Patching and the p-adic Langlands programfor GL2(Qp)
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 forGL2 (Qp ) 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 andglobal aspects of the p-adic Langlands program; in particular, it gives a new proof ofmany cases of the second author’s local-global compatibility theorem, and relaxes ahypothesis 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 GL2 (Qp ), gives rise to the p-adic local
Langlands correspondence for GL2 (Qp ), 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 BackgroundWe 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 Qp , and let
GF 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 GF . Let E be
another finite extension of Qp , which will be our field of coefficients, assumed large enough, withring of integers O, uniformizer $ and residue field F. To a continuous Galois representationr : GF → GLn (E) one would like to attach an admissible unitary E-Banach space representation
Π(r) of G := GLn (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 thecompleted 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–Tateweights, then the subspace of locally algebraic vectors Π(r)l.alg in Π(r) should be isomorphic toπsm (r) ⊗ πalg (r) as a G-representation, where πsm (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 πalg (r) is an algebraic representation of G, whose highest weight vector isdetermined by the Hodge–Tate weights of r.
1.2 Example. If F = Qp , n = 2 and r is crystalline with Hodge–Tate weights a < b, then πsm (r) isa smooth unramified principal series representation, whose Satake parameters can be calculated interms of the trace and determinant of Frobenius on Dcris (r), and πalg (r) = Symb−a−1 E 2 ⊗det1−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 = Qp by the works of
Breuil, Colmez and others, see [Bre08], [Col10a] as well as the introduction to [Col10b]. Moreover,when n = 2 and F = Qp , 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 forsuch 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 : GF → GLn (F) be a continuous representation and let Rp 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∞ [[GLn (OF )]],
where R∞ is a complete local noetherian Rp -algebra with residue field F. If y ∈ Spec R∞ is an
E-valued point thenΠy := Homcont
O (M∞ ⊗R∞ ,y O, E)
yis an admissible unitary E-Banach space representation of G. The composition Rp → R∞ →E defines an E-valued point x ∈ Spec Rp and thus a continuous Galois representation rx :GF → GLn (E). We expect that the Banach space representation Πy depends only on x andthat it should be related to rx by the hypothetical p-adic Langlands correspondence; see §6of [CEG+ 16] for a detailed discussion. We show in [CEG+ 16, Theorem 4.35] that if πsm (rx ) is
generic and x lies on an automorphic component of a potentially crystalline deformation ring∼of r̄ then Πl.alg= πsm (rx ) ⊗ πalg (rx ) as expected; moreover, the points x such that πsm (rx ) isygeneric 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 GLn (F ) that wecannot answer at the moment and that appear to be genuinely deep, as they are intertwinedwith questions about local-global compatibility for p-adically completed cohomology, with theBreuil–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 Πy depends
only on x, it is not clear that Πy is non-zero for an arbitrary y, and that furthermore Πyl.alg is
non-zero if rx 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.

2

Patching and the p-adic Langlands program for GL2 (Qp )1.3 The present paper
In this paper, we specialize the construction of [CEG+ 16] to the case F = Qp and n = 2 (so that
G := GL2 (Qp ) and K := GL2 (Zp ) 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 featuresof the GL2 (Qp ) setting beyond the use of (ϕ, Γ)-modules: the classification of irreducible mod p
representations of GL2 (Qp ) in terms of Serre weights and Hecke operators, and the fact that theWeil–Deligne representation and the Hodge–Tate weights determine a (irreducible) 2-dimensional
crystalline representation of GQp uniquely (up to isomorphism).
When combined with the results of [Paš13] (which do rely on Colmez’s functor V̌), we obtainthat 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 relationshipbetween 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 assumingthat they have
only scalar endomorphisms, so that EndGQp (r̄) = F, and that r̄ 6∼= ω0 ∗1 ⊗ χ for any characterχ : GQp → F× . For simplicity, let us assume in this introduction that r̄ is irreducible and let Rp
be its universal deformation ring. Then Rp is formally smooth over Rp . Moreover, (as F = Qp
and n = 2) we may also assume that R∞ is formally smooth over Rp , and thus over Rp .
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∞ := Rp [[x1 , . . . , xd ]] 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◦ d d
M (σ ◦ ) := Homcont
O[[K]] (M, (σ ) )factors through the action of R∞ (σ) := Rp (σ)[[x1 , . . . , xd ]]. Here Rp (σ) is the quotient of
Rp constructed by Kisin, which parameterizes crystalline representations with Hodge–Tateweights (a, b), σ ◦ is a K-invariant O-lattice in σ := Symb−a−1 E 2 ⊗ det1−a and (∗)d :=Homcont
O (∗, 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 isa non-zero G-equivariant map
πsm (rx ) ⊗ πalg (rx ) → Πl.alg
ywhere x is the image of y in Spec Rp .

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A. Caraiani et al.
The last condition says that M encodes the classical local Langlands correspondence. This iswhat 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 forthe sake of the reader familiar with Kisin’s proof of the Fontaine–Mazur conjecture [Kis09] thatthe 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 wewill ultimately be making use of patching arguments for algebraic automorphic forms on formsof 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 isirreducible.
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 GL2 (Qp ).
(ii) Let π ,→ J be an injective envelope of the above π in the category of smooth locallyadmissible representations of G on O-torsion modules. Let Pe be the Pontryagin dual of J.
Then Pe carries a unique arithmetic action of Rp and, moreover,b Rp R∞M∼= Pe ⊗
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 Πy depends only on the image of y in Spec Rp , as expected.
Let us sketch the proof of Theorem 1.7 assuming for simplicity that d = 0. The first step is toshow that M ∨ is an injective object in the category of smooth locally admissible representationsof G on O-torsion modules and that its G-socle is isomorphic to π. This is done by computing∨ ) vanishes for all irreducible F-representations π 0
HomG (π 0 , M ∨ ) and showing that Ext1G (π 0 , M∞
of G; see Proposition 4.2 and Theorem 4.15. The arguments here use the foundational resultsof 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 therings Rp (σ) are formally smooth over O, whenever σ is of the form Symb−a−1 E 2 ⊗ det1−a with
1 6 b − a 6 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 Rp are isomorphic as G-representations. Therefore, itremains to show that any two arithmetic actions of Rp on Pe coincide. As Rp is O-torsion free, it
is enough to show that two such actions induce the same action on the unitary E-Banach spaceΠ := Homcont
O (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 Rp on Π are continuous it is enough to show4

Patching and the p-adic Langlands program for GL2 (Qp )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 subspacesin 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 Rp on M (σ ◦ )[1/p] for all σ as above.
At this point we use another special feature of 2-dimensional crystalline representations of
GQp : the associated Weil–Deligne representation together with Hodge–Tate weights determine a2-dimensional crystalline representation of GQp 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) :=◦◦EndG (c-IndG
K σ) on M (σ )[1/p] completely determines the action of Rp (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 twoarithmetic actions are the same. The reduction from the case when d is arbitrary to the casewhen 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 thisproject was to find out how much of the p-adic Langlands for GL2 (Qp ) correspondence can one
recover from the patched module M∞ without using Colmez’s functors, as these constructions
are not available for groups other than GL2 (Qp ), while our patched module M∞ is. Along thesame lines, in section §5 we show that to a large extent we can recover a fundamental theoremof 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 Πy depends only on the image of yin Spec Rp . However, we are still not able to deduce using only our methods that Πy is non-zerofor an arbitrary y ∈ m-Spec R∞ [1/p]. Since M∞ is not a finitely generated module over R∞
theoretically it could happen that Πy 6= 0 for a dense subset of m-Spec R∞ [1/p], but Πy = 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 Rp on Pe to the results of [Paš13], where an actionof Rp 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 theresults of [Paš13] we show that Πy and rx 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 theimage of y in m-Spec Rp [1/p].
It follows from the construction of M∞ that after quotienting out by a certain ideal of R∞ weobtain a dual of completed cohomology, see [CEG+ 16, Corollary 2.11]. This property combinedwith 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 ρ|GQp is isomorphic
to 10 ω∗ ⊗ 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 existingideas related to the p-adic Langlands program, we have not striven for maximal generality, andwe expect that some of our hypotheses on r̄ : GQp → GL2 (Fp ) could be relaxed. In particular,

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A. Caraiani et al.
it should be possible to prove results when p = 2 by using results of Thorne [Tho] to redo thepatching 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 argumentsof [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 GL2 (F )for any finite extension F/Qp , or indeed to GLn (F ), it is natural to ask whether any of ourarguments 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 ouraxioms (AA1)-(AA4) hold, even in the generality of GLn (F ). Unfortunately, the prospects forproving analogues of our main theorems are less rosy, as it seems that none of the main inputs toour arguments will hold. Indeed, already for the case of GL2 (Qp2 ) there is no analogue availableof the classification in [Bre03a] of the irreducible F-representations of GL2 (Qp ), and it is clear
from the results of [BP12] that any such classification would be much more complicated.
Furthermore, beyond the case of GL2 (Qp ) it is no longer the case that crystalline representations are (essentially) determined by their underlying Weil–Deligne representations, so there isno possibility of deducing that a p-adic correspondence is uniquely determined by the classicalcorrespondence 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 GL2 (Qp2 )the patched module M∞ is not a projective GL2 (Qp2 )-module.
1.11 Outline of the paper
In Section 2 we recall some well-known results about Hecke algebras and crystalline deformation
rings for GL2 (Qp ). 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, andshow 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 GL2 (Qp ). It begins by showing that M∞ isa projective GL2 (Qp )-module (Theorem 4.15), before making a category-theoretic argument thatallows 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 useof (ϕ, Γ)-modules) that certain locally algebraic principal series representations admit at mostone unitary completion. Section 6 combines our results with those of [Paš13] to show that ourconstruction is compatible with Colmez’s correspondence, and as a byproduct extends someresults 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).

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Patching and the p-adic Langlands program for GL2 (Qp )1.12 Notation
We fix an odd prime p, an algebraic closure Qp of Qp , and a finite extension E/Qp in Qp , which
will be our coefficient field. We write O = OE for the ring of integers in E, $ = $E for a
uniformiser, and F := O/$ 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 ,→ Qp are contained in E.
1.12.1 Galois-theoretic notation If K is a field, we let GK denote its absolute Galois group.
Let ε denote the p-adic cyclotomic character, and ε = ω the mod p cyclotomic character. If K isa finite extension of Qp for some p, we write IK for the inertia subgroup of GK . If R is a local
ring we write mR for the maximal ideal of R. If F is a number field and v is a finite place of Fthen we let Frobv denote a geometric Frobenius element of GFv .
∼ab for the Artin map normalized
If K/Qp is a finite extension, we write ArtK : K × −→ WK
to send uniformizers to geometric Frobenius elements. To avoid cluttering up the notation, we×will use ArtQp to regard characters of Q×
p , Zp as characters of GQp , IQp respectively, without
explicitly mentioning ArtQp when we do so.

If K is a p-adic field and ρ a de Rham representation of GK over E and if τ : K ,→ E
then we will write HTτ (ρ) for the multiset of Hodge–Tate numbers of ρ with respect to τ .
bGK . Thus for
By definition, the multiset HT (ρ) contains i with multiplicity dim (ρ ⊗
K(i))
τ

E

τ,K

example HTτ (ε) = {−1}. If ρ is moreover crystalline then we have the associated filtered ϕmodule Dcris (ρ) := (ρ ⊗Qp Bcris )GK , where Bcris is Fontaine’s crystalline period ring.
1.12.2 Local Langlands correspondence Let n ∈ Z>1 , let K be a finite extension of Qp , andlet rec denote the local Langlands correspondence from isomorphism classes of irreducible smooth
representations of GLn (K) over C to isomorphism classes of n-dimensional Frobenius semisimple
Weil–Deligne representations of WK defined in [HT01]. Fix an isomorphism ı : Qp → C. We
define the local Langlands correspondence recp over Qp by ı ◦ recp = rec ◦ ı. Then rp (π) :=recp (π ⊗ | det |(1−n)/2 ) is independent of the choice of ı. In this paper we are mostly concernedwith the case that n = 2 and K = Qp .
1.12.3 Notation for duals If A is a topological O-module, we write A∨ := Homcont
O (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 Ad := Homcont
O (A, O) for its
Schikhof dual.
If F is a free module of finite rank over a ring R, then we write F ∗ := HomR (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 Ad , 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 −→ Ad ⊗R F ∗ .

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1.12.4 Group-theoretic notation Throughout the paper we write G = GL2 (Qp ) and K =GL2 (Zp ), 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 (IndG
B 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 regularaction. If χ is in fact a smooth character, then we may also form the smooth induction (IndG
B χ)sm ;G
this is the E-subspace of (IndB χ)cont consisting of smooth functions, and is a G-subrepresentationof the continuous induction.
×If χ1 and χ2 are continuous characters of Q×
p , then the character χ1 ⊗ χ2 : T → E is defined
via a0 d0 7→ χ1 (a)χ2 (d). Any continuous E-valued character χ of T is of this form, and χ issmooth if and only if χ1 and χ2 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 : GQp → GL2 (F), where as before F/Fp is a finite extension. Possibly enlarging F, we fix
a sufficiently large extension E/Qp with ring of integers O and residue field F.
We will make the following assumption from now on.
ω ∗2.2 Assumption. Assume that EndGQp (r̄) = F, and that r̄ ∼6=
0 1 ⊗ χ, for any character χ :GQp → F× .
In particular this assumption implies that r̄ has a universal deformation O-algebra Rp , and thateither 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 therepresentation theory of G := GL2 (Qp ) and K := GL2 (Zp ). Given a pair of integers a ∈ Z andb ∈ Z>0 , we let σa,b be the absolutely irreducible E-representation deta ⊗Symb E 2 of K. Notethat 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 : GQp → GL2 (Qp ) is crystalline of Hodge type σ = σa,b if it iscrystalline with Hodge–Tate weights (1 − a, −a − b),1 we write Rp (σ), for the reduced, p-torsion
free quotient of Rp 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 GL2 (Qp ).
Set σ = σa,b , and let H(σ) := EndG (c-IndG
K σ). The action of K on σ extends to the actionof G. This gives rise to the isomorphism of G-representations:G∼(c-IndG
K 1) ⊗ σ = c-IndK σ,

f ⊗ v 7→ [g 7→ f (g)σ(g)v].

The map
H(1) → H(σ),
1

φ 7→ φ ⊗ idσ

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

8

(2.4)

Patching and the p-adic Langlands program for GL2 (Qp )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 p0 p0 , with 
value p2a+b at p0 p0 , and let T ∈ H(σ) be the function supported on the double coset2 of p0 01 ,with value pa+b at p0 01 . 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 thetwo-dimensional filtered ϕ-module that has e1 , e2 as a basis of its underlying E-vector space, has
its E-linear Frobenius endomorphism ϕ being given byϕ(e1 ) = e2 , ϕ(e2 ) = −de1 + te2 ,and has its Hodge filtration given by
Fili D = D if i 6 r, Fili D = Ee1 if r + 1 6 i 6 s, and Fili 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 determineduniquely 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 GQp over E
with distinct Hodge–Tate weights (s, r) then there exists a unique pair (t, d) ∈ E × E × such that
Dcris (V ) ∼= D(r, s, t, d). Moreover, vp (d) = r + s and vp (t) > r.
Proof. This is well known, and is a straightforward computation using the fact that Dcris (V ) is
weakly admissible. For the sake of completeness, we sketch the proof; the key fact one employs isthat V 7→ Dcris (V ) is a fully faithful embedding of the category of crystalline representations of
GQp 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 e1 to be a basis for Fils Dcris (V ); the assumption that V is indecomposable implies that
Fils Dcris (V ) is not stable under ϕ, and so if we write e2 := ϕ(e1 ) then e1 , e2 is a basis for Dcris (V ),and ϕ has a matrix of the required form for a uniquely determined t and d. The asserted relations
between vp (t), vp (d), r, and s follow from the weak admissibility of Dcris (V ).
In fact, it will be helpful to state a generalisation of the previous result to the context offinite dimensional E-algebras. (Note that the definition of D(r, s, t, d) extends naturally to thecase 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 0 , and if VA is a crystalline
representation of rank two over A whose associated residual representation VE 0 := E 0 ⊗A VA
is indecomposable with distinct Hodge–Tate weights (s, r), and if Dcris (VA ) denotes the filtered
ϕ-module associated to VA , then there exists a unique pair (t, d) ∈ A × A× such that Dcris (VA ) ∼=D(r, s, t, d).
2
The function supported onPthe 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.

9A. Caraiani et al.
Proof. Choose a basis e1 for Fils Dcris (VE 0 ), and choose e1 ∈ Fils Dcris (VA ) lifting e1 . By Nakayama’s lemma, e1 generates Fils Dcris (VA ), and by considering the length of Fils Dcris (VA ) as an
E-vector space, we see that Fils Dcris (VA ) is a free A-module of rank one.
Let e2 = φ(e1 ), and write e2 for the image of e2 in Dcris (VE 0 ). As in the proof of Lemma 2.6,
e1 , e2 is a basis of Dcris (VE 0 ), and thus by another application of Nakayama’s lemma, e1 , e2 arean A-basis of Dcris (VA ). 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 GQp overE with distinct Hodge–Tate weights (s, r), for which Dcris (V ) ∼= D(r, s, t0 , d0 ), then the formal
crystalline deformation ring of V is naturally isomorphic to E[[t − t0 , d − d0 ]].
Proof. This is immediate from Lemma 2.7, and the fact that V 7→ Dcris (V ) is an equivalence of
categories.
Suppose that r̄ has a crystalline lift of Hodge type σ. By [CEG+ 16, Thm. 4.1] there isa natural E-algebra homomorphism η : H(σ) → Rp (σ)[1/p] interpolating a normalized local
Langlands correspondence rp (introduced in Section 1.12). In order to characterise this map, one
considers the composite
anη
H(σ) −→ Rp (σ)[1/p] ,→ Rp (σ) ,anwhere Rp (σ)
denotesanthe ring of rigid analytic functions on the rigid analytic generic fibre of
Spf Rp (σ). Over Rp (σ) , we may consider the universal filtered ϕ-module, and the underlying
universal Weil group representation (given by forgetting the filtration). The trace and determiannant of Frobenius on this representation are certain elements of Rp (σ)(which in fact lie in
Rp (σ)[1/p]), and η is characterised by the fact that it identifies appropriately chosen generatorsof H(σ) with this universal trace and determinant.
It is straightforward to give explicit formulas for these generators of H(σ), but we have foundit 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 ) ∈ Rp (σ)[1/p] are characterised by the following
property: if x : Rp (σ)[1/p] → Qp is an E-algebra morphism, and Vx is the corresponding twodimensional Qp -representation of GQp , then x(η(T )) = pa+b t, and x(η(S)) = p2a+b−1 d, where t,
d are respectively the trace and the determinant of ϕ on Dcris (Vx ).
Proof. Lemma 2.7 implies that there are uniquely determined t, d ∈ Qp , such that Dcris (Vx ) ∼=D(r, s, t, d), where r = −a − b and s = 1 − a. The Weil–Deligne representation associatedto D(r, s, t, d) is an unramified 2-dimensional representation of WQp , on which the geometric
Frobenius Frobp acts by the matrix of crystalline Frobenius on D(r, s, t, d), which is 10 −dt .
Thus
WD(Dcris (Vx )) = recp (χ1 ) ⊕ recp (χ2 ),×
where χ1 , χ2 : Q×p → Qp are unramified characters, such that χ1 (p)+χ2 (p) = t and χ1 (p)χ2 (p) =d.
−1/2 ∼ ιG (χ ⊗ χ ), where ιG denotes smoothIf π = (IndG= B 1
2
B |  |χ1 ⊗ χ2 )sm then π ⊗ | det |Bnormalized parabolic induction. Thenrp (π) = recp (ιG
B (χ1 ⊗ χ2 )) = recp (χ1 ) ⊕ recp (χ2 ).

10

Patching and the p-adic Langlands program for GL2 (Qp )
The action of H(1) on π K is given by sending [K p0 01 K] to p|p|χ1 (p) + χ2 (p) = t and [K p0 p0 K]
to |p|χ1 (p)χ2 (p) = p−1 d. By [CEG+ 16, Thm. 4.1] and the fact that the evident isomorphism
between π K = HomK (1, 1 ⊗ π) and HomK (σ, σ ⊗ π) is equivariant with respect to the actionsby H(1) and H(σ) via the isomorphism (2.4), we see thatx(η(T )) = p−r t = pa+b t,x(η(S)) = p−r−s d = p2a+b−1 d.

(2.10)

Since Rp (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 Rp (σ) in Rp (σ)[1/p].
Proof. It follows from (2.10) and Lemma 2.6 that for all closed points x : Rp (σ)[1/p] → Qp , wehave x(η(S)), x(η(T )) ∈ Zp . 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 GQp , is that the morphism
an(Spf R(σ))an → Spec H(σ)
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 thefollowing result.
2.12 Lemma. Let σ = σa,b with a ∈ Z and b ∈ Z>0 . Thendimκ(y) κ(y) ⊗H(σ) Rp (σ)[1/p] 6 1,

∀y ∈ m-Spec H(σ).

Proof. Let us assume that A := κ(y)⊗H(σ) Rp (σ)[1/p] is non-zero. If x, x0 ∈ m-Spec A then Frobenius on Dcris (Vx ) and Dcris (Vx0 ) 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 Dcris (Vx ) ∼= Dcris (Vx0 ) and hence x = x0 . Since D(r, s, t, d) can beconstructed over κ(y) (as t and d lie in κ(y)), so can Vx and thus κ(x) = κ(y). To complete theproof of the lemma it is enough to show that the map my → mx /m2x is surjective. Since we knowthat Rp (σ)[1/p] is a regular ring of dimension 2 by [Kis08, Thm. 3.3.8], it is enough to constructa 2-dimensional family of deformations of Dcris (Vx ) to the ring of dual numbers κ(y)[], 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 = σa,b with a ∈ Z and b ∈ Z>0 . Let y ∈ m-Spec H(σ) be the image ofx ∈ m-Spec Rp (σ)[1/p] under the morphism induced by η : H(σ) → Rp (σ)[1/p]. Then η induces
an isomorphism of completions:
∼
=
[ −→
H(σ)
Rp\
(σ)[1/p]mx .
myProof. 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 Rp (σ)[1/p] is a regular ring of dimension 2 as in the preceding proof, bothcompletions are regular rings of dimension 2. It follows from Lemma 2.12 that κ(y) = κ(x) andthe map induces a surjection on tangent spaces. Hence the map is an isomorphism.
◦ , whichIf 0 6 b 6 p − 1, then σa,b has a unique (up to homothety) K-invariant lattice σa,bis isomorphic to deta ⊗Symb O2 as a K-representation. We let σ a,b be its reduction modulo $.

11

A. Caraiani et al.
Then σ a,b is the absolutely irreducible F-representation deta ⊗Symb F2 of GL2 (Fp ); note that every(absolutely) irreducible F-representation of GL2 (Fp ) is of this form for some uniquely determineda, b with 0 6 a < p − 1. We refer to such representations as Serre weights.
If σ = σ a,b is a Serre weight with the property that r̄ has a lift r : GQp → GL2 (Zp ) that is
crystalline of Hodge type σ = σa,b , then we say that σ is a Serre weight of r̄.
Again we consider σ = σa,b with any a ∈ Z and b ∈ Z>0 . Letσ ◦ := deta ⊗ Symb O2 ,σ := deta ⊗ Symb F2

so that σ ◦ /$ = σ. We let◦H(σ ◦ ) := EndG (c-IndG
K σ ),

H(σ) := EndG (c-IndGK σ).

◦
Note that H(σ ◦ ) is p-torsion free, since c-IndG
K 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 ◦ )/$ ,→ H(σ). Furthermore, the O-subalgebra O[S ±1 , T ] of H(σ) is contained inH(σ ◦ ).
(2) If, in addition σ = σa,b with 0 6 b 6 p − 1, then O[S ±1 , T ] = H(σ ◦ ), and there is a natural
isomorphism H(σ ◦ )/$ ∼= H(σ).
◦
Proof. The isomorphism of (1) follows immediately from the fact that c-IndG
K σ is a finitely
G ◦generated O[G]-module. To see the claimed inclusion, apply HomG (c-IndK σ , −) to the exact
sequence$◦
G ◦G
0 → c-IndG
K σ → c-IndK σ → c-IndK σ → 0

so as to obtain an injective map◦
G∼H(σ ◦ )/$ ,→ HomG (c-IndG
K σ , c-IndK σ) = H(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) ' HomK (σ, c-IndG
K σ);
the image of φ ∈ H(1) under the composite map sends v ∈ σ to the function g 7→ 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 HomK (σ ◦ , c-IndG
K 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 ◦ )/$ → H(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 thesecond claim of (2). Furthermore, this shows that the inclusion O[S ±1 , T ] ,→ H(σ ◦ ) of (1) becomes an isomorphism both after reducing modulo $ as well as after inverting $ (because H(σ)
is generated by S ±1 and T by Proposition 2.5). Thus it is an isomorphism, completing the proofof (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 σ = σ a,b be a Serre weight of r̄ then the following hold:(i) The deformation ring Rp (σ) is formally smooth of relative dimension 2 over O.
(ii) The morphism of E-algebras η : H(σ) → Rp (σ)[1/p] induces a morphism of O-algebrasH(σ ◦ ) → Rp (σ).

12Patching and the p-adic Langlands program for GL2 (Qp )(iii) The character ω 1−2a−b det r̄ is unramified, and if we let– µ = (ω 1−2a−b det r̄)(Frobp ), and– if r̄ is irreducible, then λ = 0, and


χ1
∗
a+b
∼
– if r̄ is reducible, then we can write r̄ = ω⊗
for unramified characters
0 χ2 ω −b−1
χ1 , χ2 , and let λ = χ1 (Frobp ),then the composition
α : H(σ ◦ ) → Rp (σ) → F
maps T 7→ λ and S 7→ µ.
◦ ) be the completion of H(σ ◦ ) with respect to the kernel of α. Then the map\(iv) Let H(σ
∼
=
◦) →
\
H(σ ◦ ) → Rp (σ) induces an isomorphism of local O-algebras H(σRp (σ). In coordinates,e
we have Rp (σ) = O[[S − µe, T − λ]], where the tilde denotes the Teichmüller lift.

(v) If we set π := (c-IndG
K σ) ⊗H(σ),α F, then π is an absolutely irreducible representation of G,
and is independent of the choice of Serre weight 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 theseweights 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
!
n+1
0
ω
2
r̄|IQp ∼= ω m−1 ⊗p(n+1) ,0
ω2
where ω2 is a fundamental character of niveau 2, and 0 6 m < p − 1, 0 6 n 6 p − 2. Then the
Serre weights of r̄ are σ m,n and σ m+n,p−1−n (with m + n taken modulo p − 1). If r̄ is reducible,
then we may write


1
∗
m+n
∼
r̄|IQp = ω,⊗
0 ω −n−1where 0 6 m < p − 1, 0 6 n < p − 1. Then (under Assumption 2.2) if n 6= 0, the unique Serreweight of r̄ is σ m,n , while if n = 0, then σ m,0 and 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 σ is not of the form σ a,p−1 , the claims about Rp (σ) are a standard consequenceof (unipotent) Fontaine–Laffaille theory; for example, the irreducible case with O = Zp 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 anddeterminant of ϕ, by Lemma 2.6. Concretely, if r̄ is irreducible, then the crystalline lifts of r̄
of Hodge type σa,b correspond exactly to the weakly admissible modules D(−(a + b), 1 − a, t, d)where vp (t) > −a − b and p2a+b−1 d = µ. The claimed description of the deformation ring thenfollows.
Similarly, if r̄ is reducible, then it follows from Fontaine–Laffaille theory (and Assumption 2.2)that any crystalline lift of Hodge type 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 vp (t) = −a − b, pa+b t = λ, and p2a+b−1 d = µ.
This leaves only the case that σ is of the form σ a,p−1 . In this case the result is immediatefrom the main result of [BLZ04], which shows that the above description of the weakly admissible
modules continues to hold.

13

A. Caraiani et al.
Finally, (5) is immediate from the main results of [BL94, Bre03a], together with the explicit
description of 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 σ is [Bre03a, Thm.
1.3]. If r̄ is reducible and has only a single Serre weight, then the absolute irreducibility of πis [BL94, Thm. 33(2)]. In the remaining case that r̄ is reducible and has two Serre weights, thenAssumption 2.2 together with the explicit description of λ, µ above implies that λ2 6= µ, and theabsolute irreducibility of π is again [BL94, Thm. 33(2)]. The independence of π of the choice of σis [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 Serreweight 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) Rp 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 GL2 (Qp ). 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 GL2 (Qp ) in global settings other than those considered in [Eme11]
or [CEG+ 16]).
3.1 Axioms
b O O[[x1 , . . . , xd ]]. Then an O[G]-module with an arithFix an integer d > 0, and set R∞ := Rp ⊗metic 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 = σa,b . Setd◦ d
M∞ (σ ◦ ) := Homcont,
O[[K]] (M∞ , (σ ) )where we are considering continuous homomorphisms for the profinite topology on M∞ and thep-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) := Rp (σ)[[x1 , . . . , xd ]]. 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].

14

Patching and the p-adic Langlands program for GL2 (Qp )(AA4) For any σ, the action of H(σ) on M∞ (σ ◦ )[1/p] is given by the compositeηH(σ) → Rp (σ)[1/p] → Rp (σ)[[x1 , . . . , xd ]][1/p],ηwhere H(σ) → Rp (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 GLn (K) for any
finite extension K/Qp ; however, our arguments in the rest of the paper will only apply to thecase of GL2 (Qp ).)In these examples, axioms (AA1) and (AA2) essentially follow from the facts that spaces ofautomorphic forms are finite-dimensional, and that the cohomology of zero-dimensional Shimuravarieties is concentrated in a single degree (degree zero). Axioms (AA3) and (AA4) come fromthe existence of Galois representations attached to automorphic forms on unitary groups, andfrom 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 thep-adic Langlands correspondence in terms of unitary completions of locally algebraic vectors; seealso 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) := Rp (σ)[[x1 , . . . , xd ]]. If y is a maximal idealof R∞ (σ)[1/p] which lies in the support of M∞ (σ ◦ )[1/p], then we writeΠy := Homcont
M∞ ⊗R∞ ,y Oκ(y) , E .
O
We write x for the corresponding maximal ideal of Rp (σ)[1/p], rx for the deformation of r̄ corresponding to x, and set πsm (rx ) := rp−1 (WD(rx )F −ss ), which is the smooth representation of G
corresponding to the Weil–Deligne representation associated to rx by the classical Langlands correspondence rp (normalised as in Section 1.12). We write πalg (rx ) for the algebraic representationof 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
πsm (rx ) ⊗ πalg (rx ) → Πl.alg
y .
That (AA1)-(AA4) imply (AA4’) is a straightforward consequence of the defining property ofthe map η. Conversely, assume (AA1)-(AA3) and (AA4’), and write
R∞ (σ) := R∞ (σ)/Ann(M∞ (σ ◦ )).
It follows from (AA3) that the natural map R∞ (σ)[1/p] → EndR∞ (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 0 : H(σ) → 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 0 agree modulo every
maximal ideal of Rp (σ)[1/p] in the support of M∞ (σ ◦ ). It follows from (AA3) that 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 R∞ (σ)[1/p] is reduced, thus the intersection of all maximal ideals is equal to zero. Hence(AA4) holds.

15

A. Caraiani et al.
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 GL2 (Qp ). (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 groupsof 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 thestart 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 theweight 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 choiceof 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 Rp , so all ofthese rings are absorbed into the power series ring O[[x1 , . . . , xd ]]. The module M∞ is patchedfrom 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 σ = σ a,b be a Serre weight, and let σ ◦ be a K-stable O-lattice in σa,b , so that∨ ∨σ ◦ /$σ ◦ = σ. We define M∞ (σ) = Homcont
O[[K]] (M∞ , (σ) ) , so that by (AA2) we have M∞ (σ) =◦◦M∞ (σ )/$M∞ (σ ). By definition, the deformation ring Rp (σ) = Rp (σa,b ) is non-zero if and only
if σ is a Serre weight of r̄. Set R∞ (σ) = Rp (σ)[[x1 , . . . , xd ]].
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 ◦ ) 6= 0 if and only if σ is a Serre weight of r̄, in whichcase M∞ (σ ◦ ) is a free R∞ (σ)-module of rank one.
(ii) If σ is a Serre weight of r̄ then the action of H(σ) on M∞ (σ) factors through the natural
map Rp (σ)/$ → R∞ (σ)/$, and M∞ (σ) is a flat H(σ)-module.
(iii) If π 0 is an irreducible smooth F-representation of G then we have∨
HomG (π 0 , M∞
) 6= 0if and only if π 0 is isomorphic to π.
Proof. It follows from (AA3) that M∞ (σ ◦ ) 6= 0 only if Rp (σ)[1/p] 6= 0, which is equivalent toσ being a Serre weight of r̄. In this case, since σ = σa,b with 0 6 b 6 p − 1, R∞ (σ) is formallysmooth over O by Lemma 2.15, so it follows from (AA3) and the Auslander–Buchsbaum theorem

16Patching and the p-adic Langlands program for GL2 (Qp )that M∞ (σ ◦ ) is a free R∞ (σ)-module of finite rank, and that M∞ (σ ◦ )[1/p] is a locally freeR∞ (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 ◦ ) 6= 0 if and only if M∞ (σ) 6= 0, so we may assume that M∞ (σ ◦ ) 6= 0.
The first part of (2) follows from (AA4) together with Lemmas 2.15 (2) and 2.14 (2). For theremaining part of (2), note that Rp (σ)/$ is flat over H(σ) by Lemmas 2.14 and 2.15, and M∞ (σ)
is flat over Rp (σ)/$ 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 irreducible0
π as we may enlarge the field F. Let us assume that π 0 is absolutely irreducible and let σ 0 be anirreducible representation of K contained in the socle of π 0 . It follows from [BL94, Bre03a] that00
the surjection c-IndGK σ  π factors through the map
00(c-IndG
K σ ) ⊗H(σ 0 ),α0 F  π(4.3)

where α0 : H(σ 0 ) → F is given by the action of H(σ 0 ) on the one dimensional F-vector space
HomK (σ 0 , π 0 ). Moreover, (4.3) is an isomorphism unless π 0 is a character or special series. Since
∨ ∨ ∼
0
∨ ∨
) = HomG (c-IndG
M∞ (σ 0 ) ∼= HomK (σ 0 , M∞
K σ , M∞ ) ,from (4.3) we obtain a surjection of R∞ (σ 0 )-modules
0
∨ ∨
0
∨ ∨
M∞ (σ 0 ) ⊗H(σ0 ),α0 F ∼= HomG (c-IndGK σ ⊗H(σ 0 ),α0 F, M∞ )  HomG (π , M∞ ) ,∨)which moreover is an isomorphism if π 0 is not a character or special series. Thus if HomG (π 0 , M∞
is non-zero then we deduce from the previous displayed expression that M∞ (σ 0 ) ⊗H(σ0 ),α0 F 6= 0.
In particular, M∞ (σ 0 ) 6= 0 and hence σ 0 is a Serre weight for r̄ by the only if part of (1).
0
We claim that α0 coincides with the morphism α of Lemma 2.15 (3) (with σ in place of σ). To
0
see this, note that by the only if part of (1), we have that Rp (σ )/$ ⊗H(σ0 ),α0 F 6= 0, and hence
0) ⊗
0
\
\
0
by Lemma 2.15 (4), we find that H(σ0 F 6= 0, where H(σ ) denotes the completion ofH(σ ),αH(σ 0 ) with respect to the kernel of the morphism α. This proves that α and α0 coincide.
0Part (5) of Lemma 2.15 now implies that π ∼= (c-IndG
K σ ) ⊗H(σ 0 ),α0 F. Hence, (4.3) gives us aG-equivariant surjection π  π 0 , which is an isomorphism as π is irreducible.
Conversely, it follows from from (AA1) that there is an irreducible smooth F-representation π 0∨ ) is non-zero; we have just seen that π ∼ π 0 , so that Hom (π, M ∨ ) 6=
of G such that HomG (π 0 , M∞=G∞0, as required.
Finally, suppose that σ is a Serre weight of r̄. Then as above we have an isomorphism of
R∞ (σ)-modules
M∞ (σ) ⊗H(σ),α F ∼= HomG (π, M ∨ )∨ 6= 0∞

so that M∞ (σ) 6= 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 Modsm
G (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 =V H [mn ],H,nwhere the union is taken over open compact subgroups H ⊂ G and over positive integers n.

17

A. Caraiani et al.
Hn
We say that an object V of Modsm
G (R) is admissible if V [m ] is a finitely generated Rmodule for every compact open subgroup H ⊂ G and every n > 1. Moreover, V is called locallyadmissible if, for every v ∈ V , the smallest R[G]-submodule of V containing v is admissible.
We let Modl.adm(R) denote the full subcategory of Modsm
G
G (R) consisting of locally admissible
representations.
l.adm
The categories Modsm(R) are abelian (see [Eme10] for the second one) andG (R) and ModG
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 0 ⊂ M , ι(N ) ∩ M 0 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
l.adm
of [Paš13], the category Modsm(R) also
G (R) admits injective envelopes. The category ModG
admits injective envelopes. (This follows from [Paš10, Lem. 3.2] and the fact that the inclusionof Modl.adm(R) into Modsm
G
G (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 socG (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 socG (V ). Since socG (V ) ∩ W = socG (W ), it suffices to show thatany 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 0 → 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∞∨ is an injective locally admissible representation of G.
Our first aim is to show that M∞∨ is an admissible object of Modsm (R ), and thus lies in Modl.adm (O).
4.9 Lemma. M∞∞
GG

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(σ) with residue field κ(m). Then
H(σ)

Tori

(c-IndG
K σ, κ(m)) = 0,∀i > 0.

Proof. Since the map F → F is faithfully flat, we can and do assume that F is algebraicallyclosed. Since H(σ) = F[S ±1 , T ], we have m = (S − µ, T − λ) for some µ ∈ F× , λ ∈ F. Since thesequence S − µ, T − λ is regular in H(σ), the Koszul complex K• associated to it is a resolution18

Patching and the p-adic Langlands program for GL2 (Qp )of κ(m) by free H(σ)-modules, [Mat89, Thm. 16.5 (i)]. Thus the complex K• ⊗H(σ) c-IndG
Kσcomputes the Tor-groups we are after, and to verify the claim it is enough to show that thesequence S − µ, T − λ is regular on c-IndGK σ.
GIf f ∈ c-IndK σ then (Sf )(g) = f (gz), where z = p0 p0 . Since such an f is supported only on
finitely many cosets K\G, we deduce that the map
S−µG
c-IndGK σ −→ c-IndK σ

is injective. The quotient is isomorphic to c-IndG
ZK σ, where z acts on σ by µ. It follows from theproof of [BL94, Thm. 19] that c-IndG
σ
is
a
free
F[T ]-module. Thus the map
ZK
T −λ

G
c-IndGZK σ −→ c-IndZK σ

is injective, and the sequence S − µ, T − λ is regular on c-IndG
K σ, as required.
4.11 Lemma. Let m be a maximal ideal of H(σ). Then∨
ExtiG κ(m) ⊗H(σ) c-IndG
K σ, M∞ = 0,∀i > 1,

where the Ext-groups are computed in Modsm
G (O).
•∨∨
Proof. We first prove that ExtiG c-IndG
K σ, M∞ = 0, for all i > 1. Let M∞ ,→ J be an injectivesm∨
resolution of M∞ in ModG (O). Since
HomK (τ, J|K ) ∼= HomG (c-IndG τ, J)

K
G
and the functor c-IndK is exact, the restriction of an injective object in Modsm
G (O) to K issm•∨injective in ModK (O). Thus (J )|K is an injective resolution of M∞ |K in Modsm
K (O). Since• ) ∼ Hom (σ, (J • )| ), we conclude that we have natural isomorphisms
HomG (c-IndG
σ,
J
=
K
K
K



∨ ∼
i
∨
ExtiG c-IndG
K σ, M∞ = ExtK σ, M∞ ,∀i > 0.

∨ is injective in Modsm (O), and thus the
Since M∞ is a projective O[[K]]-module by (AA2), M∞
K
Ext-groups vanish as claimed.
Let F•  κ(m) be a resolution of κ(m) by finite free H(σ)-modules. Lemma 4.10 implies thatG
the complex F• ⊗H(σ) c-IndG
K σ is a resolution of κ(m) ⊗H(σ) c-IndK σ by acyclic objects for the∨ ). We conclude that the cohomology of the complex
functor HomG (∗, M∞∨
HomG (F• ⊗H(σ) c-IndG
K σ, M∞ )∨computes the groups ExtiG κ(m) ⊗H(σ) c-IndG
K σ, M∞ . We may think of the transition maps
∨ )∨ transposes these matrices
in F• as matrices with entries in H(σ). The functor HomG (∗, M∞
twice, thus we get an isomorphism of complexes:
HomG (F• ⊗H(σ) c-IndG σ, M ∨ )∨ ∼= F• ⊗H(σ) HomG (c-IndG σ, M ∨ )∨ ∼= F• ⊗H(σ) M∞ (σ).
K

∞

K

∞

The above isomorphism induces a natural isomorphism

 ∨

H(σ)
∨
∼
ExtiG κ(m) ⊗H(σ) c-IndGσ,M
κ(m), M∞ (σ) ,= ToriK∞∀i > 0.

The isomorphism implies the assertion, as M∞ (σ) is a flat H(σ)-module by Proposition 4.2 (2).
4.12 Lemma. Let y : H(σ) → F0 be a homomorphism of F-algebras, where F0 is a finite field
extension of F. Let λ0 := F0 ⊗H(σ),y c-IndG
K σ and let λ be an absolutely irreducible F-representationof G, which is either principal series or supersingular. If λ is a subquotient of λ0 then λ0 is
isomorphic to a direct sum of finitely many copies of λ.

19

A. Caraiani et al.
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:(4.13)
HomG (A, B) ⊗F F ∼= HomG (A ⊗F F, B ⊗F F),
where F denotes the algebraic closure of F, see [Paš13, Lem. 5.1]. ThenM∼
F ⊗F λ0 ∼F ⊗H(σ),ι◦y c-IndG= F ⊗F F0 ⊗H(σ),y c-IndG
Kσ =
K σ,

(4.14)

ι:F0 →F

where the sum is taken over F-algebra homomorphisms ι : F0 → F. By the classification theoremsof Barthel–Livne [BL94] and Breuil [Bre03a], each representation F ⊗H(σ),ι◦y c-IndG
K σ is either
irreducible, an extension of a special series by a character, or an extension of a character by aspecial series. Since λ is a subquotient of λ0 by assumption, λ ⊗F F is a subquotient of λ0 ⊗F F,
and since λ is neither special series nor a character, we deduce thatλ ⊗F F ∼= F ⊗H(σ),ι◦y c-IndG σ,
K

for some embedding ι : F0 ,→ F. For every τ ∈ Gal(F/F), we haveG∼∼∼
F ⊗H(σ),τ ◦ι◦y c-IndG
K σ = F ⊗F,τ (F ⊗H(σ),ι◦y c-IndK σ) = F ⊗F,τ (F ⊗F λ) = F ⊗F λ.

Hence all the summands in (4.14) are isomorphic to λ ⊗F F. It follows from (4.13) that λ0 is
isomorphic to a direct sum of copies of λ, as required.
∨ is an injective object in Modl.adm (O).
4.15 Theorem. M∞
G∨ ,→ J be an injective envelope of M ∨ in Modl.adm (O). Lemma 4.6 shows that
Proof. Let M∞∞
G∨ ,→ M ∨ ,→ J is an essential monomorphism, and thus induces an
the composition socG M∞∞∨ and soc J. Proposition 4.2 (3) shows that soc M ∨ , and thus
isomorphism between socG M∞
GG∞also socG J, is isomorphic to a direct sum of copies of the representation π associated to r̄ via
Lemma 2.15 (5).
∨ is non-zero; then there is a smooth irreducible KLet us assume that the quotient J/M∞∨
∨ generated by σ. Since
subrepresentation σ ⊂ J/M∞ . Let κ be the G-subrepresentation of J/M∞∨
J/M∞ is locally admissible, and σ is finitely generated as a K-representation, κ is an admissible∼
representation of G. Thus HomG (c-IndG
K σ, κ) = HomK (σ, κ) is a finite dimensional F-vector
space.
Let m be any irreducible H(σ)-submodule of HomG (c-IndG
K σ, κ). Since m is finite dimensional0over F, Schur’s lemma implies that F := EndH(σ) (m) is a finite dimensional division algebra over
F. Since F is a finite field we deduce that F0 is a finite field extension of F. Since H(σ) is
commutative we further deduce that m is a one dimensional F0 -vector space and thus obtain asurjective homomorphism of F-algebras y : H(σ)  F0 . Moreover, by the construction of m, we
obtain a non-zero G-equivariant map:∨
π 0 := F0 ⊗H(σ),y c-IndG
K σ → κ ⊂ J/M∞ .
∨ ) = 0 by Lemma 4.11, by applying Hom (π 0 , ∗) to the exact sequence 0 →Since Ext1G (π 0 , M∞
G∨∨
M∞ → J → J/M∞ → 0, we obtain a short exact sequence∨∨
0 → HomG (π 0 , M∞
) → HomG (π 0 , J) → HomG (π 0 , J/M∞
) → 0.
∨ ) is non-zero. Hence, Hom (π 0 , J) is non-zero.
Moreover, we know that HomG (π 0 , J/M∞
G00
Fix a non-zero G-equivariant map ϕ : π → J; then ϕ(π ) ∩ socG J 6= 0. Since socG J isisomorphic to a direct sum of copies of π, we find that π is an irreducible subquotient of π 0 . It

20

Patching and the p-adic Langlands program for GL2 (Qp )follows from Lemma 4.12 that π 0 is then isomorphic to a finite direct sum of copies of π, and so∨ ,→ J induces an isomorphismin particular is semi-simple. As we’ve already noted, the map M∞∨
0∨∼socG M∞ = socG J, and so the map HomG (π , M∞ ) → HomG (π 0 , J) is an isomorphism. This∨ ) = 0, contradicting the assumption J/M ∨ 6= 0. Hence M ∨ = J is
implies HomG (π 0 , J/M∞∞∞
injective, as required.
4.16 Removing the patching variables
We now show that we can pass from M∞ to an arithmetic action of Rp on a projective envelopeof 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 Modl.adm(A), where,Gfor 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 = GL2 (Qp ).
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  cosocC(F) P is essential. Assume
that all irreducible subquotients of cosocC(F) P are isomorphic to some given object S, for which
EndC(F) (S) = F. If HomC(F) (P, S)∨ is a free A-module of rank 1 then there is an isomorphismb F Proj(S) ∼A⊗= 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 sumof copies of λ := S ∨ . This means that we have natural isomorphisms:(cosoc P )∨ ∼= λ ⊗F HomG (λ, (cosoc P )∨ ) ∼= λ ⊗F HomC(F) (cosoc P, S).
Taking Pontryagin duals we get a natural isomorphism in C(F):bF S ∼
HomC(F) (cosoc P, 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):b F S.
P  HomC(F) (cosoc P, S)∨ ⊗The surjection Proj(S)  S induces a surjectionb F Proj(S)  HomC(F) (cosoc P, S)∨ ⊗b F S,
HomC(F) (cosoc P, S)∨ ⊗with trivial A-action on Proj(S). The source of this surjection is projective in C(A), since ingeneral for a compact A-module mb F Proj(S), −) ∼HomC(A) (m ⊗(4.18)= HomA (m, HomC(F) (Proj(S), −)),and HomC(F) (cosoc P, S)∨ = HomC(F) (P, S)∨ is projective since it is a free A-module of rank 1.
b F Proj(S) → P in C(A), such that the diagram
Hence there is a map HomC(F) (cosoc P, S)∨ ⊗/P

b F Proj(S)HomC(F) (cosoc P, S)∨ ⊗bF S
HomC(F) (cosoc P, S)∨ ⊗∼=/ 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), andhence also an isomorphism in C(A).

21

A. Caraiani et al.
4.19 Proposition. Let A be a complete local noetherian O-algebra with residue field F, thatis O-flat. Let P in C(A) be such that P is projective in C(O) and the map P  cosocC(O) P
is essential. Assume that all the irreducible subquotients of cosocC(O) P are isomorphic to S,
and EndC(O) (S) = F. If HomC(O) (P, S)∨ is a free A/$A-module of rank 1 then there is an
isomorphism in C(A):b O Proj(S) ∼A⊗= P,
where Proj(S)  S is a projective envelope of S in C(O).
b O Proj(S) and P modulo
Proof. A special case of Lemma 4.17 implies that the reductions of A ⊗b$ are isomorphic in C(F). Arguing as in (4.18) we deduce that A ⊗O Proj(S) is projective in
C(A). Thus there is a map in C(A),b O Proj(S) → P,
A⊗
which is an isomorphism modulo $. If V is a cokernel of this map then V /$V = 0, andNakayama’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 $, if U is the kernel of this map thenb O Proj(S) ∼the decomposition A ⊗= U ⊕P implies that U/$U = 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 homob A,x O is a projective envelope of S in C(O).
morphism x : A → O, P ⊗Proof. It follows from Proposition 4.19 thatb A,x O ∼b O A) ⊗b A,x O ∼P⊗
= (Proj(S) ⊗= Proj(S).
4.21 Remark. If we replace the assumption in Proposition 4.19 that HomC(O) (P, S)∨ is a free
A/$A-module of rank 1 with the assumption that it is free of rank n, then we have an isomorb O Proj(S) ∼phism A⊕n ⊗= P in C(A).
Indeed, a generalization of Lemma 4.17 to the rank n case gives an isomorphism in C(F) of⊕nb O Proj(S) and P modulo $. (In fact, the statement of Lemma 4.17 can be strengthenedA ⊗as follows: if HomC(F) (P, S)∨ is a projective object in the category of compact A-modules, thenb F Proj S ' P . The key is again the projectivity of
we have an isomorphism HomC(F) (P, S)∨ ⊗b F Proj S, which follows from our assumption andthe completed tensor product HomC(F) (P, S)∨ ⊗from (4.18).) We then upgrade the isomorphism modulo $ 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 = GL2 (Qp ).
4.22 Proposition. Let A = O[[x1 , . . . , xd ]] and choose a homomorphism of local O-algebrasbO A ∼A → R∞ , that induces an isomorphism Rp ⊗= R∞ . Then there is an isomorphism in C(A):b O A,
M∞ ∼= Pe ⊗
where Pe  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∨ is locally admissible, it follows from Lemma 4.6 that soc M ∨ ,→ M ∨that theorem, since M∞
G∞∞
is essential, and hence that M∞  cosocC(O) M∞ is essential. Proposition 4.2 (3) implies that
all the irreducible subquotients of cosocC(O) M∞ are isomorphic to π ∨ . It is therefore enough toshow that∼ HomG (π, M ∨ )∨M∞ (π) := HomC(O) (M∞ , π ∨ )∨ =∞22

Patching and the p-adic Langlands program for GL2 (Qp )is a free A/$-module of rank 1, since the assertion then follows from Proposition 4.19.
As in the proof of Proposition 4.2, we haveM∞ (π) ∼= F ⊗H(σ◦ ) M∞ (σ ◦ ).
= F ⊗H(σ) M∞ (σ) ∼It follows from Proposition 4.2 (1) that M∞ (π) is a free F ⊗H(σ◦ ) R∞ (σ)-module of rank 1. Sinceb O A and the map H(σ) → R∞ (σ)/$ factors through Rp (σ)/$ by ProposiR∞ (σ) ∼= Rp (σ) ⊗tion 4.2 (2), we conclude that the map A → R∞ induces an isomorphism
A/$ ∼= F ⊗H(σ◦ ) R∞ (σ).
(Recall that F ⊗H(σ◦ ) Rp (σ)/$ = F, by Lemma 2.15 (4).) Thus M∞ (π) is a free A/$-module ofrank 1, as required.
4.23 Corollary. Let A → R∞ be as in Proposition 4.22 and let x : A → O be a homomorphismb A,x O is a projective envelope of π ∨ in C(O) with a continuousof local O-algebras. Then M∞ ⊗Rp ∼= 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 Pe  π ∨ be a projective envelope of π ∨ in C(O).
4.25 Proposition. Pe can be endowed with an arithmetic action of Rp (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 Rp on
b A,x O ∼M∞ ⊗= Pe. Since the action of R∞ on M∞ is an arithmetic action, it follows immediatelyfrom the definitions that this induced action of Rp on Pe 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 ofinterest to us. We note that analogues of these results are valid for general choices of G, and inparticular do not use either Colmez’s functor nor p-adic Langlands correspondence for GL2 (Qp ).
Let M be a compact linear-topological O[[K]]-module, and let {Vi }i∈I be a set of continuous
K-representations on finite-dimensional E-vector spaces.
4.27 Definition. We say that {Vi }i∈I captures M if for any proper quotient M  Q, we havecont∗∗
Homcont
O[[K]] (M, Vi ) 6= HomO[[K]] (Q, Vi ) for some i ∈ I.
This definition is used only in the proof of the following result.
cont
e
e ∗
4.28 Proposition. Suppose that φ ∈ Endcont
O[[K]] (P ) kills each HomO[[K]] (P , σa,b ) for a ∈ Z andb ∈ Z>0 . Then φ = 0.

Proof. Since Pe is projective in C(O), it follows from [CDP14, Prop. 2.12] that the set {σa,b }captures Pe. The result follows from [CDP14, Lem. 2.9] (that is, from an application of thedefinition of capture to the cokernel of φ).
de, (σ ◦ )d ) .
Set M (σ ◦ ) := Homcont(
PO[[K]]23

A. Caraiani et al.
4.29 Lemma. Let σ = σa,b with a ∈ Z and b ∈ Z>0 and let my be a maximal ideal of H(σ). Thenκ(y) ⊗H(σ) M (σ ◦ )[1/p] 6= 0 if and only if mx := η(my )Rp (σ)[1/p] is a maximal ideal of Rp (σ)[1/p]in the support of M (σ ◦ )[1/p] for some (equivalently, any) arithmetic action of Rp on Pe.
Proof. Since M (σ ◦ ) is a finitely generated Rp (σ)-module and the action of H(σ) on M (σ ◦ )[1/p]factors through the action of Rp (σ)[1/p] via η, we deduce that κ(y) ⊗H(σ) M (σ ◦ )[1/p] is a finitely
generated κ(y) ⊗H(σ) Rp (σ)[1/p]-module. If κ(y) ⊗H(σ) M (σ ◦ )[1/p] 6= 0 then we deduce from
Lemma 2.12 that mx is a maximal ideal of Rp (σ)[1/p] in the support of M (σ ◦ ). Conversely, if yis the image of x then using Lemma 2.12 we obtain κ(x) = κ(y) ⊗H(σ) Rp (σ)[1/p] and hence:κ(x) ⊗Rp (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 Rp on Pe.
Proof. The existence of such an action follows from Proposition 4.25.
Let σ = σa,b with a ∈ Z and b ∈ Z>0 . Let y ∈ m-Spec H(σ), such that κ(y)⊗H(σ) M (σ ◦ )[1/p] 6=
0. Lemma 4.29 implies that y is the image of x ∈ m-Spec Rp (σ)[1/p], which lies in the supportof M (σ ◦ ). Proposition 2.13 implies that◦ )[1/p]◦\\M (σmy = M (σ )[1/p]mx ,◦ )[1/p]\as Rp\(σ)[1/p]mx -modules. Moreover, the action of Rp\(σ)[1/p]mx on M (σmy does not depend
on a given arithmetic action of Rp on Pe, 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:
Y
Y
cm ,
M ,→
Mm ,→
M
m

m

where the product is taken over all the maximal ideals in R. In fact it is enough to take theproduct over finitely many maximal ideals: if p1 , . . . , pn are minimal associated primes of M ,just pick any maximal ideals m1 ∈ V (p1 ), . . . , mn ∈ V (pn ). (The second injection follows from
[Mat89, Thm. 8.9].) This observation applied to R = Rp (σ)[1/p] and M = M (σ ◦ )[1/p] together
with Lemma 4.29 implies that we have an injection of Rp (σ)[1/p]-modules:
Y◦ )[1/p] .
\M (σ ◦ )[1/p] ,→M (σmyy∈m-Spec H(σ)

Since the map and the action of Rp (σ)[1/p] on the right hand side are independent of thearithmetic action of Rp on Pe, we deduce that the action of Rp (σ)[1/p] on M (σ ◦ )[1/p] is also
independent of the arithmetic action of Rp on Pe.
If θ : Rp → EndC(O) (Pe) and θ0 : Rp → EndC(O) (Pe) are two arithmetic actions and r ∈ Rp◦ ) for all a ∈ Z and b ∈ Z .
then it follows from the above that θ(r) − θ0 (r) will annihilate M (σa,b>0Proposition 4.28 implies that θ(r) = θ0 (r).
4.31 Remark. A different proof of the Theorem could be given using [CDP14, Prop. 2.19].
4.32 Theorem. If Pe  π ∨ is a projective envelope of π ∨ in C(O) equipped with an arithmetic
action of Rp , then there is an isomorphism in C(R∞ )b Rp R∞ ∼Pe ⊗= M∞ .

24

Patching and the p-adic Langlands program for GL2 (Qp )Proof. Let A = O[[x1 , . . . , xd ]], and choose a homomorphism of local O-algebras A → R∞ whichbO A ∼
induces an isomorphism Rp ⊗= R∞ . Proposition 4.22 implies that there is an isomorphism∼eb
P ⊗O A = M∞ in C(A); it is therefore enough to show that this isomorphism is Rp -linear. Anyb O A) ⊗b A,x O ∼b A,x O
O-algebra homomorphism x : A → O induces an isomorphism (Pe ⊗= M∞ ⊗bin C(O). We get two actions of Rp on M∞ ⊗A,x O: one of them coming from the action of Rp
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 actionsb O A) ⊗b A,x O ∼b A,x O is Rp -linear.
coincide; thus the isomorphism (Pe ⊗= M∞ ⊗We have a commutative diagram in C(A):
∼
=

bO A
Pe ⊗

Q
x:A→O



b O A) ⊗b A,x O
(Pe ⊗

∼
=

/

/ M∞

Q
x:A→O



b A,x OM∞ ⊗where the product is taken over all O-algebra homomorphisms x : A → O.
Q We know that both
O is injective (seevertical and the lower horizontal arrows are Rp -linear. The map A →x:A→Ob O − is exact, hence
for example [Paš13, Lem. 9.22]). Since Pe is O-torsion free the functor Pe ⊗the first vertical arrow is injective, which implies that the top horizontal arrow is Rp -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 GL2 (Qp ). More precisely, the isomorphismb Rp R∞ ∼Pe ⊗= M∞
exhibits M∞ as the extension of scalars of Pe, and this latter object, with its arithmetic Rp -action,is independent of all choices by Theorem 4.30. Thus, the only ambiguity in the constructionof 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, andas 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 thatthe 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 theprevious sections, namely the projective envelope Pe and the purely local map Rp → EndC(O) (Pe),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 unitarycompletion satisfying certain properties, and that such a completion comes from some Pe.
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 = σa,b = deta ⊗Symb E 2 . Let θ : H(σ) → E be a homomorphism,and set Ψ := (c-IndG
K σ) ⊗H(σ),θ E; so Ψ is a locally algebraic principal series representation of G.

25

A. Caraiani et al.
5.1 Theorem. If Ψ is irreducible, then Ψ admits at most one non-zero admissible unitaryb with the following property: for an open bounded G-invariant lattice Θ in Ψ,bcompletion ΨG−1×(Θ/$) ⊗F Fp contains no subquotient of the form (IndB χ ⊗ χω )sm , for any character χ : Qp →×

Fp , 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 inthe 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 0 , since if Π ⊗E E 0 is an absolutely
irreducible E 0 -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
Π1 . If we let Π0 denote the quotient Π/Π1 , then we must show that Π0 is zero. For the moment,
we note simply that if Π0 is non-zero, then since the composite Ψ → Π → Π0 has dense image,
we see that Π0 is another non-zero admissible unitary completion of Ψ.
Let Θ be an open bounded G-invariant lattice in Π, and let Θ1 := Π1 ∩ Θ. Since Π1 is also
admissible, Θ1 /$ will contain an irreducible subquotient π. Since we are allowed to enlarge E, we
may assume that π is absolutely irreducible. Let Pe  π ∨ be a projective envelope of π ∨ in C(O).
By the assumption on the subquotients of Θ/$, there is a Galois representation r̄ satisfying
Assumption 2.2 that corresponds to π via Lemma 2.15 (5). We equip Pe with the arithmetic
action of Rp provided by Theorem 4.30.
We let (for any admissible unitary E-Banach space representation Π of G)
cont ed
M (Π) := HomC(O) (Pe, Θd )d [1/p] ∼= HomcontG (Π, HomO (P , E)) .

The projectivity of Pe implies that Π 7→ M (Π) is an exact covariant functor from the category ofadmissible unitary E-Banach space representations of G to the category of Rp [1/p]-modules. In
particular, we have an injection M (Π1 ) ,→ M (Π). Since Pe is projective and π is a subquotientof Θ1 /$, we have M (Π1 ) 6= 0 by [Paš13, Lem. 4.13], and hence M (Π) 6= 0.
Similarly, we letd ∼◦e
M (Ψ) := HomG (Ψ, Homcont
O (P , E)) = M (σ )[1/p] ⊗H(σ),θ E.
η

Recall that H(σ) acts on M (σ ◦ )[1/p] through the composite of the homomorphism H(σ) −→Rp (σ)[1/p] and the Rp [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 θb : Rp (σ)[1/p] → E(so that then θ = θb ◦ η), in which case there is a natural isomorphism∼

M (σ ◦ )[1/p] ⊗H(σ),θ E −→ M (σ ◦ )[1/p] ⊗Rp (σ)[1/p],θb E
of E-vector spaces of dimension at most 1.
The map Ψ → Π induces a continuous homomorphism
cont econt eHomcont
G (Π, HomO (P , E)) → HomG (Ψ, HomO (P , E)).

Since the image of Ψ in Π is dense this map is injective, and by taking duals we obtain a surjectionof Rp [1/p]-modules
M (Ψ)  M (Π).

26

Patching and the p-adic Langlands program for GL2 (Qp )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 (Π1 ) is a non-zero subspace of M (Π), we therefore have induced∼∼
isomorphisms M (Π1 ) −→ M (Π) −→ M (Ψ). Since, as was noted above, M is an exact functor,
we find that M (Π0 ) = 0.
We digress for a moment in order to establish that Π1 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 (Π1 ) 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 Π1 is absolutely irreducible by [Paš13, Lem. 4.2].
Suppose now that Π0 6= 0. We may then apply the above argument with Π0 in place of Π, andfind an absolutely irreducible subrepresentation Π2 of Π0 , a GQp -representation r̄0 , an associatedirreducible GL2 (Qp )-representation π 0 , for which the projective envelope Pe2 of (π 0 )∨ gives rise to
an exact functor M 0 such that M 0 (Ψ) = M 0 (Π0 ) = M 0 (Π2 ), with all three being one-dimensional.
Since both M (Ψ) and M 0 (Ψ) are non-zero, we find that each of r̄ and r̄0 admits a lift thatis a lattice in a crystalline representation V of Hodge–Tate weights (1 − a, −(a + b)) determined(via Lemma 2.6) by the homomorphism θ̂ : Rp [1/p] → E. (Strictly speaking, for this to makesense we need to know that V is indecomposable; but this is automatic, since each of r̄ and r̄0 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 (IndG
B 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 completionof Ψ, see [BE10, Prop. 2.2.1], and the theorem is true in this case. Thus, for the remainderof the argument, we suppose that V is irreducible, or, equivalently, that Ψ does not admit anembedding into the continuous parabolic induction of a unitary character.
We now consider separately two cases, according to whether or not r̄ and r̄0 are themselves
isomorphic. If they are, then M and M 0 are isomorphic functors, and we obtain a contradiction
from the fact that M (Π0 ) = 0 while M 0 (Π0 ) is one-dimensional, implying that in fact Π0 = 0, as
required.
If r̄ and r̄0 are not isomorphic, but have isomorphic semi-simplifications, then they musteach consist of an extension of the same two characters, but in opposite directions. In this case
G
×
0 ∼
×
π∼
= (IndG
B ωχ1 ⊗ χ2 )sm , π = (IndB ωχ2 ⊗ χ1 )sm , for some smooth characters χ1 , χ2 : Qp → F .
0
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 0 .
We first show that Π2 = Π0 . To this end, set Π00 := Π0 /Π2 . If Π00 6= 0, then, running throughthe above argument another time, we find r̄00 , etc., such that (r̄00 )ss ∼= (r̄0 )ss ∼= r̄ss , and such that
00
00
00
00
0
00
M (Π ) 6= 0. But either r̄ ∼= r̄ or r̄ ∼= r̄ . Thus the functor M is isomorphic to either M 0
or to M . On the other hand, M 0 (Π00 ) = 0, and also M (Π00 ) = 0 (since Π00 is a quotient of Π0and M (Π0 ) = 0). This contradiction shows that Π00 = 0, and thus that Π0 = Π2 is absolutely
irreducible.
Since M (Π0 ) = 0, we see that Θ0 /$ does not contain a copy of π as a subquotient (here Θ0
denotes some choice of G-invariant lattice in Π0 ). Since (as we have just shown) Π0 is absolutely
irreducible, it follows from [Paš13, Cor. 8.9] that Π0 ∼= (IndG
B χ)cont for some unitary characterχ : T → E × , where T denotes the diagonal torus contained in the upper triangular Borell.adm
subgroup B of G. (The proof of this result uses various Ext computations in ModG
(O), but27

A. Caraiani et al.
does not use either of Colmez’s functors. The basic idea is that the extension of π 0 by π given by
the atome automorphe induces an embedding Pe ,→ Pe0 whose cokernel is dual to the induction
from B to G of a character; since M (Π0 ) = 0, we see that Θ0 embeds into this induction, and so isitself such an induction.) Thus Ψ admits an embedding into the continuous parabolic inductionof a unitary character, contradicting our hypothesis that V is irreducible. Thus we conclude thatin fact Π0 = 0, as required.
If Ψ were to admit two non-isomorphic admissible irreducible unitary completions Π1 and Π2
satisfying the assumptions of the theorem, the image of the diagonal map Ψ → Π1 ⊕ Π2 would
be dense in Π1 ⊕ Π2 . This yields a contradiction to what we have already proved, as Π1 ⊕ Π2 is
not irreducible.
b of the kind considered
5.2 Remark. It follows from the proof of Theorem 5.1 that if a completion Ψ
there exists, then there is a Galois representation r̄ satisfying Assumption 2.2 such that θ extends
to a homomorphism θb : Rp (σ)[1/p] → E. Furthermore, for any M∞ as in Section 3, we haveM∞ (σ ◦ )[1/p] ⊗H(σ),θ E 6= 0.
Conversely, if there is an r̄ satisfying Assumption 2.2 such that θ extends to a homomorphism
b is immediate from the main theorem
θb : Rp (σ)[1/p] → E, then the existence of a completion Ψof [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 forcrystabelline 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 resultsof this section therefore also depend on this functor. In Section 6.24 we briefly discuss how toprove 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 thatthe axioms of Section 3.1 and arguments of Section 4 admit obvious analogues in the setting ofa 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 ∨ ina 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), andshow that this satisfies the axioms of Section 3.1. This requires us to make a careful study ofvarious twisting constructions; the payoff is that we prove a stronger result than that which
would follow from the first approach.

28

Patching and the p-adic Langlands program for GL2 (Qp )6.1 Deformation rings and twistingLet Λ be the universal deformation ring of the trivial 1-dimensional representation of GQp andlet 1univ be the universal deformation. Then 1univ is a free Λ-module of rank 1 with a continuous
GQp -action. We let G act on 1univ via the inverse determinant (composed with the Artin map).
We let Λur be the quotient of Λ unramified deformations. We note that Λ and Λur are formallysmooth over O of relative dimensions 2 and 1, respectively, and in particular are O-torsion free.
Let ψ : GQp → O× be a continuous character such that ψε−1 is congruent to det r̄ modulo $.
(By Remark 2.17, this implies that ψ modulo $ considered as a character of Q×p coincides with
ψ
the central character of π.) We let Rp denote the quotient of Rp parameterising deformations withb O 1univ
determinant ψε−1 . Let runiv,ψ be the tautological deformation of r̄ to Rpψ . Then runiv,ψ ⊗ψ b
is a deformation of r̄ to Rp ⊗O Λ. Since p > 2, this induces an isomorphism of local O-algebras∼=b O Λ.
Rp −→ Rpψ ⊗(6.2)

Let Rpψ (σ) denote the quotient of Rp (σ) corresponding to deformations of determinant ψε−1 ;note that Rpψ (σ) = 0 unless ψ|Z×
is equal to the central character of σ. If ψ|Z×
is equal to thepp∼=b O Λ induces an isomorphism
central character of σ, then the isomorphism Rp −→ Rpψ ⊗∼=b O Λur .
Rp (σ) −→ Rpψ (σ) ⊗(6.3)

Let δ : GQp → O× be a character that is trivial modulo $. Twisting by δ induces isomorphisms of O-algebras∼twδ : Rp −→ Rp ,∼2twδ : Rpψδ −→ Rpψ .

(In terms of the deformation functor Dr̄ pro-represented by Rp , and the deformation functors
2
2Dr̄ψ and Dr̄ψδ pro-represented by Rpψ and Rpψδ , these isomorphisms are induced by the naturalbijections∼

Dr̄ (A) −→ Dr̄ (A),2∼Dr̄ψ (A) −→ Dr̄ψδ (A)defined by rA 7→ rA ⊗ δ.) Similarly we obtain isomorphisms∼

twδ : Λ −→ Λ,∼twδ : Λ(δ)ur −→ Λur ,and we have a commutative diagramRp∼= / ψδ 2
bO Λ
Rp ⊗
(6.2)

id



Rp



(6.4)

b tw −1
twδ ⊗
δ

∼
=
/ Rpψ ⊗
bO Λ
(6.2)

6.5 Unfixing the central characterl.adm,ψ
Let ModG(O) be the full subcategory of Modl.adm(O), consisting of those representations,Gl.adm,ψ
where Z acts by the central character ψ. Let Cψ (O) be the Pontryagin dual of ModG
(O), soψthat we can identify C (O) with a full subcategory of C(O) consisting of those objects on which
Z acts by ψ −1 .
Let Peψ  π ∨ be a projective envelope of π ∨ in Cψ (O). By [Paš13, Prop. 6.3, Cor. 8.7, Thm.

29

A. Caraiani et al.
10.71] there is a natural isomorphism∼

Rpψ −→ EndC(O) (Peψ ).

(6.6)

In Corollary 6.23 below we will prove a version of the isomorphism (6.6) for Rp .
6.7 Lemma. Let δ : GQp → O× be a character that is trivial modulo $. There is an isomorphismin C(O):2∼
ϕ : Peψδ −→ Peψ ⊗ δ −1 ◦ det .
Moreover, the following diagram commutes:
Rpψδ

twδ
∼
=

2

(6.6) ∼
=



/ Rpψ
∼
= (6.6)2

EndC(O) (Peψδ )∼=/ EndC(O) (Pe ψ )where the lower horizontal arrow is given by α 7→ ϕ ◦ α ◦ ϕ−1 .
Proof. The claimed isomorphism follows from the fact that twisting by δ −1 ◦ det induces an2
equivalence of categories between Cψ (O) and Cψδ (O), so that the twist of a projective envelope of2
π ∨ in Cψ (O) is a projective envelope of π ∨ in Cψδ (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 : GQp → O× be a character that is trivial modulo $. There is an evident isomorphismof pseudocompact O[[GQp ]]-modules:∼θ : 1univ ⊗ δ −→ 1univ ,and a commutative diagram
Λ


twδ−1
∼
=

(6.8)

∼
=

∼
=

univ )
Endcont
GQp (1/Λ∼=/ Endcont (1univ )GQpwhere the lower horizontal arrow is given by α 7→ θ ◦ α ◦ θ−1 . (In terms of the deformation functor∼
D1 pro-represented by Λ, the isomorphism twδ−1 is induced by the bijection D1 (A) −→ D1 (A)defined by χ 7→ χδ −1 .)× be a character that is trivial modulo $. Then there is an
6.9 Lemma. Let δ : Q×
p → O
isomorphism2∼ eψ
b O 1univ −→b O 1univPeψδ ⊗
P ⊗in the category C(Rp ), where Rp acts on both sides by the isomorphism of (6.2).
Proof. Using Lemma 6.7 and (6.8) we obtain isomorphisms in C(O):
b id
2
⊗
∼
b O 1univ ϕ−→b O 1univ −→Peψδ ⊗(Peψ ⊗ δ −1 ◦ det) ⊗bθ
⊗∼b O (1univ ⊗ δ ◦ det) id
b O 1univ .
−→ Peψ ⊗−→ Peψ ⊗30

Patching and the p-adic Langlands program for GL2 (Qp )∼ eψb θ : Peψδ2 ⊗b O 1univ −→b O 1univ , which
The composition of these isomorphisms is equal to φ ⊗P ⊗is an isomorphism in C(O). It follows from Lemma 6.7 and (6.8) that the following diagram
commutes:bO ΛRpψδ ⊗b tw −1
twδ ⊗
δ
∼
=

 2

∼
=

2

b O 1univ )EndC(O) (Peψδ ⊗/ Rpψ ⊗bO Λ/ EndC(O) (Pe ψ ⊗b O 1univ )b θ) ◦ α ◦ (ϕ ⊗b θ)−1 . It follows from thewhere the lower horizontal arrow is given by α 7→ (ϕ ⊗b θ is an isomorphism in C(Rp ).
above diagram and (6.4) that ϕ ⊗b O 1univ is projective in the category of pseudocompact O[[K]]-modules.
6.10 Proposition. Peψ ⊗
Proof. Let I1 := {g ∈ K : g ≡ 10 1∗ (mod $)}. Then I1 is a pro-p Sylow subgroup of K andb O 1univ is projective in the category of pseudocompact O[[I1 ]]it is enough to show that Peψ ⊗modules. Since I1 is a pro-p group there is only one indecomposable projective object in the
category, namely O[[I1 ]], and thus a pseudocompact
Q O[[I1 ]]-module is projective if and only if itis pro-free, which means that it is isomorphic to j∈J O[[I1 ]] for some indexing set J.
Let Γ := 1 + pZp . We identify Γ with the image in Gab
Qp of the wild inertia subgroup of GQp .
We may identify Λ with the completed group algebra of the pro-p completion of Gab
Qp , which isisomorphic to Γ × Zp . There is an isomorphism of O-algebras Λ ∼O[[Γ]][[x]],andinparticular=univ
1
is a pro-free and hence projective O[[Γ]]-module.
The restriction of Peψ 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 I1 , we deduce that Peψ is projective in the category of pseudocompact O[[I1 ]]-modules, whereZ1 := I1 ∩ 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 characterspreserves projectivity, so that 1univ ⊗ δ is a projective in the category of pseudocompact O[[Γ]]modules and Peψ ⊗ (δ ◦ det) is projective in the category of pseudocompact O[[I1 ]]-modules on
which Z1 acts trivially. We may identify this last category with the category of pseudocompact
O[[I1 /Z1 ]]-modules.
We have an isomorphism of I1 -representationsb O 1univ ∼b O (1univ ⊗O δ),Peψ ⊗= (Peψ ⊗ δ ◦ det) ⊗where I1 acts on 1univ ⊗O δ via the homomorphism I1 → Γ, g 7→ (det g)−1 . We may therefore
assume that ψ|Z1 is trivial, so that Peψ is projective in the category of pseudocompact O[[I1 /Z1 ]]modules. Thus there are indexing sets J1 and J2 such thatYY
O[[Γ]]
Peψ |I1 ∼O[[I1 /Z1 ]], 1univ ∼
=
=
j1 ∈J

j2 ∈J2where the first isomorphism is an isomorphism of pseudocompact O[[I1 ]]-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[[I1 ]]-modules:
Y Y
b O 1univ ) ∼b O O[[Γ]].
(Peψ ⊗
O[[I1 /Z1 ]] ⊗=j1 ∈J1 j2 ∈J2Since p > 2, the determinant induces an isomorphism between Z1 and Γ. Thus the map I1 →I1 /Z1 × Γ, g 7→ (gZ1 , (det g)−1 ) is an isomorphism of groups. The isomorphism induces a natural

31

A. Caraiani et al.
b O O[[Γ]]. We conclude that Peψ ⊗b O 1univ isisomorphism of O[[I1 ]]-modules, O[[I1 ]] ∼= O[[I1 /Z1 ]] ⊗a pro-free and hence a projective O[[I1 ]]-module.
b O 1univ is a projective object
6.11 Remark. We will see in Theorem 6.18 below that in fact Peψ ⊗of C(O). It is not so difficult to prove this directly, but we have found it more convenient todeduce it as part of the general formalism of arithmetic actions.
×
6.12 Proposition. Let σ = σa,b , and let δ : Q×p → O be a character that is trivial modulo
2◦$, chosen so that ψδ |Z×
is the central character of σ . Then there is a natural isomorphism ofp
Rp -modules
∼
=

2

univ
eψδ , (σ ◦ )d )d ⊗eψ bb O Λur ,, (σ ◦ )d )d −→ Homcont
Homcont
O[[K]] (PO[[K]] (P ⊗O 1
ψ bwhere Rp acts on the left hand side via the isomorphism Rp ∼= Rp ⊗
O Λ and on the right hand2ψδ b∼side via the isomorphism Rp = Rp ⊗O Λ.

Proof. Using Lemma 6.9 we may assume that ψ|K∩Z is the central character of σ. We note that
1univ ⊗Λ Λur is the largest quotient of 1univ on which K∩Z acts trivially. Since the central characterof (σ ◦ )d is ψ −1 |Z×, and the central character of Peψ is ψ −1 , we have a natural isomorphism
p
∼
=

univ
univ
eψ b
eψ b
⊗Λ Λur ), (σ ◦ )d )d .
Homcont, (σ ◦ )d )d −→ Homcont
O[[K]] (P ⊗O (1O[[K]] (P ⊗O 1

Since K acts trivially on 1univ ⊗Λ Λur , we have an isomorphism:b O Λur , (σ ◦ )d )d ,b O (1univ ⊗Λ Λur ), (σ ◦ )d )d ∼Homcont (Peψ ⊗= Homcont (Peψ ⊗O[[K]]

O[[K]]

ψ bwhere Λur carries a trivial K-action and Rp acts by the isomorphism Rp ∼= Rp ⊗
O Λ. To finishb O Λ-modules:
the proof we need to construct a natural isomorphism of Rpψ ⊗urur◦ d d ∼cont
◦ d d beψeψ b
Homcont
O[[K]] (P ⊗O Λ , (σ ) ) = HomO[[K]] (P , (σ ) ) ⊗O Λ .

(6.13)b O Λurb O Λur -modules. The m-adic topology on Rpψ ⊗Both sides of (6.13) are finitely generated Rpψ ⊗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 sidesof (6.13). To ease the notation we let A = Peψ , B = Λur , C = (σ ◦ )d . Since for a pseudo-compactn∼O-module D, we have Dd = Homcont
Homcont
O (D, O) = lim
O (D, O/$ ), using the adjointness←−nb O and Homcont
between ⊗(see [Bru66, Lem. 2.4]), we obtain natural isomorphisms
O
d b
d ∼
cont
cont
d d
(Homcont
O (A, C) ⊗O B) = HomO (B, (HomO (A, C) ) )
cont
cont
∼
∼
b
= HomcontO (B, HomO (A, C)) = HomO (A ⊗O B, C),and hence a natural isomorphism
d ∼
cont
d b
b
Homcont
O (A ⊗O B, C) = HomO (A, C) ⊗O B.

Since the isomorphism is natural and K acts trivially on B we obtain a natural isomorphismb O B, C)d ∼b O B.
Homcont (A ⊗= Homcont (A, C)d ⊗O[[K]]

O[[K]]

Since the action of Rpψ commutes with the action of K and the isomorphism is natural, we deducethat (6.13) holds.
ψ b6.14 Proposition. For any σ, the action of Rp ∼= Rp ⊗O Λ onuniveψ b
M 0 (σ ◦ ) := Homcont, (σ ◦ )d )dO[[K]] (P ⊗O 1

32

Patching and the p-adic Langlands program for GL2 (Qp )factors through Rp (σ). Moreover, M 0 (σ ◦ ) is a maximal Cohen–Macaulay Rp (σ)-module andM 0 (σ ◦ )[1/p] is a locally free Rp (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∼b O 1univ , thus also that M (σ ◦ ) ' M 0 (σ ◦ ).
6.18 below that Pe −→ Peψ ⊗Proof. If the central character of σ is not congruent to ψ|Z×
modulo $, then both Rp (σ) andp×0◦
M (σ ) are zero. Otherwise, there is a character δ : Qp → O× trivial modulo $ such thatis equal to the central character of σ. Proposition 6.12 gives us an isomorphism of(ψδ 2 )|Z×p
Rp -modules
2b O Λur ,M 0 (σ ◦ ) ∼(6.16)= M ψδ (σ ◦ ) ⊗2eψδ2 , (σ ◦ )d )d , and the action of Rp on the right hand side is given
where M ψδ (σ ◦ ) := Homcont
O[[K]] (P22ψδ 2 bψδ 2
on M ψδ (σ ◦ ) factors through the action of Rpψδ (σ)
by Rp ∼
= Rp ⊗
O Λ. The action of Rp
2

and makes it into a maximal Cohen–Macaulay Rpψδ (σ)-module [Paš15, Cor. 6.4, 6.5]. Since22b O Λur b O Λur is a maximal Cohen–Macaulay Rpψδ (σ) ⊗Λur ∼= O[[x]], we conclude that M ψδ (σ ◦ ) ⊗module. Using (6.3) we see that M 0 (σ ◦ ) is a maximal Cohen–Macaulay Rp (σ)-module. Since
Rp (σ)[1/p] is a regular ring a standard argument using the Auslander–Buchsbaum theorem
shows that M 0 (σ ◦ )[1/p] is a locally free Rp (σ)[1/p]-module. It follows from [Paš15, Prop. 4.14,
2.22] that2dimκ(x) M ψδ (σ ◦ ) ⊗ ψδ2 κ(x) = 1,
Rp2

∀x ∈ m-Spec Rpψδ (σ)[1/p].

This together with (6.16) gives usdimκ(x) M 0 (σ ◦ ) ⊗Rp κ(x) = 1,

∀x ∈ m-Spec Rp (σ)[1/p].

Hence, M 0 (σ ◦ )[1/p] is a locally free Rp (σ)[1/p]-module of rank 1.
The natural action of H(σ ◦ ) on M 0 (σ ◦ ) commutes with the action of Rp (σ), and hence induces
an action of H(σ) on M 0 (σ ◦ )[1/p]. Since M 0 (σ ◦ )[1/p] is locally free of rank 1 over Rp (σ)[1/p] by
Proposition 6.14, we obtain a homomorphism of E-algebras:α : H(σ) → EndRp (σ)[1/p] (M 0 (σ ◦ )[1/p]) ∼= Rp (σ)[1/p].
6.17 Proposition. The map α : H(σ) → Rp (σ)[1/p] coincides with the map η : H(σ) →Rp (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
ψ bdense subset of m-Spec Rp (σ)[1/p]. The isomorphism Rp ∼= Rp ⊗O Λ maps x to a pair (y, z),ψ
univwhere y ∈ m-Spec Rp [1/p] and z ∈ m-Spec Λ[1/p], so that if rx , ryuniv,ψ and 1univ
are Galoiszrepresentations corresponding to x, y and z respectively thenrxuniv ∼.
= ryuniv,ψ ⊗ 1univ
zLetuniv beψ b
Πx := Homcont) ⊗Rp Oκ(x) , E),
O ((P ⊗O 1eψ bΠy := Homcont
O (P ⊗Rψ Oκ(y) , E).
p

Then both are unitary G-Banach space representations and we haveΠx ∼◦ det).
= Πy ⊗ (1univz33

A. Caraiani et al.
It follows from [Paš15, Prop. 2.22] that HomK (σ, Πx ) 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) → Endκ(x) (M 0 (σ ◦ ) ⊗R κ(x)) ∼= κ(x).
p

Since σ is an algebraic representation of K, we have
HomK (σ, Πx ) ∼= HomK (σ, Πl.alg
x ),where Πl.alg
is the subspace of locally algebraic vectors in Πx .
x
It follows from the main result of [Paš13] that the specialisation Πx of Peψ coincides with the
Banach space representation attached to rxuniv 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 Πx ) that there is an embedding
πsm (rxuniv ) ⊗ πalg (rxuniv ) ,→ Πxl.alg ,where πsm (rxuniv ) = rp−1 (WD(rxuniv )F −ss ) is the smooth representation of G corresponding to theWeil–Deligne representation associated to rxuniv by the classical Langlands correspondence rp(normalised as in Section 1.12), and πalg (rxuniv ) is the algebraic representation of G whose restriction to K is equal to σ. Indeed, if the representation rxuniv is irreducible, then (for a Zariski
dense set of x) Πx is a completion of πsm (rxuniv ) ⊗ πalg (rxuniv ) by the main result of [BB10] (see inparticular [BB10, Thm. 4.3.1]), while in the case that rxuniv is reducible, the result follows fromthe explicit description of Πx in [BE10].
Since we have already noted that HomK (σ, Πx ) is one-dimensional, we find that in fact
HomK (σ, Πx ) ∼= HomK σ, πsm (rxuniv ) ⊗ πalg (rxuniv ) ∼= πsm (rxuniv )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.
ψ bb O 1univ via the isomorphism Rp ∼Let Rp act on Peψ ⊗= Rp ⊗O Λ, where (as throughout this
ψ
ψ
ψ
∼
e
section) the action of Rp on P is via the isomorphism Rp = EndCψ (O) (Peψ ) constructed in[Paš13].

b O 1univ is a projective envelope of π ∨ in C(O), and the action of Rp on
6.18 Theorem. Peψ ⊗b O 1univ is arithmetic.
Peψ ⊗b O 1univ satisfies the axioms (AA1) – (AA4)
Proof. We will show that the action of Rp on Peψ ⊗b O 1univ is a projective envelopewith d = 0; then the action is arithmetic by definition, and Peψ ⊗b O 1univ ).
of π ∨ in C(O) by Theorem 4.32 (applied with d = 0 and with M∞ taken to be Peψ ⊗b ψ Peψ is a finitely generated O[[K]]-module,
It is shown in [Paš15, Propositition 6.1] that F ⊗Rp
so the topological version of Nakayama’s lemma implies that Peψ is a finitely generated Rpψ [[K]]b O 1univ is a finitely generated module overmodule. Since 1univ is a free Λ-module of rank 1, Peψ ⊗ψ b(Rp ⊗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 0 (σ ◦ ) is all of Rp (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 fromQp -representations on compact O-modules. This is a
modification of the functor introduced by Colmez in [Col10b], see [Paš13, §5.7] for details (weCψ (O) to the category of continuous G

34

Patching and the p-adic Langlands program for GL2 (Qp )additionally have to twist the functor in [Paš13] by the inverse of the cyclotomic characterto get the desired relationship between the determinant of the Galois representations and thecentral character of GL2 (Qp )-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 inΠ then the Schikhof dual of Θ is an object of Cψ (O) and V̌(Π) := V̌(Θd )[1/p] does not dependon the choice of Θ.
The representation Peψ satisfies the conditions (N0), (N1), (N2) of [Paš15, §4] by [Paš15,
ψProp. 6.1]. In particular, we have V̌(Peψ ) ∼= runiv,ψ as Rp [[GQp ]]-modules.
Let R∞ = Rp [[x1 , . . . , xd ]] 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,b
Π∞,x := Homcont
O (M∞ ⊗R∞ Oκ(x) , E). The map Rp → R∞ induces a map Rp → κ(x) and we let
univ
univ
rx := r⊗Rp κ(x).
6.19 Corollary. We have an isomorphism of Galois representations V̌(Π∞,x ) ∼= rxuniv . In
particular, Π∞,x 6= 0 for all x ∈ m-Spec R∞ [1/p].
Proof. Theorem 4.32 allows us to assume that R∞ = Rp and M∞ = Pe is a projective envelopeof π ∨ in C(O). Since an arithmetic action of Rp on Pe is unique by Theorem 4.30, using Theoremb O 1univ . Then, with the notationb O Λ and M∞ = Peψ ⊗6.18 we may assume that R∞ = Rpψ ⊗introduced in the course of the proof of Proposition 6.17, x corresponds to a pair (y, z), Π∞,x =Πx ∼◦ det) as in the proof of Proposition 6.17. It follows from [Paš15, Lem. 4.3]
= Πy ⊗ (1univ
zuniv,ψ
∼that V̌(Πy ) = ry
. Since V̌ is compatible with twisting by characters we have V̌(Πx ) ∼
=
univ
univ
∼
V̌(Πy ) ⊗ 1z = rx , 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 Rp and hence R∞ are reduced we deduce that the action is faithful.
b Rp Pe. We will use this result in Corollary 7.5
We now use the results of [Paš13] to describe F ⊗below to describe the m-torsion in the completed cohomology of a modular curve, where m is a
maximal ideal in a Hecke algebra.
b Rp Pe with multiplicity
6.21 Proposition. The representation π ∨ occurs as a subquotient of F ⊗b Rp Pe)∨ then the G-socle filtration of κ(r̄) is describedone. More precisely, if we let κ(r̄) := (F ⊗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 andof the other principal series representation in the block of π.
(iii) If r̄ ∼= 1 ∗ ⊗χ then the G-socle filtration of κ(r̄) has length three and the graded pieces are0ωπ, 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 Theoremb Rp Pe ∼b ψ Peψ . Since we can identify the endomorphism ring of Peψ with Rpψ , see
6.18 that F ⊗
= F⊗
R
p

(6.6), it follows from Lemma 3.7 in [Paš13] applied with S = π ∨ that any Q in Cψ (O) satisfyingb ψ Peψ , so that κ(r̄) ∼the hypotheses (H1)–(H4) of [Paš13, §3] is isomorphic to F ⊗= Q∨ . (We leaveR
p

35

A. Caraiani et al.
the reader to check that (H5) is not used to prove this part of Lemma 3.7 in [Paš13].) In all thecases 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 Rp −→ EndC(O) (Pe).
∼b O 1univ as Rp [G]-modules via
Proof. Theorems 4.30 and 6.18 yield an isomorphism Pe −→ Peψ ⊗∼ψbcont univ) = Λ, while (6.6) gives
the isomorphism Rp −→ Rp ⊗O Λ given by (6.2). Now EndG (1∼a natural isomorphism Rpψ −→ EndC(O) (Peψ ). Thus the corollary amounts to proving that thenatural homomorphismb O 1univ )b O Endcont (1univ ) → EndC(O) (Peψ ⊗EndC(O) (Peψ )⊗Gis an isomorphism. The map is an injection of pseudo-compact O-algebras. This makes the ring
EndC(O) (Pe) into a compact Rp -module. By the topological version of Nakayama’s lemma, in
b Rp EndC(O) (Pe) is a oneorder to show that the map is surjective it is enough to show that F ⊗dimensional F-vector space. Since Pe is projective, we have an isomorphism:b Rp EndC(O) (Pe) ∼b Rp Pe).
F⊗= HomC(O) (Pe, F ⊗b Rp Pe with multiplicity one. Since Pe
By Proposition 6.21, π ∨ occurs as a subquotient of F ⊗b Rp Pe) is a
is a projective envelope of π ∨ and EndC(O) (π ∨ ) = F, this implies that HomC(O) (Pe, F ⊗one dimensional F-vector space, as required.
6.24 Endomorphism rings and deformation rings
We maintain the notation of the previous sections; in particular, Pe denotes a projective envelope
e := EndC(O) (Pe), then the arithmetic action of Rp on Pe provided byof π ∨ in C(O). If we write R
e which Corollary 6.23 shows is an isomorphism. The
Theorem 4.30 gives a morphism Rp → R,
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 GL2 (Qp )-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 wecan.
We begin by noting that since Pe is a projective envelope of the absolutely irreducible repree is a local ring. We will furthermore give a proof that it is commutative,
sentation π ∨ , the ring R
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]), anotherproof 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 presenta slightly rephrased version of the argument here, in order to illustrate how it fits naturally into
our present perspective.
e is commutative.
6.25 Proposition. The ring R

36

Patching and the p-adic Langlands program for GL2 (Qp )e lies in the centre of R.
e To see this, suppose
Proof. We first prove that the image of Rp in Rethat φ ∈ R. By Proposition 4.28, to show that φ commutes with the action of Rp , it suffices toshow that φ commutes with the action of Rp (σ)[1/p] on M (σ ◦ )[1/p] for each σ. Since the actionof H(σ) on M (σ ◦ )[1/p] depends only on the G-action on Pe, we see that φ commutes with theH(σ)-action on M (σ ◦ )[1/p]. The desired result then follows from Proposition 2.13.
e is commutative, we again apply Proposition 4.28, by which it suffices to show
To see that Rethat R acts on each M (σ ◦ )[1/p] through a commutative quotient. This follows from the fact thateach M (σ ◦ )[1/p] is locally free of rank one over its support in Spec Rp (σ)[1/p], and the fact thate(by the result of the previous paragraph) the R-action
commutes with the Rp -action.
e is an isomor6.26 Remark. As for proving the stronger result that the canonical map Rp → R
phism, in the forthcoming paper [EP] two of us (ME and VP) will establish the injectivity of thee (In fact we will prove a result in the more general context of [CEG+ 16]; inmorphism Rp → R.
particular, our arguments won’t rely on any special aspects of the GL2 (Qp ) 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 GL2 (Qp )-representations to GQp -representations.
6.27 Speculations in the residually scalar semi-simplification caseχ ∗Suppose for the rest of this section that r̄ ∼= 0 χ 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 inthis case; we give a speculative answer below. By twisting we may assume that χ is the trivial∨echaracter. Let π = (IndG
B ω ⊗ 1)sm , and let P be a projective envelope of π . We first give a
conjectural description of EndC(O) (Pe), under the assumption p > 2.
Let Dps 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) : GQp → A,
where:– d : GQp → A× is a continuous group homomorphism, congruent to det r̄ modulo mA ,– t : GQp → A is a continuous function with t(1) = 2, and,– for all g, h ∈ GQp , we have:(i) t(g) ≡ tr r̄(g) (mod mA );(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], Dps (A) is the set of pseudocharacters
deforming the pseudocharacter (tr r̄, det r̄) associated to r̄.) This functor is representable by acomplete local noetherian O-algebra Rps . Let (tuniv , duniv ) : GQp → Rps be the universal object.
We expect that there is a natural isomorphism of O-algebras
e := EndC(O) (Pe) ∼E= (Rps [[GQp ]]/J)op ,(6.28)

where J is the closed two sided ideal of Rps [[GQp ]] generated by all the elements of the formg 2 − tuniv (g)g + duniv (g) for all g ∈ GQp , and the superscript op indicates the opposite algebra.
We note that such an isomorphism has been established in [Paš13, §9], when the central characteris fixed, and we expect that one can deduce (6.28) from this using the twisting techniques of theprevious subsection.
Let Rp be the framed deformation ring of r̄ and let M∞ be the patched module constructed37

A. Caraiani et al.
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 Rp -algebra, and themap Rp → R∞ gives rise to a Galois representation r∞ : GQp → GL2 (R∞ ) lifting r̄. The pair
(tr r∞ , det r∞ ) gives us a point in Dps (R∞ ) and hence a map Rps → R∞ . Hence we obtain ahomomorphism of Rps -algebras Rps [[GQp ]] → M2×2 (R∞ ).
The Cayley–Hamilton theorem implies that this map is zero on J, so we obtain a left actionof Rps [[GQp ]]/J on the standard module R∞ ⊕ R∞ . If we admit (6.28) then we get a right action
e on R∞ ⊕ R∞ . We expect that there are isomorphisms in C(R∞ )of E
b e Pe ∼b R (Rp ⊕ Rp ) ⊗b e Pe.
M∞ ∼= (R∞ ⊕ R∞ ) ⊗= R∞ ⊗
EEp(6.29)

We note that the representation appearing on the right hand side of this equation has beenstudied by Fabian Sander in his thesis [San], in the setting where the central character is fixed.
b e Pe to be projective in the category of
Motivated by [San, Thm. 2] we expect (R∞ ⊕ R∞ ) ⊗Eb e Pe to be projective in C(O), so
pseudocompact O[[K]]-modules. We do not expect (R∞ ⊕ R∞ ) ⊗E
the methods of Section 4 cannot directly be applied to this case. However, it might be possible toprove (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 ofthe local-global compatibility theorem of [Eme11] (under the hypotheses that we have imposedin 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 forease of comparison to [Eme11], we instead briefly discuss the output of Taylor–Wiles patchingfor 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 ρ : GQ → GL2 (F) be an absolutely irreducible odd (so modular, by Serre’s conjecture)
representation, and assume that p > 5 and that ρ|GQ(ζp ) is irreducible. Write r̄ := ρ|GQp , and assume that r̄ satisfies Assumption 2.2. Write runiv : GQp → GL2 (Rp ) for the universal deformationof r̄. Let N (ρ) be the prime-to-p conductor of ρ; that is, the level of ρ in the sense of [Ser87]. We
assume that if q|N (ρ) with q ≡ −1 (mod p) and ρ|GQq is irreducible then ρ|IQq 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 tomodular forms of level N (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 thissimple case.
We caution the reader that while the use of types would also allow us to work at certain non-38

Patching and the p-adic Langlands program for GL2 (Qp )minimal 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 ofprimes, 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 ofmod 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 ingeneral 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 aring R∞ that was no longer formally smooth over Rp , 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 Γ1 (N (ρ)) and O-coefficients; so T is an O-algebra, generated by theoperators Tl , Sl with l - N p. Let m(ρ) be the maximal ideal of T corresponding to ρ (so thatTl − tr ρ(Frobl ) and lSl − det ρ(Frobl ) are both zero in T/m(ρ)). Let RQ,N (ρ) be the universal
deformation ring for deformations of ρ that are minimally ramified at primes l 6= p, in the sensethat they have the same conductor as ρ|GQl (and in particular are unramified if l - N (ρ)). Let
ρuniv
Q,N (ρ) : GQ → GL2 (RQ,N (ρ) ) denote the corresponding universal deformation of ρ.
We now use the notation introduced in Section 3, so that in particular we write R∞ :=b O O[[x1 , . . . , xd ]] for some d > 0. Patching the completed étale homology of the modular
Rp ⊗curves Y1 (N (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∞ −→ RQ,N (ρ) , and an isomorphism of
RQ,N (ρ) [G × GQ ]-modules∗ ∼ e(M∞ /a∞ ) ⊗RQ,N (ρ) (ρuniv(7.2)
Q,N (ρ) ) −→ H1,ét Y1 (N (ρ)), O m(ρ) .
e 1,ét Y1 (N (ρ)), O denotes completed étale homology, as described for example in [CE12].
Here H∗The action of GQ on the left hand side is via its action on (ρuniv
Q,N (ρ) ) , which as in (1.12.3) denotes
the RQ,N (ρ) -linear dual of ρuniv
Q,N (ρ) .
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 thering R∞ occurring in the patching argument is formally smooth over Rp .
7.4 Theorem. Let p > 3 be prime, and let ρ : GQ → GL2 (F) be an absolutely irreducible odd
representation, with the property that ρ|GQ(ζp ) is irreducible, N (ρ) is not divisible by any vexing
primes, and ρ|GQp satisfies Assumption 2.2. Then there is an isomorphism of RQ,N (ρ) [G × GQ ]modules∼∗e 1,ét Y1 (N (ρ)), Ob Rp (ρuniv
H−→ Pe ⊗Q,N (ρ) ) ,m(ρ)∗
where the completed tensor product on the right-hand side is computed by regarding (ρuniv
Q,N (ρ) )as a GQ -representation on an Rp -module via the natural morphism Rp → RQ,N (ρ) .

39

A. Caraiani et al.
Proof. As noted before Remark 7.1, 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
b O O[[x1 , . . . , xd ]] ∼b Rp R∞ .
M∞ ∼= Pe ⊗= Pe ⊗
Quotienting out by a∞ yields an isomorphismb Rp RQ,N (ρ) .
M∞ /a∞ ∼= Pe ⊗∗The result now follows by tensoring both sides with (ρuniv
Q,N (ρ) ) and applying (7.2).

We now show how tocompute the m(ρ)-torsion in the completed étale cohomology of modular
e 1 Y1 (N (ρ)), O as a GL2 (Qp )-representation.
curves Hét
7.5 Corollary. Under the assumptions of Theorem 7.4, we have an isomorphism of F[G × GQ ]modules
e 1 Y1 (N (ρ)), F [m(ρ)] ' κ(ρ|G ) ⊗F ρ,
H
ét

Qp

where κ(ρ|GQp ) is the representation defined in Proposition 6.21.
e 1,ét Y1 (N (ρ)), O ⊗b RQ,N (ρ) F. By TheoProof. The Pontryagin dual of the left hand side is H
rem 7.4, we have an isomorphism of F[G × GQ ]-modulese 1,ét Y1 (N (ρ)), O ⊗b F ρ∗ ,b RQ,N (ρ) F ' (F ⊗b Rp Pe) ⊗b F ρ∗ ∼H= κ(ρ|GQp )∨ ⊗the last isomorphism following from the definition of κ(ρ|GQp ). Passing to Pontryagin duals (andnoting that since ρ is an F-representation, we have ρ∗ = ρ∨ ) gives the result.
7.6 Remark. Corollary 7.5 together
with Proposition 6.21 gives a description of the G-socle
1efiltration of Hét Y1 (N (ρ)), F [m(ρ)]. Even more is true.
Since in Corollary 6.23 we have identified the endomorphism ring of Pe with Rp and κ(r̄) is byb Rp Pe, a completely formal argument (see the proof of Proposition 2.8 in [Paš16])
definition F ⊗l.admshows that κ(r̄) is up to isomorphism the unique representation in ModG
(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 representationin Modl.adm(O) satisfying these two properties.)
G
Corollary 7.5 shows that (after factoring out the GQ -action) the same characterisation carriese 1 Y1 (N (ρ)), F [m(ρ)]. However, we warn the reader that a simple-minded application
over to Hét
of this recipe will not work in general, and in particular it fails if ρ|GQp ∼= ω0 ∗1 ⊗ 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

ω ∗
⊗
χ
and
be recovered from π, which in turn determines κ(r̄). If on the other hand r̄ ∼= 01
EndGQp (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 thedeterminant of r̄ but it does not determine r̄ up to isomorphism, as it does not carry the information about the extension class in Ext1GQp (χ, χω) corresponding to r̄. The maximal representation satisfying (1) and (2) above will contain the atome automorphe corresponding to r̄ as40

Patching and the p-adic Langlands program for GL2 (Qp )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-zeroextensions in Ext1GQp (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 Pe realises the usual p-adic local Langlands correspondence for GL2 (Qp )),prove a local-global
 compatibility result for completed cohomology. They are new in the casethat ρ|GQp ∼= 10 ω∗ ⊗ 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 canbe deduced from Theorem 6.4.6 in [Eme11] with Pe replaced by a deformation of κ(r̄)∨ to Rp ,such that one obtains the universal deformation of r̄ after applying Colmez’s functor V̌ to it. If
ρ|GQp ∼
= 10 ω∗ ⊗ χ then it can be shown that Pe is not flat over Rp , and that is why the approachof [Eme11] does not work in this case.
7.8 Remark. Theorem 7.4 and Corollary 7.5 have analogues in more general settings when thegroup at p is essentially GL2 (Qp ) (or a product of copies of GL2 (Qp )). For example, taking M∞
to be the patched module of Section 3.4 (as constructed in [CEG+ 16] for n = 2), we obtainstatements 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 tothe 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 tothis 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 thetrivial character. In this case we can think of our patched modules as modules for PGL2 (Qp ), andnatural 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 PGL2 (Qp ), which is convenient when there is morethan one place lying over p; accordingly, we work below with cohomology that is completed at allprimes 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 awayfrom 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 inparticular this requires [F : Q] to be even).
Let ρ : GF → GL2 (F) be absolutely irreducible, and assume that ρ|GF (ζp ) is irreducible, andthat det ρ = ω −1 . Suppose that ρ has no vexing primes; that is, if v - p is a finite place at which ρ41

A. Caraiani et al.
is ramified and Nv ≡ −1 (mod p), and ρ|GFv is irreducible, then ρ|IFv is also irreducible. Finally,
suppose that for each place v|p, ρ|GFv satisfies Assumption 2.2.
Since D splits at all finite places, we can consider the tame level subgroup U1 (N (ρ)) ⊂∞,p∞,p
PGL1 (D ⊗ A∞,pF ) ' PGL2 (AF ) given by the image of those matrices in GL2 (AF ) that are
e 0 (U1 (N (ρ)), O) (resp. H
e 0 (U1 (N (ρ)), O))
unipotent and upper triangular modulo N (ρ). Let H
denote the completed homology (resp. cohomology) of the tower of locally symmetric spaces
associated to PGL1 (D) with tame level U1 (N (ρ)). (Note that at finite level, the locally symmetricspaces are just finite sets of points.)
We assume that ρ is modular, in the following sense: ρ determines a maximal ideal m(ρ) in
the spherical Hecke algebra (generated by Hecke operators at places not dividing p at which ρ ise 0 (U1 (N (ρ)), O) and we assume that H
e 0 (U1 (N (ρ)), O)m(ρ) 6= 0.
unramified) acting on H
Let Rρ be the universal deformation ring for deformations of ρ that are minimally ramified atplaces not dividing p, and which have determinant ε−1 . For each place v|p, let Rv be the universal
b v|p Rv .
deformation ring for deformations of ρ|GFv with determinant ε−1 , and set Rp := ⊗Q
Q
Set G = v|p PGL1 (Dv ), which we identify with v|p PGL2 (Qp ) via a fixed isomorphism.
e 0 (U1 (N (ρ)), O)m(ρ) (with a variation of the argument
By patching the completed homology H
3in [Sch15, §9]), we obtain a ring R∞ which is a power series ring over Rp , and an R∞ [G]-module
M∞ with an arithmetic action in the sense of [GN16, §5.2], together with an ideal a∞ ⊂ R∞ suche 0 (U1 (N (ρ)), O)m(ρ) as Rρ [G]-modules.
that R∞ /a∞ ∼= Rρ as local O-algebras and M∞ /a∞ ∼=H(Note that we can ensure the smoothness of R∞ over Rp 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 isomorphisme 0 (U1 (N (ρ)), O)m(ρ) ' (⊗b v|p Pe1 (ρ|GF )) ⊗b Rp Rρ ,Hv
where Pe1 (ρ|GFv ) denotes the projective envelope considered in Section 6 in the case that r̄ = ρ|GFv
and ψ = 1. Arguing as in the proof of Corollary 7.5, we obtain a G-equivariant isomorphisme 0 (U1 (N (ρ)), F)[m(ρ)] ' ⊗b v|p κ(ρ|GF ),Hv
where κ(ρ|GFv ) 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 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 GF -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 tothank 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 ofthe 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 like3

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

42

Patching and the p-adic Langlands program for GL2 (Qp )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
David Geraghty david.geraghty@bc.edu
Department of Mathematics, 301 Carney Hall, Boston College, Chestnut Hill, MA 02467, USA
Vytautas Paškūnas paskunas@uni-due.de
Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany
Sug Woo Shin sug.woo.shin@berkeley.edu
Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA / Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea

45