ann(M_{∞}(σ^{∘})).

It follows from (AA3) that the natural map \overline{R}_{∞}(σ)[1/p] → End_{R_{∞}(σ)[1/p]}(M_{∞}(σ^{∘})[1/p]) is an isomorphism, as it is injective and the cokernel is not supported on any maximal ideal of R_{∞}(σ)[1/p]. In particular the action of H(σ^{∘}) on M_{∞}(σ^{∘}) induces a homomorphism η′ : H(σ) → \overline{R}_{∞}(σ)[1/p]. We have to show that this agrees with the map induced by η.

It follows from (AA4’) and the defining property of η that η and η′ agree modulo every maximal ideal of R_{p}(σ)[1/p] in the support of M_{∞}(σ^{∘}). It follows from (AA3) that \overline{R}_{∞}(σ)[1/p] is a union of irreducible components of R_{∞}(σ)[1/p]. Since R_{∞}(σ)[1/p] is reduced we conclude that \overline{R}_{∞}(σ)[1/p] is reduced, thus the intersection of all maximal ideals is equal to zero. Hence (AA4) holds.

3.4 Existence of a patched module M_{∞} We now briefly recall some of main results of [CEG^{+}16], specialised to the case of two-dimensional representations. We emphasise that these results use only the Taylor–Wiles–Kisin patching method, and use nothing about the p-adic Langlands correspondence for GL_{2}(ℚ_{p}). (We should perhaps remark, though, that we do make implicit use of the results of [BLGGT14] in the globalisation part of the argument, and thus of the Taylor–Wiles–Kisin method for unitary groups of rank 4, and not just for U(2).) We freely use the notation of [CEG^{+}16].

Enlarging 𝔽 if necessary, we see from [CEG^{+}16, Lem. 2.2] that the hypotheses on r̄ at the start of [CEG^{+}16, §2.1] are automatically satisfied. We fix the choice of weight ξ and inertial type τ in [CEG^{+}16, §2.3] in the following way: we take τ to be trivial, and we take ξ to be the weight corresponding to a Serre weight of r̄, as in Lemma 2.15.

With this choice, the modification of the Taylor–Wiles–Kisin method carried out in [CEG^{+}16, §2.6] produces for some d > 0 an R_{∞}-module M_{∞} with an action of G. (Note that for our choice of r̄, ξ and τ, the various framed deformation rings appearing in [CEG^{+}16] are formal power series rings over O, and the framed deformation ring of r̄ is formally smooth over R_{p}, so all of these rings are absorbed into the power series ring O[[x_{1}, …, x_{d}]]. The module M_{∞} is patched from the cohomology of a definite unitary group over some totally real field in which p splits completely.)

This R_{∞}[G]-module automatically satisfies the axioms (AA1)–(AA4) above. Indeed, (AA1) and (AA2) follow from [CEG^{+}16, Prop. 2.10], and (AA3) follows from [CEG^{+}16, Lem. 4.17(1), 4.18(1)]. Finally, (AA4) is [CEG^{+}16, Thm. 4.19].

4. Existence and uniqueness of arithmetic actions

We fix an O[G]-module M_{∞} with an arithmetic action of R_{∞} in the sense of Section 3.1.

4.1 Serre weights and cosocles Now let \overline{σ} = \overline{σ}_{a,b} be a Serre weight, and let σ^{∘} be a K-stable O-lattice in σ_{a,b}, so that σ^{∘}/ϖσ^{∘} = \overline{σ}. We define M_{∞}(\overline{σ}) = Hom_{O[[K]]}^{cont}(M_{∞}, (\overline{σ})^{∨})^{∨}, so that by (AA2) we have M_{∞}(\overline{σ}) = M_{∞}(σ^{∘})/ϖM_{∞}(σ^{∘}). By definition, the deformation ring R_{p}(σ) = R_{p}(σ_{a,b}) is non-zero if and only if \overline{σ} is a Serre weight of r̄. Set R_{∞}(σ) = R_{p}(σ)[[x_{1}, …, x_{d}]].

We let π denote the absolutely irreducible smooth 𝔽-representation of G associated to r̄ via Lemma 2.15 (5).

4.2 Proposition. (i) We have M_{∞}(σ^{∘}) ≠ 0 if and only if \overline{σ} is a Serre weight of r̄, in which

case M_{∞}(σ^{∘}) is a free R_{∞}(σ)-module of rank one. (ii) If \overline{σ} is a Serre weight of r̄ then the action of H(\overline{σ}) on M_{∞}(\overline{σ}) factors through the natural

map R_{p}(σ)/ϖ → R_{∞}(σ)/ϖ, and M_{∞}(\overline{σ}) is a flat H(\overline{σ})-module. (iii) If π′ is an irreducible smooth 𝔽-representation of G then we have

Hom_{G}(π′, M_{∞}^{∨}) ≠ 0

if and only if π′ is isomorphic to π.

Proof. It follows from (AA3) that M_{∞}(σ^{∘}) ≠ 0 only if R_{p}(σ)[1/p] ≠ 0, which is equivalent to \overline{σ} being a Serre weight of r̄. In this case, since σ = σ_{a,b} with 0 ⩽ b ⩽ p − 1, R_{∞}(σ) is formally smooth over O by Lemma 2.15, so it follows from (AA3) and the Auslander–Buchsbaum theorem that M_{∞}(σ^{∘}) is a free R_{∞}(σ)-module of finite rank, and that M_{∞}(σ^{∘})[1/p] is a locally free R_{∞}(σ)[1/p]-module of rank 1. Thus M_{∞}(σ^{∘}) is free of rank one over R_{∞}(σ). This proves the “only if” direction of (1).

For (2), note that M_{∞}(σ^{∘}) ≠ 0 if and only if M_{∞}(\overline{σ}) ≠ 0, so we may assume that M_{∞}(σ^{∘}) ≠ 0. The first part of (2) follows from (AA4) together with Lemmas 2.15 (2) and 2.14 (2). For the remaining part of (2), note that R_{p}(σ)/ϖ is flat over H(\overline{σ}) by Lemmas 2.14 and 2.15, and M_{∞}(\overline{σ}) is flat over R_{p}(σ)/ϖ by the only if part of (1), as required.

To prove (3) we first note that it is enough to prove the statement for absolutely irreducible π′ as we may enlarge the field 𝔽. Let us assume that π′ is absolutely irreducible and let \overline{σ}′ be an irreducible representation of K contained in the socle of π′. It follows from [BL94, Bre03a] that the surjection c-Ind_{K}^{G} \overline{σ}′ ↠ π′ factors through the map

(c-Ind_{K}^{G} \overline{σ}′) ⊗_{H(\overline{σ}′),α′} 𝔽 ↠ π′  (4.3)

where α′ : H(\overline{σ}′) → 𝔽 is given by the action of H(\overline{σ}′) on the one dimensional 𝔽-vector space Hom_{K}(\overline{σ}′, π′). Moreover, (4.3) is an isomorphism unless π′ is a character or special series. Since

M_{∞}(\overline{σ}′) ≅ Hom_{K}(\overline{σ}′, M_{∞}^{∨})^{∨} ≅ Hom_{G}(c-Ind_{K}^{G} \overline{σ}′, M_{∞}^{∨})^{∨},

from (4.3) we obtain a surjection of R_{∞}(\overline{σ}′)-modules

M_{∞}(\overline{σ}′) ⊗_{H(\overline{σ}′),α′} 𝔽 ≅ Hom_{G}(c-Ind_{K}^{G} \overline{σ}′ ⊗_{H(\overline{σ}′),α′} 𝔽, M_{∞}^{∨})^{∨} ↠ Hom_{G}(π′, M_{∞}^{∨})^{∨},

which moreover is an isomorphism if π′ is not a character or special series. Thus if Hom_{G}(π′, M_{∞}^{∨}) is non-zero then we deduce from the previous displayed expression that M_{∞}(\overline{σ}′) ⊗_{H(\overline{σ}′),α′} 𝔽 ≠ 0. In particular, M_{∞}(\overline{σ}′) ≠ 0 and hence \overline{σ}′ is a Serre weight for r̄ by the only if part of (1).

We claim that α′ coincides with the morphism α of Lemma 2.15 (3) (with \overline{σ}′ in place of σ). To see this, note that by the only if part of (1), we have that (R_{p}(σ′)/ϖ) ⊗_{H(\overline{σ}′),α′} 𝔽 ≠ 0, and hence by Lemma 2.15 (4), we find that { \atop H(\overline{σ}′)} ⊗_{H(\overline{σ}′),α′} 𝔽 ≠ 0, where { \atop H(\overline{σ}′)} denotes the completion of H(\overline{σ}′) with respect to the kernel of the morphism α. This proves that α and α′ coincide.

Part (5) of Lemma 2.15 now implies that π ≅ (c-Ind_{K}^{G} \overline{σ}′) ⊗_{H(\overline{σ}′),α′} 𝔽. Hence, (4.3) gives us a G-equivariant surjection π ↠ π′, which is an isomorphism as π is irreducible.

Conversely, it follows from from (AA1) that there is an irreducible smooth 𝔽-representation π′ of G such that Hom_{G}(π′, M_{∞}^{∨}) is non-zero; we have just seen that π ≅ π′, so that Hom_{G}(π, M_{∞}^{∨}) ≠ 0, as required.

Finally, suppose that \overline{σ} is a Serre weight of r̄. Then as above we have an isomorphism of R_{∞}(\overline{σ})-modules

M_{∞}(\overline{σ}) ⊗_{H(\overline{σ}),α} 𝔽 ≅ Hom_{G}(π, M_{∞}^{∨})^{∨} ≠ 0

so that M_{∞}(\overline{σ}) ≠ 0. This completes the proof of the “if” direction of (1).

4.4 Smooth and admissible representations We record a few definitions, following Section 2 of [Paš13]. Let (R, 𝔪) be a complete local noetherian O-algebra with residue field 𝔽. Then Mod_{G}^{sm}(R) is the full subcategory of the category of R[G]-modules consisting of smooth objects. More precisely, these are objects V such that

V = ⋃_{H,n} V^{H}[𝔪^{n}],

where the union is taken over open compact subgroups H ⊂ G and over positive integers n.

We say that an object V of Mod_{G}^{sm}(R) is admissible if V^{H}[𝔪^{n}] is a finitely generated Rmodule for every compact open subgroup H ⊂ G and every n ⩾ 1. Moreover, V is called locally admissible if, for every v ∈ V, the smallest R[G]-submodule of V containing v is admissible. We let Mod_{G}^{l.adm}(R) denote the full subcategory of Mod_{G}^{sm}(R) consisting of locally admissible representations.

The categories Mod_{G}^{sm}(R) and Mod_{G}^{l.adm}(R) are abelian (see [Eme10] for the second one) and have enough injectives.

4.5 Definition. (i) A monomorphism ι : N ↪ M in an abelian category is called essential

if, for every non-zero subobject M′ ⊂ M, ι(N) ∩ M′ is non-zero. (ii) An injective envelope of an object N of an abelian category is an essential monomorphism

ι : N ↪ I with I an injective object of the abelian category.

If they exist, injective envelopes are unique up to (non-unique) isomorphism. By Lemma 2.3 of [Paš13], the category Mod_{G}^{sm}(R) admits injective envelopes. The category Mod_{G}^{l.adm}(R) also admits injective envelopes. (This follows from [Paš10, Lem. 3.2] and the fact that the inclusion of Mod_{G}^{l.adm}(R) into Mod_{G}^{sm}(R) has a right adjoint, namely the functor to which any smooth G-representation associates its maximal locally admissible subrepresentation.)

4.6 Lemma. If V is a locally admissible representation of G, then the inclusion soc_{G}(V) ↪ V is essential.

Proof. Any non-zero subrepresentation of V contains a non-zero finitely generated subrepresentation. Thus it suffices to show that any non-zero finitely generated subrepresentation W of V has a non-zero intersection with soc_{G}(V). Since soc_{G}(V)∩W = soc_{G}(W), it suffices to show that any such subrepresentation has a non-zero socle. This follows from the fact that every finitely generated admissible representation of G is of finite length by [Eme10, Thm. 2.3.8].

4.7 Definition. (i) An epimorphism q : M ↠ N in an abelian category is called essential if

a morphism s : M′ → M is an epimorphism whenever q ∘ s is an epimorphism. (ii) A projective envelope of an object N of an abelian category is an essential epimorphism

q : P ↠ N with P a projective object in the abelian category.

Pontryagin duality reverses arrows, so it exchanges injective and projective objects as well as injective and projective envelopes.

4.8 Projectivity of M_{∞} Our first aim is to show that M_{∞}^{∨} is an injective locally admissible representation of G.

4.9 Lemma. M_{∞}^{∨} is an admissible object of Mod_{G}^{sm}(R_{∞}), and thus lies in Mod_{G}^{l.adm}(O).

Proof. Dually it is enough by [Eme10, Lem. 2.2.11] to show that M_{∞} is a finitely generated R_{∞}[[K]]-module, which is (AA1).

4.10 Lemma. Let 𝔪 be a maximal ideal of H(\overline{σ}) with residue field κ(𝔪). Then

Tor_{i}^{H(\overline{σ})}(c-Ind_{K}^{G} \overline{σ}, κ(𝔪)) = 0, ∀i > 0.

Proof. Since the map 𝔽 → \overline{𝔽} is faithfully flat, we can and do assume that 𝔽 is algebraically closed. Since H(\overline{σ}) = 𝔽[S^{±1}, T], we have 𝔪 = (S − μ, T − λ) for some μ ∈ 𝔽^{×}, λ ∈ 𝔽. Since the sequence S − μ, T − λ is regular in H(\overline{σ}), the Koszul complex K_{•} associated to it is a resolution of κ(𝔪) by free H(\overline{σ})-modules, [Mat89, Thm. 16.5 (i)]. Thus the complex K_{•} ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ} computes the Tor-groups we are after, and to verify the claim it is enough to show that the sequence S − μ, T − λ is regular on c-Ind_{K}^{G} \overline{σ}.

If f ∈ c-Ind_{K}^{G} \overline{σ} then (Sf)(g) = f(gz), where z = ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ). Since such an f is supported only on finitely many cosets K\G, we deduce that the map

c-Ind_{K}^{G} \overline{σ} ⟶^{S−μ} c-Ind_{K}^{G} \overline{σ}

is injective. The quotient is isomorphic to c-Ind_{ZK}^{G} \overline{σ}, where z acts on \overline{σ} by μ. It follows from the proof of [BL94, Thm. 19] that c-Ind_{ZK}^{G} \overline{σ} is a free 𝔽[T]-module. Thus the map

c-Ind_{ZK}^{G} \overline{σ} ⟶^{T −λ} c-Ind_{ZK}^{G} \overline{σ}

is injective, and the sequence S − μ, T − λ is regular on c-Ind_{K}^{G} \overline{σ}, as required.

4.11 Lemma. Let 𝔪 be a maximal ideal of H(\overline{σ}). Then

Ext_{G}^{i}(κ(𝔪) ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨}) = 0, ∀i ⩾ 1,

where the Ext-groups are computed in Mod_{G}^{sm}(O).

Proof. We first prove that Ext_{G}^{i}(c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨}) = 0, for all i ⩾ 1. Let M_{∞}^{∨} ↪ J^{•} be an injective resolution of M_{∞}^{∨} in Mod_{G}^{sm}(O). Since

Hom_{K}(τ, J|_{K}) ≅ Hom_{G}(c-Ind_{K}^{G} τ, J)

and the functor c-Ind_{K}^{G} is exact, the restriction of an injective object in Mod_{G}^{sm}(O) to K is injective in Mod_{K}^{sm}(O). Thus (J^{•})|_{K} is an injective resolution of M_{∞}^{∨}|_{K} in Mod_{K}^{sm}(O). Since Hom_{G}(c-Ind_{K}^{G} \overline{σ}, J^{•}) ≅ Hom_{K}(\overline{σ}, (J^{•})|_{K}), we conclude that we have natural isomorphisms

Ext_{G}^{i}(c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨}) ≅ Ext_{K}^{i}(\overline{σ}, M_{∞}^{∨}), ∀i ⩾ 0.

Since M_{∞} is a projective O[[K]]-module by (AA2), M_{∞}^{∨} is injective in Mod_{K}^{sm}(O), and thus the Ext-groups vanish as claimed.

Let F_{•} ↠ κ(𝔪) be a resolution of κ(𝔪) by finite free H(\overline{σ})-modules. Lemma 4.10 implies that the complex F_{•} ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ} is a resolution of κ(𝔪) ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ} by acyclic objects for the functor Hom_{G}(∗, M_{∞}^{∨}). We conclude that the cohomology of the complex

Hom_{G}(F_{•} ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨})

computes the groups Ext_{G}^{i}(κ(𝔪) ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨}). We may think of the transition maps in F_{•} as matrices with entries in H(\overline{σ}). The functor Hom_{G}(∗, M_{∞}^{∨})^{∨} transposes these matrices twice, thus we get an isomorphism of complexes:

Hom_{G}(F_{•} ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨})^{∨} ≅ F_{•} ⊗_{H(\overline{σ})} Hom_{G}(c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨})^{∨} ≅ F_{•} ⊗_{H(\overline{σ})} M_{∞}(\overline{σ}).

The above isomorphism induces a natural isomorphism

(Ext_{G}^{i}(κ(𝔪) ⊗_{H(\overline{σ})} c-Ind_{K}^{G} \overline{σ}, M_{∞}^{∨}))^{∨} ≅ Tor_{i}^{H(\overline{σ})}(κ(𝔪), M_{∞}(\overline{σ})), ∀i ⩾ 0.

The isomorphism implies the assertion, as M_{∞}(\overline{σ}) is a flat H(\overline{σ})-module by Proposition 4.2 (2).

4.12 Lemma. Let y : H(\overline{σ}) → 𝔽′ be a homomorphism of 𝔽-algebras, where 𝔽′ is a finite field extension of 𝔽. Let λ′ := 𝔽′⊗_{H(\overline{σ}),y}c-Ind_{K}^{G} \overline{σ} and let λ be an absolutely irreducible 𝔽-representation of G, which is either principal series or supersingular. If λ is a subquotient of λ′ then λ′ is isomorphic to a direct sum of finitely many copies of λ.

Proof. In the course of the proof we will use the following fact repeatedly: if A and B are 𝔽-representations of G and A is finitely generated as an 𝔽[G]-module then:

Hom_{G}(A, B) ⊗_{𝔽} \overline{𝔽} ≅ Hom_{G}(A ⊗_{𝔽} \overline{𝔽}, B ⊗_{𝔽} \overline{𝔽}),  (4.13)

where \overline{𝔽} denotes the algebraic closure of 𝔽, see [Paš13, Lem. 5.1]. Then

\overline{𝔽} ⊗_{𝔽} λ′ ≅ \overline{𝔽} ⊗_{𝔽} 𝔽′ ⊗_{H(\overline{σ}),y} c-Ind_{K}^{G} \overline{σ} ≅ ⨁_{ι:𝔽′→\overline{𝔽}} \overline{𝔽} ⊗_{H(\overline{σ}),ι∘y} c-Ind_{K}^{G} \overline{σ},  (4.14)

where the sum is taken over 𝔽-algebra homomorphisms ι : 𝔽′ → \overline{𝔽}. By the classification theorems of Barthel–Livne [BL94] and Breuil [Bre03a], each representation \overline{𝔽} ⊗_{H(\overline{σ}),ι∘y} c-Ind_{K}^{G} \overline{σ} is either irreducible, an extension of a special series by a character, or an extension of a character by a special series. Since λ is a subquotient of λ′ by assumption, λ ⊗_{𝔽} \overline{𝔽} is a subquotient of λ′ ⊗_{𝔽} \overline{𝔽}, and since λ is neither special series nor a character, we deduce that

λ ⊗_{𝔽} \overline{𝔽} ≅ \overline{𝔽} ⊗_{H(\overline{σ}),ι∘y} c-Ind_{K}^{G} \overline{σ},

for some embedding ι : 𝔽′ ↪ \overline{𝔽}. For every τ ∈ Gal(\overline{𝔽}/𝔽), we have

\overline{𝔽} ⊗_{H(\overline{σ}),τ∘ι∘y} c-Ind_{K}^{G} \overline{σ} ≅ \overline{𝔽} ⊗_{\overline{𝔽},τ} (\overline{𝔽} ⊗_{H(\overline{σ}),ι∘y} c-Ind_{K}^{G} \overline{σ}) ≅ \overline{𝔽} ⊗_{\overline{𝔽},τ} (\overline{𝔽} ⊗_{𝔽} λ) ≅ \overline{𝔽} ⊗_{𝔽} λ.

Hence all the summands in (4.14) are isomorphic to λ ⊗_{𝔽} \overline{𝔽}. It follows from (4.13) that λ′ is isomorphic to a direct sum of copies of λ, as required.

4.15 Theorem. M_{∞}^{∨} is an injective object in Mod_{G}^{l.adm}(O).

Proof. Let M_{∞}^{∨} ↪ J be an injective envelope of M_{∞}^{∨} in Mod_{G}^{l.adm}(O). Lemma 4.6 shows that the composition soc_{G} M_{∞}^{∨} ↪ M_{∞}^{∨} ↪ J is an essential monomorphism, and thus induces an isomorphism between soc_{G} M_{∞}^{∨} and soc_{G} J. Proposition 4.2 (3) shows that soc_{G} M_{∞}^{∨}, and thus also soc_{G} J, is isomorphic to a direct sum of copies of the representation π associated to r̄ via Lemma 2.15 (5).

Let us assume that the quotient J/M_{∞}^{∨} is non-zero; then there is a smooth irreducible Ksubrepresentation \overline{σ} ⊂ J/M_{∞}^{∨}. Let κ be the G-subrepresentation of J/M_{∞}^{∨} generated by \overline{σ}. Since J/M_{∞}^{∨} is locally admissible, and \overline{σ} is finitely generated as a K-representation, κ is an admissible representation of G. Thus Hom_{G}(c-Ind_{K}^{G} \overline{σ}, κ) ≅ Hom_{K}(\overline{σ}, κ) is a finite dimensional 𝔽-vector space.

Let m be any irreducible H(\overline{σ})-submodule of Hom_{G}(c-Ind_{K}^{G} \overline{σ}, κ). Since m is finite dimensional over 𝔽, Schur’s lemma implies that 𝔽′ := End_{H(\overline{σ})}(m) is a finite dimensional division algebra over 𝔽. Since 𝔽 is a finite field we deduce that 𝔽′ is a finite field extension of 𝔽. Since H(\overline{σ}) is commutative we further deduce that m is a one dimensional 𝔽′-vector space and thus obtain a surjective homomorphism of 𝔽-algebras y : H(\overline{σ}) ↠ 𝔽′. Moreover, by the construction of m, we obtain a non-zero G-equivariant map:

π′ := 𝔽′ ⊗_{H(\overline{σ}),y} c-Ind_{K}^{G} \overline{σ} → κ ⊂ J/M_{∞}^{∨}.

Since Ext_{G}^{1}(π′, M_{∞}^{∨}) = 0 by Lemma 4.11, by applying Hom_{G}(π′, ∗) to the exact sequence 0 → M_{∞}^{∨} → J → J/M_{∞}^{∨} → 0, we obtain a short exact sequence

0 → Hom_{G}(π′, M_{∞}^{∨}) → Hom_{G}(π′, J) → Hom_{G}(π′, J/M_{∞}^{∨}) → 0.

Moreover, we know that Hom_{G}(π′, J/M_{∞}^{∨}) is non-zero. Hence, Hom_{G}(π′, J) is non-zero.

Fix a non-zero G-equivariant map φ : π′ → J; then φ(π′) ∩ soc_{G} J ≠ 0. Since soc_{G} J is isomorphic to a direct sum of copies of π, we find that π is an irreducible subquotient of π′. It follows from Lemma 4.12 that π′ is then isomorphic to a finite direct sum of copies of π, and so in particular is semi-simple. As we’ve already noted, the map M_{∞}^{∨} ↪ J induces an isomorphism soc_{G} M_{∞}^{∨} ≅ soc_{G} J, and so the map Hom_{G}(π′, M_{∞}^{∨}) → Hom_{G}(π′, J) is an isomorphism. This implies Hom_{G}(π′, J/M_{∞}^{∨}) = 0, contradicting the assumption J/M_{∞}^{∨} ≠ 0. Hence M_{∞}^{∨} = J is injective, as required.

4.16 Removing the patching variables We now show that we can pass from M_{∞} to an arithmetic action of R_{p} on a projective envelope of π^{∨}, where as always π is the representation.associated to r̄ via Lemma 2.15 (5). Let (A, 𝔪) be a complete local noetherian O-algebra. Let ℭ(A) be the Pontryagin dual of Mod_{G}^{l.adm}(A), where, for the moment, we allow G to be any p-adic analytic group. There is a forgetful functor from ℭ(A) to ℭ(O). In this subsection we prove a structural result about objects P of ℭ(A) that are projective in ℭ(O). We will apply this result to P = M_{∞} and G = GL_{2}(ℚ_{p}).

4.17 Lemma. Let (A, 𝔪) be a complete local noetherian 𝔽-algebra with residue field 𝔽. Let P ∈ ℭ(A) be such that P is projective in ℭ(𝔽) and the map P ↠ cosoc_{ℭ(𝔽)} P is essential. Assume that all irreducible subquotients of cosoc_{ℭ(𝔽)} P are isomorphic to some given object S, for which End_{ℭ(𝔽)}(S) = 𝔽. If Hom_{ℭ(𝔽)}(P, S)^{∨} is a free A-module of rank 1 then there is an isomorphism A ˆ⊗_{𝔽} Proj(S) ≅ P in ℭ(A), where Proj(S) ↠ S is a projective envelope of S in ℭ(𝔽).

Proof. The assumption on the cosocle of P implies that (cosoc P)^{∨} is isomorphic to a direct sum of copies of λ := S^{∨}. This means that we have natural isomorphisms:

(cosoc P)^{∨} ≅ λ ⊗_{𝔽} Hom_{G}(λ, (cosoc P)^{∨}) ≅ λ ⊗_{𝔽} Hom_{ℭ(𝔽)}(cosoc P, S).

Taking Pontryagin duals we get a natural isomorphism in ℭ(𝔽):

Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} S ≅ cosoc P.

Since the isomorphism is natural, it is an isomorphism in ℭ(A) with the trivial action of A on S. Hence, we get a surjection in ℭ(A):

P ↠ Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} S.

The surjection Proj(S) ↠ S induces a surjection

Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} Proj(S) ↠ Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} S,

with trivial A-action on Proj(S). The source of this surjection is projective in ℭ(A), since in general for a compact A-module m

Hom_{ℭ(A)}(m ˆ⊗_{𝔽} Proj(S), −) ≅ Hom_{A}(m, Hom_{ℭ(𝔽)}(Proj(S), −)),  (4.18)

and Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} = Hom_{ℭ(𝔽)}(P, S)^{∨} is projective since it is a free A-module of rank 1. Hence there is a map Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} Proj(S) → P in ℭ(A), such that the diagram

\xymatrix{Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} Proj(S) \ar@{->>}[d] \ar[r] & P \ar@{->>}[d] \\ Hom_{ℭ(𝔽)}(cosoc P, S)^{∨} ˆ⊗_{𝔽} S \ar[r]^{≅} & cosoc P}

commutes. If we forget the A-action then we obtain a map in ℭ(𝔽) between projective objects, which induces an isomorphism on their cosocles. Hence the map is an isomorphism in ℭ(𝔽), and hence also an isomorphism in ℭ(A).

4.19 Proposition. Let A be a complete local noetherian O-algebra with residue field 𝔽, that is O-flat. Let P in ℭ(A) be such that P is projective in ℭ(O) and the map P ↠ cosoc_{ℭ(O)} P is essential. Assume that all the irreducible subquotients of cosoc_{ℭ(O)} P are isomorphic to S, and End_{ℭ(O)}(S) = 𝔽. If Hom_{ℭ(O)}(P, S)^{∨} is a free A/ϖA-module of rank 1 then there is an isomorphism in ℭ(A):

A ˆ⊗_{O} Proj(S) ≅ P,

where Proj(S) ↠ S is a projective envelope of S in ℭ(O).

Proof. A special case of Lemma 4.17 implies that the reductions of A ˆ⊗_{O} Proj(S) and P modulo ϖ are isomorphic in ℭ(𝔽). Arguing as in (4.18) we deduce that A ˆ⊗_{O} Proj(S) is projective in ℭ(A). Thus there is a map in ℭ(A),

A ˆ⊗_{O} Proj(S) → P,

which is an isomorphism modulo ϖ. If V is a cokernel of this map then V/ϖV = 0, and Nakayama’s lemma for compact O-modules implies that V = 0. Since P is projective this surjection must split. Since the map is an isomorphism modulo ϖ, if U is the kernel of this map then the decomposition A ˆ⊗_{O} Proj(S) ≅ U ⊕P implies that U/ϖU = 0, and so U = 0, as required.

4.20 Corollary. Let A, P, and S be as in Proposition 4.19. Then for any O-algebra homomorphism x : A → O, P ˆ⊗_{A,x} O is a projective envelope of S in ℭ(O).

Proof. It follows from Proposition 4.19 that

P ˆ⊗_{A,x} O ≅ (Proj(S) ˆ⊗_{O} A) ˆ⊗_{A,x} O ≅ Proj(S).

4.21 Remark. If we replace the assumption in Proposition 4.19 that Hom_{ℭ(O)}(P, S)^{∨} is a free A/ϖA-module of rank 1 with the assumption that it is free of rank n, then we have an isomorphism A^{⊕n} ˆ⊗_{O} Proj(S) ≅ P in ℭ(A).

Indeed, a generalization of Lemma 4.17 to the rank n case gives an isomorphism in ℭ(𝔽) of A^{⊕n} ˆ⊗_{O} Proj(S) and P modulo ϖ. (In fact, the statement of Lemma 4.17 can be strengthened as follows: if Hom_{ℭ(𝔽)}(P, S)^{∨} is a projective object in the category of compact A-modules, then we have an isomorphism Hom_{ℭ(𝔽)}(P, S)^{∨} ˆ⊗_{𝔽} Proj S ≃ P. The key is again the projectivity of the completed tensor product Hom_{ℭ(𝔽)}(P, S)^{∨} ˆ⊗_{𝔽} Proj S, which follows from our assumption and from (4.18).) We then upgrade the isomorphism modulo ϖ to an isomorphism in ℭ(A) as in the proof of Proposition 4.19, again relying on (4.18).

We now apply the results above in the special case of P = M_{∞} and G = GL_{2}(ℚ_{p}).

4.22 Proposition. Let A = O[[x_{1}, …, x_{d}]] and choose a homomorphism of local O-algebras A → R_{∞}, that induces an isomorphism R_{p} ˆ⊗_{O} A ≅ R_{∞}. Then there is an isomorphism in ℭ(A):

M_{∞} ≅ P̃ ˆ⊗_{O} A,

where P̃ ↠ π^{∨} is a projective envelope of π^{∨} in ℭ(O).

Proof. Theorem 4.15 implies that M_{∞} is projective in ℭ(O). As we already noted in the proof of that theorem, since M_{∞}^{∨} is locally admissible, it follows from Lemma 4.6 that soc_{G} M_{∞}^{∨} ↪ M_{∞}^{∨} is essential, and hence that M_{∞} ↠ cosoc_{ℭ(O)} M_{∞} is essential. Proposition 4.2 (3) implies that all the irreducible subquotients of cosoc_{ℭ(O)} M_{∞} are isomorphic to π^{∨}. It is therefore enough to show that

M_{∞}(π) := Hom_{ℭ(O)}(M_{∞}, π^{∨})^{∨} ≅ Hom_{G}(π, M_{∞}^{∨})^{∨}

is a free A/ϖ-module of rank 1, since the assertion then follows from Proposition 4.19. As in the proof of Proposition 4.2, we have

M_{∞}(π) ≅ 𝔽 ⊗_{H(\overline{σ})} M_{∞}(\overline{σ}) ≅ 𝔽 ⊗_{H(σ^{∘})} M_{∞}(σ^{∘}).

It follows from Proposition 4.2 (1) that M_{∞}(π) is a free 𝔽 ⊗_{H(σ^{∘})} R_{∞}(σ)-module of rank 1. Since R_{∞}(σ) ≅ R_{p}(σ) ˆ⊗_{O} A and the map H(\overline{σ}) → R_{∞}(σ)/ϖ factors through R_{p}(σ)/ϖ by Proposition 4.2 (2), we conclude that the map A → R_{∞} induces an isomorphism

A/ϖ ≅ 𝔽 ⊗_{H(σ^{∘})} R_{∞}(σ).

(Recall that 𝔽 ⊗_{H(σ^{∘})} R_{p}(σ)/ϖ = 𝔽, by Lemma 2.15 (4).) Thus M_{∞}(π) is a free A/ϖ-module of rank 1, as required.

4.23 Corollary. Let A → R_{∞} be as in Proposition 4.22 and let x : A → O be a homomorphism of local O-algebras. Then M_{∞} ˆ⊗_{A,x} O is a projective envelope of π^{∨} in ℭ(O) with a continuous R_{p} ≅ R_{∞} ⊗_{A,x} O-action, which commutes with the action of G.

Proof. This follows from Corollary 4.20.

4.24 Uniqueness of arithmetic actions As in the statement of Proposition 4.22, we let P̃ ↠ π^{∨} be a projective envelope of π^{∨} in ℭ(O).

4.25 Proposition. P̃ can be endowed with an arithmetic action of R_{p} (in the sense of Section 3.1 when d = 0).

Proof. Making any choice of morphism x : A → O in Corollary 4.23, we obtain an action of R_{p} on M_{∞} ˆ⊗_{A,x} O ≅ P̃. Since the action of R_{∞} on M_{∞} is an arithmetic action, it follows immediately from the definitions that this induced action of R_{p} on P̃ is also an arithmetic action.

4.26 Recapping capture We now very briefly recall the theory of capture from [CDP14, §2.4], specialised to the case of interest to us. We note that analogues of these results are valid for general choices of G, and in particular do not use either Colmez’s functor nor p-adic Langlands correspondence for GL_{2}(ℚ_{p}).

Let M be a compact linear-topological O[[K]]-module, and let {V_{i}}_{i∈I} be a set of continuous K-representations on finite-dimensional E-vector spaces.

4.27 Definition. We say that {V_{i}}_{i∈I} captures M if for any proper quotient M ↠ Q, we have Hom_{O[[K]]}^{cont}(M, V_{i}^{∗}) ≠ Hom_{O[[K]]}^{cont}(Q, V_{i}^{∗}) for some i ∈ I.

This definition is used only in the proof of the following result.

4.28 Proposition. Suppose that ϕ ∈ End_{O[[K]]}^{cont}(P̃) kills each Hom_{O[[K]]}^{cont}(P̃, σ_{a,b}^{∗}) for a ∈ ℤ and b ∈ ℤ_{⩾0}. Then ϕ = 0.

Proof. Since P̃ is projective in ℭ(O), it follows from [CDP14, Prop. 2.12] that the set {σ_{a,b}} captures P̃. The result follows from [CDP14, Lem. 2.9] (that is, from an application of the definition of capture to the cokernel of ϕ).

Set M(σ^{∘}) := (Hom_{O[[K]]}^{cont}(P̃, (σ^{∘})^{d}))^{d}.

4.29 Lemma. Let σ = σ_{a,b} with a ∈ ℤ and b ∈ ℤ_{⩾0} and let 𝔪_{y} be a maximal ideal of H(σ). Then κ(y) ⊗_{H(σ)} M(σ^{∘})[1/p] ≠ 0 if and only if 𝔪_{x} := η(𝔪_{y})R_{p}(σ)[1/p] is a maximal ideal of R_{p}(σ)[1/p] in the support of M(σ^{∘})[1/p] for some (equivalently, any) arithmetic action of R_{p} on P̃.

Proof. Since M(σ^{∘}) is a finitely generated R_{p}(σ)-module and the action of H(σ) on M(σ^{∘})[1/p] factors through the action of R_{p}(σ)[1/p] via η, we deduce that κ(y)⊗_{H(σ)} M(σ^{∘})[1/p] is a finitely generated κ(y) ⊗_{H(σ)} R_{p}(σ)[1/p]-module. If κ(y) ⊗_{H(σ)} M(σ^{∘})[1/p] ≠ 0 then we deduce from Lemma 2.12 that 𝔪_{x} is a maximal ideal of R_{p}(σ)[1/p] in the support of M(σ^{∘}). Conversely, if y is the image of x then using Lemma 2.12 we obtain κ(x) = κ(y) ⊗_{H(σ)} R_{p}(σ)[1/p] and hence:

κ(x) ⊗_{R_{p}(σ)[1/p]} M(σ^{∘})[1/p] ≅ κ(y) ⊗_{H(σ)} M(σ^{∘})[1/p],

which implies that κ(y) ⊗_{H(σ)} M(σ^{∘})[1/p] is non-zero.

4.30 Theorem. There is a unique arithmetic action of R_{p} on P̃.

Proof. The existence of such an action follows from Proposition 4.25.

Let σ = σ_{a,b} with a ∈ ℤ and b ∈ ℤ_{⩾0}. Let y ∈ m-Spec H(σ), such that κ(y)⊗_{H(σ)}M(σ^{∘})[1/p] ≠ 0. Lemma 4.29 implies that y is the image of x ∈ m-Spec R_{p}(σ)[1/p], which lies in the support of M(σ^{∘}). Proposition 2.13 implies that

{ \atop M(σ^{∘})[1/p]}_{𝔪_{y}} = { \atop M(σ^{∘})[1/p]}_{𝔪_{x}},

as { \atop R_{p}(σ)[1/p]}_{𝔪_{x}}-modules. Moreover, the action of { \atop R_{p}(σ)[1/p]}_{𝔪_{x}} on { \atop M(σ^{∘})[1/p]}_{𝔪_{y}} does not depend on a given arithmetic action of R_{p} on P̃, as it acts via the isomorphism in Proposition 2.13. If M is a finitely generated module over a noetherian ring R then we have injections:

M ↪ ∏_{𝔪} M_{𝔪} ↪ ∏_{𝔪} M̂_{𝔪},

where the product is taken over all the maximal ideals in R. In fact it is enough to take the product over finitely many maximal ideals: if 𝔭_{1}, …, 𝔭_{n} are minimal associated primes of M, just pick any maximal ideals 𝔪_{1} ∈ V (𝔭_{1}), …, 𝔪_{n} ∈ V (𝔭_{n}). (The second injection follows from [Mat89, Thm. 8.9].) This observation applied to R = R_{p}(σ)[1/p] and M = M(σ^{∘})[1/p] together with Lemma 4.29 implies that we have an injection of R_{p}(σ)[1/p]-modules:

M(σ^{∘})[1/p] ↪ ∏_{y∈m-Spec H(σ)} { \atop M(σ^{∘})[1/p]}_{𝔪_{y}}.

Since the map and the action of R_{p}(σ)[1/p] on the right hand side are independent of the arithmetic action of R_{p} on P̃, we deduce that the action of R_{p}(σ)[1/p] on M(σ^{∘})[1/p] is also independent of the arithmetic action of R_{p} on P̃.

If θ : R_{p} → End_{ℭ(O)}(P̃) and θ′ : R_{p} → End_{ℭ(O)}(P̃) are two arithmetic actions and r ∈ R_{p} then it follows from the above that θ(r) − θ′(r) will annihilate M(σ_{a,b}^{∘}) for all a ∈ ℤ and b ∈ ℤ_{⩾0}. Proposition 4.28 implies that θ(r) = θ′(r).

4.31 Remark. A different proof of the Theorem could be given using [CDP14, Prop. 2.19].

4.32 Theorem. If P̃ ↠ π^{∨} is a projective envelope of π^{∨} in ℭ(O) equipped with an arithmetic action of R_{p}, then there is an isomorphism in ℭ(R_{∞})

P̃ ˆ⊗_{R_{p}} R_{∞} ≅ M_{∞}.

Proof. Let A = O[[x_{1}, …, x_{d}]], and choose a homomorphism of local O-algebras A → R_{∞} which induces an isomorphism R_{p} ˆ⊗_{O} A ≅ R_{∞}. Proposition 4.22 implies that there is an isomorphism P̃ ˆ⊗_{O} A ≅ M_{∞} in ℭ(A); it is therefore enough to show that this isomorphism is R_{p}-linear. Any O-algebra homomorphism x : A → O induces an isomorphism (P̃ ˆ⊗_{O} A) ˆ⊗_{A,x} O ≅ M_{∞} ˆ⊗_{A,x} O in ℭ(O). We get two actions of R_{p} on M_{∞} ˆ⊗_{A,x} O: one of them coming from the action of R_{p} on M_{∞}, the other transported by the isomorphism. Both actions are arithmetic: the first one by Proposition 4.25, the second one by assumption. Theorem 4.30 implies that the two actions coincide; thus the isomorphism (P̃ ˆ⊗_{O} A) ˆ⊗_{A,x} O ≅ M_{∞} ˆ⊗_{A,x} O is R_{p}-linear.

We have a commutative diagram in ℭ(A):

\xymatrix{P̃ ˆ⊗_{O} A \ar[d] \ar[r]^{≅} & M_{∞} \ar[d] \\ ∏_{x:A→O}(P̃ ˆ⊗_{O} A) ˆ⊗_{A,x} O \ar[r]^{≅} & ∏_{x:A→O} M_{∞} ˆ⊗_{A,x} O}

where the product is taken over all O-algebra homomorphisms x : A → O. We know that both vertical and the lower horizontal arrows are R_{p}-linear. The map A → ∏_{x:A→O} O is injective (see

for example [Paš13, Lem. 9.22]). Since P̃ is O-torsion free the functor P̃ ˆ⊗_{O} − is exact, hence the first vertical arrow is injective, which implies that the top horizontal arrow is R_{p}-linear.

4.33 Remark. The preceding result shows that, in particular, the construction of M_{∞} in [CEG^{+}16] is independent of the choices made in the case of GL_{2}(ℚ_{p}). More precisely, the isomorphism

P̃ ˆ⊗_{R_{p}} R_{∞} ≅ M_{∞}

exhibits M_{∞} as the extension of scalars of P̃, and this latter object, with its arithmetic R_{p}-action, is independent of all choices by Theorem 4.30. Thus, the only ambiguity in the construction of M_{∞} is in the number of power series variables in R_{∞}, and in their precise action. As one is free to choose as many Taylor–Wiles primes as one wishes in the patching construction, and as the presentations of global deformation rings as quotients of power series rings over local deformation rings are non-canonical, it is evident that this is the exact degree of ambiguity that the construction of M_{∞} is forced to permit.

5. Unitary completions of principal series representations

In this section, we record some arguments related to the paper [BB10], which proves using (φ, Γ)- module techniques that the locally algebraic representations associated to crystabelline Galois representations admit a unique unitary completion. We will use the machinery developed in the previous sections, namely the projective envelope P̃ and the purely local map R_{p} → End_{ℭ(O)}(P̃), to independently deduce (without using (φ, Γ)-module techniques) that the locally algebraic representations corresponding to crystalline types of regular weight admit at most one unitary completion satisfying certain properties, and that such a completion comes from some P̃.

We will use this result in Section 7 below to show, assuming the existence results of [BB10], that certain of these representations occur in completed cohomology. This gives an alternative approach to proving modularity results in the crystalline case.

As in Section 2, we write σ = σ_{a,b} = det^{a} ⊗Sym^{b}E^{2}. Let θ : H(σ) → E be a homomorphism, and set Ψ := (c-Ind_{K}^{G} σ)⊗_{H(σ),θ} E; so Ψ is a locally algebraic principal series representation of G.

5.1 Theorem. If Ψ is irreducible, then Ψ admits at most one non-zero admissible unitary completion Ψ̂ with the following property: for an open bounded G-invariant lattice Θ in Ψ̂, (Θ/ϖ)⊗_{𝔽} \overline{𝔽}_{p} contains no subquotient of the form (Ind_{B}^{G} χ⊗χω^{−1})_{sm}, for any character χ : ℚ_{p}^{×} → \overline{𝔽}_{p}^{×}, and no special series or characters.

If a completion satisfying this property exists, then it is absolutely irreducible.

Proof. Let Π be a non-zero admissible unitary completion of Ψ that satisfies the property in the statement of the theorem. We will first show that Π is absolutely irreducible (and, indeed, most of the work of the proof will be in showing this). We note that in the course of the proof we are allowed to replace E by a finite field extension E′, since if Π ⊗_{E} E′ is an absolutely irreducible E′-Banach space representation of G then Π is an absolutely irreducible E-Banach space representation of G.

Since Π is admissible, it will contain an irreducible closed sub-Banach space representation Π_{1}. If we let Π′ denote the quotient Π/Π_{1}, then we must show that Π′ is zero. For the moment, we note simply that if Π′ is non-zero, then since the composite Ψ → Π → Π′ has dense image, we see that Π′ is another non-zero admissible unitary completion of Ψ.

Let Θ be an open bounded G-invariant lattice in Π, and let Θ_{1} := Π_{1} ∩ Θ. Since Π_{1} is also admissible, Θ_{1}/ϖ will contain an irreducible subquotient π. Since we are allowed to enlarge E, we may assume that π is absolutely irreducible. Let P̃ ↠ π^{∨} be a projective envelope of π^{∨} in ℭ(O). By the assumption on the subquotients of Θ/ϖ, there is a Galois representation r̄ satisfying Assumption 2.2 that corresponds to π via Lemma 2.15 (5). We equip P̃ with the arithmetic action of R_{p} provided by Theorem 4.30.

We let (for any admissible unitary E-Banach space representation Π of G)

M(Π) := Hom_{ℭ(O)}(P̃, Θ^{d})^{d}[1/p] ≅ Hom_{G}^{cont}(Π, Hom_{O}^{cont}(P̃, E))^{d}.

The projectivity of P̃ implies that Π ↦ M(Π) is an exact covariant functor from the category of admissible unitary E-Banach space representations of G to the category of R_{p}[1/p]-modules. In particular, we have an injection M(Π_{1}) ↪ M(Π). Since P̃ is projective and π is a subquotient of Θ_{1}/ϖ, we have M(Π_{1}) ≠ 0 by [Paš13, Lem. 4.13], and hence M(Π) ≠ 0.

Similarly, we let

M(Ψ) := Hom_{G}(Ψ, Hom_{O}^{cont}(P̃, E))^{d} ≅ M(σ^{∘})[1/p] ⊗_{H(σ),θ} E.

Recall that H(σ) acts on M(σ^{∘})[1/p] through the composite of the homomorphism H(σ) ⟶^{η} R_{p}(σ)[1/p] and the R_{p}[1/p]-action on M(σ^{∘})[1/p], and that, by Lemma 4.29, either the fibre module M(σ^{∘})[1/p]⊗_{H(σ),θ} E vanishes, or else θ extends to a homomorphism θ̂: R_{p}(σ)[1/p] → E (so that then θ = θ̂∘ η), in which case there is a natural isomorphism

M(σ^{∘})[1/p] ⊗_{H(σ),θ} E ⟶^{∼} M(σ^{∘})[1/p] ⊗_{R_{p}(σ)[1/p],θ̂} E

of E-vector spaces of dimension at most 1.

The map Ψ → Π induces a continuous homomorphism

Hom_{G}^{cont}(Π, Hom_{O}^{cont}(P̃, E)) → Hom_{G}(Ψ, Hom_{O}^{cont}(P̃, E)).

Since the image of Ψ in Π is dense this map is injective, and by taking duals we obtain a surjection of R_{p}[1/p]-modules

M(Ψ) ↠ M(Π).

Since the target is non-zero, and the source is an E-vector space of dimension at most 1, this must be an isomorphism. Since M(Π_{1}) is a non-zero subspace of M(Π), we therefore have induced isomorphisms M(Π_{1}) ⟶^{∼} M(Π) ⟶^{∼} M(Ψ). Since, as was noted above, M is an exact functor, we find that M(Π′) = 0.

We digress for a moment in order to establish that Π_{1} is in fact absolutely irreducible. Since it is irreducible and admissible, its endomorphism ring is a division algebra over E. On the other hand, since M is a functor, and since M(Π_{1}) is one-dimensional over E, we see that this division algebra admits a homomorphism to E. Thus this division algebra is in fact equal to E, and this implies that Π_{1} is absolutely irreducible by [Paš13, Lem. 4.2].

Suppose now that Π′ ≠ 0. We may then apply the above argument with Π′ in place of Π, and find an absolutely irreducible subrepresentation Π_{2} of Π′, a G_{ℚ_{p}}-representation r̄′, an associated irreducible GL_{2}(ℚ_{p})-representation π′, for which the projective envelope P̃_{2} of (π′)^{∨} gives rise to an exact functor M′ such that M′(Ψ) = M′(Π′) = M′(Π_{2}), with all three being one-dimensional.

Since both M(Ψ) and M′(Ψ) are non-zero, we find that each of r̄ and r̄′ admits a lift that is a lattice in a crystalline representation V of Hodge–Tate weights (1 − a, −(a + b)) determined (via Lemma 2.6) by the homomorphism θ̂ : R_{p}[1/p] → E. (Strictly speaking, for this to make sense we need to know that V is indecomposable; but this is automatic, since each of r̄ and r̄′ is indecomposable and has non-scalar semisimplification.)

We note that V is reducible if and only if Ψ may be identified with the locally algebraic subrepresentation of a continuous induction (Ind_{B}^{G} χ)_{cont}, for some unitary character χ : T → E^{×}, where T denotes the diagonal torus contained in the upper triangular Borel subgroup B of G. In this case it is well-known that this continuous induction is the universal unitary completion of Ψ, see [BE10, Prop. 2.2.1], and the theorem is true in this case. Thus, for the remainder of the argument, we suppose that V is irreducible, or, equivalently, that Ψ does not admit an embedding into the continuous parabolic induction of a unitary character.

We now consider separately two cases, according to whether or not r̄ and r̄′ are themselves isomorphic. If they are, then M and M′ are isomorphic functors, and we obtain a contradiction from the fact that M(Π′) = 0 while M′(Π′) is one-dimensional, implying that in fact Π′ = 0, as required.

If r̄ and r̄′ are not isomorphic, but have isomorphic semi-simplifications, then they must each consist of an extension of the same two characters, but in opposite directions. In this case π ≅ (Ind_{B}^{G} ωχ_{1} ⊗ χ_{2})_{sm}, π′ ≅ (Ind_{B}^{G} ωχ_{2} ⊗ χ_{1})_{sm}, for some smooth characters χ_{1}, χ_{2} : ℚ_{p}^{×} → 𝔽^{×}. In the terminology of [Col10b], π and π′ are the two constituents of an atome automorphe, which by definition is the unique (up to isomorphism) non-split extension between π and π′.

We first show that Π_{2} = Π′. To this end, set Π′′ := Π′/Π_{2}. If Π′′ ≠ 0, then, running through the above argument another time, we find r̄′′, etc., such that (r̄′′)^{ss} ≅ (r̄′)^{ss} ≅ r̄^{ss}, and such that M′′(Π′′) ≠ 0. But either r̄′′ ≅ r̄ or r̄′′ ≅ r̄′. Thus the functor M′′ is isomorphic to either M′ or to M. On the other hand, M′(Π′′) = 0, and also M(Π′′) = 0 (since Π′′ is a quotient of Π′ and M(Π′) = 0). This contradiction shows that Π′′ = 0, and thus that Π′ = Π_{2} is absolutely irreducible.

Since M(Π′) = 0, we see that Θ′/ϖ does not contain a copy of π as a subquotient (here Θ′ denotes some choice of G-invariant lattice in Π′). Since (as we have just shown) Π′ is absolutely irreducible, it follows from [Paš13, Cor. 8.9] that Π′ ≅ (Ind_{B}^{G} χ)_{cont} for some unitary character χ : T → E^{×}, where T denotes the diagonal torus contained in the upper triangular Borel subgroup B of G. (The proof of this result uses various Ext computations in Mod_{G}^{l.adm}(O), but does not use either of Colmez’s functors. The basic idea is that the extension of π′ by π given by the atome automorphe induces an embedding P̃ ↪ P̃′ whose cokernel is dual to the induction from B to G of a character; since M(Π′) = 0, we see that Θ′ embeds into this induction, and so is itself such an induction.) Thus Ψ admits an embedding into the continuous parabolic induction of a unitary character, contradicting our hypothesis that V is irreducible. Thus we conclude that in fact Π′ = 0, as required.

If Ψ were to admit two non-isomorphic admissible irreducible unitary completions Π_{1} and Π_{2} satisfying the assumptions of the theorem, the image of the diagonal map Ψ → Π_{1} ⊕ Π_{2} would be dense in Π_{1} ⊕ Π_{2}. This yields a contradiction to what we have already proved, as Π_{1} ⊕ Π_{2} is not irreducible.

5.2 Remark. It follows from the proof of Theorem 5.1 that if a completion Ψ̂ of the kind considered there exists, then there is a Galois representation r̄ satisfying Assumption 2.2 such that θ extends to a homomorphism θ̂ : R_{p}(σ)[1/p] → E. Furthermore, for any M_{∞} as in Section 3, we have M_{∞}(σ^{∘})[1/p] ⊗_{H(σ),θ} E ≠ 0.

Conversely, if there is an r̄ satisfying Assumption 2.2 such that θ extends to a homomorphism θ̂: R_{p}(σ)[1/p] → E, then the existence of a completion Ψ̂ is immediate from the main theorem of [BB10] (which shows that some completion exists) together with the main theorem of [Ber10] (which implies the required property of the subquotients).

5.3 Remark. As is true throughout this paper, the results of this section are equally valid for crystabelline representations, but for simplicity of notation we have restricted ourselves to the crystalline case.

6. Comparison to the local approach

We now examine the compatibility of our constructions with those of [Paš13], which work with fixed central characters. The arguments of [Paš13] make use of Colmez’s functor, and the results of this section therefore also depend on this functor. In Section 6.24 we briefly discuss how to prove some of the more elementary statements in [Paš13] using the results of the previous section (and in particular not using Colmez’s functor). We assume throughout this section that p ⩾ 5, as this assumption is made in various of the results of [Paš13] that we cite.

There are two approaches that we could take to this comparison. One would be to note that the axioms of Section 3.1 and arguments of Section 4 admit obvious analogues in the setting of a fixed central character, and thus show that if we pass to a quotient of M_{∞} with fixed central character, we obtain a uniquely determined p-adic Langlands correspondence. These axioms are satisfied by the purely local object constructed in [Paš13] (which is a projective envelope of π^{∨} in a category of representations with fixed central character), and this completes the comparison.

While this route would be shorter, we have preferred to take the second approach, and go in the opposite direction: we promote the projective envelope from [Paš13] to a representation with non-constant central character by tensoring with the universal deformation of the trivial 1-dimensional representation (which has a natural Galois action by local class field theory), and show that this satisfies the axioms of Section 3.1. This requires us to make a careful study of various twisting constructions; the payoff is that we prove a stronger result than that which would follow from the first approach.

6.1 Deformation rings and twisting Let Λ be the universal deformation ring of the trivial 1-dimensional representation of G_{ℚ_{p}} and let 1^{univ} be the universal deformation. Then 1^{univ} is a free Λ-module of rank 1 with a continuous G_{ℚ_{p}}-action. We let G act on 1^{univ} via the inverse determinant (composed with the Artin map). We let Λ^{ur} be the quotient of Λ unramified deformations. We note that Λ and Λ^{ur} are formally smooth over O of relative dimensions 2 and 1, respectively, and in particular are O-torsion free.

Let ψ : G_{ℚ_{p}} → O^{×} be a continuous character such that ψε^{−1} is congruent to det r̄ modulo ϖ. (By Remark 2.17, this implies that ψ modulo ϖ considered as a character of ℚ_{p}^{×} coincides with the central character of π.) We let R_{p}^{ψ} denote the quotient of R_{p} parameterising deformations with determinant ψε^{−1}. Let r^{univ,ψ} be the tautological deformation of r̄ to R_{p}^{ψ}. Then r^{univ,ψ} ˆ⊗_{O} 1^{univ} is a deformation of r̄ to R_{p}^{ψ} ˆ⊗_{O} Λ. Since p > 2, this induces an isomorphism of local O-algebras

R_{p} ⟶^{≅} R_{p}^{ψ} ˆ⊗_{O} Λ.  (6.2)

Let R_{p}^{ψ}(σ) denote the quotient of R_{p}(σ) corresponding to deformations of determinant ψε^{−1}; note that R_{p}^{ψ}(σ) = 0 unless ψ|_{ℤ_{p}^{×}} is equal to the central character of σ. If ψ|_{ℤ_{p}^{×}} is equal to the

central character of σ, then the isomorphism R_{p} ⟶^{≅} R_{p}^{ψ} ˆ⊗_{O} Λ induces an isomorphism

R_{p}(σ) ⟶^{≅} R_{p}^{ψ}(σ) ˆ⊗_{O} Λ^{ur}.  (6.3)

Let δ : G_{ℚ_{p}} → O^{×} be a character that is trivial modulo ϖ. Twisting by δ induces isomorphisms of O-algebras

tw_{δ} : R_{p} ⟶^{∼} R_{p}, tw_{δ} : R_{p}^{ψδ^{2}} ⟶^{∼} R_{p}^{ψ}.

(In terms of the deformation functor Dr̄ pro-represented by R_{p}, and the deformation functors Dψ ; r̄ and Dψδ^{2} ; r̄ pro-represented by R_{p}^{ψ} and R_{p}^{ψδ^{2}}, these isomorphisms are induced by the natural bijections

Dr̄(A) ⟶^{∼} Dr̄(A), Dψ ; r̄(A) ⟶^{∼} Dψδ^{2} ; r̄(A)

defined by r_{A} ↦ r_{A} ⊗ δ.) Similarly we obtain isomorphisms

tw_{δ} : Λ ⟶^{∼} Λ, tw_{δ} : Λ(δ)^{ur} ⟶^{∼} Λ^{ur},

and we have a commutative diagram

\xymatrix{R_{p} \ar[d]_{id} \ar[r]^{≅}_{(6.2)} & R_{p}^{ψδ^{2}} ˆ⊗_{O} Λ \ar[d]^{_{tw_{δ} ˆ⊗ tw_{δ^{−1}}}} \\ R_{p} \ar[r]^{≅}_{(6.2)} & R_{p}^{ψ} ˆ⊗_{O} Λ}  (6.4)

6.5 Unfixing the central character Let Mod_{G}^{l.adm,ψ}(O) be the full subcategory of Mod_{G}^{l.adm}(O), consisting of those representations, where Z acts by the central character ψ. Let ℭ^{ψ}(O) be the Pontryagin dual of Mod_{G}^{l.adm,ψ}(O), so that we can identify ℭ^{ψ}(O) with a full subcategory of ℭ(O) consisting of those objects on which Z acts by ψ^{−1}.

Let P̃^{ψ} ↠ π^{∨} be a projective envelope of π^{∨} in ℭ^{ψ}(O). By [Paš13, Prop. 6.3, Cor. 8.7, Thm.

10.71] there is a natural isomorphism

R_{p}^{ψ} ⟶^{∼} End_{ℭ(O)}(P̃^{ψ}).  (6.6)

In Corollary 6.23 below we will prove a version of the isomorphism (6.6) for R_{p}.

6.7 Lemma. Let δ : G_{ℚ_{p}} → O^{×} be a character that is trivial modulo ϖ. There is an isomorphism in ℭ(O):

φ : P̃^{ψδ^{2}} ⟶^{∼} P̃^{ψ} ⊗ δ^{−1} ∘ det .

Moreover, the following diagram commutes:

\xymatrix{R_{p}^{ψδ^{2}} \ar[d]_{(6.6)}^{≅} \ar[r]^{^{tw^{δ}}}_{≅} & R_{p}^{ψ} \ar[d]^{(6.6)}_{≅} \\ End_{ℭ(O)}(P̃^{ψδ^{2}}) \ar[r]^{≅} & End_{ℭ(O)}(P̃^{ψ})}

where the lower horizontal arrow is given by α ↦ φ ∘ α ∘ φ^{−1}.

Proof. The claimed isomorphism follows from the fact that twisting by δ^{−1} ∘ det induces an equivalence of categories between ℭ^{ψ}(O) and ℭ^{ψδ^{2}}(O), so that the twist of a projective envelope of π^{∨} in ℭ^{ψ}(O) is a projective envelope of π^{∨} in ℭ^{ψδ^{2}}(O). The commutativity of the diagram follows from the compatibility of the constructions of [Paš13], and in particular of the isomorphism (6.6), with twisting. (This comes down to the compatibility of the functor V̌, discussed below, with twisting.)

Let δ : G_{ℚ_{p}} → O^{×} be a character that is trivial modulo ϖ. There is an evident isomorphism of pseudocompact O[[G_{ℚ_{p}}]]-modules:

θ : 1^{univ} ⊗ δ ⟶^{∼} 1^{univ},

and a commutative diagram

\xymatrix{Λ \ar[d]^{≅} \ar[r]^{^{tw^{δ^{−1}}}}_{≅} & Λ \ar[d]_{≅} \\ End_{G_{ℚp}}^{cont}(1^{univ}) \ar[r]^{≅} & End_{G_{ℚp}}^{cont}(1^{univ})}  (6.8)

where the lower horizontal arrow is given by α ↦ θ∘α∘θ^{−1}. (In terms of the deformation functor D_{1} pro-represented by Λ, the isomorphism tw_{δ^{−1}} is induced by the bijection D_{1}(A) ⟶^{∼} D_{1}(A) defined by χ ↦ χδ^{−1}.)

6.9 Lemma. Let δ : ℚ_{p}^{×} → O^{×} be a character that is trivial modulo ϖ. Then there is an isomorphism

P̃^{ψδ^{2}} ˆ⊗_{O} 1^{univ} ⟶^{∼} P̃^{ψ} ˆ⊗_{O} 1^{univ}

in the category ℭ(R_{p}), where R_{p} acts on both sides by the isomorphism of (6.2).

Proof. Using Lemma 6.7 and (6.8) we obtain isomorphisms in ℭ(O):

P̃^{ψδ^{2}} ˆ⊗_{O} 1^{univ} ⟶^{φ ˆ⊗ id}(P̃^{ψ} ⊗ δ^{−1} ∘ det) ˆ⊗_{O} 1^{univ} ⟶^{∼}
⟶^{∼} P̃^{ψ} ˆ⊗_{O}(1^{univ} ⊗ δ ∘ det) ⟶^{id ˆ⊗ θ} P̃^{ψ} ˆ⊗_{O} 1^{univ}.

The composition of these isomorphisms is equal to ϕ ˆ⊗ θ : P̃^{ψδ^{2}} ˆ⊗_{O} 1^{univ} ⟶^{∼} P̃^{ψ} ˆ⊗_{O} 1^{univ}, which is an isomorphism in ℭ(O). It follows from Lemma 6.7 and (6.8) that the following diagram commutes:

\xymatrix{R_{p}^{ψδ^{2}} ˆ⊗_{O} Λ \ar[d] \ar[r]^{^{tw^{δ} ˆ⊗ tw^{δ^{−1}}}}_{≅} & R_{p}^{ψ} ˆ⊗_{O} Λ \ar[d] \\ End_{ℭ(O)}(P̃^{ψδ^{2}} ˆ⊗_{O} 1^{univ}) \ar[r]^{≅} & End_{ℭ(O)}(P̃^{ψ} ˆ⊗_{O} 1^{univ})}

where the lower horizontal arrow is given by α ↦ (φ ˆ⊗ θ) ∘ α ∘ (φ ˆ⊗ θ)^{−1}. It follows from the above diagram and (6.4) that φ ˆ⊗ θ is an isomorphism in ℭ(R_{p}).

6.10 Proposition. P̃^{ψ} ˆ⊗_{O} 1^{univ} is projective in the category of pseudocompact O[[K]]-modules.

Proof. Let I_{1} := \{g ∈ K : g ≡ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) (mod ϖ)\}. Then I_{1} is a pro-p Sylow subgroup of K and it is enough to show that P̃^{ψ} ˆ⊗_{O} 1^{univ} is projective in the category of pseudocompact O[[I_{1}]]- modules. Since I_{1} is a pro-p group there is only one indecomposable projective object in the category, namely O[[I_{1}]], and thus a pseudocompact O[[I_{1}]]-module is projective if and only if it is pro-free, which means that it is isomorphic to ∏_{j∈J} O[[I_{1}]] for some indexing set J.

Let Γ := 1 + pℤ_{p}. We identify Γ with the image in G_{ℚ_{p}}^{ab} of the wild inertia subgroup of G_{ℚ_{p}}. We may identify Λ with the completed group algebra of the pro-p completion of G_{ℚ_{p}}^{ab}, which is isomorphic to Γ × ℤ_{p}. There is an isomorphism of O-algebras Λ ≅ O[[Γ]][[x]], and in particular 1^{univ} is a pro-free and hence projective O[[Γ]]-module.

The restriction of P̃^{ψ} to K is projective in the category of pseudocompact O[[K]]-modules on which Z ∩ K acts by the central character ψ^{−1}, [Paš15, Corollary 5.3]. By restricting further to I_{1}, we deduce that P̃^{ψ} is projective in the category of pseudocompact O[[I_{1}]]-modules, where Z_{1} := I_{1} ∩ Z acts by the central character ψ^{−1}.

Since p > 2 there is a character δ : Γ → O^{×}, such that δ^{2} = ψ. Twisting by characters preserves projectivity, so that 1^{univ} ⊗ δ is a projective in the category of pseudocompact O[[Γ]]- modules and P̃^{ψ} ⊗ (δ ∘ det) is projective in the category of pseudocompact O[[I_{1}]]-modules on which Z_{1} acts trivially. We may identify this last category with the category of pseudocompact O[[I_{1}/Z_{1}]]-modules.

We have an isomorphism of I_{1}-representations

P̃^{ψ} ˆ⊗_{O} 1^{univ} ≅ (P̃^{ψ} ⊗ δ ∘ det) ˆ⊗_{O}(1^{univ} ⊗_{O} δ),

where I_{1} acts on 1^{univ} ⊗_{O} δ via the homomorphism I_{1} → Γ, g ↦ (det g)^{−1}. We may therefore assume that ψ|_{Z_{1}} is trivial, so that P̃^{ψ} is projective in the category of pseudocompact O[[I_{1}/Z_{1}]]- modules. Thus there are indexing sets J_{1} and J_{2} such that

P̃^{ψ}|_{I_{1}} ≅ ∏_{j_{1}∈J} O[[I_{1}/Z_{1}]], 1^{univ} ≅ ∏_{j_{2}∈J_{2}} O[[Γ]]

where the first isomorphism is an isomorphism of pseudocompact O[[I_{1}]]-modules, and the second isomorphism is an isomorphism of pseudocompact O[[Γ]]-modules. Since completed tensor products commute with products, we obtain an isomorphism of pseudocompact O[[I_{1}]]-modules:

(P̃^{ψ} ˆ⊗_{O} 1^{univ}) ≅ ∏_{j_{1}∈J_{1}} ∏_{j_{2}∈J_{2}} O[[I_{1}/Z_{1}]] ˆ⊗_{O} O[[Γ]].

Since p > 2, the determinant induces an isomorphism between Z_{1} and Γ. Thus the map I_{1} → I_{1}/Z_{1} × Γ, g ↦ (gZ_{1}, (det g)^{−1}) is an isomorphism of groups. The isomorphism induces a natural isomorphism of O[[I_{1}]]-modules, O[[I_{1}]] ≅ O[[I_{1}/Z_{1}]] ˆ⊗_{O} O[[Γ]]. We conclude that P̃^{ψ} ˆ⊗_{O} 1^{univ} is a pro-free and hence a projective O[[I_{1}]]-module.

6.11 Remark. We will see in Theorem 6.18 below that in fact P̃^{ψ} ˆ⊗_{O} 1^{univ} is a projective object of ℭ(O). It is not so difficult to prove this directly, but we have found it more convenient to deduce it as part of the general formalism of arithmetic actions.

6.12 Proposition. Let σ = σ_{a,b}, and let δ : ℚ_{p}^{×} → O^{×} be a character that is trivial modulo ϖ, chosen so that ψδ^{2}|_{ℤ_{p}^{×}} is the central character of σ^{∘}. Then there is a natural isomorphism of R_{p}-modules

Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O} 1^{univ}, (σ^{∘})^{d})^{d} ⟶^{≅} Hom_{O[[K]]}^{cont}(P̃^{ψδ^{2}}, (σ^{∘})^{d})^{d} ˆ⊗_{O} Λ^{ur},

where R_{p} acts on the left hand side via the isomorphism R_{p} ≅ R_{p}^{ψ} ˆ⊗_{O} Λ and on the right hand side via the isomorphism R_{p} ≅ R_{p}^{ψδ^{2}} ˆ⊗_{O} Λ.

Proof. Using Lemma 6.9 we may assume that ψ|_{K∩Z} is the central character of σ. We note that 1^{univ}⊗_{Λ}Λ^{ur} is the largest quotient of 1^{univ} on which K∩Z acts trivially. Since the central character of (σ^{∘})^{d} is ψ^{−1}|_{ℤ_{p}^{×}}, and the central character of P̃^{ψ} is ψ^{−1}, we have a natural isomorphism

Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O} 1^{univ}, (σ^{∘})^{d})^{d} ⟶^{≅} Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O}(1^{univ} ⊗_{Λ} Λ^{ur}), (σ^{∘})^{d})^{d}.

Since K acts trivially on 1^{univ} ⊗_{Λ} Λ^{ur}, we have an isomorphism:

Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O}(1^{univ} ⊗_{Λ} Λ^{ur}), (σ^{∘})^{d})^{d} ≅ Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O} Λ^{ur}, (σ^{∘})^{d})^{d},

where Λ^{ur} carries a trivial K-action and R_{p} acts by the isomorphism R_{p} ≅ R_{p}^{ψ} ˆ⊗_{O} Λ. To finish the proof we need to construct a natural isomorphism of R_{p}^{ψ} ˆ⊗_{O} Λ-modules:

Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O} Λ^{ur}, (σ^{∘})^{d})^{d} ≅ Hom_{O[[K]]}^{cont}(P̃^{ψ}, (σ^{∘})^{d})^{d} ˆ⊗_{O} Λ^{ur}.  (6.13)

Both sides of (6.13) are finitely generated R_{p}^{ψ} ˆ⊗_{O} Λ^{ur}-modules. The 𝔪-adic topology on R_{p}^{ψ} ˆ⊗_{O} Λ^{ur} induces a topology on them, and makes them into pseudo-compact, O-torsion free O-modules. It is therefore enough to construct a natural isomorphism between the Schikhof duals of both sides of (6.13). To ease the notation we let A = P̃^{ψ}, B = Λ^{ur}, C = (σ^{∘})^{d}. Since for a pseudo-compact O-module D, we have D^{d} = Hom_{O}^{cont}(D, O) ≅ \underleftarrow{lim}_{n} Hom_{O}^{cont}(D, O/ϖ^{n}), using the adjointness between ˆ⊗_{O} and Hom_{O}^{cont} (see [Bru66, Lem. 2.4]), we obtain natural isomorphisms

(Hom_{O}^{cont}(A, C)^{d} ˆ⊗_{O} B)^{d} ≅ Hom_{O}^{cont}(B, (Hom_{O}^{cont}(A, C)^{d})^{d})
≅ Hom_{O}^{cont}(B, Hom_{O}^{cont}(A, C)) ≅ Hom_{O}^{cont}(A ˆ⊗_{O} B, C),

and hence a natural isomorphism

Hom_{O}^{cont}(A ˆ⊗_{O} B, C)^{d} ≅ Hom_{O}^{cont}(A, C)^{d} ˆ⊗_{O} B.

Since the isomorphism is natural and K acts trivially on B we obtain a natural isomorphism

Hom_{O[[K]]}^{cont}(A ˆ⊗_{O} B, C)^{d} ≅ Hom_{O[[K]]}^{cont}(A, C)^{d} ˆ⊗_{O} B.

Since the action of R_{p}^{ψ} commutes with the action of K and the isomorphism is natural, we deduce that (6.13) holds.

6.14 Proposition. For any σ, the action of R_{p} ≅ R_{p}^{ψ} ˆ⊗_{O} Λ on

M′(σ^{∘}) := Hom_{O[[K]]}^{cont}(P̃^{ψ} ˆ⊗_{O} 1^{univ}, (σ^{∘})^{d})^{d}

factors through R_{p}(σ). Moreover, M′(σ^{∘}) is a maximal Cohen–Macaulay R_{p}(σ)-module and M′(σ^{∘})[1/p] is a locally free R_{p}(σ)[1/p]-module of rank 1.

6.15 Remark. Recall that M(σ^{∘}) was defined in the last section. Soon we will see from Theorem 6.18 below that P̃ ⟶^{∼} P̃^{ψ} ˆ⊗_{O}1^{univ}, thus also that M(σ^{∘}) ≃ M′(σ^{∘}).

Proof. If the central character of σ is not congruent to ψ|_{ℤ_{p}^{×}} modulo ϖ, then both R_{p}(σ) and M′(σ^{∘}) are zero. Otherwise, there is a character δ : ℚ_{p}^{×} → O^{×} trivial modulo ϖ such that (ψδ^{2})|_{ℤ_{p}^{×}} is equal to the central character of σ. Proposition 6.12 gives us an isomorphism of R_{p}-modules

M′(σ^{∘}) ≅ M^{ψδ^{2}}(σ^{∘}) ˆ⊗_{O} Λ^{ur},  (6.16)

where M^{ψδ^{2}}(σ^{∘}) := Hom_{O[[K]]}^{cont}(P̃^{ψδ^{2}}, (σ^{∘})^{d})^{d}, and the action of R_{p} on the right hand side is given

by R_{p} ≅ R_{p}^{ψδ^{2}} ˆ⊗_{O} Λ. The action of R_{p}^{ψδ^{2}} on M^{ψδ^{2}}(σ^{∘}) factors through the action of R_{p}^{ψδ^{2}}(σ) and makes it into a maximal Cohen–Macaulay R_{p}^{ψδ^{2}}(σ)-module [Paš15, Cor. 6.4, 6.5]. Since Λ^{ur} ≅ O[[x]], we conclude that M^{ψδ^{2}}(σ^{∘}) ˆ⊗_{O} Λ^{ur} is a maximal Cohen–Macaulay R_{p}^{ψδ^{2}}(σ) ˆ⊗_{O} Λ^{ur}- module. Using (6.3) we see that M′(σ^{∘}) is a maximal Cohen–Macaulay R_{p}(σ)-module. Since R_{p}(σ)[1/p] is a regular ring a standard argument using the Auslander–Buchsbaum theorem shows that M′(σ^{∘})[1/p] is a locally free R_{p}(σ)[1/p]-module. It follows from [Paš15, Prop. 4.14, 2.22] that

dim_{κ(x)} M^{ψδ^{2}}(σ^{∘}) ⊗_{R_{p}^{ψδ^{2}}} κ(x) = 1, ∀x ∈ m-Spec R_{p}^{ψδ^{2}}(σ)[1/p].

This together with (6.16) gives us

dim_{κ(x)} M′(σ^{∘}) ⊗_{R_{p}} κ(x) = 1, ∀x ∈ m-Spec R_{p}(σ)[1/p].

Hence, M′(σ^{∘})[1/p] is a locally free R_{p}(σ)[1/p]-module of rank 1.

The natural action of H(σ^{∘}) on M′(σ^{∘}) commutes with the action of R_{p}(σ), and hence induces an action of H(σ) on M′(σ^{∘})[1/p]. Since M′(σ^{∘})[1/p] is locally free of rank 1 over R_{p}(σ)[1/p] by Proposition 6.14, we obtain a homomorphism of E-algebras:

α : H(σ) → End_{R_{p}(σ)[1/p]}(M′(σ^{∘})[1/p]) ≅ R_{p}(σ)[1/p].

6.17 Proposition. The map α : H(σ) → R_{p}(σ)[1/p] coincides with the map η : H(σ) → R_{p}(σ)[1/p] constructed in [CEG^{+}16, Thm. 4.1].

Proof. It is enough to show that the specialisations of α and η at x coincide for x in a Zariski dense subset of m-Spec R_{p}(σ)[1/p]. The isomorphism R_{p} ≅ R_{p}^{ψ} ˆ⊗_{O} Λ maps x to a pair (y, z), where y ∈ m-Spec R_{p}^{ψ}[1/p] and z ∈ m-Spec Λ[1/p], so that if r_{x}^{univ}, r_{y}^{univ,ψ} and 1_{z}^{univ} are Galois representations corresponding to x, y and z respectively then

r_{x}^{univ} ≅ r_{y}^{univ,ψ} ⊗ 1_{z}^{univ}.

Let

Π_{x} := Hom_{O}^{cont}((P̃^{ψ} ˆ⊗_{O} 1^{univ}) ˆ⊗_{R_{p}} O_{κ(x)}, E),

Π_{y} := Hom_{O}^{cont}(P̃^{ψ} ˆ⊗_{R_{p}^{ψ}} O_{κ(y)}, E).

Then both are unitary G-Banach space representations and we have

Π_{x} ≅ Π_{y} ⊗ (1_{z}^{univ} ∘ det).

It follows from [Paš15, Prop. 2.22] that Hom_{K}(σ, Π_{x}) is a one-dimensional κ(x)-vector space, and the action of H(σ) on it coincides with the specialisation of α at x, which can be written as the composite

H(σ) → End_{κ(x)}(M′(σ^{∘}) ⊗_{R_{p}} κ(x)) ≅ κ(x).

Since σ is an algebraic representation of K, we have

Hom_{K}(σ, Π_{x}) ≅ Hom_{K}(σ, Π_{x}^{l.alg}),

where Π_{x}^{l.alg} is the subspace of locally algebraic vectors in Π_{x}.

It follows from the main result of [Paš13] that the specialisation Π_{x} of P̃^{ψ} coincides with the Banach space representation attached to r_{x}^{univ} via the p-adic local Langlands correspondence. It is then a consequence of the construction of the appropriate cases of the p-adic local Langlands correspondence (that is, the construction of Π_{x}) that there is an embedding

π_{sm}(r_{x}^{univ}) ⊗ π_{alg}(r_{x}^{univ}) ↪ Π_{x}^{l.alg},

where π_{sm}(r_{x}^{univ}) = r_{p}^{−1}(WD(r_{x}^{univ})^{F −ss}) is the smooth representation of G corresponding to the Weil–Deligne representation associated to r_{x}^{univ} by the classical Langlands correspondence r_{p} (normalised as in Section 1.12), and π_{alg}(r_{x}^{univ}) is the algebraic representation of G whose restriction to K is equal to σ. Indeed, if the representation r_{x}^{univ} is irreducible, then (for a Zariski dense set of x) Π_{x} is a completion of π_{sm}(r_{x}^{univ}) ⊗ π_{alg}(r_{x}^{univ}) by the main result of [BB10] (see in particular [BB10, Thm. 4.3.1]), while in the case that r_{x}^{univ} is reducible, the result follows from the explicit description of Π_{x} in [BE10].

Since we have already noted that Hom_{K}(σ, Π_{x}) is one-dimensional, we find that in fact

Hom_{K}(σ, Π_{x}) ≅ Hom_{K}(σ, π_{sm}(r_{x}^{univ}) ⊗ π_{alg}(r_{x}^{univ})) ≅ π_{sm}(r_{x}^{univ})^{K},

and the right-hand side of this isomorphism is indeed a one-dimensional vector space on which H(σ) acts via the specialisation of η.

Let R_{p} act on P̃^{ψ} ˆ⊗_{O} 1^{univ} via the isomorphism R_{p} ≅ R_{p}^{ψ} ˆ⊗_{O} Λ, where (as throughout this section) the action of R_{p}^{ψ} on P̃^{ψ} is via the isomorphism R_{p}^{ψ} ≅ End_{ℭ^{ψ}(O)}(P̃^{ψ}) constructed in [Paš13].

6.18 Theorem. P̃^{ψ} ˆ⊗_{O} 1^{univ} is a projective envelope of π^{∨} in ℭ(O), and the action of R_{p} on P̃^{ψ} ˆ⊗_{O} 1^{univ} is arithmetic.

Proof. We will show that the action of R_{p} on P̃^{ψ} ˆ⊗_{O} 1^{univ} satisfies the axioms (AA1) – (AA4) with d = 0; then the action is arithmetic by definition, and P̃^{ψ} ˆ⊗_{O} 1^{univ} is a projective envelope of π^{∨} in ℭ(O) by Theorem 4.32 (applied with d = 0 and with M_{∞} taken to be P̃^{ψ} ˆ⊗_{O}1^{univ}).

It is shown in [Paš15, Propositition 6.1] that 𝔽 ˆ⊗_{R_{p}^{ψ}} P̃^{ψ} is a finitely generated O[[K]]-module,

so the topological version of Nakayama’s lemma implies that P̃^{ψ} is a finitely generated R_{p}^{ψ}[[K]]- module. Since 1^{univ} is a free Λ-module of rank 1, P̃^{ψ} ˆ⊗_{O} 1^{univ} is a finitely generated module over (R_{p}^{ψ} ˆ⊗_{O} Λ)[[K]], and so (AA1) holds. Proposition 6.10 implies that (AA2) holds. Proposition 6.14 implies that (AA3) holds (indeed, it shows that the support of M′(σ^{∘}) is all of R_{p}(σ)[1/p]). Proposition 6.17 implies that (AA4) holds.

Recall that for each fixed central character ψ : Z → O^{×}, there is an exact functor V̌ from ℭ^{ψ}(O) to the category of continuous G_{ℚ_{p}}-representations on compact O-modules. This is a modification of the functor introduced by Colmez in [Col10b], see [Paš13, §5.7] for details (we additionally have to twist the functor in [Paš13] by the inverse of the cyclotomic character to get the desired relationship between the determinant of the Galois representations and the central character of GL_{2}(ℚ_{p})-representations.) If Π is an admissible unitary E-Banach space representation of G with central character ψ, and Θ is an open bounded G-invariant lattice in Π then the Schikhof dual of Θ is an object of ℭ^{ψ}(O) and V̌(Π) := V̌(Θ^{d})[1/p] does not depend on the choice of Θ.

The representation P̃^{ψ} satisfies the conditions (N0), (N1), (N2) of [Paš15, §4] by [Paš15, Prop. 6.1]. In particular, we have V̌(P̃^{ψ}) ≅ r^{univ,ψ} as R_{p}^{ψ}[[G_{ℚ_{p}}]]-modules.

Let R_{∞} = R_{p}[[x_{1}, …, x_{d}]] and let M_{∞} be an R_{∞}[G]-module satisfying the axioms (AA1)– (AA4). To x ∈ m-Spec R_{∞}[1/p] we associate a unitary κ(x)-Banach space representation of G, Π_{∞,x} := Hom_{O}^{cont}(M_{∞} ˆ⊗_{R_{∞}} O_{κ(x)}, E). The map R_{p} → R_{∞} induces a map R_{p} → κ(x) and we let r_{x}^{univ} := r^{univ} ⊗_{R_{p}} κ(x).

6.19 Corollary. We have an isomorphism of Galois representations V̌(Π_{∞,x}) ≅ r_{x}^{univ}. In particular, Π_{∞,x} ≠ 0 for all x ∈ m-Spec R_{∞}[1/p].

Proof. Theorem 4.32 allows us to assume that R_{∞} = R_{p} and M_{∞} = P̃ is a projective envelope of π^{∨} in ℭ(O). Since an arithmetic action of R_{p} on P̃ is unique by Theorem 4.30, using Theorem 6.18 we may assume that R_{∞} = R_{p}^{ψ} ˆ⊗_{O} Λ and M_{∞} = P̃^{ψ} ˆ⊗_{O} 1^{univ}. Then, with the notation introduced in the course of the proof of Proposition 6.17, x corresponds to a pair (y, z), Π_{∞,x} = Π_{x} ≅ Π_{y} ⊗ (1_{z}^{univ} ∘ det) as in the proof of Proposition 6.17. It follows from [Paš15, Lem. 4.3] that V̌(Π_{y}) ≅ r_{y}^{univ,ψ}. Since V̌ is compatible with twisting by characters we have V̌(Π_{x}) ≅ V̌(Π_{y}) ⊗ 1_{z}^{univ} ≅ r_{x}^{univ}, as required.

6.20 Corollary. R_{∞} acts faithfully on M_{∞}.

Proof. It follows from Corollary 6.19 that M_{∞} ⊗_{R_{∞}} κ(x) is non-zero for all x ∈ m-Spec R_{∞}[1/p]. Since R_{p} and hence R_{∞} are reduced we deduce that the action is faithful.

We now use the results of [Paš13] to describe 𝔽 ˆ⊗_{R_{p}} P̃. We will use this result in Corollary 7.5 below to describe the 𝔪-torsion in the completed cohomology of a modular curve, where 𝔪 is a maximal ideal in a Hecke algebra.

6.21 Proposition. The representation π^{∨} occurs as a subquotient of 𝔽 ˆ⊗_{R_{p}} P̃ with multiplicity one. More precisely, if we let κ(r̄) := (𝔽 ˆ⊗_{R_{p}} P̃)^{∨} then the G-socle filtration of κ(r̄) is described as follows:

(i) If r̄ is irreducible then κ(r̄) ≅ π. (ii) If r̄ is a generic non-split extension of characters (so the ratio of the two characters is not

1, ω^{±1}), the G-socle filtration of κ(r̄) has length two, with graded pieces consisting of π and of the other principal series representation in the block of π.

(iii) If r̄ ≅ ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} )⊗χ then the G-socle filtration of κ(r̄) has length three and the graded pieces are π, the twist by χ∘det of the Steinberg representation, and two copies of the one-dimensional representation χ ∘ det.

Proof. We choose any continuous character ψ such that ψε^{−1} lifts det r̄. It follows from Theorem 6.18 that 𝔽 ˆ⊗_{R_{p}} P̃ ≅ 𝔽 ˆ⊗_{R_{p}^{ψ}} P̃^{ψ}. Since we can identify the endomorphism ring of P̃^{ψ} with R_{p}^{ψ}, see (6.6), it follows from Lemma 3.7 in [Paš13] applied with S = π^{∨} that any Q in ℭ^{ψ}(O) satisfying the hypotheses (H1)–(H4) of [Paš13, §3] is isomorphic to 𝔽 ˆ⊗_{R_{p}^{ψ}} P̃^{ψ}, so that κ(r̄) ≅ Q^{∨}. (We leave the reader to check that (H5) is not used to prove this part of Lemma 3.7 in [Paš13].) In all the cases Q has been constructed explicitly in [Paš13] and it is immediate from the construction of Q that the assertions about the socle filtration hold, see [Paš13, Propositions 6.1, 8.3, Remark 10.33].

6.22 Remark. In the first two cases of Proposition 6.21, κ(r̄) coincides with what Colmez calls the atome automorphe in [Col10b, §VII.4]. In the last case, κ(r̄) has an extra copy of χ ∘ det. This has to do with the fact that Colmez requires that his atome automorphe lift to irreducible unitary Banach space representations of G.

6.23 Corollary. There is a natural isomorphism R_{p} ⟶^{∼} End_{ℭ(O)}(P̃).

Proof. Theorems 4.30 and 6.18 yield an isomorphism P̃ ⟶^{∼} P̃^{ψ} ˆ⊗_{O}1^{univ} as R_{p}[G]-modules via the isomorphism R_{p} ⟶^{∼} R_{p}^{ψ} ˆ⊗_{O}Λ given by (6.2). Now End_{G}^{cont}(1^{univ}) = Λ, while (6.6) gives a natural isomorphism R_{p}^{ψ} ⟶^{∼} End_{ℭ(O)}(P̃^{ψ}). Thus the corollary amounts to proving that the natural homomorphism

End_{ℭ(O)}(P̃^{ψ})ˆ⊗_{O}End_{G}^{cont}(1^{univ}) → End_{ℭ(O)}(P̃^{ψ} ˆ⊗_{O}1^{univ})

is an isomorphism. The map is an injection of pseudo-compact O-algebras. This makes the ring End_{ℭ(O)}(P̃) into a compact R_{p}-module. By the topological version of Nakayama’s lemma, in order to show that the map is surjective it is enough to show that 𝔽 ˆ⊗_{R_{p}} End_{ℭ(O)}(P̃) is a one dimensional 𝔽-vector space. Since P̃ is projective, we have an isomorphism:

𝔽 ˆ⊗_{R_{p}} End_{ℭ(O)}(P̃) ≅ Hom_{ℭ(O)}(P̃, 𝔽 ˆ⊗_{R_{p}} P̃).

By Proposition 6.21, π^{∨} occurs as a subquotient of 𝔽 ˆ⊗_{R_{p}} P̃ with multiplicity one. Since P̃ is a projective envelope of π^{∨} and End_{ℭ(O)}(π^{∨}) = 𝔽, this implies that Hom_{ℭ(O)}(P̃, 𝔽 ˆ⊗_{R_{p}} P̃) is a one dimensional 𝔽-vector space, as required.

6.24 Endomorphism rings and deformation rings We maintain the notation of the previous sections; in particular, P̃ denotes a projective envelope of π^{∨} in ℭ(O). If we write R̃ := End_{ℭ(O)}(P̃), then the arithmetic action of R_{p} on P̃ provided by Theorem 4.30 gives a morphism R_{p} → R̃, which Corollary 6.23 shows is an isomorphism. The proof of that Corollary uses the analogous statement proved in [Paš13] (when the central character is fixed), a key input to the proof of which is Colmez’s functor from GL_{2}(ℚ_{p})-representation to Galois representations. It is natural to ask (especially in light of possible generalizations) whether this isomorphism can be proved using just the methods of the present paper, without appealing to Colmez’s results. In this subsection we address this question, to the extent that we can.

We begin by noting that since P̃ is a projective envelope of the absolutely irreducible representation π^{∨}, the ring R̃ is a local ring. We will furthermore give a proof that it is commutative, from the perspective of this paper. As already noted, this result is not new. Indeed, in addition to being a consequence of Corollary 6.23 (and thus, essentially, of the results of [Paš13]), another proof is given in [CDP14] (see Cor. 2.22 of that paper). This latter proof uses the capturing techniques that we are also employing in the present paper, and (since it is easy to do so) we present a slightly rephrased version of the argument here, in order to illustrate how it fits naturally into our present perspective.

6.25 Proposition. The ring R̃ is commutative.

Proof. We first prove that the image of R_{p} in R̃ lies in the centre of R̃. To see this, suppose that ϕ ∈ R̃. By Proposition 4.28, to show that ϕ commutes with the action of R_{p}, it suffices to show that ϕ commutes with the action of R_{p}(σ)[1/p] on M(σ^{∘})[1/p] for each σ. Since the action of H(σ) on M(σ^{∘})[1/p] depends only on the G-action on P̃, we see that ϕ commutes with the H(σ)-action on M(σ^{∘})[1/p]. The desired result then follows from Proposition 2.13.

To see that R̃ is commutative, we again apply Proposition 4.28, by which it suffices to show that R̃ acts on each M(σ^{∘})[1/p] through a commutative quotient. This follows from the fact that each M(σ^{∘})[1/p] is locally free of rank one over its support in Spec R_{p}(σ)[1/p], and the fact that (by the result of the previous paragraph) the R̃-action commutes with the R_{p}-action.

6.26 Remark. As for proving the stronger result that the canonical map R_{p} → R̃ is an isomorphism, in the forthcoming paper [EP] two of us (ME and VP) will establish the injectivity of the morphism R_{p} → R̃. (In fact we will prove a result in the more general context of [CEG^{+}16]; in particular, our arguments won’t rely on any special aspects of the GL_{2}(ℚ_{p}) situation, such as the existence of Colmez’s functors.) However, proving the surjectivity of this morphism seems to be more difficult, and we currently don’t know a proof of this surjectivity that avoids appealing to the theory of Colmez’s functor from GL_{2}(ℚ_{p})-representations to G_{ℚ_{p}}-representations.

6.27 Speculations in the residually scalar semi-simplification case Suppose for the rest of this section that r̄ ≅ ( \begin{matrix}χ & ∗ \\ 0 & χ \end{matrix} ) for some χ; so in particular r̄ does not satisfy Assumption 2.2. It is natural to ask what the modules M_{∞} constructed in [CEG^{+}16] look like in this case; we give a speculative answer below. By twisting we may assume that χ is the trivial character. Let π = (Ind_{B}^{G} ω ⊗ 1)_{sm}, and let P̃ be a projective envelope of π^{∨}. We first give a conjectural description of End_{ℭ(O)}(P̃), under the assumption p > 2.

Let D^{ps} be a functor from the category 𝔄 of complete local noetherian O-algebras with residue field 𝔽 to the category of sets, that assigns to A ∈ 𝔄 the set of pairs of functions (t, d) : G_{ℚ_{p}} → A, where:

– d : G_{ℚ_{p}} → A^{×} is a continuous group homomorphism, congruent to det r̄ modulo 𝔪_{A},

– t : G_{ℚ_{p}} → A is a continuous function with t(1) = 2, and,

– for all g, h ∈ G_{ℚ_{p}}, we have:

(i) t(g) ≡ tr r̄(g) (mod 𝔪_{A}); (ii) t(gh) = t(hg); (iii) d(g)t(g^{−1}h) − t(g)t(h) + t(gh) = 0.

(The “ps” is for “pseudocharacter”. By [Che14, Lem. 1.9], D^{ps}(A) is the set of pseudocharacters deforming the pseudocharacter (tr r̄, det r̄) associated to r̄.) This functor is representable by a complete local noetherian O-algebra R^{ps}. Let (t^{univ}, d^{univ}) : G_{ℚ_{p}} → R^{ps} be the universal object. We expect that there is a natural isomorphism of O-algebras

Ẽ := End_{ℭ(O)}(P̃) ≅ (R^{ps}[[G_{ℚ_{p}}]]/J)^{op},  (6.28)

where J is the closed two sided ideal of R^{ps}[[G_{ℚ_{p}}]] generated by all the elements of the form g^{2} − t^{univ}(g)g + d^{univ}(g) for all g ∈ G_{ℚ_{p}}, and the superscript op indicates the opposite algebra. We note that such an isomorphism has been established in [Paš13, §9], when the central character is fixed, and we expect that one can deduce (6.28) from this using the twisting techniques of the previous subsection.

Let R_{p}^{□} be the framed deformation ring of r̄ and let M_{∞} be the patched module constructed in [CEG^{+}16] (or the variant for the completed cohomology of modular curves that we briefly discuss in Section 7 below) and let R_{∞} be the patched ring. Then R_{∞} is an R_{p}^{□}-algebra, and the map R_{p}^{□} → R_{∞} gives rise to a Galois representation r_{∞} : G_{ℚ_{p}} → GL_{2}(R_{∞}) lifting r̄. The pair (tr r_{∞}, det r_{∞}) gives us a point in D^{ps}(R_{∞}) and hence a map R^{ps} → R_{∞}. Hence we obtain a homomorphism of R^{ps}-algebras R^{ps}[[G_{ℚ_{p}}]] → M_{2×2}(R_{∞}).

The Cayley–Hamilton theorem implies that this map is zero on J, so we obtain a left action of R^{ps}[[G_{ℚ_{p}}]]/J on the standard module R_{∞} ⊕ R_{∞}. If we admit (6.28) then we get a right action of Ẽ on R_{∞} ⊕ R_{∞}. We expect that there are isomorphisms in ℭ(R_{∞})

M_{∞} ≅ (R_{∞} ⊕ R_{∞}) ˆ⊗Ẽ P̃ ≅ R_{∞} ˆ⊗_{R_{p}^{□}}(R_{p}^{□} ⊕ R_{p}^{□}) ˆ⊗Ẽ P̃.  (6.29)

We note that the representation appearing on the right hand side of this equation has been studied by Fabian Sander in his thesis [San], in the setting where the central character is fixed. Motivated by [San, Thm. 2] we expect (R_{∞} ⊕ R_{∞}) ˆ⊗Ẽ P̃ to be projective in the category of pseudocompact O[[K]]-modules. We do not expect (R_{∞} ⊕R_{∞}) ˆ⊗Ẽ P̃ to be projective in ℭ(O), so the methods of Section 4 cannot directly be applied to this case. However, it might be possible to prove (6.29) using Colmez’s functor. This would show that M_{∞} does not depend on the choices made in the patching process.

7. Local-global compatibility

In this final section, we briefly explain how the results of this paper give a simple new proof of the local-global compatibility theorem of [Eme11] (under the hypotheses that we have imposed in this paper, which differ a little from those of [Eme11]: locally at p, we have excluded the case of split r̄, and have allowed a slightly different collection of indecomposable reducible r̄’s; and in the global context we consider below, we exclude the possibility of so-called vexing primes). Applying these considerations to the patched modules constructed in [CEG^{+}16] allows us to prove a localglobal compatibility result for the completed cohomology of a compact unitary group, but for ease of comparison to [Eme11], we instead briefly discuss the output of Taylor–Wiles patching for modular curves.

Patching in this context goes back to [TW95], but the precise construction we need is not in the literature. It is, however, essentially identical to that of [CEG^{+}16] (or the variant for Shimura curves presented in [Sch15]), so to keep this paper at a reasonable length we simply recall the output of the construction here.

Let \overline{ρ} : G_{ℚ} → GL_{2}(𝔽) be an absolutely irreducible odd (so modular, by Serre’s conjecture) representation, and assume that p ⩾ 5 and that \overline{ρ}|_{G_{ℚ(ζp)}} is irreducible. Write r̄:= \overline{ρ}|_{G_{ℚp}}, and assume that r̄ satisfies Assumption 2.2. Write r^{univ} : G_{ℚ_{p}} → GL_{2}(R_{p}) for the universal deformation of r̄. Let N(\overline{ρ}) be the prime-to-p conductor of \overline{ρ}; that is, the level of \overline{ρ} in the sense of [Ser87]. We assume that if q|N(\overline{ρ}) with q ≡ −1 (mod p) and \overline{ρ}|_{G_{ℚq}} is irreducible then \overline{ρ}|_{I_{ℚq}} is also irreducible.

7.1 Remark. The last condition we have imposed excludes the so-called vexing primes q. The assumption that there are no vexing primes means that the Galois representations associated to modular forms of level N(\overline{ρ}) are necessarily minimally ramified. This assumption can be removed by considering inertial types at such primes as in [CDT99]. Since the arguments using types are standard and are orthogonal to the main concerns of this paper, we restrict ourselves to this simple case.

We caution the reader that while the use of types would also allow us to work at certain nonminimal levels, the most naive analogues of Theorem 7.4 fail to hold at arbitrary tame levels. It seems that to formulate a clean statement, one should pass to infinite level at a finite set of primes, and formulate the compatibility statement in terms of the local Langlands correspondence in families of [EH14], as is done in [Eme11].

However, it does not seem to be easy to prove this full local-global compatibility statement using only the methods of the present paper; indeed, the proof in [Eme11] ultimately makes use of mod p multiplicity one theorems that rely on q-expansions, whereas in our approach, we are only using multiplicity one theorems that result from our patched modules being Cohen–Macaulay, and certain of our local deformation rings being regular (namely the minimal deformation rings at places not dividing p, and the deformation rings considered in Lemma 2.15). Note that in general the (non-minimal) local deformation rings at places away from p need not be regular (even after inverting p), so that carrying out the patching construction below would result in a ring R_{∞} that was no longer formally smooth over R_{p}, to which the results of Section 4 would not apply.

Let 𝕋 be the usual Hecke algebra acting on (completed) homology and cohomology of modular curves with (tame) level Γ_{1}(N(\overline{ρ})) and O-coefficients; so 𝕋 is an O-algebra, generated by the operators T_{l}, S_{l} with l ∤ Np. Let 𝔪(\overline{ρ}) be the maximal ideal of 𝕋 corresponding to \overline{ρ} (so that T_{l} − tr \overline{ρ}(Frob_{l}) and lS_{l} − det \overline{ρ}(Frob_{l}) are both zero in 𝕋/𝔪(\overline{ρ})). Let R_{ℚ,N(\overline{ρ})} be the universal deformation ring for deformations of \overline{ρ} that are minimally ramified at primes l ≠ p, in the sense that they have the same conductor as \overline{ρ}|_{G_{ℚ_{l}}} (and in particular are unramified if l ∤ N(\overline{ρ})). Let ρ_{ℚ,N(\overline{ρ})}^{univ} : G_{ℚ} → GL_{2}(R_{ℚ,N(\overline{ρ})}) denote the corresponding universal deformation of \overline{ρ}.

We now use the notation introduced in Section 3, so that in particular we write R_{∞} := R_{p} ˆ⊗_{O}O[[x_{1}, …, x_{d}]] for some d ⩾ 0. Patching the completed étale homology of the modular curves Y_{1}(N(\overline{ρ})) (and using an argument of Carayol [Car94], as in [Eme11, §5.5] and [EGS15, §§6.2, 6.3], to factor out the Galois action on the completed cohomology; see also [Sch15, §9] for the analogous patching construction for Shimura curves), we obtain (for some d ⩾ 0) an R_{∞}[G]-module M_{∞} with an arithmetic action, with the further property that there is an ideal 𝔞_{∞} of R_{∞}, an isomorphism of local O-algebras R_{∞}/𝔞_{∞} ⟶^{∼} R_{ℚ,N(\overline{ρ})}, and an isomorphism of R_{ℚ,N(\overline{ρ})}[G × G_{ℚ}]-modules

(M_{∞}/𝔞_{∞}) ⊗_{R_{ℚ,N(\overline{ρ})}} (ρ_{ℚ,N(\overline{ρ})}^{univ})^{∗} ⟶^{∼} H̃_{1,ét}(Y_{1}(N(\overline{ρ})), O)_{𝔪(\overline{ρ})}.  (7.2)

Here H̃_{1,ét}(Y_{1}(N(\overline{ρ})), O) denotes completed étale homology, as described for example in [CE12]. The action of G_{ℚ} on the left hand side is via its action on (ρ_{ℚ,N(\overline{ρ})}^{univ})^{∗}, which as in (1.12.3) denotes the R_{ℚ,N(\overline{ρ})}-linear dual of ρ_{ℚ,N(\overline{ρ})}^{univ}.

7.3 Remark. Here we have used implicitly that the minimally ramified local (framed) deformation rings are all smooth, which follows for example from [CHT08, Lem. 2.4.19]; this ensures that the ring R_{∞} occurring in the patching argument is formally smooth over R_{p}.

7.4 Theorem. Let p > 3 be prime, and let \overline{ρ} : G_{ℚ} → GL_{2}(𝔽) be an absolutely irreducible odd representation, with the property that \overline{ρ}|_{G_{ℚ(ζp)}} is irreducible, N(\overline{ρ}) is not divisible by any vexing primes, and \overline{ρ}|_{G_{ℚp}} satisfies Assumption 2.2. Then there is an isomorphism of R_{ℚ,N(\overline{ρ})}[G × G_{ℚ}]- modules

H̃_{1,ét}(Y_{1}(N(\overline{ρ})), O)_{𝔪(\overline{ρ})} ⟶^{∼} P̃ ˆ⊗_{R_{p}} (ρ_{ℚ,N(\overline{ρ})}^{univ})^{∗},

where the completed tensor product on the right-hand side is computed by regarding (ρ_{ℚ,N(\overline{ρ})}^{univ})^{∗} as a G_{ℚ}-representation on an R_{p}-module via the natural morphism R_{p} → R_{ℚ,N(\overline{ρ})}.

Proof. As noted before Remark 7.1, \overline{ρ} is modular by Serre’s conjecture, so in particular M_{∞} is not zero. By Theorem 4.32 there is an isomorphism of R_{∞}[G]-modules

M_{∞} ≅ P̃ ˆ⊗_{O} O[[x_{1}, …, x_{d}]] ≅ P̃ ˆ⊗_{R_{p}} R_{∞}.

Quotienting out by 𝔞_{∞} yields an isomorphism

M_{∞}/𝔞_{∞} ≅ P̃ ˆ⊗_{R_{p}} R_{ℚ,N(\overline{ρ})}.

The result now follows by tensoring both sides with (ρ_{ℚ,N(\overline{ρ})}^{univ})^{∗} and applying (7.2).

We now show how to compute the 𝔪(\overline{ρ})-torsion in the completed étale cohomology of modular curves H̃_{ét}^{1}(Y_{1}(N(\overline{ρ})), O) as a GL_{2}(ℚ_{p})-representation.

7.5 Corollary. Under the assumptions of Theorem 7.4, we have an isomorphism of 𝔽[G×G_{ℚ}]- modules

H̃_{ét}^{1}(Y_{1}(N(\overline{ρ})), 𝔽)[𝔪(\overline{ρ})] ≃ κ(\overline{ρ}|_{G_{ℚp}}) ⊗_{𝔽} \overline{ρ},

where κ(\overline{ρ}|_{G_{ℚp}}) is the representation defined in Proposition 6.21.

Proof. The Pontryagin dual of the left hand side is H̃_{1,ét}(Y_{1}(N(\overline{ρ})), O) ˆ⊗_{R_{ℚ,N(\overline{ρ})}} 𝔽. By Theorem 7.4, we have an isomorphism of 𝔽[G × G_{ℚ}]-modules

H̃_{1,ét}(Y_{1}(N(\overline{ρ})), O) ˆ⊗_{R_{ℚ,N(\overline{ρ})}} 𝔽 ≃ (𝔽 ˆ⊗_{R_{p}} P̃) ˆ⊗_{𝔽} \overline{ρ}^{∗} ≅ κ(\overline{ρ}|_{G_{ℚp}})^{∨} ˆ⊗_{𝔽} \overline{ρ}^{∗},

the last isomorphism following from the definition of κ(\overline{ρ}|_{G_{ℚp}}). Passing to Pontryagin duals (and noting that since \overline{ρ} is an 𝔽-representation, we have \overline{ρ}^{∗} = \overline{ρ}^{∨}) gives the result.

7.6 Remark. Corollary 7.5 together with Proposition 6.21 gives a description of the G-socle filtration of H̃_{ét}^{1}(Y_{1}(N(\overline{ρ})), 𝔽)[𝔪(\overline{ρ})]. Even more is true.

Since in Corollary 6.23 we have identified the endomorphism ring of P̃ with R_{p} and κ(r̄) is by definition 𝔽 ˆ⊗_{R_{p}} P̃, a completely formal argument (see the proof of Proposition 2.8 in [Paš16]) shows that κ(r̄) is up to isomorphism the unique representation in Mod_{G}^{l.adm}(O) that is maximal with respect to the following two properties:

(i) the socle of κ(r̄) is π; (ii) π occurs as a subquotient of κ(r̄) with multiplicity 1.

(It is maximal in the sense that it cannot be embedded into any other strictly larger representation in Mod_{G}^{l.adm}(O) satisfying these two properties.)

Corollary 7.5 shows that (after factoring out the G_{ℚ}-action) the same characterisation carries over to H̃_{ét}^{1}(Y_{1}(N(\overline{ρ})), 𝔽)[𝔪(\overline{ρ})]. However, we warn the reader that a simple-minded application of this recipe will not work in general, and in particular it fails if \overline{ρ}|_{G_{ℚp}} ≅ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) ⊗ χ.

More precisely, if Assumption 2.2 is in force then r̄ is determined up to isomorphism by the data of its determinant and its unique irreducible subrepresentation. This information can be recovered from π, which in turn determines κ(r̄). If on the other hand r̄ ≅ ( \begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix} ) ⊗ χ and End_{G_{ℚp}} (r̄) = 𝔽 then the G-socle of the atome automorphe associated by Colmez in [Col10b, §VII.4] to r̄, which we still call π, is the Steinberg representation twisted by χ ∘ det. This representation still carries the information about the irreducible subrepresentation of r̄ and the determinant of r̄ but it does not determine r̄ up to isomorphism, as it does not carry the information about the extension class in Ext_{G_{ℚp}}^{1} (χ, χω) corresponding to r̄. The maximal representation satisfying (1) and (2) above will contain the atome automorphe corresponding to r̄ as a subrepresentation, but it will be strictly bigger. In fact it can be shown that it is the smallest representation that contains all the atomes automorphes corresponding to different non-zero extensions in Ext_{G_{ℚp}}^{1} (χ, χω).

7.7 Remark. Theorem 7.4 and Corollary 7.5, when combined with Theorems 6.18 and 4.32 (which together show that P̃ realises the usual p-adic local Langlands correspondence for GL_{2}(ℚ_{p})), prove a local-global compatibility result for completed cohomology. They are new in the case that \overline{ρ}|_{G_{ℚp}} ≅ ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} ) ⊗ χ. In particular, they answer a question raised in Remark 1.2.9 in [Eme11] by confirming the expectation of Remark 6.1.23 of loc.cit.. In the other cases, Theorem 7.4 can be deduced from Theorem 6.4.6 in [Eme11] with P̃ replaced by a deformation of κ(r̄)^{∨} to R_{p}, such that one obtains the universal deformation of r̄ after applying Colmez’s functor V̌ to it. If \overline{ρ}|_{G_{ℚp}} ≅ ( \begin{matrix}1 & ∗ \\ 0 & ω \end{matrix} ) ⊗ χ then it can be shown that P̃ is not flat over R_{p}, and that is why the approach of [Eme11] does not work in this case.

7.8 Remark. Theorem 7.4 and Corollary 7.5 have analogues in more general settings when the group at p is essentially GL_{2}(ℚ_{p}) (or a product of copies of GL_{2}(ℚ_{p})). For example, taking M_{∞} to be the patched module of Section 3.4 (as constructed in [CEG^{+}16] for n = 2), we obtain statements about the completed cohomology of unitary groups that are compact at infinity.

Perhaps a case of greater interest is that of the completed cohomology of definite quaternion algebras over totally real fields. (One reason for this case to be of interest is its relationship to the cohomology of the Lubin-Tate tower as in [Sch15, Thm. 6.2].) We expect that our results can be extended to this setting, although there is one wrinkle: in order to carry out Taylor– Wiles patching, we need to fix a central character, and as a consequence, our patched module has a fixed central character, and no longer satisfies the axioms of Section 3. One approach to this difficulty would be to formulate analogues of those axioms with an arbitrary fixed central character, making use of the twisting constructions of Section 6 and “capture” arguments of [Paš16, §2.1], but this leads to ugly statements.

Instead, we content ourselves with considering the case that the fixed central character is the trivial character. In this case we can think of our patched modules as modules for PGL_{2}(ℚ_{p}), and natural analogues of our axioms can be formulated in this setting; this is carried out in [GN16, §5], where an analogue of the results of Section 4 is proved. In fact, the arguments there allow us to consider modules for a product of copies of PGL_{2}(ℚ_{p}), which is convenient when there is more than one place lying over p; accordingly, we work below with cohomology that is completed at all primes above p. (Of course, the case of cohomology that is completed at a single prime above p can be deduced from this by returning to finite level via taking appropriate locally algebraic vectors.)

As explained in Remark 7.1, one has to take some care with ramification at places away from p, and we therefore content ourselves with considering quaternion algebras that split at all finite places. The reader wishing to prove extensions of these results to more general quaternion algebras is advised to examine the patching arguments of [GK14, §4], which work in this setting. We further caution the reader that we have not attempted to check every detail of the expected result explained below.

Let F be a totally real field in which p ⩾ 5 splits completely, and let D be a quaternion algebra over F that is split at all finite places and definite at all infinite places (note that in particular this requires [F : ℚ] to be even).

Let \overline{ρ} : G_{F} → GL_{2}(𝔽) be absolutely irreducible, and assume that \overline{ρ}|_{G_{F (ζp)}} is irreducible, and that det \overline{ρ} = ω^{−1}. Suppose that \overline{ρ} has no vexing primes; that is, if v ∤ p is a finite place at which \overline{ρ} is ramified and Nv ≡ −1 (mod p), and \overline{ρ}|_{G_{Fv}} is irreducible, then \overline{ρ}|_{I_{Fv}} is also irreducible. Finally, suppose that for each place v|p, \overline{ρ}|_{G_{Fv}} satisfies Assumption 2.2.

Since D splits at all finite places, we can consider the tame level subgroup U_{1}(N(\overline{ρ})) ⊂ PGL_{1}(D ⊗ 𝔸_{F}^{∞,p}) ≃ PGL_{2}(𝔸_{F}^{∞,p}) given by the image of those matrices in GL_{2}(𝔸_{F}^{∞,p}) that are unipotent and upper triangular modulo N(\overline{ρ}). Let H̃_{0}(U_{1}(N(\overline{ρ})), O) (resp. H̃^{0}(U_{1}(N(\overline{ρ})), O)) denote the completed homology (resp. cohomology) of the tower of locally symmetric spaces associated to PGL_{1}(D) with tame level U_{1}(N(\overline{ρ})). (Note that at finite level, the locally symmetric spaces are just finite sets of points.)

We assume that \overline{ρ} is modular, in the following sense: \overline{ρ} determines a maximal ideal 𝔪(\overline{ρ}) in the spherical Hecke algebra (generated by Hecke operators at places not dividing p at which \overline{ρ} is unramified) acting on H̃_{0}(U_{1}(N(\overline{ρ})), O) and we assume that H̃_{0}(U_{1}(N(\overline{ρ})), O)_{𝔪(\overline{ρ})} ≠ 0.

Let R\overline{ρ} be the universal deformation ring for deformations of \overline{ρ} that are minimally ramified at places not dividing p, and which have determinant ε^{−1}. For each place v|p, let R_{v} be the universal deformation ring for deformations of \overline{ρ}|_{G_{Fv}} with determinant ε^{−1}, and set R_{p} := ˆ⊗_{v|p} R_{v}.

Set G = ∏_{v|p} PGL_{1}(D_{v}), which we identify with ∏_{v|p} PGL_{2}(ℚ_{p}) via a fixed isomorphism. By patching the completed homology H̃_{0}(U_{1}(N(\overline{ρ})), O)_{𝔪(\overline{ρ})} (with a variation of the argument in [Sch15, §9]), we obtain^{3} a ring R_{∞} which is a power series ring over R_{p}, and an R_{∞}[G]-module M_{∞} with an arithmetic action in the sense of [GN16, §5.2], together with an ideal 𝔞_{∞} ⊂ R_{∞} such that R_{∞}/𝔞_{∞} ≅ R\overline{ρ} as local O-algebras and M_{∞}/𝔞_{∞} ≅ H̃_{0}(U_{1}(N(\overline{ρ})), O)_{𝔪(\overline{ρ})} as R\overline{ρ}[G]-modules. (Note that we can ensure the smoothness of R_{∞} over R_{p} since we are assuming that D splits at all finite places and that there are no vexing primes.)

Applying [GN16, Prop. 5.2.2] now gives a G-equivariant isomorphism

H̃_{0}(U_{1}(N(\overline{ρ})), O)_{𝔪(\overline{ρ})} ≃ (ˆ⊗_{v|p} P̃^{1}(\overline{ρ}|_{G_{Fv}})) ˆ⊗_{R_{p}} R\overline{ρ},

where P̃^{1}(\overline{ρ}|_{G_{Fv}}) denotes the projective envelope considered in Section 6 in the case that r̄ = \overline{ρ}|_{G_{Fv}} and ψ = 1. Arguing as in the proof of Corollary 7.5, we obtain a G-equivariant isomorphism

H̃^{0}(U_{1}(N(\overline{ρ})), 𝔽)[𝔪(\overline{ρ})] ≃ ˆ⊗_{v|p} κ(\overline{ρ}|_{G_{Fv}}),

where κ(\overline{ρ}|_{G_{Fv}}) denotes the representation defined in Proposition 6.21.

One can obtain analogous results in the case where [F : ℚ] is odd and D is split at one infinite place of F and ramified at all the others. In this case, one works with a tower of Shimura curves. The main difference to the argument is to note that \overline{ρ} only contributes to completed homology (or cohomology) in degree 1 (since the D^{×}(𝔸^{∞})-action factors through the reduced norm in degree 0), and the G_{F} -action can be factored out by the same argument as for modular curves.

Acknowledgements

This paper has its germ in conversations between three of us (M.E., T.G., V.P.) during the 2011 Durham Symposium on Automorphic Forms and Galois Representation, and we would like to thank the organizers Fred Diamond, Payman Kassaei and Minhyong Kim as well as Durham University, EPSRC and the LMS for providing a fertile atmosphere for discussion. The ideas of the paper were developed further when the six of us participated in focussed research groups on “The p-adic Langlands program for non-split groups” at the Banff Centre and AIM; we would like

3We caution the reader that one should carefully check this claim and we have not done so.

to thank AIM and BIRS for providing an excellent working atmosphere, and for their financial support. We would also like to thank the anonymous referees for their comments on the paper.

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Ana Caraiani a.caraiani@imperial.ac.uk Department of Mathematics, Imperial College London, London SW7 2AZ, UK

Matthew Emerton emerton@math.uchicago.edu Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA

Toby Gee toby.gee@imperial.ac.uk Department of Mathematics, Imperial College London, London SW7 2AZ, UK

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