Draft: March 23, 2011
LOCAL-GLOBAL COMPATIBILITY IN THE p-ADICLANGLANDS PROGRAMME FOR GL2/Q
MATTHEW EMERTON

Contents
1. Introduction
1.1. The local-global compatibility conjecture
1.2. Statement of results
1.3. The organization of the paper
1.4. Background and additional remarks
1.5. Acknowledgments
2. Notation and conventions
3. Mod p and p-adic local Langlands
3.1. Representations of G
3.2. Colmez’s magical functor
3.3. Mod p and p-adic local Langlands
3.4. The reducible case
3.5. Weights
3.6. Lattices
3.7. Deformations
4. Local Langlands at primes ` 6= p
4.1. The Kirillov functor
4.2. Classical local Langlands for GL2 (Q` ).
4.3. Mod p local Langlands for GL2 (Q` ).
4.4. Local Langlands in p-adic families for the groups GΣ0 .
5. $-adically completed cohomology and Hecke algebrasb 1 and some related spaces
5.1. HO
5.2. Hecke algebras
b1 .
5.3. The ρ-part of H
O
5.4. A density result
b1
5.5. The Galois action on HO,ρ,Σ5.6. Ordinary forms
1
5.7. Multiplicities of weights in Hk,ρ,Σ
.
b1
6. Local-global compatibility and the structure of HO,ρ,Σ6.1. A refined local-global compatibility conjecture
6.2. Main results
6.3. The Tρ,Σ [GΣ0 ]-modules X(θ)
6.4. Evaluation maps
6.5. Proof of Theorem 6.4.20
7. Applications
7.1. Applications to the Fontaine–Mazur conjecture
The author was supported in part by NSF grants DMS-0401545 and DMS-0701315.
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7.2. Applications to a conjecture of Kisin
7.3. Promodularity
7.4. Locally algebraic vectors
Appendix A. Banach modules
Appendix B. Orthonormalizable and pro-free modules
Appendix C. Coadmissible representations
References

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95
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98
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116

1. Introduction
The aim of this paper is to establish (under certain technical hypotheses) the
local-global compatibility conjecture introduced in [38]. As one application, we
obtain results related to the Fontaine–Mazur conjecture on two-dimensional Galois
representations.
1.1. The local-global compatibility conjecture. In [38] we made the following
local-global compatibility conjecture (Conjecture 1.1.1 of that reference):
1.1.1. Conjecture. If V is an odd irreducible continuous two-dimensional representation of GQ , defined over a finite extension E of Qp , and unramified outside of ab 1 ) decomposes
finite set of primes, then the GL2 (Af )-representation HomE[GQ ] (V, H
E
as a restricted tensor productO∼0b 1E ) −→π` (V ).
HomE[GQ ] (V, H
B(V |GQp ) ⊗`6=p

b 1 denotes the p-adically completed cohoIn the statement of the conjecture, H
E
mology of the tower of modular curves (as defined in [35]; see also Subsection 5.1
below), B(V |GQp ) denotes the admissible unitary Banach space representation of
GL2 (Qp ) associated to V |GQp via the p-adic local Langlands correspondence, andfor each prime ` 6= p, π` (V ) is the admissible smooth representation of GL2 (Q` )associated to V |GQ` via (a modified version of) the classical local Langlands correspondence. (The representation that we here denote π` (V ) was denoted by π`m (V )in [38]. It coincides with the representation attached to V |GQ` by the classical local
Langlands correspondence with respect to the Tate normalization, except in thosecases in which the latter representation is not generic (i.e. not infinite-dimensional).
See Subsection 4.2 below for a precise specification of π` (V ).)1.1.2. Remark. In fact, Conjecture 1.1.1 is likely false as stated in one particularcase — namely, when the local Galois representation V |GQp is a twist of an extensionof the cyclotomic character by the trivial character; see Remark 6.1.23 below.
At the time that [38] was written, the existence of a p-adic local Langlands correspondence for two-dimensional p-adic representations of GQp was still conjectural.
However, such a correspondence has now been constructed by Colmez and Kisin[25, 61]. We rely heavily on the deformation-theoretic formulation of the correspondence due to Kisin [61], not only to give Conjecture 1.1.1 a precise sense, butalso as a key tool in our arguments. For our application to the Fontaine–Mazur
conjecture, we also rely crucially on Colmez’s results on the non-vanishing of locallyalgebraic vectors [25, Thm. VI.6.18].

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

3

1.2. Statement of results. We work in the context of promodular Galois representations. Let E be a finite extension of Qp , with ring of integers O, residue
field k, and uniformizer $. Recall that a continuous irreducible two-dimensional
representation V of GQ over E is called promodular if it is isomorphic to the Galois
representation attached to a cuspidal p-adic modular eigenform of some (possibly
non-integral) weight, or equivalently, if the pseudo-representation attached to V
lies in the Zariski closure, in an appropriate deformation space, of the set of p-adic
pseudo-representations attached to classical cuspforms. (See e.g. [38, §7.3].)
The following theorem is our main result in the direction of Conjecture 1.1.1. It
is proved in Subsection 6.2 below.
1.2.1. Theorem. Let V be a continuous, irreducible, promodular two-dimensional
representation of GQ over E, unramified outside of a finite set of primes. Let V
denote the residual representation attached to V , and assume that the following
conditions are satisfied:(a) V is absolutely irreducible.
1 ∗for any k-valued character χ of GQp (where ∗ may(b) V |GQp 6∼ χ ⊗0 εor may not be zero, and ε denotes the mod p cyclotomic character).
The following conclusions then hold:(1) For some finite set of primes Σ containing the primes at which V is ramified, together with p, there is a non-zero GL2 (Qp ) × GL2 (AΣf )-equivariant
map
O0
b 1E ).
π` (V ) → HomE[GQ ] (V, H
B(V |GQp ) ⊗`6∈ΣIf, furthermore, V |GQp is neither the direct sum of two characters, nor an
extension of a character by itself, then any
 such map
 is an embedding.
1 ∗(2) If V is p-distinguished, i.e. V |GQp 6∼ χ ⊗for any k-valued char0 1
acter χ of GQp (where ∗ may or may not be zero), then there is a GL2 (Af )equivariant isomorphismO∼0b 1E ).
B(V |GQp ) ⊗π` (V ) −→ HomE[GQ ] (V, H`6=p

In fact, in the situation of part 2 of the theorem, we prove a stronger result, whichb 1 ) for those V lifting V ,describes not only the multiplicity spaces HomE[GQ ] (V, HE
b 1 itself (and even its unit ball Hb 1 ) obtained
but the structure of the summand of H
EO
by localizing at V . (See Theorem 6.2.13 below for the precise statement.) As wellas the p-adic local Langlands correspondence of [25, 61], this description involves
the local Langlands correspondence in families developed in [43] (and recalled in
Section 4 below).
As explained in [38], Theorem 1.2.1 has the following corollary (proved in detailas Theorems 7.1.1 and 7.2.1 below). It is here that we use Colmez’s result on the
non-vanishing of locally algebraic vectors, a result which was stated as one of theconjectural properties of the p-adic local Langlands correspondence in [38].
1.2.2. Corollary. Let V be a continuous, irreducible, promodular two-dimensional
representation of GQ over E, unramified outside of a finite set of primes. Let V

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MATTHEW EMERTON

denote the residual representation attached to V , and assume that the following
conditions are satisfied:(a) V is absolutely
 irreducible.
1 ∗(b) V |GQp 6∼ χ ⊗for any k-valued character χ of GQp (where ∗ may
0 ε
or may not be zero).
The following conclusions then hold:(1) If V |GQp is trianguline, then V is a twist of the Galois representation attached to an overconvergent p-adic cuspidal eigenform of finite slope.
(2) If V |GQp is de Rham with distinct Hodge–Tate weights, and if V |GQp is not
the direct sum of two characters that have isomorphic reductions modulo $,then V is a twist of the Galois representation attached to a classical cuspidal
eigenform of weight k ≥ 2.
Our requirement that V be promodular is little restriction, in light of the following theorem, which is a consequence of the results and methods of the papers[4, 31, 53, 54, 55, 60]. (See Subsection 7.3 below for the proof.)
1.2.3. Theorem (Böckle, Diamond–Flach–Guo, Khare–Wintenberger, Kisin).
Let V be a continuous, irreducible, odd, two-dimensional representation of GQ
over E, unramified outside of a finite set of primes Σ. Assume that the following conditions hold:(a) p > 2.
(b) V |GQ(ζp ) is absolutely irreducible.




1 ∗
1 ∗
or χ ⊗, for any k-valued character χ(c) V |GQp 6∼ χ ⊗0 ε
0 1
of GQp (where ∗ may or may not be zero).
Then V is promodular, of tame level divisible only by the primes in Σ \ {p}.
Combining this result with corollary 1.2.2, we obtain the following result.
1.2.4. Theorem. Let V be a continuous, irreducible, odd, two-dimensional representation of GQ over E, unramified outside of a finite set of primes, with associatedresidual representation V . Assume that the following conditions hold:(a) p > 2.
(b) V |GQ(ζp ) is absolutely irreducible.




1 ∗
1 ∗
(c) V |GQp 6∼ χ ⊗, for any k-valued character χor χ ⊗0 1
0 ε
of GQp (where ∗ may or may not be zero).
The following conclusions then hold:(1) If V |GQp is trianguline, then V is a twist of the Galois representation attached to an overconvergent p-adic cuspidal eigenform of finite slope.
(2) If V |GQp is de Rham with distinct Hodge–Tate weights, then V is a twistof the Galois representation attached to a classical cuspidal eigenform of
weight k ≥ 2.
The first of these results confirms (under the given hypotheses) a conjecture of
Kisin. (See [56, Conj. 11.8], as well as the discussion in note (2) on p. 450 of theLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

5

same paper.) The second of these results confirms (again for V satisfying the given
hypotheses) a conjecture of Fontaine and Mazur [46, Conj. 3c].
1.2.5. Remark. A result similar to part 2 of Theorem 1.2.4, but with slightly differKisin’s result, hypothesis (c)ent hypotheses on V , has been proved by Kisin[59]. In ε ∗is replaced by the condition that V |GQp 6∼ χ ⊗for any k-valued character0 1
χ of GQp (where ∗ may or may not be zero). As a corollary of his result, Kisin has
deduced Conjecture 11.8 of [56] (under the same set of hypotheses on V ), a result
which is related to, but somewhat weaker than, part 1 of our Theorem 1.2.4.
In fact, what Kisin established in [59] is that the Fontaine–Mazur conjecture fortwo-dimensional representations of GQ (under the additional technical hypotheses
just described) follows from the existence of a p-adic local Langlands correspondence for two-dimensional de Rham representations of GQp with distinct Hodge–
Tate weights, provided that this correspondence is compatible in a suitable mannerwith the classical correspondence. Such a correspondence, satisfying the requisitecompatibility, had already been constructed by Berger and Breuil in the crystabelline setting [3] and by Colmez in the semi-stable (up to a twist) setting [22].
As we have already remarked, the p-adic local Langlands correspondence for twodimensional representations of GQp has now been constructed in full generality by
Colmez and Kisin. In the case of two-dimensional representations that are potentially crystalline, but not crystabelline, we establish the compatibility with theclassical correspondence in Theorem 3.3.22 below, building on Colmez’s resultsin [25]. This completes the proof of the properties of the p-adic local Langlands
correspondence required to deduce the Fontaine–Mazur conjecture in such cases(i.e. potentially crystalline, but not crystabelline, locally at p) via the argumentsof [59].
Some further discussion of the relationship between our approach and Kisin’s
approach to the Fontaine–Mazur conjecture is given in Subsection 1.4 below.
Our local-global compatibility result (in the strong form provided by Theorem 6.2.13) has as a corollary the following theorem, which describes the multiplicities with which a mod p Galois representation can appear in the mod p étale
cohomology of a modular curve of arbitrary level.
1
Write Hk1 := lim
Hét(Y (N ), k), the inductive limit over all levels N of the étale−→N

cohomology, with coefficients in k, of the full (geometrically disconnected) modular
curve of level N . This is naturally a representation of GQ × GL2 (Af ).
1.2.6. Theorem. Let ρ : GQ → GL2 (k) be absolutely irreducible and modular (orequivalently, odd [53, 54, 55, 60]). Suppose furthermore that p > 2, and that ρ |GQp is




1 ∗
1 ∗
not equivalent to either χ⊗or χ⊗, for any k-valued character
0 10 ε
χ of GQp (where ∗ may or may not be zero). Then there is a k-linear and GL2 (Af )equivariant isomorphism
π⊗

O

0

∼

π(ρ |GQ ) −→ Homk[GQ ] (ρ, Hk1 ),
`

`6=p

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MATTHEW EMERTON

where the representation π of GL2 (Qp ) is attached to ρ |GQp via the mod p local
Langlands correspondence of Theorem 3.3.2 below, and for each ` 6= p, the representation π(ρ |GQ ) of GL2 (Q` ) is attached to ρ |GQ via the mod p local Langlands``
correspondence for GL2 (Q` ) given in [43] (and recalled in Theorem 4.3.1 below).
1.2.7. Remark. I expect the same result to hold when p = 2, but our imperfectunderstanding of the mod 2 and 2-adic local Langlands correspondence for GL2 (Q` )(` 6= 2) leads to a less definitive result in this case; see Remark 6.1.21.
1.2.8. Remark. Theorem 1.2.6 is related to, and indeed generalizes, classical “modp multiplicity one” results due to Mazur, Ribet, Wiles, and others (see e.g. [64, 72,
65, 48, 33, 86]). These results describe the multiplicity with which ρ appears in thecohomology of modular curves of level Γ1 (N ), when N is prime-to-p, or divisible
by at most a single power of p. They may be recovered from Theorem 1.2.6 (forthose ρ to which the theorem applies) by passing to the invariants under a suitablychosen compact open subgroup of GL2 (Af ); we should point out, though, thatthese results are among the tools that we use in the proof of Theorem 1.2.6. (Note,however, that we only use the more straightforward results of [64, 72, 65, 48, 33];
we do not use either the existence of companion forms [48] or [86, Thm. 2.1 (ii)]in the more difficult case when (in the notation of that reference) ∆(p) is trivial
mod m. Indeed, Theorem 1.2.6 gives a new proof of these results, for those ρ towhich it applies; see Remark 6.2.15 below.)
1.2.9. Remark. We conjecture that
1.2.6 continues to hold in the case
 Theorem
1 ∗when ρ |GQp is equivalent to χ ⊗, for some k-valued character χ of GQp
0 1


1 ∗
(with p > 3, so that ρ |GQp is not equivalent to χ ⊗, for any k-valued
0 ε


1 ∗
character χ of GQp ). In the remaining cases, when ρ is equivalent to χ⊗,
0 εfor some k-valued character χ of GQp , we also expect an analogous result to hold.
However, the correct definition of π in this case is more problematic (and in particular, is not provided by Theorem 3.3.2 below). See Remark 6.1.23 below for afurther discussion of this point.
1.2.10. Remark. In the paper [79] Serre stated his conjecture that any odd continuous irreducible representation ρ : GQ → GL2 (k) should be modular, of minimal
weight and prime-to-p level determined by the local properties of ρ at p and awayfrom p respectively. This conjecture has now been proved [53, 54, 55, 60]. In thesame paper, Serre raised the question as to whether his conjecture might admit areformulation in terms of the mod p representation theory of the adèlic GL2 [79,Question 3.4 (2)]. The preceding theorem gives a positive answer to Serre’s question (for those ρ to which it applies). Indeed, from this representation-theoretic
statement one can determine the minimal weight of ρ (which is determined by
the socle of π as a GL2 (Zp )-representation — see Lemma 3.5.5 below, as well as
the discussion in the proof of [14, Thm. 3.15]) as well as the minimal prime-to-plevel of ρ (which is equal to the conductor of the smooth GL2 (Apf )-representationN`6=p π(ρ |GQ )).
`As was already intimated in the discussion of Remark 1.2.5, as another application of our results we prove Theorem 3.3.22 below, which for any de Rham

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

7

representation with distinct Hodge–Tate weights V of GQp over E, gives a precise
description of the locally algebraic vectors in the Banach space representation B(V )of GL2 (Qp ) associated to V via the p-adic local Langlands correspondence. This
completes the results of Colmez in [25, §§VI.6], where this question is investigated,and completely solved in the case when V becomes crystalline or semi-stable overan abelian extension of Qp .
1.3. The organization of the paper. In Section 2 we introduce some notation
which will be in force throughout the paper. The next three sections are preparatory in nature. In Section 3 we recall various recent results (primarily due toColmez and Kisin) that construct the mod p and p-adic local Langlands correspondences, and establish some additional related results. In Section 4, we recall
some of the results of [43] regarding the local Langlands correspondence at primes` 6= p in characteristic p or over complete local rings of residue characteristic p. In
Section 5 we recall the construction of (and introduce notation for) the p-adicallycompleted cohomology of modular curves, as well as some related constructions
and results (in particular, for the mod p cohomology of modular curves). We give
representation-theoretic interpretations of several results from the literature in thiscontext, including the mod p multiplicity one results of [64, 72, 65, 48, 33], as well
as some of the results from the theory of p-adically ordinary modular forms.
Section 6 presents our main results. As already noted above, in addition to
Theorem 1.2.1, which describes the multiplicities with which various Galois representations appear in p-adically completed cohomology, we prove (in many cases) astronger result (Theorem 6.2.13), which gives a complete description (as a representation of GQ × GL2 (Af )) of the space of p-adically completed cohomology itself(after localizing at an appropriately chosen two-dimensional mod p Galois representation). Finally, in Section 7, we deduce the various modularity results statedabove.
In the appendices we develop some basic functional analysis that is required in
the main body of the text.
1.3.1. Remark. It may be useful, for those readers who are primarily interested
in the proof of Theorem 1.2.4, to indicate those parts of the paper that are necessary for the proof of this result. The proof of the theorem (other than part 2of the theorem in the case when V|GQp is the direct sum of two characters) requires the material of Subsections 3.1–3.4 and 5.1–5.4; the following material from
Section 6: Definition 6.1.1, Theorem 6.2.1, Proposition 6.2.2, the proof of Theorem 1.2.1 (which is given in Subsection 6.2), Subsection 6.3 up to and includingm
Corollary 6.3.17 (always taking θ = πΣ), and Subsection 6.4 up to Remark 6.4.3;
Subsections 7.1–7.3; and the material of the appendices. The same material (butsubstituting Subsection 7.4 in place of Subsections 7.1–7.3) serves to give the proofof Theorem 3.3.22.
The remainder of the paper (including all the material in Section 4) is devoted
to proving the precise local-global compatibility of Theorem 1.2.1 (2). The mostdifficult case of the argument is when V |GQp is a direct sum of two characters,
and a large part of the paper is devoted solely to dealing with this case; this istrue in particular of Subsections 3.6, 3.7, 5.5, 5.6, and 6.5. Our proof of part 2 ofTheorem 1.2.4 (i.e. the Fontaine–Mazur conjecture) in the case when V|GQp is the
direct sum of two characters requires the full strength of all this material; on the

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MATTHEW EMERTON

other hand, from the point of view of the Fontaine–Mazur conjecture (as opposed to
local-global compatibility), this case is little interest, since Skinner and Wiles have
proved much stronger results in the direction of the Fontaine–Mazur conjecture in
the ordinary case [80, 81].
1.4. Background and additional remarks. In this section we describe someof the previous work on the problems considered in this paper, and compare ourapproach to that of the earlier work. We will begin by discussing the FontaineMazur conjecture [46, Conj. 3c]. In fact, the version of the conjecture that we
address in this paper is the following:
1.4.1. Conjecture. If V is an odd irreducible continuous two-dimensional representation of GQ , defined over a finite extension E of Qp , unramified outside ofa finite set of primes, and de Rham at p with distinct Hodge–Tate weights, thenV is a twist of the Galois representation attached to a classical modular form of
weight k ≥ 2.
1.4.2. Remark. It is believed that the condition of being odd can be omitted
from this conjecture, i.e. that it is a consequence of the other hypotheses (see [46,Conj. 3c]). Recently, this has been verified by Calegari [15] in the case when V isordinary at p (under the technical assumptions that p ≥ 7, and that the associatedresidual representation has image containing SL2 (Fp )).
The original breakthrough in the direction of Conjecture 1.4.1 was made in thework of Wiles [86] and Taylor–Wiles [83]. Further progress was made in the papers[30], [26], [11], [32], [31], [82], [57], [58], [59], [60]. Each of these papers, otherthan [59], establishes a result in the direction of Conjecture 1.4.1 of the following
form: if V is as in the statement of the theorem, and if furthermore p > 2 (except
in [32, 60], which treat the case p = 2), V is modular and V |GQ(ζp ) is irreducible(where V denotes the residual Galois representation attached to V ),1 and V satisfies
some additional restrictions, both on the Hodge–Tate weights of V |GQ(ζp ) and (withthe exception of [57] and [60]) on the amount of ramification occurring in the typeof the de Rham representation V |GQp , then V is modular.2 (For a discussion of theresult of [59], see Remark 1.2.5 above, as well as the remarks below.) Each of these
papers also follows (in broad outline) the strategy originally implemented in [86]and [83], namely, to consider a deformation ring R parametrizing deformations of
V of the same Hodge–Tate weights and type as V , as well as a Hecke algebra T
parametrizing modular deformations of V of these weights and type, and then to
prove that the natural map R → T is an isomorphism.
In their paper [47], Gouvêa and Mazur prove that, under sufficiently strong
hypotheses on V , every deformation of V is promodular, i.e. is attached to a padic modular form. More precisely, Gouvêa and Mazur considered an unrestricted
1In each of the papers [32] and [60], both of which consider the case p = 2, the irreducibility

condition on V |GQ(ζ ) — the so-called Taylor–Wiles non-degeneracy condition — is replaced bypa slight variant.
2We should mention that in the case when V is assumed to be ordinary, or nearly ordinary,at p, it is not necessary to impose any restrictions on the Hodge–Tate weights of V , essentiallybecause Hida theory [50] provides a very tight link between the p-ordinary eigenforms in anarbitrary weight k ≥ 2 and the p-ordinary eigenforms in weight 2. The papers [80] and [81] provevery strong results in the direction of Conjecture 1.4.1 in the p-ordinary case; in particular, theysucceed in eliminating the Taylor–Wiles non-degeneracy condition in this case.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

9

deformation ring R associated to V , as well as a Hecke algebra T acting on an appropriate space of p-adic modular forms, and proved that the natural map R → T
is an isomorphism. A little later, Böckle [4] was able to combine the results of[86, 83, 30] with the argument of [47] to prove in some generality that an arbitrary
deformation of a modular V is necessarily pro-modular. (Theorem 1.2.3 above provides a generalization of Böckle’s result.) The results of [47] and [4] are sometimes
called “big R equals big T” theorems, to distinguish them from the “small R equals
small T” theorems discussed in the preceding paragraph.
When they were first proved, the “big R equals big T” result of [47, 4] appeared
to have no bearing on Conjecture 1.4.1, since one seemed no closer to proving that
conjecture in the case of V associated to p-adic modular forms than in the caseof arbitrary V .3 However, in [38] we showed that Conjecture 1.1.1 for V implies
Conjecture 1.4.1 for V , provided that the p-adic local Langlands correspondence
exists, and has sufficiently nice properties. This suggested the approach to Conjecture 1.4.1 that we take in this paper, namely to prove Conjecture 1.1.1 for all
promodular deformations of a given modular V , and then to combine this resultwith a “big R equal big T” theorem to deduce Conjecture 1.4.1 for all deformationsof V . In fact, one can hope to implement this strategy for any odd V , since, as werecalled above, Serre’s celebrated conjecture on the modularity of such a V [79] has
now been proved [53, 54, 55, 60].
Of course, this strategy depends on the existence of a p-adic local Langlands
correspondence with the required properties. The original idea of the p-adic local
Langlands correspondence for two-dimensional representations of GQp is due to
Breuil [9], who conjectured the form that such a correspondence should take in thecase of de Rham representations with distinct Hodge–Tate weights. In [10], Breuil
also formulated a version of Conjecture 1.1.1, for those V associated to classicalmodular forms of weight k ≥ 2. Some results in the direction of this conjecture
were obtained in [10], [12], and [38].
To the best of my knowledge, the suggestion that the p-adic local Langlands
correspondence should exist for arbitrary two-dimensional representations of GQp(rather than just de Rham representations with distinct Hodge–Tate weights, asconsidered by Breuil) was first made by Colmez (in his talk at the Durham symposium in the summer of 2004). This in turn suggested our generalization of Breuil’s
local-global compatibility conjecture (i.e. Conjecture 1.1.1).
Prior to the workshop on p-adic representations held in Montréal in September 2005, Conjecture 1.1.1 seemed inaccessible; indeed, the p-adic local Langlands
correspondence was not yet defined in general. After the Montréal workshop, thesituation changed completely. Colmez, in his talk at the workshop, explained theconstruction of a functor from admissible smooth GL2 (Qp )-representations over Artinian Zp -algebras to (ϕ, Γ)-modules (and hence to local Galois representations); see[25] and Subsection 3.2 below. Soon after the workshop, Kisin explained to Colmez
how his functor could be used to provide a deformation-theoretic construction of thep-adic local Langlands correspondence for GL2 (Qp ), provided that a certain map
on Exts was injective. Colmez quickly saw how to prove this injectivity result [25]3The only prior result that we are aware of that considers Conjecture 1.4.1 specifically for Galois
representations arising from p-adic modular forms is the paper [56] of Kisin, which essentiallyproves the conjecture in the case of representations associated to overconvergent p-adic eigenformsof finite slope.

10

MATTHEW EMERTON(which has since been proved in greater generality by Paškūnas [67]), and as a consequence he and Kisin were able to construct the previously hypothetical p-adic local
Langlands correspondence. After learning of this construction of Colmez and Kisin,
I realized that using it (and especially, exploiting its deformation-theoretic formulation), one could deduce (many cases of) local-global compatibility for promodular
Galois representations (i.e. Theorem 1.2.1 above), and hence Theorem 1.2.4 in the
direction of the Fontaine–Mazur conjecture.
Already in his talk at the Montréal conference, Kisin had announced a resulton the Fontaine–Mazur conjecture, essentially identical to part 2 of Theorem 1.2.4
above, in the case when V |GQp is absolutely irreducible and V |GQp is crystabelline.
His argument also relied on results from the p-adic local Langlands correspondence,in particular, the results of [3]. Indeed, as we already noted in Remark 1.2.5, it
is not the triangulanity of V |GQp which is crucial to Kisin’s method, but rather
the compatibility between the p-adic and classical local Langlands correspondences
for V |GQp , which at the time of the Montréal conference was known only in thetrianguline context [3, 22]. Soon after the conference, Kisin was able to simplify
his arguments through the use of Colmez’s functor, while simultaneously generalizing them so as to include the case when V |GQp is reducible and (now that thecompatibility between the p-adic and classical local Langlands for two-dimensional
representations of GQp has been completely proved) V |GQp is not necessarily trianguline. (See [59, Thm., p. 642] and the discussion following the statement of thattheorem.)Despite the similarity of several ingredients, the approach to the Fontaine–Mazur
conjecture taken in this paper is quite different from that of [59]. The argumentof [59] involves a “small R equals small T” theorem in the spirit of the original
approach of Wiles [86] and Taylor–Wiles [83]; Colmez’s functor and the p-adic local
Langlands correspondence appear as tools for obtaining control over the complicated singularities of the deformation spaces that can arise when one makes norestrictions on the weight and type. On the other hand, the argument of this paper
relies on a “big R equals big T” theorem. It is local-global compatibility, rather
than a Taylor–Wiles-type argument, that then allows one to characterize the classical points among the promodular ones in terms of their p-adic Hodge theoretic
properties.

1.5. Acknowledgments. The debt that this paper owes to the ideas of Christophe
Breuil, Pierre Colmez, and Mark Kisin regarding the p-adic local Langlands correspondence will be evident to the reader. In particular, I would like to thank
Pierre Colmez and Mark Kisin for sharing their ideas and results with me. I would
also like to thank Mark Kisin, as well as Christophe Breuil, Kevin Buzzard, Frank
Calegari, Brian Conrad, Toby Gee, James Newton, and Robert Pollack, for theirhelpful conversations and correspondence related to this article.
I gave lectures on some of the results of this article at a workshop at the American Institute of Mathematics in February 2006, and at the special semester on
eigenvarieties at Harvard in April 2006. I would like to thank both institutions, aswell as Kiran Kedlaya and David Savitt (who organized the AIM workshop) andBarry Mazur (who organized the special semester on eigenvarieties at Harvard).

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

11

Finally, I would like to thank Adrian Iovita and Henri Darmon, as well as theCentre de Recherches Mathématiques at the University of Montréal, for organizing
and hosting the wonderful workshop on p-adic representations in September 2005.
2. Notation and conventions
Throughout the paper, we fix a prime p, as well as a finite extension E of Qp ,with ring of integers O. We let k denote the residue field of O, and $ a choiceof uniformizer of O. Let Art(O) denote the category of local Artinian O-algebras
whose residue fields are a finite extension of k (or equivalently, which are of finite
length as O-modules), and let Comp(O) denote the category of complete Noetherian
local O-algebras whose residue fields are a finite extension of k.
We let Q denote the algebraic closure of Q in C, and for each prime ` we choose an
algebraic closure Q` . As usual, we write GQ to denote the Galois group Gal(Q/Q)
and GQ` to denote the Galois group Gal(Q` /Q` ). For each prime ` we also fix∼
an isomorphism C −→ Q` (whose choice, however, will play no overt role). This
determines an embedding Q ,→ Q` , and hence an embedding GQ` ,→ GQ , via which
we identify GQ` with a decomposition group at ` in GQ . We also fix an embedding
E ,→ Qp (the choice of which, again, will play no overt role). Given this choice,∼together with our choice of isomorphism C −→ Qp , it makes sense to speak ofmodular forms being defined over E, or of a C-linear representation of a group
being defined over E.
∼ Q
b denote the profinite completion of Z, so Zb −→We let Z` prime Z` . We let Afbdenote the ring of finite adèles over Q; so Af = Q ⊗Z Z. More generally, for anyΣbΣb Σ := Q
finite set of primes Σ we write Z`6∈Σ Z` and Af := Q ⊗Z Z .
We will frequently employ the following notation related to finite sets of primes:
we will let Σ0 denote a finite set of primes that does not contain p, and then writeΣ := Σ0 ∪ {p}.
abFor each prime `, local class field theory gives an embedding Q×` ,→ GQ` (thelocal Artin map), which we normalize by mapping ` to a lift of geometric Frobenius.
Global class field theory gives an isomorphism∼× ∼abb × −→Z
Q×>0 \Af −→ GQ

b × ,→ A× ), which we(the first isomorphism being induced by the embedding Z
fnormalize to be compatible with our chosen normalization of the local Artin maps.
×
We let ε : GQp → Z×p denote the p-adic cyclotomic character, and ε : GQp → Fp
the mod p cyclotomic character (i.e. the mod p reduction of ε). We also regard ε
and ε as characters of Q×
p , via local class field theory.
For any prime ` we let | |` denote the `-adic absolute value on Q` , normalized
so that |`|` = `−1 . Note that the if ` 6= p, then the p-adic cyclotomic character of
GQ` corresponds to the character | |` of Q×` via local class field theory.
If V is any GQp -representation, then we write V ab to denote the maximal subrepresentation of V on which GQp acts through its maximal abelian quotient. In otherwords, V ab denotes the set of fixed points of the commutator subgroup of GQp .
We say that a continuous finite-dimensional representation of GQp is crystabelline
if it becomes crystalline after restricting to the Galois group of an abelian extensionof Qp .

12

MATTHEW EMERTON

We write GQ:= GL2 (Qp ). If Σ0 is a finite set of primes not containing p, then we
write GΣ0 := `∈Σ0 GL2 (Q` ). We let T denote the diagonal torus in G, namely


 a 0
T :=
| a, d ∈ Q×
⊂ G,
p
0 d
and let S denote the following subtorus of T :


 a 0
S :=
| a ∈ Q×
⊂ T.
p
0 1
×
We will identify S with Q×
p in the evident manner, i.e. via the isomorphism Qp 3


a 0
a 7→
∈ S. We let B denote the upper triangular Borel subgroup of G, with
0 1
unipotent radical N (so B = T N ), and we write P = SN ⊂ B. We let B denotethe lower triangular
Borel subgroup of G, with unipotent radical N (so B = T N,
T
of T via χ1 ⊗ χ2 , where the χiand T = B B). We typically denote a charactera0
are characters of Q×p (and χ1 ⊗ χ2 : 0 d 7→ χ1 (a)χ2 (d)).
If H is any locally compact group, and A is a ring, then we say that a representation of H on an A-module M is smooth if every element m ∈ M is fixed bysome open subgroup of H. We say that the representation is smooth admissible if0for every open subgroup H 0 of H, the A-submodule M H of M , consisting of H 0 invariant elements, is finitely generated. In the case when H = G (:= GL2 (Qp )), we
will apply some additional terminology in the case of non-smooth representations,based in part on that of [39]. For precise definitions and terminology, the reader isreferred to Subsection 3.1.
For any profinite group H, and any object A of Comp(O), having maximal idealm, we let A[[H]] denote the completed group ringA[[H]] := lim
A[H/H 0 ],
←−
H0

0

where H runs over the open subgroups of H, each of the rings A[H/H 0 ] is equippedwith its m-adic topology, and A[[H]] is equipped with the projective limit topology.
The ring A[[H]] is then a profinite topological ring.
If V is any E-vector space, by a lattice (or O-lattice) in v, we will always meana $-adically separated O-submodule of V that spans V over E. Note that if V isan E-Banach space, then an O-submodule of V is $-adically separated if and only
if it is bounded, and is furthermore $-adically complete if and only if it is closed
in V (or equivalently, open, since any Banach space is barrelled).
If V is a representation of G, then we let Vl.alg denote the G-subrepresentationof V consisting of the locally algebraic vectors (as defined in [34, Prop.-Def. 4.2.6]).
If we let Ĝ denote the set of isomorphism classes of irreducible finite-dimensional
algebraic L
representations of G over E, then Vl.alg is the image of the natural evaluaHomH (W, V ) ⊗E W → V (the inductive limit being taken overtion map W ∈Ĝ lim−→
H

the directed set of all open subgroups H of G); see [34, Prop. 4.2.4, Cor. 4.2.7]. If
H is a fixed open subgroup of G, then we write VH−alg to denote the E-subspace ofV consisting of H-algebraic
vectors, which we define to be the image of the naturalLevaluation map W ∈Ĝ HomH (W, V ) ⊗E W → V .
We normalize the local Langlands correspondence for GL2 (Q` ) (for any prime `)in the following way: if π = IndGχ ⊗ χ2 is an irreducible parabolically induced
B 1

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

13

representation (with the χi being smooth characters of Q×
p with values in the multiplicative group of some characteristic zero field), then we define WD(π), the associated Weil–Deligne representation (which in the case is simply a representationof the Weil group) to be χ1 ⊕ χ2 | |−1` , where we now regard the χi as characters of∼ab
Wp , via the reciprocity isomorphism Q×
p −→ Wp of local class field theory. (See
Subsection 4.2 for a further explication of our normalization.)
In Sections 5 and 6 below, we consider the action of GQ × GL2 (Af ) on the inductive limit of the cohomology of modular curves, and on its p-adic completion. We
follow the same conventions for this action as in the reference [38]. Unfortunately,
these conventions do not agree with those of [16]: if g ∈ GL2 (Af ), then the automorphism of the cohomology of modular curves induced by g that we consider inthis paper is equal to the automorphism induced by the inverse transposed element(g −1 )t under the action considered in [16].
3. Mod p and p-adic local Langlands
Our goal in this section is to describe certain relationships between, on the
one hand, the mod p and p-adic representation theory of G, and, on the other,
two-dimensional mod p and p-adic representations of the local Galois group GQp ,relationships which are known respectively as the mod p and p-adic local Langlands
correspondence. In the remainder of these introductory remarks we briefly recall
the history of these two correspondences.
The semi-simple mod p local Langlands correspondence (so-called because it
does not take into account possible extension classes) was constructed by Breuil[8, Déf 1.1]. The possible existence of a p-adic local Langlands correspondence wasoriginally suggested by Breuil in [8, 9]. Substantial progress on its development
was made in the papers [22, 3, 23, 2], after Colmez observed a relationship between
some of the constructions introduced by Breuil in [9] (in the context of the representation theory of G) and certain constructions in the theory of (ϕ, Γ)-modules(which provide a way of describing GQp -representations).
In his talk at the Montréal conference in September 2005, Colmez further developed this relationship by defining an actual functor from G-representations to
GQp -representations. Upon seeing Colmez’s talk, Kisin suggested that this functor
might be combined with a certain deformation-theoretic argument to construct thep-adic local Langlands correspondence in general. Kisin’s suggestion was predicatedupon the construction of a mod p correspondence that enhanced the correspondence
considered in [8, 2] by taking extensions into account. Such an enhanced mod p
correspondence has now been constructed by Colmez (in most cases — see Theorem 3.3.2 below), and building on this and Kisin’s deformation-theoretic strategy,
the p-adic local Langlands correspondence has now been constructed – see [25, 61],
as well as Theorem 3.3.13, Definition 3.3.15, and Remark 3.3.19 below.
In fact, in the paper [25] Colmez goes significantly further than he went in
his Montréal lecture, and as well as defining a functor from G-representations to
GQp -representations, he gives a construction going in the opposite direction, fromtwo-dimensional GQp -representations to G-representations. In this way he is able toconstruct the p-adic local Langlands correspondence without directly using deformation theory (although the construction of the functor still employs a version of
Kisin’s deformation-theoretic strategy). However, for the purposes of this paper, we
require Kisin’s full-blown deformation-theoretic formulation of the correspondence.

14

MATTHEW EMERTON

3.1. Representations of G. Let A be an object of Comp(O), with maximal
ideal m. In the paper [39], we introduced various categories of representations of
the group G over A.4 For the purposes of this paper, the most important of theseis the category of $-adically admissible G-representations over A, whose definition
we will recall here.
3.1.1. Definition. An A[G]-module π is called a $-adically admissible representation of G over A if the following conditions are satisfied:(1) π is $-adically complete and separated.
(2) The O-torsion subspace π[$∞ ] is of bounded exponent, i.e. π[$i ] = π[$∞ ]for sufficiently large values of i.
(3) For any i ≥ 0, the induced A[G]-action on π/$i π satisfies the following
condition: for any element v ∈ π/$i π, there is some compact open subgroup H of G and some j ≥ 0 such that v is fixed by H and annihilated
by mj .
(4) The A/m-vector space (π/$π)[m] forms an admissible smooth representation of G under the induced G-action.
(A) the category of $-adically admissible represenWe denote by Mod$−adm
Gtations of G over A. If A is in fact an object of Art(O), then an A[G]-module
is $-adically admissible if and only if it is a smooth admissible representationof G in the usual sense, and hence we will also denote this category simply by$−adm
Modadm(A) for non-Artinian A are proG (A). Examples of objects in ModG
vided by Lemma 3.1.16 below.
3.1.2. Caution. The reader who consults [39] should be aware that, in the casewhen A is not Artinian, the definitions used in [39] of a smooth G-representation
over A, and of an admissible smooth G-representation over A, are not the standard
ones. Throughout this paper, we will always use the standard terminology, as
explained in Section 2.
If π is a $-adically admissible representation of G over A, then the tensor product
E ⊗O π has a natural structure of E-Banach space (given by taking the norm whose
unit ball is the image of π in E ⊗O π). The actions of each of A and G on E ⊗O π
will then be jointly continuous. (We equip A with its m-adic topology.)
3.1.3. Proposition. If π1 and π2 are two objects of Mod$−adm(A) and π1 → π2
Gis an A[G]-linear morphism, then the induced map on E-Banach spaces(3.1.4)

E ⊗O π1 → E ⊗O π2

necessarily has closed image.
Proof. Replacing π1 and π2 by their maximal O-torsion free quotients (which again
lie in the category Mod$−adm(A), by [39, Lem. 2.4.8]), we may assume that eachG
πi is O-torsion free, and via the natural map πi → E ⊗O πi , may identify πi withthe unit ball in the Banach space E ⊗O πi . Furthermore, replacing π1 by its imagein π2 (which lies in the category Mod$−adm(A), by [39, Prop. 2.4.12]), we mayGassume that the map π1 → π2 , and hence also the map (3.1.4), is injective. From
[39, Prop. 2.4.13] we see that the map π1 → π2 has closed image, and that the
4Although we will only apply the results of this subsection in the case when G = GL (Q ), inp
2fact they make sense and are valid for any locally p-adic analytic group G.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

15$-adic topology on π2 induces the $-adic topology on π1 . This implies that themap (3.1.4) has closed image, as required.

3.1.5. Remark. In the case when A = O, the preceding proposition is a consequence of the results of [77].
3.1.6. Lemma. If π1 and π2 are two object of Mod$−adm(A) that are torsion freeG
as O-modules, and if
π1 → π2(3.1.7)

is an A[G]-linear morphism for which the induced map on E-Banach spaces (3.1.4)
is injective, then the given map (3.1.7) is an isomorphism if and only if the map(3.1.8)(π1 /$π1 )[m] → (π2 /$π2 )[m],obtained by reducing (3.1.7) modulo $ and then passing to the m-torsion parts of
its source and target, is injective.
Proof. Clearly the given condition on (3.1.8) is necessary. We will show that it isalso sufficient.
The assumptions of the lemma imply that (3.1.7) is injective, with O-torsion
cokernel. If we denote this cokernel by π3 , then since π1 and π2 are both $-adicallyadmissible smooth representations of G over A, it follows from [39, Prop. 2.4.13]
that the same is true of π3 .
The map (3.1.7) sits in the short exact sequence
0 → π1 → π2 → π3 → 0,
which induces the exact sequence(3.1.9)0 → π3 [$] → π1 /$π1 → π2 /$π2 .

Since π3 is a $-adically admissible representation of G over A that is torsion as an
O-module, we see that every element of π3 is killed by some power of m. On theother hand, if (3.1.8) is injective, then we see from a consideration of (3.1.9) that
π3 [m] = π3 [$, m] = 0. It then follows that in fact π3 = 0, and thus that (3.1.7) is
an isomorphism, as claimed.

We next recall the definition of a finitely generated augmented representation of
G over A [39, §2.1].
3.1.10. Definition. By a finitely generated augmented representation of G overA we mean an A[G]-module M equipped with a finitely generated A[[H]]-module
structure for some (equivalently, any) compact open subgroup H of G, such thatthe two induced A[H]-actions (the first induced by the inclusion A[H] ⊂ A[[H]]and the second by the inclusion A[H] ⊂ A[G]) coincide.
aug
We let Modfg(A) denote the category of finitely generated augmented repreGsentations of G over A, with morphisms being maps which are simultaneously A[G]aug
linear and A[[H]]-linear. Each object of Modfg(A) is equipped with its canonicalG
topology [39, Prop. 2.1.2], making it a profinite topological A[[H]]-module for eachcompact open subgroup H of G.
We now introduce introduce a class of representations that plays an importantrole in the deformation theory of G-representations.

16

MATTHEW EMERTON

3.1.11. Definition. We say that an A[G]-module π is an orthonormalizable admissible representation of G over A if the following conditions are satisfied:(1) π is an orthonormalizable A-module, in the sense of Definition B.1.
(2) The induced G-action on π/mi π makes this quotient an admissible smooth
representation of G over A/mi , for each i ≥ 0.
3.1.12. Proposition. The functor π 7→ HomA (π, A) induces an equivalence of categories between the category of orthonormalizable admissible representations of G
over A (with morphisms taken to be A[G]-linear maps), and the full subcategory
augof Modfg(A) consisting of finitely generated augmented representations of G on
G
pro-free A-modules. (See Definition B.10 for the notion of a pro-free A-module.)
Proof. Proposition B.11 shows that π 7→ HomA (π, A) induces an equivalence of
categories between the category of orthonormalizable A-modules and the categoryof pro-free profinite A-modules, and also that there is an isomorphism∼

HomA (π, A)/mi HomA (π, A) −→ HomA/mi (π/mi π, A/mi ),for each i ≥ 0. From this isomorphism, one sees that if π is equipped with an
A-linear G-action, then this action induces an admissible smooth action on π/mi π
if and only if the contragredient action on HomA (π, A) makes HomA (π, A)/mi afinitely generated augmented G-representation over A/mi . Since this latter condition holds for all i > 0 if and only if HomA (π, A) is a finitely generated augmented
G-representation over A, the proposition follows.

We note for the record that there is a broader class of representations containing
the orthonormalizable ones, which we now define, although we will have no need toconsider representations in this class that are not orthonormalizable.
3.1.13. Definition. We say that an A[G]-module π is an m-adically admissible
representation of G over A if the following conditions are satisfied:(1) π is m-adically complete.
(2) The action map G × π → π is continuous, when π is given its m-adic
topology. (Equivalently, the induced action of G on π/mi π is smooth foreach i ≥ 0.)(3) The induced representation of G on π/mi (which is smooth, by (2)) is admissible. (It is in fact enough to assume this for π/mπ, since a smooth extensionof admissible representations is again admissible [39, Prop. 2.2.13].)
Related to this definition, we can define the class of admissible unitarizable
Banach representations of G on modules over certain localizations of A.
3.1.14. Definition. If p ∈ Spec A is a prime ideal of A, then an admissible unitary
Banach representation of G over Ap consists of a Banach module π over Ap (inthe sense of Definition A.3), equipped with an action of G, such that Ap admits achoice of unit ball π 0 which is G-invariant, and which is m-adically admissible as a
G-representation over A.
3.1.15. Remark. Other than in Subsection 3.6, we will have no cause to consider
the preceding definition other than in the case when A = O and p is the zero ideal,
so that Ap = E. In this case, the above definition reduces to the usual notion ofan admissible unitary Banach representation of G over E.
We close this subsection with some technical lemmas.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

17

3.1.16. Lemma. If π is an orthonormalizable admissible A[G]-module, and if X is
ˆ A X is aa cofinitely generated A-module (in the sense of Definition C.1), then π ⊗$-adically admissible G-representation over A.
aug
Proof. Let M := HomA (π, A) be the object of Modfg(A) corresponding toG
π under the anti-equivalence of Proposition 3.1.12. Proposition C.5 shows that
HomO (X, O) is finitely generated over A, and hence M ⊗A HomO (X, O) is again
augan O-torsion free object of Modfg(A). It follows from [39, Prop. 2.4.11] and its
Gproof that the continuous O-dual of M ⊗A X is an object of Mod$−adm(A). OneGˆ A X.
easily verifies that this dual is isomorphic to π ⊗3.1.17. Lemma. If M is a finitely generated A-module, if π is an orthonormalizableadmissible A[G]-module, and if X is a cofinitely generated A-module (in the senseof Definition C.1), then the natural map
ˆ A π → HomA (M, X ⊗ˆ A π)
HomA (M, X)⊗
is an isomorphism.
Proof. Part 3 of Lemma B.6 shows that the natural map
HomA (M, X) ⊗A π → HomA (M, X ⊗A π)
is an isomorphism. Passing to $-adic completions, we obtain an isomorphismˆ A π → HomA (M, X ⊗A π)b.
HomA (M, X)⊗
Thus it remains to show that the natural map
ˆ A π)
HomA (M, X ⊗A π)b→ HomA (M, X ⊗is an isomorphism.
If we choose a presentation As → Ar → M → 0 of A, then we may form thediagram(3.1.18)0/ HomA (M, X ⊗A π)/ (X ⊗A π)r/ (X ⊗A π)s0/ HomA (M, X ⊗ˆ A π)

/ (X ⊗
ˆ A π)r/ (X ⊗
ˆ A π)swhose rows are exact. Thus it suffices to show that the bottom row is obtained fromˆ Aπ
the top row by passing to $-adic completions. Lemma 3.1.16 shows that X ⊗is a $-adically admissible G-representation over A, and so [39, Prop. 2.4.12] showsthat the O-torsion part of the cokernel of the upper right arrow in (3.1.18) has
exponent bounded independently of i. Since∼

ˆ A π)/$i (X ⊗
ˆ A π)(X ⊗A π)/$i (X ⊗A π) −→ (X ⊗for any i, we conclude that the sequence
0 → HomA (M, X ⊗A π)/$i HomA (M, X ⊗ π) → (X ⊗A π)/$i (X ⊗A π))r
→ (X ⊗A π)/$i (X ⊗A π))s
induced by the top row of (3.1.18) is exact modulo A-modules that are torsion
over O, of exponent bounded independently of i. Passing to the projective limit
in i, we thus obtain an exact sequenceˆ A π)r → (X ⊗
ˆ A π)s ,0 → HomA (M, X ⊗A π)b→ (X ⊗

18

MATTHEW EMERTON

which must then coincide with the bottom row of (3.1.18). This completes theproof of the lemma.

3.1.19. Lemma. Suppose that A is finite and flat over O, and reduced, and that
π is an orthonormalizable admissible A[G]-module with the property that for every
minimal prime p of A, the tensor product κ(p) ⊗A π is an absolutely topologicallyirreducible Banach space representation of G over κ(p). Then if M is any finitely
generated A-module that is flat over O, the natural map M → HomA[G] (π, M ⊗A π)
is an isomorphism.
Proof. Lemma B.7 shows that the natural map
M ⊗A End(π) → HomA (π, M ⊗A π)
is an isomorphism, and hence so is the natural map
M ⊗A EndA[G] (π) → HomA[G] (π, M ⊗A π).
∼

Thus it suffices to prove that A −→ EndA[G] (π).
∼
Our assumption on π implies that E ⊗O A −→ EndA[G]−cont (E ⊗A π) (where
the target denotes the space of continuous A[G]-module endomorphismsof the ET
Banach space E ⊗A π). Since by Lemma B.8 we have A = E ⊗O A EndA (π) (theintersection taking place in EndE⊗O A (E ⊗A π)), the present lemma follows.

3.1.20. Lemma. If π is a smooth G-representation over A, then a finitely generated
A-submodule of π is B-invariant if and only if it is G-invariant.
Proof. Any finitely generated A-submodule of π is fixed by some compact open
subgroup H of G. Since H and B together generate G, the lemma follows.

3.2. Colmez’s magical functor. Let A be an object of Art(O). As was alreadymentioned, Colmez [25] has defined an exact functor
5
MF : full subcategory of Modadm
G (A) consisting of objects of finite length−→category of continuous GQp -representations
on finitely generated A-modules.

We sketch the definition of MF; for more details, the reader should consult[25, §IV]. (See also [41].) Let A be an object of Art(O),and let π be an object
1 Zp
of Modadmon π makes π
G (A) which is of finite length. The action of
0 1
∼
an 
A[[Zp ]]-module,
and hence (via the isomorphism A[[Zp ]] −→
 A[[X]] defined

Z×
0
1 1
p 0
p
by
7→ X + 1) an A[[X]]-module. The actions of
and
0 1
0 1
0 1
furthermore give π the structure of a (ϕ, Γ)-module over A[[X]].
Let W be a finitely generated A-submodule of π, which generates π over A[G],and is invariant under GL2 (Zp )Z. Then W is a Γ-invariant A[[X]]-submoduleof π, and we define M (V, W ) to be the minimal ϕ-invariant A[[X]]-submodule of
5Equivalently [39, Thm. 2.3.8], objects that are finitely generated as A[G]-modules.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

19






p 0
1 Zp
- and
0 1
0 1
invariant subspace of π containing W .) We see that M (V, W ) is in fact a (ϕ, Γ)invariant A[[X]]-submodule of π.
The ϕ-action on M (V, W ) induces a Γ-equivariant, A[[X]]-linear map

π containing W . (Equivalently, M (V, W ) is the minimal(3.2.1)ϕ∗ M (V, W ) → M (V, W ),and one can show that the kernel and cokernel of this map are both finitely generated A-modules. If M (V, W )∗ denotes the Pontrjagin dual of M (V, W ), then we set
D(π) := A((X)) ⊗A[[X]] M (V, W )∗ . We equip M (V, W )∗ , and hence D(π), with thecontragredient Γ-action. The map (3.2.1) gives rise to a Γ-equivariant dual map
M (V, W )∗ → ϕ∗ M (V, W )∗whose source and target are again finitely generated over A. This latter map thus
induces a Γ-equivariant isomorphism∼

D(π) −→ ϕ∗ D(π).
The inverse of this isomorphism equips D(π) with an étale ϕ-structure, so that D(π)
becomes an étale (ϕ, Γ)-module over A((X)). One shows that D(π) is independent(up to natural isomorphism) of the choice of W , and that it is finitely generated
over A((X)). It thus corresponds to a representation of GQp on a finitely generated
A-module. We define MF(π) to be the Pontrjagin dual of this GQp -representation.6
Note that with these definitions, the functor π 7→ D(π) is contravariant, while thefunctor π 7→ MF(π) is covariant.
It follows directly from the definition of MF that there is an isomorphism of(ϕ, Γ)-modules(3.2.2)∼D(π) −→ D(MF(π)∗ ).

From the construction of D(π), one also easily deduces the existence of a canonicalP -equivariant map
∗(3.2.3)ψ −∞ (D(π)) → π(see [25, Prop. IV.3.2]7).
The functor MF is compatible with change of scalars. We give a table of values of
MF on the standard absolutely irreducible representations of G over k (as computedin [25]):8
MF(χ ◦ det)MF((χ ◦ det) ⊗ St)χ ⊗ χ2 )MF(IndG
B 1

G
G
r ∨
MF c−IndGL2 (Zp )Z (Sym k) /T (c−IndGL2 (Zp )Z (Symr k)∨ )

=0
=χ
= χ1
GQ
−(r+1)= IndGQp ψ2
p2

6The notation MF is chosen to stand for “magical functor”, “mysterious functor”, or “Montréalfunctor”. Colmez denotes his functor by π 7→ V(π). Note, though, that the definition of V involvesa twist by the cyclotomic character (see [25, §IV.2.3]), which does not appear in the definitionof MF.
7We use the notation ψ −∞ (D) of Colmez’s earlier papers, e.g. [22]. This corresponds to thenotation D]  Qp of [25].
8When comparing this table of values to the values of the functor V, as computed in [25], thereader should remember that MF and V differ by a cyclotomic twist.

20

MATTHEW EMERTON(where, in the last line, ψ2 denotes a fundamental character of level 2, and 0 ≤×r ≤ p − 1). Furthermore, MF is compatible with twisting: if χ : Q×
is a
p → A
adm
continuous character, and if π is a finite length object of ModG (A), then there isa natural isomorphism∼

MF(χ ◦ det ⊗A π) −→ χ ⊗A MF(π)(where, in the expression on the right hand side, χ is regarded as a character of GQp ,via local class field theory).
Let A be an object of Comp(O). In the paper [39] we defined the functor
OrdB : Mod$−adm(A) → Mod$−adm(A) from $-adically admissible smooth GG
Trepresentations over A to $-adically admissible smooth T -representations over A,
G: Mod$−adm(A) → Mod$−adm(A) given by parright adjoint to the functor IndBT
G
abolic induction. (See [39, Thm. 4.3.2].)
Our next result describes the interaction between the functor OrdB and Colmez’s
functor MF. Let A be an object of Art(O). Note that, via our identification of S
with Q×p , any finite-dimensional smooth representation of S over A may be regarded
as such a representation of Q×
p over A, and hence, via local class field theory, as acontinuous representation of Gab
Qp over A.
3.2.4. Proposition. If π is a finite length representation of G over some object A
of Art(O), then there is a natural A-linear Gab
Qp -equivariant embedding
OrdB (π) |S ,→ MF(π)ab .
(We regard the source of this embedding as a Gab
Qp -representation in the mannerindicated above.) If, furthermore, π admits a central character, and contains no
G-invariant finitely generated A-submodules, then this embedding is in fact an isomorphism.
Proof. The adjointness property of OrdB yields a natural G-equivariant map
G
IndB
OrdB (π) → π,

whose kernel is finitely generated over A. Applying MF to this map thus yields a
GQp -equivariant injection
OrdB (π) |S ,→ MF(π)(where the source is regarded as a Gab
Qp -representation in the manner indicatedabove), which evidently factors through the inclusion
MF(π)ab ⊂ MF(π),(3.2.5)yielding a GQp -equivariant injection
OrdB (π) |S ,→ MF(π)ab .

(3.2.6)

Conversely, the inclusion (3.2.5) gives rise to a surjection of Pontrjagin duals
MF(π)∗ → (MF(π)ab )∗ ,which in turn gives rise to a P -equivariant map∼ψ −∞ (D(π)) −→ ψ −∞ (D(MF(π)∗ )) → ψ −∞ (D((MF(π)ab )∗ ))LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

21(the first isomorphism being provided by (3.2.2)). Passing once more to Pontrjagin
duals, we obtain a map∗∗ψ −∞ (D((MF(π)ab )∗ )) → ψ −∞ (D(π)) .
Composing this with the canonical map (3.2.3) we obtain a P -equivariant map
∗(3.2.7)
ψ −∞ (D((MF(π)ab )∗ )) → π.
∗Lemma 3.2.8 below shows that the P -representation ψ −∞ (D((MF(π)ab )∗ )) sits
in a short exact sequence of P -representations
∗
0 → ε ◦ det ⊗MF(π)ab → ψ −∞ (D((MF(π)ab )∗ )) → Cc (Qp , MF(π)ab ) → 0.
Suppose now that π admits a central character η. We extend the P -action on eachof the members of this short exact sequence to a B (= P Z)-action, by declaring that
Z acts via the character η. The map (3.2.7) is then B-equivariant. If we suppose inaddition that π contains no non-trivial finitely generated G-invariant A-submodules,
then by Lemma 3.1.20 it similarly contains no non-trivial finitely generated Binvariant A-submodules, and so the map (3.2.7) factors through Cc (Qp , MF(π)ab ).
The resulting map Cc (Qp , MF(π)ab ) → π corresponds to a map
MF(π)ab → OrdB (π),
via the adjunction formula of [39, Cor. 4.2.6], which is immediately checked toprovide an inverse to (3.2.6). This completes the proof of the theorem.

3.2.8. Lemma. If V is a finitely generated A-modulewith a continuous action∗−∞of Gab,then
theP-representation
ψ
(D(V
))
sits
in
a short exact sequence ofQp
P -representations
∗
0 → (ε ◦ det) ⊗ V ∗ → ψ −∞ (D(V )) → Cc (Qp , V ∗ ) → 0.
(Here we regard the Gab
Qp -representation V as a P -representation via the map P →∼abS −→ Q×p → GQp , the first arrow being the canonical surjection and the third being
given by local class field theory.)

Proof. Choose n > 0 so that A is an O/$n -algebra. One easily verifies that∼
there is a P -equivariant isomorphism ψ −∞ (D(V )) −→ V ⊗O/$n ψ −∞ (D(O/$n )),where O/$n is equipped with the trivial GQp -action, and V is regarded as a P representation in the manner indicated∗ ∼ in the statement of the lemma.
∗ Thus there
is an isomorphism ψ −∞ (D(V )) −→ V ∗ ⊗O/$n ψ −∞ (D(O/$n )) . Since there∼
is also an isomorphism Cc (Qp , V ∗ ) −→ V ∗ ⊗O/$n Cc (Qp , O/$n ), it suffices to verify
the lemma in the case when V = O/$n , i.e. to show that there is a short exact
sequence∗(3.2.9)0 → ε ◦ det ⊗O/$n → ψ −∞ (D(O/$n )) → Cc (Qp , O/$n ) → 0.
The inclusion D] (O/$n ) ⊂ D(O/$n ) induces an isomorphism∼ψ −∞ (D] (O/$n )) −→ ψ −∞ (D(O/$n ))(see [24, p. 6]), while it follows from [22, Ex. 4.32] that there is an exact sequence
0 → Cc (Qp , O/$n )∗ → ψ −∞ (D] (O/$n )) → ε−1 ◦ det ⊗O/$n → 0.
The Pontrjagin dual of this sequence thus yields the required short exact sequence (3.2.9).
22

MATTHEW EMERTON

The following proposition is useful for comparing the structure of π and MF(π).
3.2.10. Proposition. If π is a finite length object of Modadm
G (A) admitting a central
character, and admitting no G-invariant subquotients that are finitely generated
over A, then MF induces an isomorphism between the lattice of G-invariant Asubmodules of π and the lattice of GQp -invariant A-submodules of MF(π).
Proof. Since MF is exact, it induces an order preserving map from the lattice of Ginvariant A-submodules of π to the lattice of GQp -invariant submodules of MF(π).
Suppose that π1 , π2 ⊂ π are G-invariant A-submodules for which MF(π1 ) andMF(π2 ) coincide, when regarded as subobjects of MF(π). If we writeπ3 = π1 + π2 ,then the exactness of MF implies that MF(π3 ) = MF(π1 ) = MF(π2 ) . Again using
the exactness of MF, we see that MF(π3 /π1 ) = 0. Since π admits no G-invariant
subquotients that are finitely generated over A, we see that π3 /π1 = 0. Similarly,
π3 /π2 = 0, and thus π1 = π2 . Hence the map on lattices of subobjects induced by
MF is an injection.
We turn to proving that the map on subobjects given by MF is a surjection.
Let M be a GQp -invariant submodule of MF(π). Choose π 0 ⊂ π maximally so that
MF(π 0 ) ⊂ M . Replacing π by π/π 0 and M by M/MF(π 0 ), we may assume that M
is a subobject of MF(π) with the property that if π 0 ⊂ π is such that MF(π 0 ) ⊂ M,
then π 0 = 0. Our goal is to show that M = 0. If in fact M is non-zero, thenit is clearly no loss of generality to replace M by an irreducible submodule, andthus we may assume that M is irreducible. In particular, letting m denote themaximal ideal of A, we may assume that M ⊂ MF(π)[m], the m-torsion submoduleof MF(π). Thus, replacing π by π[m], we may assume that A = k.
If k 0 is a finite extension of k, and if π 0 ⊂ k 0 ⊗k π is such that MF(π 0 ) ⊂ k 0 ⊗k M,
then if we let k 0 ⊗k π 00 denote the smallest subspace of k 0 ⊗k π defined over kthat contains π 0 , then π 00 is G-invariant, and MF(π 00 ) ⊂ M . Thus π 00 , and hence
π 0 , vanishes. Consequently, it is no loss of generality to extend scalars. Let 0 =π0 ⊂ π1 ⊂ · · · ⊂ πn−1 ⊂ πn = π be a Jordan–Hölder filtration of π. Extending
scalars if necessary, we may assume that each of the composition factors πi /πi−1
is absolutely irreducible. By assumption, each of these composition factors is also
infinite-dimensional over k, and thus a consideration of our table of values of MF
shows that for each i = 1, . . . , n, the GQp -representation MF(πi /πi−1 ) is absolutely
irreducible, of dimension one or two. In particular, 0 = MF(π0 ) ⊂ MF(π1 ) ⊂ · · · ⊂MF(πn−1 ) ⊂ MF(πn ) = MF(π) is a Jordan–Hölder filtration of MF(π), and so we
may assume that M is absolutely irreducible, of dimension one or two. Our goal is
to exhibit a G-subrepresentation π 0 of π such that MF(π 0 ) = M .
Suppose first that M is of dimension one, and that GQp acts on M through thecharacter χ. If η denotes the central character of π, then by Proposition 3.2.4,
together with the adjointness between OrdB and parabolic induction, we may findχ ⊗ χ−1 η → π with the property that, if
a G-equivariant homomorphism IndG
B
0
π denotes the image of this map, then MF(π 0 ) = M. Suppose, on the other
hand, that M has dimension
two. The inclusion
M ⊂ MF(π) induces an injec∗∗tion ψ −∞ (D(M ∗ )) → ψ −∞ (D(MF(π)∗ )) , which, when composed with thecanonical map (3.2.3), yields a P -equivariant map
∗(3.2.11)ψ −∞ (D(M ∗ )) → π.
If we extend the P -action on the source of this map to a B-action, by requiring Z toact via the central character η of π, then (3.2.11) becomes B-equivariant. Since M

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q23

∗is absolutely irreducible of dimension two, the B-representation ψ −∞ (D(M ∗ ))is in fact the B-representation underlying an absolutely irreducible supersingular
representation of G over k [2, Thm. 2.2.1]. The map (3.2.11) is then necessarilyG-equivariant, by [67, Thm. 1.1]. Thus, if we denote its image by π 0 , then again π 0
is a G-subrepresentation of π with the property that MF(π 0 ) = M. This completes
the proof of the proposition.
In the remainder of this subsection, we discuss how to extend the domain of MF
to certain G-representations over more general coefficient rings.
To being with, suppose that A is an object of Comp(O), with maximal ideal m.
If π is an m-adically admissible representation of G over A (in the sense of Definition 3.1.13), with the additional property that π/mπ (and hence π/mi π for alli ≥ 0) has finite length as a G-representation, then we defineMF(π) := lim
MF(π/mi π).
←−iSince MF is exact, it takes admissible smooth G-representations on free A/mi modules to GQp -representations on free A/mi -modules. Thus, with the preceding
definition, we see that MF provides an exact covariant functor
MF : {orthonormalizable admissible G-representations π over A
such that π/mπ has finite length}−→ {GQp -representations on finite rank free A-modules}.
We may extend the definition of MF further. Namely, let p ∈ Spec A be a prime
ideal of A, and suppose that π is an admissible unitary Banach representation of
G over Ap , in the sense of Definition 3.1.14, satisfying the following assumption:
3.2.12. Assumption. The representation π contains a G-invariant m-adically admissible unit ball π 0 with the additional property that π 0 /mπ 0 is of finite length.
We then defineMF(π) := Ap ⊗A MF(π 0 );this evidently depends on π 0 only up to commensurability.
We will typically apply this construction in the case when A = O and p is thezero ideal, so that Ap = E, and we are considering admissible unitary E-Banach
representations of G with the additional property that the reduction mod $ of aG-invariant unit ball is of finite length.
However, in Subsection 3.6, we will have to consider a more general context, andbp
in fact will have to iterate the previous ind-pro extension of MF. Namely, let Abdenote the p-adic completion of Ap , and suppose given a p-adically complete Ap module π, equipped with an action of G, with the property that for each i ≥ 0, thequotient π/pi π is an admissible unitary Banach representation over Ap /pi Ap (notethat this quotient is isomorphic to the localization (A/pi )p , so that the notionof admissible unitary Banach representation over this quotient is defined) which
satisfies Assumption 3.2.12. We then defineMF(π) := lim
MF(π/pi π).
←−
ibp , and that we are given a Banach
Finally, suppose that q is a prime ideal of Ab
module π over (Ap )q , equipped with a G-action, and admitting a unit ball π 0 which

24

MATTHEW EMERTON

is G-invariant, with the property that for each i ≥ 0, the quotient π 0 /pi π 0 is anadmissible unitary Banach representation over Ap /pi Ap which satisfies Assumption 3.2.12. We then definebp )q ⊗ b MF(π 0 ).
MF(π) := (AAp0This only depends on π up to commensurability.
3.3. Mod p and p-adic local Langlands. Let k be finite field of characteristic p, and let ρ : GQp → GL2 (k) be a continuous representation. Throughout thissubsection we make the following technical assumption regarding ρ:
3.3.1. Assumption. ρ is not equivalent to a representation of the form


1 ∗
χ⊗
0 ε
(for some k-valued character χ of GQp , where ∗ may or may not be zero).
The following result is due to Colmez [24].
3.3.2. Theorem. If ρ satisfies Assumption 3.3.1, then there is a finite length object
π of Modadm
G (k), unique up to isomorphism, such that:(1) MF(π) ∼= ρ.
(2) π has central character equal to det(ρ)ε.
(3) π has no finite-dimensional G-invariant subobject or quotient.
Since MF is compatible with extension of scalars, the same is true of the formation of π.
3.3.3. Remark. We briefly describe the structure of π (assuming that π is either
reducible or absolutely irreducible, a condition that can always be attained bymaking a finite extension of scalars):χ1 0(1) Suppose that ρ ∼, and that χ1 χ2−1 6= ε±1 . Then
0 χ2


G
π := IndGχ ⊗ χ2 ε ⊕ IndB
χ2 ⊗ χ1 ε .
B 1Our assumption on χ1 χ−1ensures that both direct summands are irre2
ducible.
χ1 ∗(2) Suppose that ρ ∼, where ∗ denotes a non-trivial extension. If
0 χ2±1χ1 χ−1, then π is an extension
2 6= ε0 → IndGχ ⊗ χ2 ε → π → IndGχ ⊗ χ1 ε → 0.
B 1
B 2
(Our hypothesis on χ1 χ2−1 implies that each of these induced representations is irreducible.)χ∗(3) Suppose that ρ ∼, where ∗ denotes a non-trivial extension.
0 χε−1
Suppose also that p > 3, so that ε 6= ε−1 . The π has a unique Jordan–Hölder filtration
0 ⊂ π1 ⊂ π2 ⊂ π∼∼∼Gsuch that π 1 −→ (χ◦det)⊗St, π 2 /π 1 −→ χ◦det, and π/π 2 −→ IndB
χε−1 ⊗
χε. (Our assumption on p ensures that this induced representation is irreducible.)LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

25(4) If ρ is absolutely irreducible, then π is an irreducible supersingular representation of G.
3.3.4. Remark. Note that the semi-simplification of π (as a G-representation) coincides with the G-representation associated to ρ via (a suitably normalized versionof) the semi-simple mod p local Langlands correspondence of [8].
3.3.5. Remark. If ρ |Ip is not the twist of an extension of 1 by itself or ε±1 , thenanother construction of the associated representation π is given in [13, §20]. In
this reference, π is not characterized by its behaviour under application of thefunctor MF, but rather by the structure of its invariants under the principal congruence subgroup of GL2 (Zp ) (together with the fact that it is generated by theseinvariants).
Our goal now is to describe the deformation-theoretic approach to the construction of the p-adic local Langlands correspondence (for two-dimensional GQp representations lifting ρ) due to Colmez and Kisin.
3.3.6. Definition. We let Def(ρ) denote the following category fibred in groupoids
over Comp(O): for any object A of Comp(O), with maximal ideal m, the groupoid
Def(A) has as objects free rank two A-modules V equipped with a continuous action∼of GQp , as well as an A/m-linear and GQp -equivariant isomorphism ı : V /mV −→A/m ⊗k ρ, and as morphisms the A-linear and GQp -equivariant isomorphisms thatare compatible with the given maps ı.
We typically regard Def(ρ) as a groupoid-valued functor on Comp(O), and thus
refer to Def(ρ) as the deformation functor associated to ρ.
Let π be the object of Modadm
G (k) attached to ρ via Theorem 3.3.2.
3.3.7. Definition. We let Def(π) denote the following category fibred in groupoids
over Comp(O): for any object A of Comp(O), with maximal ideal m, the groupoid
Def(A) has as objects orthonormalizable admissible representations π of G over∼A equipped with an A/m-linear and G-equivariant isomorphism ı : π/mπ −→A/m ⊗k π, and as morphisms the A-linear and G-equivariant isomorphisms thatare compatible with the given maps ı.
As with Def(ρ), we typically regard Def(π) as a groupoid-valued functor onComp(O), and thus refer to Def(π) as the deformation functor associated to π.
The functor MF induces a natural morphism of deformation functors(3.3.8)

MF : Def(π) → Def(ρ).

(as follows from the exactness of MF; see [61, §2.2]).
We next introduce some subgroupoids of Def(π) and Def(ρ), and some variantsof the morphism (3.3.8).
3.3.9. Definition. We define Def ∗ (π) to be the subgroupoid of Def(π) consisting×of those deformations π such that the centre of G acts
 on π via the A -valued
character det MF(π) ε (where we regard det MF(π) ε as a character of Q×
p via
local class field theory).
The morphism (3.3.8) induces a morphism(3.3.10)

MF : Def ∗ (π) → Def(ρ).

26

MATTHEW EMERTON

3.3.11. Definition. We define Def crys (ρ) to be the full subgroupoid of Def(ρ) obtained as the Zariski closure in Def(ρ) of the set of crystalline points in the generic fibre of Def(ρ). We define Def crys (π) to be the fibre product of Def ∗ (π) and Def crys (ρ)
over Def(ρ) (where the former maps to Def(ρ) via (3.3.10), and the latter maps toDef(ρ) via the natural fully faithful embedding).
The map (3.3.10) induces a tautological map(3.3.12)

MF : Def crys (π) → Def crys (ρ).

We now state the p-adic local Langlands correspondence, in its deformationtheoretic formulation due to Kisin [61].
3.3.13. Theorem. If ρ satisfies Assumption 3.3.1, then (3.3.10) is a fully faithfulembedding Def ∗ (π) ,→ Def(ρ), while its restriction (3.3.12) is in fact an equivalence∼of deformation functors Def crys (π) −→ Def crys (ρ).
3.3.14. Remark. In this remark we discuss some results related to the preceding
definitions and theorem which, however, we won’t need in the present paper.
We first remark that whenever Assumption 3.3.1 holds, it is known that there isan equality Def crys (ρ) = Def(ρ). Indeed, if p > 3, or p = 3 and ρ Ip is not isomorphic
to a twist of ω22 ⊕ ω26 , then this has been proved by Kisin [61, Cor. 1.11]; if p = 3
and ρ Ip is isomorphic to a twist of ω22 ⊕ ω26 , this has been proved by Böckle (seethe discussion following the statement of [5, Thm. 1.1]); and if p = 2 this has been
proved by Chenevier (see footnote (7) on p. 292 of [25]). However, since we do notneed the equality of Def crys (ρ) and Def(ρ) in this paper, and since the result for
p = 2 remains unpublished, we have decided to maintain the a priori distinction
between Def crys (ρ) and Def(ρ) in this paper.
We also mention that if Def(ρ) and Def(π) are representable, i.e. if ρ and π
have only scalar endomorphisms (note that the functor MF induces an isomorphism between their endomorphism algebras [61, Lem. 2.1.2]), and if p ≥ 3 when
π is supersingular, then one finds that Def ∗ (π) = Def(π), and hence (taking into
account the discussion of the preceding paragraph) that (3.3.8) is actually an equiv∼
alence Def(π) −→ Def(ρ). This is most difficult to establish in the supersingularcase, where it is due to Paškūnas [69]. (Note that although Paškūnas’s results areproved under the assumption that p ≥ 3, we expect the equality Def ∗ (π) = Def(π)
to continue to hold in the supersingular case when p = 2.) More generally, we
should mention that the complications caused by the central character in relating
the deformations of π and ρ were first observed by Paškūnas (see [69] and theintroduction to [61]).
3.3.15. Definition. Let VE be a two-dimensional E-vector space equipped with acontinuous representation of GQp , and suppose that VE contains a GQp -invariant
O-lattice V such that ρ := V /$V satisfies Assumption 3.3.1, and such that V
lies in Def crys (ρ). Theorem 3.3.13 then yields a representation π of G on an orthonormalizable O-module lifting π such that MF(π) = V , and we define B(VE ) :=E ⊗O π. Thus B(VE ) is an E-Banach space equipped with an admissible unitary
G-representation, which is easily seen to be independent of the choice of V (since
any two such choices are commensurable).
The association VE 7→ B(VE ) is the p-adic local Langlands correspondence.
3.3.16. Remark. The preceding definition applies in particular to any VE which
is absolutely irreducible, provided that p > 3. Indeed in this case we have that

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

27

Def crys (ρ) = Def(ρ), while ε 6= ε−1 , so that — applying a suitable analogue of [71,Prop. 2.1] in the case when ρss is isomorphic to a twist of 1 ⊕ ε — we see that VE
contains a V satisfying the requirements of the definition.
3.3.17. Remark. In the paper [25], using his functor from two-dimensional GQp representations to G-representations, Colmez has defined the p-adic local Langlands
correspondence for every absolutely irreducible representation of GQp .
3.3.18. Definition. We say that a continuous two-dimensional representation of
GQp over E is exceptional if it is a twist of a crystalline representation whose associated Dieudonné module is not Frobenius semi-simple (or equivalently, for which
the characteristic polynomial of the Frobenius endomorphism on the associatedDieudonné module has a double root).
3.3.19. Remark. Let VE be a two-dimensional E-vector space equipped with acontinuous representation of GQp which is absolutely irreducible and triangulinein the sense of [23], and not exceptional in the sense of the preceding definition.
The constructions of [3, 22, 23] give rise to an admissible unitary Banach space
representation B(VE ) of G associated to VE .9 It is an important aspect of thep-adic local Langlands correspondence of Definition 3.3.15 that it coincides with
explicit correspondence of [3, 22, 23] in the case when both are defined. (Indeed,this explicit correspondence is one of the tools used in the proof of Theorem 3.3.13,
and hence in the construction of the correspondence of Definition 3.3.15.)
3.3.20. Remark. In the case when VE is a reducible two-dimensional continuous
representation of GQp , there is an extensive discussion in [38, §6] of the expectedstructure of the Banach space representation B(VE ) of G that should be attached
to VE by the p-adic local Langlands correspondence. Using the results of [40] (and,in particular, applying the methods of computation from §4 of that reference), it
is straightforward to verify the various conjectures of [38, §§6.2, 6.3, 6.4, 6.5], andthus B(VE ) can indeed be defined following the prescriptions of [38, §6]. We will
not do this here, however, since we do not need this for our purposes. Rather, in
Subsection 3.4 below we will establish some somewhat weaker results about B(VE )in the the reducible case, using the deformation-theoretic view-point adopted here,
and the functor MF.
If VE is a reducible two-dimensional continuous representation of GQp which isde Rham with distinct Hodge–Tate weights, then one expects that the associatedp-adic Banach representation B(VE ) of G should have the properties originally
posited by Breuil in [8, 9]. In this direction, one has the following result, due toBerger, Breuil, and Colmez in the non-exceptional trianguline case, and to Colmez
for those VE that are not trianguline.
3.3.21. Theorem. If VE is a continuous two-dimensional representation of GQp
over E and if VE is de Rham with distinct Hodge–Tate weights a > b, then B(VE )contains non-zero locally algebraic vectors.
9When p > 2, Paškūnas [68] has shown that the results of [3] extend to cover the exceptional
cases, and it is expected that they will similarly extend to cover these cases when p = 2. Using
Paškūnas’s result would allow us to simplify some of the arguments of the present paper in thecase when p > 2, e.g. the proof of Theorem 6.4.7. However, since we wish to include the casep = 2 in our arguments, we haven’t incorporated these simplifications.

28

MATTHEW EMERTON

Proof. If VE is absolutely irreducible, trianguline, and not exceptional (in the senseof Definition 3.3.18), then this follows from the explicit construction of B(VE ) given
in [3, 22]. If VE is absolutely reducible, then one may describe B(VE ) explicitly(see e.g. the description of B(VE ) that we give in Subsection 3.4 below), and so
verify the theorem. The theorem for arbitrary VE (satisfying the assumptions of
the theorem) has been proved by Colmez [25, Thm. VI.6.18]
Colmez has shown that, conversely, if B(VE )l.alg is non-zero, then VE is de Rhamwith distinct Hodge–Tate weights [25, Prop. VI.5.1, Thm. VI.6.13]. Furthermore,
we have the following strengthening of Theorem 3.3.21, which precisely describes
the locally algebraic vectors in B(VE ) in the case when they are non-zero.
3.3.22. Theorem. If VE is a continuous two-dimensional representation of GQp
over E, and if VE is de Rham with distinct Hodge–Tate weights a > b, then there∼
is a G-equivariant isomorphism B(VE )l.alg −→ Syma−b−1 E 2 ⊗E detb+1 ⊗E πp (VE ).
In particular, B(VE ) contains non-zero locally algebraic vectors.
Proof. In the case when VE is crystabelline, or is the twist of a semi-stable representation, this has been proved by Colmez [25, Thm. VI.6.50]. In the remainingcase (i.e. when VE is potentially crystalline but not crystabelline), it follows from[25, Thm. VI.6.42], together with a global calculation using the theory developedin this paper. The proof is presented in Subsection 7.4 below.

3.3.23. Remark. In fact Colmez has already shown in [25, Prop. VI.5.1] that if
VE is de Rham with distinct Hodge–Tate weights a > b, then the locally algebraic
vectors of B(VE ) are necessarily isomorphic (as a G-representation) to the tensor
product of Syma−b−1 E 2 ⊗E detb+1 with a smooth representation of G. The pointof the preceding theorem is that it gives a precise description of the smooth factor
in this tensor product.
The following result is also important in applications. (In [25], Colmez has given
a proof of this result using his functor from two-dimensional GQp -representations
to G-representations; see [25, Thm. 0.17 (ii)]. We give a proof here using just theproperties of the functor MF.)
3.3.24. Proposition. If VE is a continuous two-dimensional representation of GQp
for which B(VE ) is defined, and if VE is (absolutely) irreducible, then B(VE ) is also
(absolutely) topologically irreducible.
Proof. Suppose that VE is irreducible. Choose a G-invariant bounded open lattice
π in B(VE ) which deforms a representation π satisfying the conditions of TheoT
rem 3.3.2. Let W ⊂ B(VE ) be a closed subrepresentation, and write W0 = π W .
Since W0 is saturated in π, we see that MF(W0 ) is saturated in MF(π) (since MF
is exact). Combining this with the fact that E ⊗O MF(π) ∼= VE is irreducible, wesee (again using the exactness of MF) that either MF(W0 ) = 0 or MF(π/W0 ) = 0.
Thus either W0 is a finitely generated O-submodule of π, or else π/W0 is a finitely
generated O-quotient module of π. However, since π admits no finite-dimensional
G-invariant subrepresentations or quotient representations, we see that one of W0
or π/W0 must vanish. Thus either W = 0 or else W = B(VE ), and so B(VE ) is
topologically irreducible, as claimed. Since the formation of B(VE ) is compatible
with extending scalars, we conclude that if VE is absolutely irreducible, then B(VE )is absolutely topologically irreducible.
LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

29

3.4. The reducible case. If VE is a continuous reducible two-dimensional representation of GQp over E satisfying the conditions of Remark 3.3.19 (i.e. containing
an O-lattice V in VE whose reduction V satisfies Assumption 3.3.1), then as inRemark 3.3.19, we may associate a Banach space representation B(VE ) of G to VE .
In this subsection we will establish some results regarding the structure of B(VE )in this situation. (By employing the methods of [40, §4], it is possible in fact toverify all the conjectures related to the structure of B(VE ) that we made in [38].
We won’t do this here, however, since the somewhat weaker statements that weestablish below will suffice for our present purposes.)
Write VE as an extension
0 → χ1 → VE → χ2 → 0,

(3.4.1)
×

where each χi : GQp → O ⊂ E × is a continuous character.
3.4.2. Proposition. If χ1 χ−1
2 6= ε, then B(VE ) is an extensionGχ1 ⊗ χ2 ε → B(VE ) → IndGχ ⊗ χ1 ε → 0,
0 → IndB
B 2

which is split or not according to whether or not (3.4.1) is. Furthermore, the GG
χ1 ⊗ χ2 ε and IndGχ ⊗ χ1 ε are topologically irreducible.
representations IndB
B 2
Proof. Choose a GQp -invariant lattice V in VE whose reduction V satisfies Assumption 3.3.1. Theorem 3.3.13 shows that we may find a lift π of π over O, with central∼
character equal to χ1 χ2 ε, and such that MF(π) −→ V. Since π contains no finitedimensional G-invariant subspaces, we see that π contains no finitely generated
G-invariant O-submodules. Proposition 3.2.4 (and the known value of the central
character of π) thus yields an embedding
χ1 ⊗ χ2 ε ,→ OrdB (π),
which in turn, by virtue of the adjointness property of OrdB , gives rise to a Gequivariant map(3.4.3)Gχ1 ⊗ χ2 ε → π.
IndB

Since χ1 6= χ2 ε, by assumption, the source of this map becomes topologicallyirreducible after tensoring with E over O. Consequently, (3.4.3) must be injective.
If we let π1 denote the saturation in π of its image, then π1 contains this image
with finite index, and consequently∼Gχ1 ⊗ χ2 ε) = χ1 .
MF(π1 ) −→ MF(IndB∼∼

Thus, writing π2 = π/π1 , we compute that MF(π2 ) −→ MF(π)/MF(π1 ) −→V /χ1 = χ2 . Since π contains at most one finite-dimensional Jordan–Hölder factor,of dimension one if it exists at all, we see that either π2 contains no finitely generated
G-invariant O-submodules, or else that it contains such a submodule of rank one,the quotient of π2 by which then contains no such submodules. Letting π3 = π2 inthe first case, or the quotient of π2 by the rank one G-invariant submodule in thesecond case, we see that∼MF(π3 ) −→ MF(π2 ) = χ2 .
We thus deduce from Proposition 3.2.4 an isomorphism∼

OrdB (π3 ) −→ χ2 ,

30

MATTHEW EMERTON

and hence a map(3.4.4)GIndB
χ2 ⊗ χ1 ε → π3 .

Again by assumption we have that χ2 6= χ1 ε, and thus this map must be an
embedding. If we let π4 denote the cokernel of this embedding, then MF(π4 ) = 0.
Thus if we write π 4 := π4 /$π4 , then π 4 is a G-invariant quotient of π for whichMF(π 4 ) = 0. On the other hand, π has no finite-dimensional G-invariant quotient.
We conclude that π 4 = 0, and thus that π4 = 0. Thus we conclude that (3.4.4) is
an isomorphism.
G
Now the Jordan–Hölder factors of π are precisely the union of those of IndB
χ1 ⊗G
G
χ2 ε and those of IndB χ2 ⊗ χ1 ε. Also, π 1 := π1 /$π1 and IndB χ1 ⊗ χ2 ε havethe same Jordan–Hölder factors. Combining this observation with the isomorphism (3.4.4), we conclude that the Jordan–Hölder factors of π are precisely theunion of those of π 1 and those of π 3 := π3 /$π3 . Consequently it must be the case∼χ ⊗ χ1 ε) by π1 .
that π2 = π3 , and thus that π is an extension of π3 (−→ IndG
B 2
Tensoring with E over O (and recalling that π1 contains the image of (3.4.3) withfinite index), we find that B(VE ) is an extension of the required form.
∼
Clearly if B(VE ) is a split extension, then the isomorphism VE −→ E ⊗O M F (π)
shows that VE is itself split. On the other hand, if VE is split, then we may reverse
the roles of χ1 and χ2 , and, obtaining a corresponding description of B(VE ) asG
χ1 ⊗ χ2 ε by IndGχ ⊗ χ1 ε, conclude in turn that B(VE ) isan extension of IndB
B 2
isomorphic to the direct sum of these two representations. This completes the proofof the proposition.
−1
3.4.5. Proposition. If χ1 χ−1, then B(VE ) admits a unique
2 = ε, i.e. χ2 = χ1 εthree step Jordan–Hölder filtration (in the category of admissible Banach representations of G) 0 = B0 ⊂ B1 ⊂ B2 ⊂ B3 = B(VE ), such that:b(1) B1 ∼= (χ1 ◦ det) ⊗ St;∼(2) B2 /B1 = χ1 ◦ det;GG −1(3) B3 /B1 ∼χ1 ε−1 ⊗ χ1 ε ∼ε ⊗ ε) .
= IndB= (χ1 ◦ det) ⊗ (IndB

Proof. The proof of this result is similar to that of the preceding proposition. As
in that proposition we choose lattices V in VE and π in B(VE ), related by anisomorphism∼
MF(π) −→ V,
and we obtain a map(3.4.6)GIndB
χ1 ⊗ χ2 ε → π.

G
Now χ1 = χ2 ε by assumption, and thus IndGχ ⊗ χ2 ε = IndB
χ1 ⊗ χ1 is not
B 1
b by χ1 ◦ det. Since π contains
irreducible, but rather is an extension of (χ1 ◦ det) ⊗ St
no finite-dimensional G-invariant subspaces, we see that (3.4.6) must factor to yieldan embedding(3.4.7)b ,→ π.
Stbb is irreducible, the image of this embedding is saturated.
Also, since St(= St/$
St)
Let π2 denote the cokernel of (3.4.7). We see that MF(π2 ) = χ1 ⊗ ε, and (using
the known structure of π, namely that is an extension by (χ1 ◦det)⊗St of a non-split

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

31

G
extension of IndB
χ1 ε ⊗ χ1 ε−1 by χ1 ◦ det) we see that π 2 := π2 /$π2 is a non-split
extension

0 → χ1 ◦ det → π 2 → IndGχ ε ⊗ χ1 ε−1 → 0.
B 1

(3.4.8)If we let π3 denote the quotient of π2 by its maximal finitely generated G-invariant
O-submodule (the latter is of rank at most one), then we conclude as in the proofof the preceding proposition that∼IndG
χ ε−1 ⊗ χ1 ε −→ π3 ,B 1
and thus that∼G
π 3 := π3 /$π3 −→ IndB
χ1 ε−1 ⊗ χ1 ε.
A comparison with (3.4.8) shows that π2 must be an extension of π3 by a rank one
G-invariant O-module. Tensoring with E over O, we conclude that B(VE ) has a
Jordan–Hölder filtration of the type described in the statement of the proposition.
The uniqueness of this filtration follows from the uniqueness of the Jordan–Hölder
filtration of π.

3.4.9. Remark. In the situation of Proposition 3.4.5, one can show that the onedimensional representation B2 /B1 is necessarily isomorphic to the character χ1 ◦det.
Indeed, as already mentioned, one can confirm the speculations of the discussion ofb that embeds[38, §6.5], namely that the extension B2 of χ1 ◦ det by (χ1 ◦ det) ⊗ St
as a closed subrepresentation of B(VE ) is isomorphic to (χ1 ◦ det) ⊗ B(2, L), where
L is the L-invariant of the extension
0 → χ1 → VE → χ1 ε−1 → 0.
3.5. Weights. In this subsection we recall the notion of Serre weights and its
relation to the mod p local Langlands correspondence.
3.5.1. Definition. A Serre weight is an (isomorphism class of) irreducible representation(s) of GL2 (Fp ) over k. The Serre weights are precisely the (isomorphism classes of the) representations (Symr k 2 )∨ ⊗k dets , where 0 ≤ r ≤ p − 1, and0 ≤ s ≤ p − 2.
3.5.2. Definition. If V is a Serre weight, then we write
H(V ) := EndG (c−IndG
GL2 (Zp ) V ).
It follows from [1] that H(V ) = k[T, Z, Z −1 ], where T is the endomorphism corp 0
p 0
, while Z is the element
lying in the
responding to the double coset
0 1
0 p
centre of G.
3.5.3. Remark. If π is any smooth representation of G over k, then there is a∼
canonical isomorphism HomGL2 (Zp ) (V, π) −→ HomG (c−IndG
GL2 (Zp ) V, π), and thus
HomGL2 (Zp ) (V, π) is endowed with a canonical action of H(V ).
To any continuous representation ρ : GQp → GL2 (k) we may attach a set W (ρ)
of Serre weights, as follows. (We essentially follow the recipe of [14, §3], but use acohomological normalization.)L∼
If ρ is absolutely irreducible, then (k ⊗k ρ) |Ip −→ ψ2a+pb ψ2pa+b , for someuniquely determined 0 ≤ b < a ≤ p − 1, and we set
W (ρ) := {(Syma−b−1 k 2 )∨ ⊗k deta , (Symp−a+b k 2 )∨ ⊗k detb+1 }.

32

MATTHEW EMERTON

Thus W (ρ) is a set of two weights.
If ρ is absolutely reducible, then the definition of W (ρ) is given via a slightly
more involved rule: If 0 < r ≤ p − 1, and 0 ≤ s < p − 1, then the representation(Symr k 2 )∨ ⊗k dets lies in W (ρ) if and only if ρ |Ip sits in a short exact sequence ofthe form
0 → εs → ρ |Ip → εs−1−r → 0,
while dets lies in W (ρ) if and only if ρ |Ip sits in a short exact sequence of the form
0 → εs → ρ |Ip → εs−1 → 0
for which the extension class is furthermore peu ramifiée in the sense of [79].
Thus, if ρ is reducible, then W (ρ) consists of one, two, three or even (in somecases when p = 3) four weights. (See the table in the proof of [14, Thm. 3.15].)
3.5.4. Definition. If V ∈ W (ρ), then we define an H(V ) module m(V, ρ) as follows:(1) If ρ is absolutely irreducible, then we define m(V, ρ) to be one dimensional,
with T acting by 0 and Z acting by det(ρ)(p). (Here we think of det(ρ) as×××
a homomorphism Gab
Qp → k , and hence as a homomorphism Qp → k , bylocal class field theory.)(2) If ρ is absolutely reducible, and V = (Symr k 2 )∨ ⊗k dets , then we setm(V, ρ) := HomIp (εs , ρ) as a k-vector space. Note that HomIp (εs , ρ) is
naturally a representation of the quotient group GQp /Ip , and so in particular is equipped with an action of the geometric Frobenius at p. We
equip m(V, ρ) with an H(V )-module structure by declaring that T act via
geometric Frobenius, and that Z act via the scalar det(ρ)(p).
Suppose now that ρ satisfies Assumption 3.3.1, and let π denote the smooth
G-representation over k attached to ρ via Theorem 3.3.2. If V is a Serre weight,
then, by Remark 3.5.3, the space HomGL2 (Zp ) (V, π) is naturally an H(V )-module.
3.5.5. Lemma.
(1) If V 6∈ W (ρ), then HomGL2 (Zp ) (V, π) = 0.
(2) If V ∈ W (ρ), then there is an isomorphism of H(V )-modules m(V, ρ) ∼=HomGL2 (Zp ) (V, π).
Proof. This is relatively straightforward, based on the explicit description of π given
in Remark 3.3.3, or in [25]. In the case when ρ |Ip is not the twist of an extensionof 1 by either itself or ε±1 , it is an immediate consequence of the description of π
given in [13, §20].

3.6. Lattices. This subsection is devoted to the proof of two technical, but important, propositions. To begin with we suppose that ρ : GQp → GL2 (O) is continuous,±1
that ρE := E ⊗O ρ is irreducible, and that ρss = χ1 ⊕ χ2 , where χ1 χ−1.
2 6= 1, εadm
Let π be the object of ModG (A) associated to ρ via Theorem 3.3.2, and let π
be the deformation of π associated to the deformation ρ of ρ via Theorem 3.3.13.
Following the notation of Remark 3.3.19, we write B(ρE ) := E ⊗O π. Our first goalin this subsection is establish a proposition that allows us to distinguish π among
the various $-adically complete and separated, G-invariant O-lattices in B(ρE ).
Before stating the stating the proposition, we will make a definition. Note that∼the composite of the isomorphism S −→ Q×
p with the local Artin map defines aninjection S ,→ Gab
,
and
hence
an
anti-diagonalembeddingQp(3.6.1)

S ,→ Gab
Qp × S

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

33(defined as the local Artin map on the first factor, and the map s 7→ s−1 on thesecond factor).
3.6.2. Definition. If V is a representation of GQp × S, and if V ab denote themaximal subobject of V on which GQp acts through its maximal abelian quotientabGab
is a GQp × S-subrepresentation of V ), then we let V ab,S denote
Qp (note that V
the subspace of V ab consisting of S-fixed vectors, where S acts through the antiab
diagonal embedding (3.6.1) and the action of Gab.
Qp × S-on V
We can now state and prove our first proposition.
3.6.3. Proposition. If π0 is a G-invariant, $-adically complete and separated, Ginvariant lattice in B(ρE ), with the property that, for each n > 0, the action of GQp
on the quotient(ρ/$n ρ) ⊗O/$n OrdB (π0 /$n π0 )ab,S(ρ/$n ρ) ⊗O/$n OrdB (π0 /$n π0 )factors through Gab
Qp , then π0 is equal to a scalar multiple of π.
Proof. According to Proposition 3.2.10, the functor MF induces a bijection between
the set of $-adically complete and separated, G-invariant lattices in B(ρE ), andthe set of GQp -invariant lattices in ρE . Thus it similarly induces a bijection between
the set of $-adically complete and separated, G-invariant lattices in B(ρE ), modulo
scaling, and the set of GQp -invariant lattices in ρE , modulo scaling.
Denote these latter two sets by Lπ and Lρ . Since χ1 and χ2 are distinct, and ρE
is irreducible, the set Lρ is a finite (positive) length segment in the tree of all (notnecessarily G-invariant) lattices in ρE . We let ρ1 and ρ2 denote the two endpointsof this segment, the labelling chosen so that ρ1 is a non-split extension of χ2 byχ1 , while ρ2 is a non-split extension of χ1 by χ2 . If ρ0 is any (lattice representingan) element of Lρ , then we write d1 (ρ0 ) (resp. d2 (ρ0 )) to denote the distance ofρ0 from ρ1 (resp. ρ2 ). Note that d1 (ρ0 ) + d2 (ρ0 ) is constant (i.e. independentof ρ0 ), equal to the length d of the segment Lρ . For i = 1, 2, the natural numberdi (ρ0 ) admits the following characterization: for any n ≥ 0, the quotient ρ0 /$n ρ0
contains a (necessarily abelian) GQp -subrepresentation χei (ρ0 )n which is a successive
extension of min(di (π0 ), n) copies of χi , and no longer such subrepresentation (i.e.
no subrepresentation which is a successive extension of more than this many copiesof χi ). The representations χ
ei (ρ0 )n are compatible in an evident way if n varies,∼
namely: if m ≤ n, then χei (ρ0 )n /$m χei (ρ0 )n −→ χei (ρ0 )m . Note also that χei (ρ0 )n∼ei (ρ0 )di (ρ0 ) foris annihilated by $di (ρ0 ) , for any value of n, and so χ
ei (ρ0 )n −→ χ
any n ≥ di (ρ0 ). If n ≤ di (π0 ), then the quotient (ρ0 /$n ρ0 )/e
χi (ρ0 )n is an abelian
representation of GQp (it is a successive extension of n copies of χ3−i ), while ifn > di (π0 ) then it is not (it contains a non-split extension of χi by χ3−i ).
Now choose ρ0 = MF(π0 ). Proposition 3.2.4 gives rise to a Q×
p -equivariant
embeddingχe1 (ρ0 )d ⊕ χ
e2 (ρ0 )d ,→ OrdB (π0 /$d π0 ),and thus to an embedding(ρ/$d1 (ρ0 ) ρ) ⊗O/$d1 (ρ0 ) O χ
e1 (ρ0 )d ⊕ (ρ/$d2 (ρ0 ) ρ) ⊗O/$d2 (ρ0 ) O χ
e2 (ρ0 )d,→ (ρ/$d ρ) ⊗O/$d OrdB (π0 /$d π0 ).

34

MATTHEW EMERTON

Taking n = d in the hypothesis of the theorem, we find that for each choice of
i = 1, 2, the cokernel of the map
ab,S(ρ/$di (ρ0 ) ρ) ⊗O/$di (ρ0 ) χei (ρ0 )d,→ (ρ/$di (ρ0 ) ρ) ⊗O/$di (ρ0 ) χei (ρ0 )d
is again an abelian representation of GQp .
Now (ρ/$di (ρ0 ) ρ)ab = χ
e1 (ρ)di (ρ0 ) ⊕ χ
e2 (ρ)di (ρ0 ) , and soab,S(ρ/$di (ρ0 ) ρ) ⊗O/$di (ρ0 ) χei (ρ0 )d=χei (ρ)di (ρ0 ) ⊗O/$di (ρ0 ) χei (ρ0 )d .
Thus we conclude that the action of GQp on the quotient (ρ/$di (ρ0 ) ρ)/eχi (ρ)di (ρ0 )is abelian. This implies that di (ρ0 ) ≤ di (ρ) for i = 1, 2. However, since d1 (ρ0 ) +d2 (ρ0 ) = d1 (ρ) + d2 (ρ) = d, we conclude that di (ρ0 ) = di (ρ) for i = 1, 2, and thusthat ρ0 is a scalar multiple of ρ. Consequently, π0 is a scalar multiple of π, as
claimed.

Our next result is a variant of the preceding one, in which we work with moregeneral coefficients, but draw a weaker conclusion. To this end we suppose that
A is an object of Comp(O) that is reduced and flat over O, and that ρ : GQp →GL2 (A) is a continuous representation. If p is a point of Spec A[1/p] then we write
ρ(p) := κ(p) ⊗A ρ, where κ(p) is the residue field of p.
We assume that ρ is generically irreducible, i.e. is such that ρ(a) is irreduciblefor each minimal prime a of A. We also suppose given an orthonormalizable admissible representation θ of G over A, with the property that the characters of
E ⊗O MF(θ/pθ) and ρ(p) coincide for any closed point p ∈ Spec A[1/p]. This implies that, if ρ(p) is irreducible, then it is isomorphic to E ⊗O MF(θ/pθ). The pointof the following proposition is to provide a criterion for such an isomorphism incertain cases when ρ(p) is reducible.
3.6.4. Proposition. Suppose that p is a closed point of Spec A[1/p] with the property that ρ(p)ss = χ1 ⊕ χ2 , where χ1 , χ2 are characters χi : GQp → κ(p)× such±1
that χ1 χ−1, and such that E ⊗O (θ/pθ) contains no finite-dimensional G2 6= 1, εinvariant subquotients. If there exists a faithful cofinitely generated A-module (inthe sense of Definition C.1) such that, for each n > 0, the action of GQp on thequotient(ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ) ⊗A/pn Y [ pn ]ab,S(ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ) ⊗A/pn Y [ pn ]∼factors through Gab
Qp , then E ⊗O MF(θ/pθ) = ρ(p).
Proof. We wish to reduce ourselves to the consideration of lattices in a representation over a discrete valuation ring, as in the context of the preceding proposition. To this end, we choose a finite morphism A → A0 , where A0 is an object ofComp(O) that is normal of dimension two, containing a prime p0 which maps to
p under the induced map Spec A0 → Spec A. We furthermore can and do choose
A0 so that ρ0 := A0 ⊗A ρ is generically irreducible. We write θ0 := A0 ⊗A θ, andnote that it suffices to prove that E ⊗O MF(θ0 /p0 θ) ∼= κ(p0 ) ⊗A0 ρ0 . Indeed, there∼
are natural isomorphisms E ⊗O MF(θ0 /p0 θ) −→ κ(p0 ) ⊗κ(p) E ⊗O MF(θ/pθ) and∼κ(p0 ) ⊗A0 ρ0 −→ κ(p0 ) ⊗κ(p) ρ(p). Thus E ⊗O MF(θ/pθ) and ρ(p) are representations of GQp over κ(p) that become isomorphic over κ(p0 ), and hence are themselves
isomorphic.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

35

We now set Y 0 := HomA (A0 , Y ); note that Y 0 is cofinitely generated over A0 , by
Proposition C.13 (1). For any n > 0, there are natural isomorphisms∼(θ0 /p0 n θ0 ) ⊗A0 /p0 n Y 0 [p0n ] −→ (θ/pn θ) ⊗A/pn Y 0 [pn ]and∼Y 0 [pn ] −→ HomA (A0 /p0 n , Y [pn ]),and hence (e.g. by Lemma 3.1.17, applied with A taken to be A/pn , π taken to beθ/pn θ, which is orthonormalizable over A/pn by Lemma B.6 (4), and M taken to be
A0 /p0 n ; note that there is no need to pass to completed tensor products, since A/pn
is finite over O, so that the usual tensor products are already $-adically complete)a natural isomorphism∼(θ0 /p0 n θ0 ) ⊗A0 /p0 n Y 0 [p0n ] −→ HomA A0 /p0n , (θ/pn θ) ⊗A/pn Y [ pn ] .
There are evident isomorphisms(ρ0 /p0 n ρ0 ) ⊗A0 /p0n OrdB HomA A0 /p0n , (θ/pn θ) ⊗A/pn Y [ pn ]∼−→ HomA A0 /p0n , OrdB (ρ/pn ρ) ⊗A/pn (θ/pn θ) ⊗A/pn Y [ pn ]andab,S(ρ0 /p0n ρ0 ) ⊗A0 /p0n OrdB HomA A0 /p0n , (θ/pn θ) ⊗A/pn Y [ pn ]ab,S ∼.
−→ HomA A0 /p0n , (ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ) ⊗A/pn Y [ pn ]Since HomA (A0 /p0 n , –) is a left-exact functor, we deduce from the hypothesis of thetheorem that for each n > 0, the action of GQp on the quotient(ρ0 /p0 n ρ0 ) ⊗A0 /p0 n OrdB (θ0 /p0 n θ0 ) ⊗A0 /p0 n Y 0 [p0 n ]ab,S(ρ0 /p0 n ρ0 ) ⊗A0 /p0 n OrdB (θ0 /p0 n θ0 ) ⊗A0 /p0 n Y 0 [p0 n ]0factors through Gab
Qp . Note also that since A is reduced (being normal), the module
Y 0 is faithful over A0 , by Proposition C.13 (2). Combining this with the conclusionof the preceding paragraph, we see that we may replace A, p, ρ, and θ by A0 , p0 , ρ0 ,and θ0 , and hence assume that A itself is normal of dimension two.
Let Yctf denote the maximal cotorsion free cofinitely generated A-submodule of Y
(as in Definition C.39, with Γ taken to be the trivial group). Proposition C.40 showsthat Yctf is a faithful A-module. For any n > 0, the embedding Yctf [pn ] ,→ Y [pn ]induces a map(θ/pn θ) ⊗A/pn Yctf [pn ] ,→ (θ/pn θ) ⊗A/pn Y [pn ]which is again an embedding (by Lemma B.6 (1), applied to the orthonormalizable
A/pn -module θ/pn θ), and it follows that the action of GQp on the quotient(ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ) ⊗A/pn Yctf [ pn ]ab,S(ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ) ⊗A/pn Yctf [ pn ]factors through Gab
Qp . Thus, replacing Y by Yctf , we may assume that Y is cotorsion
free. Then, since Y is cotorsion free and the localization Ap is a discrete valuation
ring (being normal and of dimension one), we find that E ⊗O Y [pn ] is free as an(Ap /pn Ap )-module, for any n > 0. (Indeed, if we write N := HomO (Y, O), then N

36

MATTHEW EMERTON

is torsion free over A, by assumption. Applying Proposition C.11, with X taken to
be Y and M taken to be A/pn , we find that there is an isomorphism∼(Ap /pn Ap ) ⊗A N −→ HomO (E ⊗O Y [pn ], E).
Since Ap ⊗A N is finitely generated and torsion free, and hence free, over the discrete
valuation ring Ap , we see that the source of this isomorphism is free over Ap /pn Ap ,and hence so is the source. Passing to E-duals, we find that E ⊗O Y [pn ] is itself
free over (Ap /pn Ap ), as claimed.) Thus we conclude that the GQp -action on(ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ)ab,S(ρ/pn ρ) ⊗A/pn OrdB (θ/pn θ)factors through Gab
Qp , for any n > 0.
bp denote the completion of the localization Ap , let K denote the fraction
Let A
bp , and consider θb := A
bp ⊗
ˆ A θ, the p-adic completion of Ap ⊗A θ (which may
field of Ab We
also be described as the projective limit lim E ⊗O (θ/pn θ)), as well as K ⊗ b θ.
Ap

←−
n

are in the context of the discussion at the end of Subsection 3.2, and we observe thatb is an Ab ∼bp -lattice in MF(K ⊗ b θ)
bp is a complete discreteMF(θ)= K ⊗A ρ. Since AApb θb ∼
valuation ring, and since by assumption θ/p= E ⊗O θ/pθ contains no finitedimensional G-invariant subquotients, we may apply the same argument as thatb is a scalar multiple
used to prove Proposition 3.6.3 above to conclude that MF(θ)
b ∼bp ⊗A ρ, and so in particular that MF(θ)
bp ⊗A ρ. Reducing(in K ⊗Abp ρ) of A
=A∼
this isomorphism modulo p, we find that E ⊗O MF(θ/pθ) = ρ(p), as claimed. 
3.7. Deformations. In this subsection we prove some technical results which willallow us to construct certain deformations of G-representations. These results will
be applied only in the proof of Proposition 6.5.11.
We suppose that A is a finite, flat, reduced, local O-algebra, with maximal∼
ideal m. Let p be a minimal prime ideal of A such that O −→ A/p, and let J
denote the intersection of the minimal primes of A that are distinct from p. Thus
J is a saturated ideal of A (i.e. A/J is flat over O), and J ∩ p = 0. There is a short
exact sequence(3.7.1)0 −→ Aa7→(a mod p,a mod J)−→

A/p × A/J

(a1 ,a2 )7→a1 −a2−→A/(p + J) −→ 0.

∼

Let s > 0 be such that O/$s O −→ A/(J + p). If we let ı : A/J → O/$s O
denote the composition of the surjection A/J → A/(J + p) with the inverse of thisisomorphism, then a consideration of the short exact sequence (3.7.1) shows that
there is an isomorphism(3.7.2)∼A −→ {(x, y) ∈ A/J × O | ı(x) ≡ y mod $s }.

We may use this description of A to construct orthonormalizable A-modules bygluing together orthonormalizable modules over O and A/J. Suppose to begin withthat X is an orthonormalizable A-module. Write XJ := X/JX and Xp := X/pX.
It follows from Lemma B.6 (4) that XJ is an orthonormalizable A/J-module, while
Xp is an orthonormalizable O-module. The transitivity of tensor products showsthat there are canonical isomorphisms∼

XJ /pXJ −→ X/(p + J)XLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

and

37

∼

Xp /$s Xp −→ X/(p + J)X,
and we define∼

φ : Xp /$s Xp −→ XJ /pXJ
to be the composite of the second of these with the inverse of the first.
3.7.3. Lemma. There is a natural isomorphism∼

X −→ {(u, v) ∈ Xp × XJ | φ(u) = v}.
Proof. This follows by tensoring the short exact sequence (3.7.1) with X over A,
taking into account the flatness of X over A (Part 1 of Lemma B.6).

Conversely, we now suppose given an orthonormalizable O-module Xp , an orthonormalizable A/J-module XJ , and an isomorphism∼

φ : Xp /$s Xp −→ XJ /pXJ .
Define(3.7.4)

X := {(u, v) ∈ Xp × XJ | φ(u) = v}.

3.7.5. Lemma. The A-module X is orthonormalizable, and there are natural iso∼∼morphisms X/pX −→ Xp and X/JX −→ XJ .
Proof. Choose a basis {ei }i∈I of Xp /$s Xp , and lift {ei }i∈I (resp. {φ(ei }i∈I toan orthonormal basis {epi }i∈I of Xp (resp. an orthonormal basis {eJi }i∈I of XJ ).
One easily verifies, taking into account the isomorphism (3.7.2), that {(epi , eJi )}i∈I
forms an orthonormal basis for X. We leave the (easy) verification of the claimed
isomorphisms to the reader.

We now present a variant of the gluing procedure just discussed. To begin with,write A0 := HomO (A, O) (and define (A/J)0 analogously). The description (3.7.2)
of A gives rise to a corresponding description of A0 . Before stating it, note that∨ ∼ 1
1∼(A/J)0 /(A/J)0 [p] −→ A/(p + J) −→ s O/O.
(3.7.6)s$$
The inverse of this isomorphism gives an embedding

1
1
1
∼
(A/J)0 /(A/J)0 [p] ,→ s (A/J)0 /(A/J)0 , : s O/O −→s$$$
and we have the isomorphism
1
1∼
A0 −→ {(u0 , v 0 ) ∈ s O × s (A/J)0 | (u0 mod O) ≡ v 0 mod (A/J)0 };$$if we regard A as a subring of O × A/J via (3.7.2), then the pairing between A and
A0 is defined viah(u, v), (u0 , v 0 )i := hu, u0 i − hv, v 0 i.
We may re-express the preceding isomorphism via the following exact sequence:11(u0 ,v 0 )7→(u0 )−v 0 1−→(A/J)0 /(A/J)0 .
(3.7.7)0 −→ A0 −→ s O × s (A/J)0$$$s
Suppose now that we are given an orthonormalizable A/J-module XJ . If wedefine VJ := (A/J)0 ⊗A/J XJ , then the isomorphism (3.7.6) gives rise to a natural
isomorphism111∼( s VJ /VJ )[p] −→ ( s XJ )/p( s XJ ).
$
$
$

38

MATTHEW EMERTON

Thus if we are given an orthonormalizable O-module Xp and an isomorphism φ :∼Xp /$s Xp −→ XJ /pXJ , we may equally well regard φ an an isomorphism
1
1
1∼
φ : s Xp /Xp −→ ( s VJ /VJ )[p] ⊂ s VJ /VJ .
$$$3.7.8. Lemma. In the context of the preceding discussion, if we write11
V := {(u, v) ∈ s Xp × s VJ | φ(u mod Xp ) = v mod VJ },$$∼then there is an isomorphism A0 ⊗A X −→ V, for some orthonormalizable Amodule X.
Proof. If we define the A-module X via (3.7.4), then Lemma 3.7.5 shows that X is
orthonormalizable. Now tensoring X over A with the exact sequence (3.7.7), andtaking into account the flatness of X over A (Lemma B.6 (1), and the isomorphisms∼∼
X/pX −→ Xp and X/JX −→ XJ of Lemma 3.7.5, we obtain an exact sequence
1
1
1
0 → A0 ⊗A X → s Xp × s VJ → s VJ /VJ ,$$$where the third arrow is defined by (u, v) 7→ φ(u) − v. The lemma follows.

The next result gives a characterization of those A-modules of the type constructed in the preceding lemma. Before stating it, we note that passing to transposes induces a natural isomorphism∼

A = HomA (A, A) −→ HomA (A0 , A0 ).
Thus if X is any orthonormalizable A-module, it follows from Lemma B.6 (3) that
there is a natural isomorphism∼

X −→ HomA (A0 , A0 ⊗A X).

(3.7.9)3.7.10. Lemma. If V is an A-module, then the following are equivalent:(1) There exists an orthonormalizable A-module X and an isomorphism∼

A0 ⊗A X −→ V.
(2) HomA (A0 , V ) is an orthonormalizable A-module, and the natural map
A0 ⊗A HomA (A0 , V ) → V
is an isomorphism.
Furthermore, any isomorphism as in 1 induces an isomorphism∼

X −→ HomA (A0 , V ).
Proof. Obviously 2 implies 1. Conversely, any isomorphism as in 1 induces thesecond in the sequence of isomorphisms∼∼

X −→ HomA (A0 , A0 ⊗A X) −→ Hom(A0 , V ),the first being the isomorphism (3.7.9). Thus 1 implies 2, and the final statementof the lemma also holds.

We are finally ready to explain how the previous results can be used to constructcertain deformations of G-representations.
3.7.11. Proposition. Let V be a G-representation on an m-adically complete Amodule, and let π be a semi-simple object of Modadm
G (A). Suppose that:LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

39∼(1) There exists a deformation XJ of π over A/J such that (A/J)0 ⊗A/J XJ −→V [J].
(2) V [p] is a deformation of π over O.
∼(3) For each Serre weight U , the embedding π −→ (VJ /$VJ )[m] ,→ (V /$V )[m]induces an isomorphism
∼socH(U ) HomGL2 (Zp ) (U, π) −→ socH(U ) HomGL2 (Zp ) U, (VJ /$VJ )[m] .
(We refer to Definition 3.5.1 for the definition of Serre weight, to Definition 3.5.2 for the definition of H(U ), and to Remark 3.5.3 for the definitionof the action of H(U ) on the functor HomGL2 (Zp ) (U, – ). If W is a GL2 (Qp )representation, then we write socH(U ) HomGL2 (Zp ) (U, W ) to denote the socleof HomGL2 (Zp ) (U, W ) as a H(U )-representation.)
Then there is a deformation X of π over A and a G-equivariant A-linear isomor∼
phism A0 ⊗A X −→ V.
Proof. Write C to denote the cokernel of the inclusion V [J] ,→ V, so that we havea short exact sequence
0 → V [J] → V → C → 0(3.7.12)of A[G]-modules. Since∼

O/$r −→ A/(p + J),
it follows that the class in Ext1A[G] (C, V [J]) represented by (3.7.12) is annihilated
by multiplication by $s , and hence that there is a map of A[G]-modules φ : C →L[J]/$s L[J] such that (3.7.12) is obtained by pulling back the short exact sequence
1
1(3.7.13)
0 → V [J] → s V [J] → s V [J]/V [J] → 0$$
via φ.
Let K denote the kernel of φ. Note that K may also be described as the maximal submodule of C such that pull-back of (3.7.12) along the inclusion K ,→ C
splits; this alternate description of K shows that surjection V → C restricts to an
isomorphism∼

V [p] −→ K.

(3.7.14)

Choose t to be maximal such that K ⊂ $t C; note that $s C ⊂ K, and thusthat s ≥ t. There is an embedding φ0 : $−t K/K ,→ V [J]/$t V [J], defined as thecomposite∼φ$−t K/K −→ (C/K)[$t ] ,→ (V [J]/$s V [J])[$t ]∼∼−→ $s−t V [J]/$s V [J] −→ V [J]/$t V [J].
Pulling back (3.7.12) via the inclusion $−t K ,→ C yields a short exact sequence(3.7.15)

0 → V [J] → W → $−t K → 0,

obtained by pulling back the short exact sequence (3.7.13) along the embedding φ0 .
∼Suppose now that s > t. We claim that (W/$W )[m] −→ π ⊕ π, and (consequently) that the exact sequence
0 → (V [J]/$V [J])[m] → (W/$W )[m] → ($−t K/$−t+1 K)
is then in fact short exact (i.e. exact on the right as well), and split.

40

MATTHEW EMERTON

To see this, suppose first that t = 0. In this case (3.7.15) is split, and the claim
is evident. Suppose next that s > t > 0. Write(3.7.16)∼

B −→ {(x, y) ∈ A/J × O | ı(x) ≡ y mod $t }.

It follows from Lemma 3.7.8 that there is an orthonormalizable B-module Y such∼that B 0 ⊗B Y −→ W. We then compute that∼

(W/$W )[m] −→ (B/mB)∨ ⊗k (Y /mY ).
∼Since s > t, there is an isomorphism B/mB −→ k ⊕ k. On the other hand,∼∼∼

Y /mY −→ (Y /J)/m(Y /J) −→ XJ /mXJ −→ π.
This proves the claim when t > 0.
Continuing to suppose that s > t, we see that by virtue of our choice of t, themap ($−t K/$−t+1 K) → C/$C is not identically zero, and thus we find that theimage of the natural map (W/$W )[m] → (V /$V )[m] has a non-zero projection
into C/$C. Since π is semi-simple as a G-representation, so is (W/$W )[m], andthus so is its image in (V /$V )[m]. Thus this image contains an irreducible Gsubrepresentation, say Y , which embeds into C/$C. If U is a Serre weight lying
in the GL2 (Zp )-socle of Y , then the composite
socH(U ) HomGL2 (Zp ) (U, Y ) ,→ socH(U ) HomGL2 (Zp ) U, (V /$V )[m]
→ socH(U ) HomGL2 (Zp ) (U, C/$C)
is injective, and consequently socH(U ) HomGL2 (Zp ) (U, Y ) does not lie in the imageof the embedding
socH(U ) HomGL2 (Zp ) U, (VJ /$VJ )[m] ,→ socH(U ) HomGL2 (Zp ) U, (V /$V )[m] .
This contradicts the third hypothesis in the statement of the proposition, and hence
we conclude that necessarily s = t. Thus K = $s C, hence $−s K = C and φ0 = φ,and so in fact W = V. The proposition now follows from Lemma 3.7.8.

4. Local Langlands at primes ` 6= p
Q
For any finite set Σ0 of primes distinct from p, write GΣ0 := `∈Σ0 GL2 (Q` ).
This section is devoted to the representation theory of the groups GΣ0 with p-adic
and mod p coefficients, or more generally with coefficients in objects of Comp(O).
More precisely, after recalling the definition of the Kirillov functor, we describe
various aspects of the local Langlands correspondence in these contexts. Our mainreference is the paper [43].
4.1. The Kirillov functor. In this subsection we recall the definition of the Kirillov functor on smooth GΣ0 -representations.
4.1.1. Definition. Write


Y
Y
Y
a b
P0 := {|a ∈Z×,b∈Z` } ⊂
GL2 (Z` ) ⊂ GΣ0 .
`
0 1
`∈Σ0

`∈Σ0

`∈Σ0

For each ` ∈ Σ0 , define the operator U` ∈ O[GL2 (Q` )] ⊂ O[GΣ0 ] via the formula

`−1 
X
` i
U` =
.
0 1
i=0

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

41

If X is a smooth representation of GΣ0 over an object A of Comp(O), then one easily
checks that the operators U` preserve the A-submodule of P0 -invariant vectors X P0 .
DefineFΣ0 (X) = {x ∈ X P0 | U` x = 0 for each ` ∈ Σ0 } ⊂ X P0 .
The formation of FΣ0 (X) is evidently functorial in X (with values in the categoryof A-modules), and we refer to FΣ0 as the Kirillov functor.
The following result summarizes the key properties of FΣ0 .
4.1.2. Theorem.
(1) The functor FΣ0 is exact on the category of smooth GΣ0 representations over A.
(2) If X is finitely generated over A[G], then FΣ0 (X) is finitely generated
over A.
(3) If X is an absolutely irreducible representation of GΣ0 over k, then FΣ0 (X)is at most one-dimensional over k.
4.1.3. Definition. We say that a representation X of GΣ0 over k is generic if X
contains no non-zero subrepresentations W for which FΣ0 (W ) = 0.
4.1.4. Lemma. Let f : W → X be a GΣ0 -equivariant map from one smooth admissible GΣ0 -representation over k to another. Suppose that:(1) W is generic.
(2) The map f induces an injection FΣ0 (W ) ,→ FΣ0 (X) after applying the
Kirillov functor FΣ0 .
Then FΣ0 is injective.
Proof. Let U denote the kernel of f . Since FΣ0 is exact, we see from condition 2that FΣ0 (U ) = 0. Condition 1 then implies that U = 0, as required.

4.2. Classical local Langlands for GL2 (Q` ). Let A be an object of Comp(O).
Suppose that A is furthermore a domain, with field of fractions K, and that
K is of characteristic zero. Let ` 6= p be prime, and consider a representationρ` : GQ` → GL2 (K) which is continuous, in the sense that it is obtained from a continuous representation GQ` → GL2 (A) (the target being equipped with its m-adic
topology) via extension of scalars. In [43] we define a local Langlands correspondence ρ` 7→ π(ρ` ), where π(ρ` ) is an admissible smooth representation of GL2 (Q` )over K. As usual, this correspondence is defined by first passing from the Galois representation ρ` to a Weil–Deligne representation (see [27, §8] and [43]), then
Frobenius semi-simplifying, and then applying a suitably normalized version of thelocal Langlands correspondence relating Frobenius semi-simple Weil–Deligne representations and admissible smooth GL2 (Q` )-representations. There is one caveatthat we should draw attention to, however: in [43] we apply a generic version of thiscorrespondence. As a consequence, π(ρ` ) is always generic, but in certain situations
is not irreducible.
To be clear, let us recall a more explicit description of the correspondence of [43]:L−1(1) If ρ` = χ1 χ2 , labelled so that χ1 χ−1
2 6= | |` , thenGL (Q` )π(ρ` ) := IndB 2

χ1 | |` ⊗ χ2 .

(This representation is irreducible if χ1 χ−1
6= | |` , and is a non-trivial
2extension of χ1 ◦ det by (χ1 ◦ det) ⊗ St otherwise.)

42

MATTHEW EMERTON(2) If ρ` is a non-split extension of χ2 by χ1 , then either χ1 = χ2 = χ, in whichcase
GL (Q )π(ρ` ) := IndB 2 ` χ| |` ⊗ χ,or else χ1 χ2−1 = | |` , in which caseπ(ρ` ) := (χ1 ◦ det) ⊗ St.
(3) If ρ` is absolutely irreducible, then π(ρ` ) is a cuspidal representation, withcentral character equal to det(ρ` )| |` .
Given ρ` as above, we let πe(ρ` ) denote the smooth contragredient representationto π(ρ` ).
4.3. Mod p local Langlands for GL2 (Q` ). In [43], we construct a mod p local
Langlands correspondence, whose key properties are summarized in the following
theorem:
4.3.1. Theorem. There is a map ρ` 7→ π(ρ` ) from the set of isomorphism classesof continuous representations ρ` : GQ` → GL2 (k) to the set of isomorphism classesof finite length admissible smooth GL2 (Q` )-representations satisfying the following
conditions:(1) For any ρ` , the associated GL2 (Q` )-representation π(ρ` ) is generic, in thesense of Definition 4.1.3 (equivalently, π(ρ` ) contains no finite-dimensional
subrepresentations).
(2) If K is any finite extension of Qp with residue field k, and if ρ` : GQ` →GL2 (K) is a continuous representation lifting ρ` : GQ` → GL2 (k), thenthere is a GL2 (Q` )-invariant integral lattice π(ρ` )◦ contained in π(ρ` ) (theadmissible smooth GL2 (Q` )-representation over K attached to ρ` via thelocal Langlands correspondence of the preceding section), whose mod $ reduction π(ρ` )◦ admits a GL2 (Q` )-equivariant embedding π(ρ` )◦ ,→ π(ρ` ).
Furthermore, the lattice π(ρ` )◦ is uniquely determined up to multiplication
by an element of K × .
(3) The representation π(ρ) is minimal with respect to satisfying conditions (1)and (2), i.e. given any representation π of GL2 (Q` ) satisfying these two
conditions with respect to ρ, there is a GL2 (Q` )-equivariant embedding
π(ρ) ,→ π.
Furthermore, the correspondence ρ` 7→ π(ρ` ) is characterized by these conditions.
e(ρ` ) denote the smooth
Given ρ` as in the statement of the theorem, we let π
contragredient representation to π(ρ` ).
4.4. Local Langlands in p-adic families for the groups GΣ0 . Let A be an
object of Comp(O) with maximal ideal m and suppose that A is reduced and flatover O (so that every minimal prime of A has residue characteristic zero). For eachpoint p ∈ Spec A, let κ(p) denote the fraction field of A/p.
Let Σ0 be a finite set of primes not containing p, and suppose given a continuous
representation ρ : GQ → GL2 (A). If p ∈ Spec A, then we write ρ(p) := κ(p) ⊗A ρ.
In the case of the maximal ideal m, we also write ρ := ρ(m) = k ⊗A ρ.
In [43] we prove the following result:4.4.1. Theorem. There is (up to isomorphism)at most one coadmissible (in theQsense of Definition C.23) smooth GΣ0 := `∈Σ0 GL2 (Q` )-representation X satisfying the following conditions:

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

43(1) The GΣ0 -representation (X/$X)[m]is generic, and furthermore the k
vector space FΣ0 (X/$X)[m] is at most one-dimensional.
(2) There is a Zariski dense subset of closed points Π ⊂ Spec A[1/p], such that:(a) For each p ∈ Π, if for some ` ∈ Σ0 , the Galois representation ρ(p)GQ`
is a non-generic principal series representation, then p is a minimal
prime of A.
(b) For each p ∈ Π, there is a κ(p)-linear, GΣ0 -equivariant isomorphismO∼
E ⊗O X[p] −→π(ρ(p) |GQ` ).
`∈Σ0(c) The closure in X (in the sense of Definition
P C.28) of the saturationin X (in the sense of Definition C.6) of p∈Π X[p] coincides with X
itself.
Any such X satisfies the following additional conditions:(3) X is cotorsion free as an A-module (in the sense of Definition C.37).
e is κ(a)-linearly(4) For each minimal prime a of A, the tensor product κ(a)⊗A X
and GL2 (Q` )-equivariantly isomorphic to ⊗`∈Σ0 πe(ρ` (a)).
(5) For every closed point p of Spec A[1/p], there is a GΣ0 -equivariant embeddingO
π(ρ(p) |GQ` ).
E ⊗O X[p] ,→`∈Σ04.4.2. Definition. If a coadmissible smooth representation X of GΣ0 over A satisfying the conditions of Theorem 4.4.1 exists, then we denote it by πΣ0 (ρ), and welet πeΣ0 (ρ) denote the smooth contragredient to πΣ0 (ρ).
In the case when S = {`} consists of a single prime, we write π` (ρ) and πe` (ρ)
rather than π{`} (ρ) and πe{`} (ρ) (assuming that they exist). In [43] we prove thatπΣ0 (ρ) exists if and only if π` (ρ) exists for each ` ∈ Σ0 , and that πeΣ0 (ρ) may be
identified with the maximal A-torsion free quotient of ⊗`∈Σ0 πe` (ρ).
The preceding discussion applies in particular to the case when A = O. In thiscase, we have the following result.
4.4.3. Proposition. Suppose that ρ : GQ → GL2 (O) is continuous, and write
ρE := E ⊗O ρ for the associated E-valued representation. In this case the admissible
smooth GΣ0 -representations πΣ0 (ρ) does exist, and is an admissible smooth GΣ0 representation over O, which is characterized by the following properties:(1) There is an E-linear, G-equivariant isomorphismO∼
E ⊗O πΣ0 (ρ) −→π(ρE |GQ` )`∈Σ0(where π(ρE |GQ` ) is the GL2 (Q` )-representation over E attached to ρE |GQ`
via the local Langlands correspondence discussed in Subsection 4.2).
(2) The quotient πΣ0 (ρ)/$πΣ0 (ρ) is generic, in the sense of Definition 4.1.3.
Proof. For each ` ∈ Σ0 , let π(ρE|GQ` )◦ denote the GL2 (Q` )-invariant O-lattice inπ(ρE|GQ` ) satisfying condition 2 of Theorem 4.3.1, and setπΣ0 (ρ) := ⊗`∈Σ0 π(ρE |G` )◦ .

44

MATTHEW EMERTON

The representation πΣ0 (ρ) satisfies the conditions of Theorem 4.4.1 with respect to ρ
(taking Π to consist of the zero ideal of O). Thus πΣ0 (ρ) exists. By construction, it
satisfies conditions 1 and 2 of the proposition, and is easily seen to be characterized
by those conditions.

5. $-adically completed cohomology and Hecke algebras
In this section we recall the basic definitions and results related to the $-adicallycompleted cohomology of modular curves, and the associated $-adically completed
Hecke algebras. For further details the reader may consult [36] and [38].
b 1 and some related spaces. For any compact open subgroup Kf of the
5.1. HO
adèlic group GL2 (Af ), we let Y (Kf ) denote the modular curve
Y (Kf ) := GL2 (Q)\ (C \ R) × GL2 (Af ) /Kf .
As is well-known, Y (Kf ) admits a canonical model as an algebraic curve over Q,
which we again denote by Y (Kf ). We write1
H 1 (Kf )A := Hét
Y (Kf )/ Q , A ,where the subscript ét denotes étale cohomology, and where A denotes one of E, O,
or O/$s O for some s > 0. (Mostly we will have s = 1, in which case O/$O = k.)
There is a natural action of GQ on H 1 (Kf )A .
If K p is some fixed compact open subgroup of GL2 (Apf ), then we write
H 1 (K p )A := lim
H 1 (Kp K p )A ,−→Kp

where the inductive limit is taken over all compact open subgroups Kp of G :=GL2 (Qp ), and where again A denotes one of E, O, or O/$s O. There are naturalcommuting actions of G and GQ on H 1 (K p )A .
For each s > 0 there is a natural G × GQ -equivariant isomorphism∼

H 1 (K p )O /$s H 1 (K p )O −→ H 1 (K p )O/$s O .

(5.1.1)
We write∼b 1 (K p )O := lim H 1 (K p )O /$s H 1 (K p )O −→lim
H 1 (K p )O/$s O
H
←−
←−
s

s

to denote the $-adic completion of H 1 (K p )O . Since the formation of $-adic comb 1 (K p )O extends to a $-adicallypletions is functorial, the GQ × G-action on H1pbb 1 (K p )O makes it acontinuous GQ × G-action on H (K )O . The G-action on H$-adically admissible G-representation over O, in the sense of Definition 3.1.1.
(See [36].) The isomorphisms (5.1.1) induce a natural GQ × G-equivariant isomorphism(5.1.2)∼b 1 (K p )O /$s H
b 1 (K p )O −→H
H 1 (K p )O/$s O

for each s > 0.
We writeb 1 (K p )E := E ⊗O H
b 1 (K p )O .
H
b 1 (K p )O as unit ball, and equipped with aThis is an E-Banach space, having H
continuous action of GQ × G. The G-action makes it a unitary admissible Banach
space representation of G over E.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

45

If A is one of E or O, then we writeb 1 := lim H
b 1 (K p )A ,
H
A

−→
Kp

and equip it with the O-linear inductive limit topology.10 It is also equipped witha continuous action of GQ × GL2 (Af ). For any s > 0, we also write
1
1
p
1
HO/$
s O := lim H (K )O/$ s O = lim H (Kf )O/$ s O
−→
−→
Kp

Kf

(the first inductive limit being taken over all compact open subgroups of GL2 (Apf ),and the second over all compact open subgroups of GL2 (Af )). It is equippedwith a smooth action of GQ × GL2 (Af ). (The case when s = 1, with coefficients
O/$O = k, will be of most interest.) The isomorphism (5.1.2) induces a natural
isomorphism
1 ∼
1
1
bO
bO
−→ HO/$
H
/$s H
sO ,
which is GQ × GL2 (Af )-equivariant.11
Suppose that Σ0Qis a finite set of primes not containingp, write Σ := Σ0 ∪ {p},Qand write GΣ0 := `∈Σ0 GL2 (Q` ). Let K0Σ := `6∈Σ GL2 (Z` ). If KΣ0 is an open
subgroup of GΣ0 , then we writeb 1 (KΣ )A := Hb 1 (KΣ K0Σ )A ,H
00
for A = E or O, andH 1 (KΣ0 )O/$s O := H 1 (KΣ0 K0Σ )O/$s O ,for any s > 0. In particular, when s = 1, we have
H 1 (KΣ0 )k := H 1 (KΣ0 K0Σ )k .
We also write
1b A,Σ
b 1 (KΣ )A ,H:= limH
0−→KΣ
0for A = E or O, and11HO/$s O,Σ := lim H (KΣ0 )O/$ s O ,−→KΣ
0

b 1 are equipped with a continuous action of GQ ×G×GΣ ,for s > 0. The modules H
0A,Σ
1
for A = E or O, while the modules HO/$s O,Σ are equipped with a smooth actionof the same group.
There are analogues of all the above constructions with compactly supported
cohomology in place of usual cohomology; we denote the resulting spaces by inb 1 , etc. There are also analogues
cluding an extra subscript c (as usual), thus Hc,Ob0 ,with cohomology in degree zero rather than degree one, which we denote by H
O
etc. (See [36, §4] and [38, §7.2].)
We finish this section by noting the following lemma (which can already be foundin the proof of [38, Thm. 7.10.7]).
10I.e. we take as a basis of neighbourhoods of the origin in H
b 1 those O-submodules U suchA
T b1 pb 1 (K p )A for each tame level K p . When A = E, this isthat U H(K )A is $-adically open in H
the usual locally convex inductive limit topology.
11As remarked in Section 2, the G and GL (A )-actions that we consider in this paper are
2
fobtained from those considered in [16] by applying the automorphism g 7→ (g −1 )t .

46

MATTHEW EMERTON

b 1 contains no one-dimensional G-subrepresentations.
5.1.3. Lemma. The space HE
b 1 , then cupping U with
Proof. If U is a one-dimensional G-invariant subspace of HE0b
an appropriately chosen element of HE , if necessary, we may assume that the actionof GL2 (Zp ) on U is trivial, and hence in particular that U is locally algebraic. (Cf.
the proof of [38, Lem. 7.3.19].) However, it follows from [38, Thm. 7.4.2], together
with the fact that the local factor at p of the GL2 (Af )-representation attached to ab 1 cannot contain a one-dimensional
modular newform is generic [51, p. 354], that H
E
locally algebraic G-subrepresentation. This proves the lemma.

5.1.4. Remark. Although we will not need the result in this paper, we note thatb 1 contains no non-zero
the proof of the preceding lemma extends to show that H
Efinite-dimensional G-subrepresentation. (Use the fact that any finite-dimensional
representation of SL2 (Qp ) over E is locally algebraic, as follows from [78, LG 5.42,
Thm. 2].)5.2. Hecke algebras. Fix a compact open subgroup K p of GL2 (Apf ). If Kp is acompact open subgroup of G, then we let T(Kp K p ) denote the O-algebra of GQ ×Gequivariant endomorphisms of H 1 (Kp K p )E generated by the Hecke operators S`
and T` for those primes ` distinct from p and unramified in K p . If Kp0 ⊂ Kp is
an inclusion of compact open subgroups of G, then there is a natural surjection
T(Kp0 K p ) → T(Kp K p ).
5.2.1. Definition. We define T(K p ) := lim
T(Kp K p ), and we equip T(K p ) with its←−
Kpprojective limit topology, each of the O-algebras T(Kp K p ) being equipped with its$-adic topology.
The O-algebra T(K p ) is then topologically generated by the Hecke operators S`
b 1 (K p )O , and so also on H
b 1 (K p )E .12
and T` , and acts faithfully on Hp
The O-algebra T(K ) is reduced, commutative, and complete with respect to its
natural topology, and in fact decomposes as the product of finitely many complete
local O-algebras, which are in bijection with the (finitely many) Galois conjugacy
classes of systems of Hecke eigenvalues arising from modular forms of level K p
defined over k. Furthermore, its natural topology coincides with the product of them-adic topologies on each of its finitely many local factors, and so in particular, it is
a compact topological ring. (These claims are easily deduced from Definition 5.2.1.)
For any inclusion K p0 ⊂ K p , there is induced a continuous morphism T(K p0 ) →T(K p ) taking S` and T` in the source to S` and T` in the target, for ` 6= p andunramified in K p0 , and compatible with respect to the natural closed embeddingb 1 (Kp )E ,→ H
b 1 (K p0 )E and the action of T(K p ) (resp. T(K p0 )) on its source (resp.
Htarget).
We now fix a continuous absolutely irreducible representation ρ : GQ → GL2 (k),and a finite set Σ0 of primes, not containing p, chosen so that ρ is unramified
outside Σ := Σ0 ∪ {p}. Assume that ρ is modular. (As conjectured by Serre, andproved by Khare, Wintenberger, and Kisin [53, 54, 55, 60], this is implied by the
12The topological O-algebra T(K p ) admits the following alternative definition: namely, whenb 1 (K p )E , it coincides with the weakly closed Oregarded as an algebra of endomorphisms of Hb 1 (K p )E topologically generated by the Hecke
algebra of GQ × G-equivariant endomorphisms of H
operators S` and T` for those primes ` distinct from p and unramified in K p . However, we won’t
need this interpretation in what follows.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

47

a priori weaker assumption that ρ is odd.) If KΣ0 is a compact open subgroupof GΣ0 , and if K0Σ has the same meaning as in the preceding subsection, then we
write T(KΣ0 ) := T(KΣ0 K0Σ ).
5.2.2. Definition. We say that a compact open subgroup KΣ0 ⊂ GΣ0 is an allowable level for ρ if we may find a maximal ideal m of T(KΣ0 ), having residue field k,which is associated to ρ in the usual sense, namelyT` mod m = trace ρ(Frob` ) , `S` mod m = det ρ(Frob` )for ` 6∈ Σ (or equivalently, if there exists a newform of tame level KΣ0 whose
associated Galois representation lifts ρ).
If KΣ0 is an allowable level for ρ, then we write T(KΣ0 )ρ := T(KΣ0 )m to denote the completion of T(KΣ0 ) at m; note that T(KΣ0 )ρ is simply a direct factorof T(KΣ0 ), which by construction has residue field k. Since ρ is assumed to bemodular, any sufficiently small KΣ0 ⊂ GΣ0 is an allowable level for ρ.
5.2.3. Remark. The preceding claim merits some explanation. Indeed, if modular
is simply understood to mean that ρ arises from some modular form, then a priori
one could think that it might be necessary to enlarge Σ0 first. However, it is known
that ρ arises from a modular form of level equal to the Artin conductor of ρ, and so
in particular of level divisible only by primes in Σ. (This follows from the proof of
Serre’s conjecture in [53, 54, 55, 60]. Of course, for p > 2, and for most cases whenp = 2, it was known earlier, under the rubric “the weak Serre’s conjecture implies
the strong Serre’s conjecture” – see [74] and the references therein.)
As is well-known, for any allowable level KΣ0 , we may construct a deformation
ρ(KΣ0 ) : GQ → GL2 (T(KΣ0 )ρ ) of ρ, unramified outside of Σ, uniquely determinedby the requirement
trace ρ(KΣ0 )(Frob` ) = T` , det ρ(KΣ0 )(Frob` ) = S`
for ` 6∈ Σ. (For example, for any compact open subgroup Kp of G, if we write
T(Kp KΣ0 )ρ := T(KΣ0 )ρ ⊗T(KΣ0 ) T(Kp KΣ0 K0Σ ), then [18, Thm. 3] constructs adeformation ρ(Kp KΣ0 ) : GQ → GL2 (T(Kp KΣ0 )ρ ), whose characteristic polynomial
on Frob` equals X 2 − T` X + `S` , for ` 6∈ Σ. Since T(KΣ0 )ρ := lim
T(Kp KΣ0 )ρ (by←−Kp

Definition 5.2.1), we may construct the deformation ρ(KΣ0 ) as the projective limitof the deformations ρ(Kp KΣ0 ).)Let Rρ,Σ denote the complete local Noetherian O-algebra with residue field kthat parametrizes deformations of ρ, unramified outside of Σ, over complete local
O-algebras. Let ρu denote the universal deformation of ρ to a continuous representation ρu : GQ → GL2 (Rρ,Σ ). If we writet` := trace ρu (Frob` ) ∈ Rρ,Σ , s` := `−1 det ρu (Frob` ) ∈ Rρ,Σ(` 6∈ Σ), then Rρ,Σ is topologically generated by the elements t` [18]. For eachallowable level KΣ0 , the deformation ρ(KΣ0 ) of ρ determines, and is determined by,a local O-algebra homomorphisms φ(KΣ0 ) : Rρ,Σ → T(KΣ0 )ρ . This latter map is
uniquely determined by the requirement that φ(KΣ0 )(t` ) = T` for ` 6∈ Σ. One hasin addition that φ(KΣ0 )(s` ) = S` , and hence that φ(KΣ0 ) is surjective, since theelements S` and T` topologically generate T(KΣ0 )ρ . (And one concludes that in factT(KΣ0 )ρ is topologically generated by the elements T` alone; this is well-known,
and also follows directly from the results of [18].)

48

MATTHEW EMERTON

5.2.4. Lemma.
(1) If KΣ0 0 ⊂ KΣ0 is an inclusion of allowable levels, thenthe induced map T(KΣ0 0 )ρ → T(KΣ0 )ρ is a surjection of complete local
Noetherian O-algebras.
(2) If furthermore KΣ0 is a sufficiently small allowable level, then the induced
map T(KΣ0 0 )ρ → T(KΣ0 )ρ of part 1 is an isomorphism.
Proof. Claim 1 follows from the fact that the image of T(KΣ0 0 )ρ in T(KΣ0 )ρ isclosed (since T(KΣ0 0 )ρ is compact) and contains a set of topological generators ofT(KΣ0 )ρ (namely, the elements T` , for ` 6∈ Σ). (Alternatively, one may regard eachof T(KΣ0 )ρ and T(KΣ0 0 )ρ as quotients of Rρ,Σ — see diagram (5.2.6) below.) As
for claim 2, recall that, as is explained in [17] and [63], there is a positive integerNΣ0 , divisible only by the primes in Σ0 , with the property that for any modular lift
ρ of ρ, unramified away from Σ, the prime-to-p-part of the conductor of ρ divides
NΣ0 . Thinking of both T(KΣ0 0 )ρ and T(KΣ0 )ρ as quotients of Rρ,Σ , we thus see thatthey coincide as quotients of Rρ,Σ , provided (for example) that KΣ0 is containedin the congruence subgroup of level NΣ0 .
T(KΣ0 )ρ , the projective limit being taken
5.2.5. Definition. We write Tρ,Σ = lim←−
KΣ0over the allowable levels KΣ0 of GΣ0 . If ` 6∈ Σ, we write T` to denote the elementof Tρ,Σ uniquely determined by the requirement that it map to the element T` ineach of the algebras T(KΣ0 )ρ .
Part 1 of Lemma 5.2.4 shows that the transition maps in the projective limit
defining Tρ,Σ are surjective, while part 2 of the lemma shows that they are eventually isomorphisms. In particular, we see that Tρ,Σ is a complete local Noetherian
O-algebra.
If KΣ0 0 ⊂ KΣ0 is an inclusion of allowable levels, then we obtain a commutative
diagram(5.2.6)Rρ,Σ

LL

LLLφ(KΣ0 )
0
φ(KΣ
) LLL
0


T(KΣ0 0 )ρLL&
/ T(KΣ0 )ρ ,Passing to the limit over all KΣ0 , we obtain a local surjection of complete local
O-algebras φΣ : Rρ,Σ → Tρ,Σ , uniquely characterized by the requirement thatφΣ (t` ) = T` for ` 6∈ Σ.
5.2.7. Definition. We let ρmΣ denote the deformation of ρ over Tρ,Σ associatedto φΣ .
Referring again to Lemma 5.2.4, one sees that ρmΣ coincides with ρ(KΣ0 ) for
sufficiently small KΣ0 .
As usual, for a point p ∈ Spec Tρ,Σ , we will write κ(p) to denote the residue fieldof p. We recall in particular that if p is a closed point of Spec Tρ,Σ [1/p], then κ(p)is a finite extension of E.
5.2.8. Definition. We say that a closed point p ∈ Spec Tρ,Σ [1/p] is a classicalclosed point if the system of Hecke eigenvalues Spec Tρ,Σ [1/p] → κ(p) determinedby p arises from a classical cuspform of weight k ≥ 2.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

49m
5.2.9. Definition. For any p ∈ Spec Tρ,Σ , we write ρ(p)◦ := ρm
Σ /pρΣ , and ρ(p) :=◦m ∼κ(p) ⊗Tρ,Σ ρΣ = κ(p) ⊗Tρ,Σ /p ρ(p) .

Thus ρ(p) is a two-dimensional representation of GQ over κ(p), and ρ(p)◦ is a
GQ -invariant lattice in ρ(p), free of rank two over the integral domain Tρ,Σ /p.
b 1 . We continue to fix ρ, Σ0 , and Σ := Σ0 ∪ {p} as in the
5.3. The ρ-part of H
O
preceding subsection.
5.3.1. Definition. For each allowable level KΣ0 ⊂ GΣ0 , and for A = E or O,
b 1 (KΣ )A . Similarly, if Kp is a fixedb 1 (KΣ )A,ρ := T(KΣ )ρ ⊗T(K ) H
we write H000Σ0compact open subgroup of G, and if Kf = Kp KΣ0 K0Σ , then we write H 1 (Kf )A,ρ :=T(KΣ0 )ρ ⊗T(KΣ0 ) H 1 (Kf )A .
b 1 (KΣ )A,ρ is natuSince T(KΣ0 )ρ is a direct factor of T(KΣ0 ), we see that H
01bb 1 (KΣ )A is
rally a direct summand of H (KΣ0 )A . Since the T(KΣ0 )-action on H01
b (KΣ )A,ρ is GQ × G-invariant. (Here, as in DefiGQ × G-equivariant, we see that H
0nition 5.3.1, A = E or O.) If m denotes the maximal ideal in T(KΣ0 ) correspondingb 1 (KΣ )O,ρ may
to ρ (so that T(KΣ0 )ρ is the m-adic completion of T(KΣ0 )), then H01balso be described as the m-adic completion of H (O, KΣ0 ).
If KΣ0 0 ⊂ KΣ0 is an inclusion of allowable levels of GΣ0 , then the closed emb 1 (KΣ )A ,→ H
b 1 (K 0 )A restricts to a closed embedding H
b 1 (KΣ )A,ρ ,→bedding H
0
0
Σ0
1
0
b
H (KΣ0 )A,ρ (again for A = E or O).
b 1 (KΣ )A,ρ , where theb1
H
5.3.2. Definition. For A = E or O, write H
0A,ρ,Σ := lim−→
KΣ0inductive limit is taken over all allowable levels KΣ0 ⊂ GΣ0 .
b1
One immediately sees that H
A,ρ,Σ is a GQ × G × GΣ0 -invariant direct summandb 1 . The natural surjection Tρ,Σ → T(KΣ )ρ (for each allowable level KΣ ⊂
of H
0
0
A,Σ
b 1 (KΣ )A,ρ as a Tρ,Σ -module, for any allowable levelGΣ0 ) allows us to regard H0b 1 (KΣ )A,ρ ,→KΣ0 ⊂ GΣ0 , and this structure is compatible with the embeddings H
0
1
0
1
b
b
H (KΣ0 )A,ρ . Thus HA,ρ,Σ is also naturally a Tρ,Σ -module, and the GQ × G × GΣ0 b1
action on H
A,ρ,Σ is evidently Tρ,Σ -linear.
We also have a mod $s analogue of the preceding constructions for any s > 0,
which we denote by H 1 (KΣ0 )O/$s O,ρ (for any allowable level KΣ0 ⊂ GΣ0 ) and1HO/$s O,ρ,Σ , defined in the evident manner. The isomorphism (5.1.2) gives rise to
an isomorphism(5.3.3)∼b 1 (KΣ )O,ρ /$s H
b 1 (KΣ )O,ρ −→H
H 1 (KΣ0 )O/$s O,ρ ,00and hence also an isomorphism(5.3.4)∼1
1
1
b O,ρ,Σ
b O,ρ,Σ
H/$s H−→ HO/$s O,ρ,Σ ,for each s > 0. (In the particular case when s = 1, which will be of the most interest,
∼
1
b1
b1
the preceding isomorphism may be written as HO,ρ,Σ /$ HO,ρ,Σ −→ Hk,ρ,Σ .)b 1 (KΣ )O,ρ5.3.5. Lemma. For any allowable level KΣ0 ⊂ GΣ0 for ρ, the module H0
is a $-adically admissible representation of G over Tρ,Σ .

50

MATTHEW EMERTON

b 1 (KΣ )O,ρ is a direct summand of Hb 1 (KΣ )O , it is a $-adically
Proof. Since H00admissible representation of G over O. In order to prove the lemma, it then suffices
to observe (taking into account the isomorphism (5.3.3)) that for any i ≥ 0, eachi b1b1element of H
O,ρ,Σ /$ HO,ρ,Σ is annihilated by some power of the maximal idealof Tρ,Σ .
5.3.6. Lemma. Let KΣ0 ⊂ GΣ0 be an allowable level.
(1) The natural map
KΣ01
b E,ρ,Σ
b 1 (KΣ )E,ρ → H
H0
is an isomorphism.
(2) If Kp is a compact open subgroup of G, then, writing Kf := Kp KΣ0 K0Σ ,the natural mapb 1 (KΣ )E,ρ Kp
H 1 (Kf )E,ρ → H
0is an isomorphism.
Proof. Claim 1 follows from [36, Prop. 2.2.13]. Claim 2 follows from the isomorphism [36, (4.3.4)], taking the representation W appearing there to be the trivial
representation.

5.3.7. Definition. If τ ∈ C \ R, then let GL2 (Q)τ denote the stabilizer of τin GL2 (Q). We say thatT a compact open subgroup Kf of GL2 (Af ) is neat if theintersection GL2 (Q)τ gKf g −1 = {1} for each τ ∈ C \ R and each g ∈ GL2 (Af ).
(This condition ensures that GL2 (Q) acts on (C \ R) × GL2 (Af )/Kf without fixed
points.)5.3.8. Lemma. Let KΣ0 ⊂ GΣ0 be an allowable level.
(1) The natural map
KΣ01
b O,ρ,Σ
b 1 (KΣ )O,ρ → H
H0

is an isomorphism.
(2) If Kp is a compact open subgroup of G, and if Kf := Kp KΣ0 K0Σ is neat,then the natural mapb 1 (KΣ )O,ρ Kp
H 1 (Kf )O,ρ → H
0is an isomorphism, as is the natural map
1
H 1 (Kf )O/$s O,ρ → HO/$s O,ρ,Σ

Kp KΣ0,for any s > 0.
Proof. We begin by putting ourselves in the situation of 2. Thus we choose acompact open subgroup Kp ⊂ G, and write Kf := Kp KΣ0 K0Σ . We furthermore
assume that Kf is neat. If Kp0 ⊂ Kp and KΣ0 0 ⊂ KΣ0 are normal open subgroups,
and if we write Kf0 := Kp0 KΣ0 0 K0Σ , then the Hochschild–Serre spectral sequencegives, for any s > 0, a short exact sequence
0 → H 1 Kf /Kf0 , H 0 (Y (Kf0 ), O/$s O) → H 1 Y (Kf ), O/$s O
K→ H 1 Y (Kf0 ), O/$s O f → 0.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

51(Note that our assumption that Kf is neat implies that Kf /Kf0 acts freely on
Y (Kf0 ), so that we in a context to which the Hochschild–Serre spectral sequence
applies.) Since ρ is assumed to be irreducible, localizing at ρ yields an isomorphism
K∼(5.3.9)H 1 (Kf )O/$s O,ρ −→ H 1 (Kf0 )O/$s O,ρ f .
Passing to the inductive limit over Kp0 and KΣ0 0 in (5.3.9), one obtains the secondof the isomorphisms in 2.
Taking Kp0 = Kp in (5.3.9), then passing to the inductive limit over Kp , andthen to the projective limit over s, one obtains an isomorphism∼b 1 (KΣ )O,ρ −→b 1 (KΣ0 )O,ρ KΣ0 .
H
H
00Passing to the inductive limit over KΣ0 0 yields the isomorphism of part 1.
Passing to the inductive limit over Kp0 in (5.3.9), then to the projective limit
over s, and then to the inductive limit over KΣ0 0 , one obtains an isomorphism
Kp KΣ0∼1
b O,ρ,Σ
H 1 (Kf )O,ρ −→ H.
In combination with the isomorphism of part 1, this yields the first of the isomorphisms of part 2.

5.3.10. Remark. A lemma of Carayol [17] shows that even when Kf is not neat,claim 2 of the preceding result holds as long as ρ is not induced from a characterof GQ(√−1) or GQ(√−3) .
1
5.3.11. Proposition. For any s > 0, the space HO/$s O,ρ,Σ is an admissible smooth0srepresentation of G × GΣ0 over O/$ O.

Proof. This follows from part 2 of the preceding lemma, and the fact that the space
H 1 (Kf )O/$s O,ρ is finitely generated over O/$s O.
5.3.12. Remark. In fact, the preceding proposition holds generally, without lo1
calizing at ρ ; i.e. HO/$is an admissible smooth representation of G × GΣ0 .
s O,Σ
0However, the analogue of Lemma 5.3.8 does not hold in general.
Kpb1
5.3.13. Proposition. If Kp is any compact open subgroup of G, then (HE,ρ,Σ )
is finitely generated as an E[GΣ0 ]-module.

Proof. If ρ is any lift of ρ, then for any prime ` 6= p, the conductor of ρ at ` isbounded in terms of the conductor of ρ of ` [17, 63]. Thus if we fix some power pnof p, any classical lift ρ of ρ that is unramified outside Σ, and whose conductor at
p is bounded by pn , in fact has bounded conductor. It follows that there are only
finitely many such non-isomorphic lifts of any given weight k ≥ 2.
In particular, there are only finitely many lifts ρ of ρ that are classical of weight 2,
unramified away from Σ, and for which π(ρ |GQp ) (the representation of GL2 (Qp )associated to ρ |GQp via the classical local Langlands correspondence at p) has nonzero Kp -invariants. Replacing E by a finite extension, if necessary, we may assume
that all such lifts ρ are defined over E. It then follows from [38, Thm. 7.4.2] andthe main theorem of [16] that there is an isomorphismM∼1b E,ρ,Σ
)Kp −→
πp (ρ |GQp )Kp ⊗ πΣ0 (ρ),
(H
ρ

52

MATTHEW EMERTON

where the direct sum runs over all such lifts ρ. Since the direct sum is finite,since each of the spaces πp (ρ |GQp )Kp is finite-dimensional, and since each of theGΣ0 -representations πΣ0 (ρ) is finitely generated, the proposition follows.
b 1 )Kp is5.3.14. Corollary. If Kp is any compact open subgroup of G, then (Hk,ρ,Σ
finitely generated as a k[GΣ0 ]-module.
Proof. It follows from Lemma 5.3.8 that reduction modulo $ induces an isomorphism∼
111b O,ρ,Σ
b O,ρ,Σ
)Kp /$(H)Kp −→ (Hk,ρ,Σ
)Kp .
(H
1
)Kp is the reduction modulo $ of aThus Proposition 5.3.13 shows that (Hk,ρ,ΣGΣ0 -invariant O-lattice in a finitely generated admissible smooth representation ofGΣ0 over E. Since any such lattice is finitely generated over O[GΣ0 ], we conclude1)Kp is finitely generated over k[GΣ0 ], as required.
that (Hk,ρ,Σ

5.3.15. Proposition. If Kp is a compact open subgroup of G, and if KΣ0 ⊂ GΣ0
is an allowable level, chosen so that Kp KΣ0 K0Σ is neat, then, for each s > 0,H 1 (KΣ0 )O/$s O,ρ is injective as a smooth representation of Kp over O/$s O.
Proof. Let M be any finitely generated smooth representation of Kp over O/$s O,
and write M ∨ := HomO/$s O (M, O/$s O) to denote the Pontrjagin dual of M .
Since Kp KΣ0 K0Σ is neat, the representation M ∨ induces a local system M∨ oneach of the modular curves Y (Kp0 KΣ0 K0Σ ), as Kp0 ranges over the normal open
subgroups of Kp . Since M is finitely generated and smooth, if Kp0 is sufficiently
small, then Kp0 acts trivially on M , and so M∨ is the constant local system. Thus,
if we write∼

H 1 (KΣ0 )O/$s O −→ lim
H 1 (Y (Kp0 KΣ0 K0Σ ), O/$s O),−→0
Kp

where Kp0 runs over all sufficiently small normal open subgroups of Kp , then wefind that(5.3.16) HomKp M, H 1 (KΣ0 )O/$s O∼−→ lim
HomKp M, H 1 (Y (Kp0 KΣ0 K0Σ ), O/$s O)
−→
0
Kp

∼

−→ limH 1 Y (Kp0 KΣ0 K0Σ ), M∨
−→

Kp

.

0
Kp

On the other hand, the Hochschild-Serre spectral sequence yields, after passing to
the limit over all Kp0 , a short exact sequence
H 1 Kp /Kp0 , H 0 (Y (Kp KΣ0 K0Σ ), M∨ ) → H 1 Y (Kp KΣ0 K0Σ ), M∨
0 → lim
−→
0
Kp

→ lim
H 1 Y (Kp0 KΣ0 K0Σ ), M∨
−→

Kp

→ 0,

0
Kp

which, when combined with (5.3.16), may be rewritten as the short exact sequence
0 → lim
H 1 Kp /Kp0 , H 0 (Y (Kp KΣ0 K0Σ ), M∨ ) → H 1 Y (Kp KΣ0 K0Σ ), M∨
−→
0
Kp


→ HomKp M, H 1 (KΣ0 )O/$s O → 0.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

53

If we localize at the maximal ideal in the Hecke algebra corresponding to ρ, theH 0 terms in this short exact sequence vanish, since ρ is absolutely irreducible byassumption, and so we obtain a natural isomorphism ∼
(5.3.17)H 1 Y (Kp KΣ0 K0Σ ), M∨ ρ −→ HomKp M, H 1 (KΣ0 )O/$s O,ρ .
Now if 0 → M0 → M1 → M2 → 0 is any short exact sequence of finitely generated
smooth Kp -representations over O/$s O, then we obtain a corresponding short
exact sequence of sheaves∨∨
0 → M∨
2 → M1 → M0 → 0,

and hence a long exact sequence of cohomology


0
Σ
∨
0 → H 0 Y (Kp KΣ0 K0Σ ), M∨
2 → H Y (Kp KΣ0 K0 ), M1


1
Σ
∨
→ H 0 Y (Kp KΣ0 K0Σ ), M∨
0 → H Y (Kp KΣ0 K0 ), M2


1
Σ
∨
→ H 1 Y (Kp KΣ0 K0Σ ), M∨
1 → H Y (Kp KΣ0 K0 ), M0 → 0.
Again, localizing at the maximal ideal associated to ρ annihilates the H 0 terms,
and hence we obtain a short exact sequence


1
Σ
∨
0 → H 1 Y (Kp KΣ0 K0Σ ), M∨
2 ρ → H Y (Kp KΣ0 K0 ), M1 ρ
→ H 1 Y (Kp KΣ0 K0Σ ), M∨
0 ρ → 0.
The natural isomorphism (5.3.17) then shows that the corresponding exact sequenceof Homs
0 → HomKp M2 , H 1 (KΣ0 )O/$s O,ρ → HomKp M1 , H 1 (KΣ0 )O/$s O,ρ→ HomKp M0 , H 1 (KΣ0 )O/$s O,ρ
is also exact on the right. Thus HomKp –, H 1 (KΣ0 )O/$s O,ρ is an exact functor(on the category of finitely generated smooth Kp -representations over O/$s O,
and hence on the category of all smooth Kp -representations over O/$s O), and so
H 1 (KΣ0 )O/$s O,ρ is indeed injective as a smooth Kp -representation over O. This
completes the proof of the proposition.

5.3.18. Remark. The method of proof of the preceding proposition is a standard
technique in the study of the cohomology of sheaves on modular curves, which canbe summarized by the statement that H 1 becomes an exact functor after localizing
at a non-Eisenstein maximal ideal of the Hecke algebra. The interpretation in termsof injectivity is a useful way to express this technique (and its many consequences)
via a simple structural statement.
5.3.19. Corollary. If Kp is a pro-p open subgroup of G, and if KΣ0 ⊂ GΣ0 is anallowable level, chosen so that Kp KΣ0 K0Σ is neat, then, for some r > 0, there is anb 1 (KΣ )O,ρ ∼isomorphism H= C (Kp , O)r of $-adically admissible Kp -representations
0
over O.
Proof. It suffices to show that there is an isomorphism of admissible smooth representations of Kp over O,(5.3.20)b 1 (KΣ )O,ρ ∼b 1 (KΣ )O,ρ /$s H
H= C (K p , O/$s O)r ,
0
0

54

MATTHEW EMERTON

for some r > 0, and each s > 0. It will then follow that r is independent of s, and we
can pass to the projective limit in s to establish the isomorphism in the statementof the corollary.
Since Kp is a pro-p group, the completed group ring (O/$s O)[[Kp ]] is a (noncommutative) local ring. Thus, if M is any non-zero finitely generated projective(O/$s O)[[Kp ]]-module, then M is isomorphic to (O/$s O)[[Kp ]]r for some r > 0.
Dualizing, we find that if π is a smooth admissible Kp -module over O/$s O thatis also injective, then there is an isomorphism π ∼= C (K p , O/$s O)r of admissible
smooth representations of Kp over O, for some r > 0. Consequently, to obtain theb 1 (KΣ )O,ρ /$s H
b 1 (KΣ )O,ρ is injecisomorphism (5.3.20), it suffices to show that H
00tive as an admissible smooth representation (or equivalently, by [40, Prop. 2.1.6],
simply as a smooth representation) of Kp over O/$s O. Taking into account theisomorphism (5.3.3), we must equivalently show that H 1 (KΣ0 )O/$s O,ρ is injective,
for each s > 0. This follows from the preceding proposition.

5.4. A density result. Recall that for any G-representation V , we write Vl.alg
to denote the G-subrepresentation of V consisting of locally algebraic vectors, andthat VGL2 (Zp )−alg denotes the subspace of V consisting of vectors which are algebraic
under the action of GL2 (Zp ).
5.4.1. Proposition. If KΣ0 ⊂ GΣ0 is an allowable level, then the space of GL2 (Zp )
b 1 (KΣ )E,ρb 1 (KΣ )E,ρ .
algebraic vectors H
is dense in H00
GL (Z )−alg2pProof. Let Kp be a compact open pro-p-subgroup of G, chosen to be normal in
GL2 (Zp ), and also chosen so small that Kp KΣ0 K0Σ is neat. Corollary 5.3.19 thenb 1 (KΣ )E,ρ ∼shows that H= C (Kp , E)r for some r > 0, and hence that the topological010bdual H (KΣ0 )E,ρ is free as a module over E ⊗O O[[Kp ]].
b 1 (KΣ )0
This implies in turn that H
0 E,ρ is projective as an E ⊗O O[[GL2 (Zp )]]module. Indeed, we have a natural isomorphism of functors
b 1 (KΣ )0E,ρ , –HomE⊗O O[[GL2 (Zp )]] H0∼b 1 (KΣ )0E,ρ , – GL2 (Zp )/Kp−→ HomE⊗O O[[Kp ]] H
0
(where the superscript indicates passage to invariants under the natural action ofb 1 (KΣ )0 , – ), the target of which is exact (byGL2 (Zp )/Kp on HomE⊗O O[[Kp ]] H
0 E,ρb 1 (KΣ )0 over E ⊗O O[[Kp ]], together with the fact that
virtue of the freeness of H
0 E,ρpassage to invariants under the finite group GL2 (Zp )/Kp is exact).
Since any projective module is a direct summand of a free module, undoing theb 1 (KΣ )E,ρ may be GL2 (Zp )-equivariantly embedded as aduality we find that H0stopological direct summand of
 C GL2 (Zp ), E for some s > 0, and thus it suffices
to show that C GL2 (Zp ), E) GL (Z )−l.alg is dense in C GL2 (Zp ), E GL (Z )−l.alg .
2
p
2
p

The space C GL2 (Zp ), E GL (Z )−alg is evidently equal to the space of polynomial2pfunctions (with coefficients in E) on GL2 (Zp ) (regarded as the Zp -points of theaffine group scheme GL2 ). Since the theory of Mahler expansions shows that theseare dense in the space C GL2 (Zp ), E , the proposition follows.

5.4.2. Remark. In fact the proceeding result remains true even without localizing at ρ. That is, for any compact open subgroup K p of GL2 (Apf ), the spaceb 1 (K p )E,GL (Z )−alg is dense in the E-Banach space H
b 1 (K p )E . To see this, noteH2p

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

55

that it suffices to prove the proposition after replacing K p by an open subgroup K p0 ,since averaging over a set of coset representatives of K p0 in K p induces a continuousb 1 (K p0 )E onto H
b 1 (K p )E which commutes with the G-action on
projection from H
these spaces (by [36, Prop. 2.2.13] and its proof). Thus we may shrink K p if necessary so that Kf := GL2 (Zp )K p is neat. The proof of [36, Thm. 2.1.5] — together
with the fact that each modular curve Y (Kf ) is homotopic to a finite simplicial
b 1 (K p )E fits into a right exact sequence
complex of dimension one — shows that H
of GL2 (Zp )-representations on E-Banach spacesrsb 1 (K p )E → 0
C GL2 (Zp ), E → C GL2 (Zp ), E → Hsb 1 (K p )E induces a map(for some r, s ≥ 0), and the surjection C GL2 (Zp ), E → Hsb 1 (K p )E,GL (Z )−alg .
C GL2 (Zp ), E GL2 (Zp )−alg → H2pAs we already observed in the preceding proof, the space C GL2 (Zp ), E GL (Z )−alg2p
b 1 (K p )E,GL (Z )−alg is indeed
is dense in C GL2 (Zp ), E , and so it follows that H2pb 1 (K p )E,GL (Z )−alg , as claimed.
dense in H2pb 1 (K p )E may be regarded as a cohomological analogue
5.4.3. Remark. The space Hof the space of p-adic modular forms of tame level Kp with coefficients in E. The
preceding result is then
L analogous pto the result of Katz [52, Thm. 2.1], whichshows that the spacek≥0 Mk (K , E) is dense in the space of p-adic modularpforms. (Here Mk (K , E) denotes the space of weight k modular forms of levelGL2 (Zp )K p .)We fix ρ, Σ0 , and Σ := Σ0 ∪ {p} as in the preceding subsections.
5.4.4. Definition. If KΣ0 ⊂ GΣ0 is any allowable level for ρ, then we let C(KΣ0 )denote the subset of closed points p ∈ Spec T(KΣ0 )ρ [1/p] that are classical andwhose associated Galois representations are crystalline locally at p. Similarly, we
let C denote the subset of closed points p ∈ Spec Tρ,Σ [1/p] that are classical andwhose associated Galois representations are crystalline locally at p.
5.4.5. Corollary.
(1) If KΣ0 ⊂ GΣ0 is any allowable level for ρ, then the directL
b 1 (KΣ )E,ρ [p]l.alg is dense in H
b 1 (KΣ )E,ρ .
sum p∈C(KΣ ) H
0
0
0
L
b1
b1
(2) The direct sum
H[p]l.alg is dense in H
.
p∈C

E,ρ,Σ

E,ρ,Σ

b 1 (KΣ )E,GL (Z )−alg is dense H
b 1 (KΣ )E .
Proof. Proposition 5.4.1 shows that H
0
0
2
p
1
b
Tensoring with T(KΣ0 )ρ over T(KΣ0 ), we find that H (KΣ0 )E,ρ,GL2 (Zp )−alg is denseb 1 (KΣ )E,ρ . Thus E[G]H
b 1 (KΣ )E,ρ,GL (Z )−alg (i.e. the E[G]-representation
in H
0
0
2
p
1
b (KΣ )E,ρ,GL (Z )−alg ) is also dense in H
b 1 (KΣ )E,ρ . From [38,generated by H
002pThm. 7.4.2] we deduce thatMb 1 (KΣ )E,ρ,GL (Z )−alg =b 1 (KΣ )E,ρ [p]l.alg ,
E[G]H
H
0
0
2
p
p

where the sum is taken over all classical closed points p ∈ Spec Tρ,Σ corresponding
to modular forms of weight k ≥ 2 and prime-to-p conductor. The main result of[75] shows that the set of such p is precisely the set C(KΣ0 ), and so part 1 of thecorollary is proved. Part 2 then follows by passing to the inductive limit in KΣ0 . 

56

MATTHEW EMERTON

The next corollary is a well-known result, which follows (for example) from theresult of Katz recalled in Remark 5.4.3. It is also easily deduced from the precedingcorollary.
5.4.6. Corollary. The set C is Zariski dense in Spec Tρ,Σ .
T
L
b1
Proof. Suppose that t ∈ p∈C p. Then t annihilates p∈C H
E,ρ,Σ [p]. The preceding1bcorollary shows that this subspace of H
E,ρ,Σ (in fact, even its subspace of locally1balgebraic vectors) is dense in all of HE,ρ,Σ . It follows that t thus annihilates all
b1
b1
of H
, and hence (since Tρ,Σ,ρ acts faithfully on H
, by virtue of its very
E,ρ,Σ

E,ρ,Σ

definition) that t = 0.



The final result of this subsection gives a technical strengthening of the preceding result. Before stating it, we define an important class of closed points inSpec Tρ,Σ [1/p].
5.4.7. Definition. We say that a closed point p ∈ Spec Tρ,Σ [1/p] is allowable if it isa classical closed point for which the local representation ρ(p) |GQp is crystabelline
and absolutely irreducible, and is not exceptional in the sense of Definition 3.3.18.
5.4.8. Remark. It is expected (and is a consequence of Tate’s conjecture [21]) thatin fact every point p of C for which ρ(p) |GQp is absolutely irreducible is allowable.
We now state our result.
5.4.9. Lemma.
(1) If KΣ0 ⊂ GΣ0 is any allowable level for ρ, then the set ofallowable points p ∈ Spec T(KΣ0 )ρ [1/p] is Zariski dense in Spec T(KΣ0 )ρ .
(2) The set of allowable points p ∈ Spec Tρ,Σ [1/p] is Zariski dense in Spec Tρ,Σ .
Proof. If p ∈ C(KΣ0 ) is not allowable, then the theory of the eigencurve allows us towrite it as the limit of a sequence of points pn ∈ C(KΣ0 ) which are allowable. The
first claim of the lemma follows from this observation together with the preceding∼corollary. The second claim follows from the first and the fact that T(KΣ0 )ρ −→Tρ,Σ for a sufficiently small choice of KΣ0 .

b1
5.5. The Galois action on H
O,ρ,Σ . We continue to fix ρ, Σ0 , and Σ := Σ0 ∪ {p}as in the preceding subsections. In this subsection we show that it is possible tob1“factor out” the Galois action from H
O,ρ,Σ0 . To this end, we defineb1UΣ := HomTS,ρ [GQ ] (ρmΣ , HO,ρ,Σ ).
There is a natural evaluation map(5.5.1)b1
ρmΣ ⊗Tρ,Σ UΣ → HO,ρ,Σ .

Reducing each side of this map modulo $, and passing to m-torsion parts, we obtain
a map(5.5.2)1
ρ ⊗k (UΣ /$UΣ )[m] → Hk,ρ,Σ[m].

5.5.3. Proposition. The evaluation map (5.5.1) is an isomorphism.
Proof. If KΣ0 is an allowable level for ρ, then Lemma 5.3.8 provides a natural
isomorphism ∼1b 1 (KΣ )O,ρ −→b O,ρ,Σ
(5.5.4)H(H)KΣ0 ,
0LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

57

from which we in turn deduce a natural isomorphism(5.5.5)

K∼
b1
UΣ Σ0 −→ HomTρ,Σ [GQ ] ρm
Σ , H (KΣ0 )O,ρ .

Passing to KΣ0 -invariants in the source and target of (5.5.1) thus yields a map(5.5.6)K

Σ0
b 1 (KΣ )O,ρ .
ρm→HΣ ⊗Tρ,Σ UΣ
0

To prove the proposition, it suffices to show that this map is an isomorphism foreach allowable level KΣ0 .
We begin by considering the map(5.5.7)KΣ0
b 1 (KΣ )E,ρ .
E ⊗O ρm→H
Σ ⊗Tρ,Σ UΣ
0obtained by tensoring (5.5.6) with E over O. If p ∈ Spec Tρ,Σ [1/p] is any classicalb 1 (KΣ )E,ρ [p]l.alg (which the isomorphism (5.5.4) shows is natclosed point, then H0KΣ0b1urally isomorphic to (H[p]l.alg ) is isomorphic (as a GQ -representation) toE,ρ,Σ )
a direct sum of copies of ρ(p) [38, Thm. 7.4.2]. Thus the map
K

b 1 (KΣ )E,ρ [p],ρ(p) ⊗κ(p) (E ⊗O UΣ Σ0 [p]) → H
0
obtained by passing to p-torsion parts in the source and target of (5.5.7), contains
b 1 (KΣ )E,ρ [p]l.alg in its image.
H0b1
Now KΣ0 acts smoothly on H
E,ρ,Σ , and so averaging over KΣ0 induces a continuous projection onto the space of KΣ0 -invariants, which by (5.5.4) we identify withb 1 (KΣ )E,ρ . It thus follows from Corollary 5.4.5 that L H
b 1 (KΣ )E,ρ [p]l.alg (whereH00pb 1 (KΣ )E,ρ [p],p ranges over the classical closed points of Spec Tρ,Σ [1/p]) is dense in H
0and hence we see that the map (5.5.7) has dense image.
There are natural isomorphisms
∼b1(UΣ )KΣ0 −→ HomTρ,Σ [GQ ] ρm
Σ , H (KΣ0 )O,ρ
GQ
∼
∨
b1
−→ (ρm.
Σ ) ⊗TS,ρ H (KΣ0 )O,ρb 1 (KΣ )O,ρ is a $-adically admissible representation of G over O, the same
Since H0∨b1
is true of (ρmΣ ) ⊗TS,ρ H (KΣ0 )O,ρ (which is isomorphic as a G-representation to ab 1 (KΣ )O,ρ ), and thus of its closed and saturated subdirect sum of two copies of H
0
GQ
∨
1
b
, by [39, Prop. 2.4.13]. Hence (UΣ )KΣ0
H
(K)⊗
representation (ρmΣ0 )O,ρ
TS,ρ
Σ
K

Σ0
is an admissible G-representation over O, and thus so is ρm(which is isoΣ ⊗Tρ,Σ UΣ

K

morphic as a G-representation to a direct sum of two copies of UΣ Σ0 ). Thus (5.5.7)
is a continuous G-equivariant E-linear map between admissible continuous representations of G over E, and so necessarily has closed image (as follows from theresults of [77]; see [34, Prop. 6.2.9] or Proposition 3.1.3 above for explicit statements). Since we have already seen that the map (5.5.1) has dense image, we
conclude that it is in fact surjective.
Thus, in order to show that (5.5.6) is an isomorphism, it suffices, by Lemma 3.1.6,
to show that the map(5.5.8)1
ρ ⊗k (UΣ /$UΣ )[m] → Hk,ρ,Σ[m],obtained by reducing (5.5.6) modulo $ and passing to m-torsion parts of the source
and target, is injective. A consideration of the isomorphism (5.5.5) shows that there

58

MATTHEW EMERTON

is a natural embedding
K

K(UΣ Σ0 /$UΣ Σ0 )[m] ,→ Homk[GQ ] (ρ, H 1 (KΣ0 )k,ρ ),such that the natural evaluation mapρ ⊗k Homk[GQ ] (ρ, H 1 (KΣ0 )k,ρ ) → H 1 (KΣ0 )k,ρextends (5.5.8). The injectivity of (5.5.8) is thus proved once we note that thisevaluation map is injective, since ρ is an absolutely irreducible GQ -representation.

5.5.9. Corollary. The evaluation map (5.5.2) is an isomorphism.
Proof. This follows by reducing the isomorphism (5.5.1) mod $ and passing tom-torsion parts.
5.5.10. Remark. The analogue of Proposition 5.5.3 for cohomology at finite levels
has been proved by Carayol [18, Thm. 4].
5.5.11. Remark. In Section 6, we will use a similar method to furthermore “facb1
tor out” the G-action from H
O,ρ,Σ (under some appropriate additional hypotheseson ρ).
5.6. Ordinary forms. We present an interpretation of one of the main resultsof the theory of p-adically ordinary modular forms, namely the reducibility of theassociated Galois representations locally at p, from the representation-theoretic
view-point of this note. (See Remark 5.6.10 below for an indication of the connection between the statements we give and the more traditional formulation in termsof p-adically ordinary eigenforms.)
To begin with, suppose that π is a smooth representation of G over E, whichcontains a $-adically separable G-invariant O-submodule π 0 (more succinctly, aG-invariant O-lattice) which is admissible, in the sense that π 0 /$π 0 (and hence
π 0 /$n π 0 for any n ≥ 0) is an admissible smooth representation of G. Note that
π itself is then necessarily admissible. Although the G-representations π or π 0 do
not quite fit into the context considered in [39] (since the coefficients of π are E,
while the O-module π 0 will typically not be p-adically complete), we will begin byobserving that the basic definitions
 paper apply to them.
 of that
1 Zp
Indeed, if we write N0 :=⊂ GL2 (Zp ), and then set T + := {t ∈0 1
T tN0 t−1 ⊂ N0 }, then [39, Def. 3.1.3] applies perfectly well to π and π0 , and so
we may define the Hecke T + -action on π N0 and π0N0 (and, of course, the Hecke
T + -action on the latter is the restriction of the Hecke T + -action on the former).
5.6.1. Remark. Since π is an admissible smooth G-representation over E, it isin particular an admissible locally analytic G-representation over G, and so thediscussion of [35, §3.4] applies to define an action of T + on π N0 . This action isclosely related to the Hecke T + -action on π N0 defined in the preceding paragraph.
Indeed, it differs from it simply by a twist by the modulus character | |p ⊗ | |−1p .
+(Since the definition of the T -action in [35, §3.4] involves an averaging which does
not appear in the definition of the Hecke T + -action given by [39, Def. 3.1.3].)

o
n
1 ∗
For each integer r ≥ 0, write Ir := g ∈ GL2 (Zp ) | g ≡. Note that if
0 1
t ∈ T + , then tIr t−1 ⊂ Ir for each r ≥ 0, and so the Hecke T + -action on π (resp. π N0 )

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

59

preserves π Ir (resp. π0Ir ) for each r ≥ 0. We define Ar to be the image of O[T + ] in
EndO (π Ir ) under the mapping induced by the Hecke T + -action. Since Ar preserves
the separated sublattice π0Ir of the finite-dimensional E-vector space π Ir , we see∼that Ar is a finite O-algebra, and so there is an isomorphism Ar −→ Ar,m , where mranges over the finitely many maximal ideals of Ar . We say that aQmaximal ideal
m is ordinary if the image of T + is disjoint from m, write Ar,ord := ordinary m Ar,m(a direct factor of Ar ), and let er,ord denote the idempotent in A which projects
onto Aord .
S
S
5.6.2. Definition. Define OrdB (π) := r er,ord π Ir and OrdB (π0 ) := r er,ord π0Ir .
Note that OrdB (π) (resp. OrdB (π0 )) is naturally an E[T ]-module (resp. an O[T ]module).
∼

Evidently E ⊗O OrdB (π0 ) −→ OrdB (π). The following lemma shows that theformation of OrdB is similarly compatible with reduction modulo powers of $.
5.6.3. Lemma. For any integer n ≥ 0, the surjection π0 → π0 /$n π0 induces an∼embedding OrdB (π0 )/$n OrdB (π0 ) −→ OrdB (π/$n π) (where OrdB (π0 ) is definedaccording to Definition 5.6.2, while OrdB (π0 /$n π0 ) is defined according to [39,Def. 3.1.9]). If the natural map π0N0 → (π0 /$r π0 )N0 is furthermore a surjection,then this embedding is in fact an isomorphism.
Proof. Clearly∼

OrdB (π0 )/$n OrdB (π0 ) −→(5.6.4)[er,ord (π0Ir /$n π0Ir )r(the union taking place in π0 /$r π0 ). The proof of [39, Lem. 3.1.5] provides anatural isomorphism∼er,ord (π0Ir /$r π0Ir ) −→ Hom(O/$n O)[T + ] (O/$n O)[T ], π0Ir /$r π0Ir

= Hom(O/$n O)[T + ] (O/$n O)[T ], π0Ir /$r π0Ir T −finfor each r ≥ 0 (where the equality holds since π0Ir /$r π0Ir is a finite O/$n Omodule), and thus a natural isomorphism[(5.6.5)er,ord (π0Ir /$n π0Ir )r∼
Hom(O/$n O)[T + ] (O/$n O)[T ], π0Ir /$r π0Ir T −fin .
−→ lim−→r

S
S
Since π0N0 = r π0Ir , and hence π0N0 /$n π0N0 = r (π0 /$n π0 )Ir
place in π0 /$r π0 ), it follows from [39, Lem. 3.2.2] that
(5.6.6)(the union taking

lim
Hom(O/$n O)[T + ] (O/$n O)[T ], π0Ir /$r π0Ir T −fin
−→
r


∼
−→ Hom(O/$n O)[T + ] (O/$n O)[T ], π0N0 /$r π0N0 T −fin,→ Hom(O/$n O)[T + ] (O/$n O)[T ], (π0 /$n π0 )N0 =: OrdB (π0 /$n π0 ).
The isomorphisms (5.6.4) and (5.6.5), together with the embedding (5.6.6), provide the required embedding, which is clearly an isomorphism if π0N0 /$π0N0 =(π0 /$π0 )N0 .
60

MATTHEW EMERTON

5.6.7. Lemma. If π is absolutely irreducible as a G-representation, then OrdB (π)
is either zero or one-dimensional. In the latter case, if α ⊗ β denotes the character through which T acts on OrdB (π), and WD(π) denotes the representationof the Weil–Deligne group attached to π via the local Langlands correspondence(normalized as in Section 2), then there is a Weil–Deligne-equivariant embedding
α ,→ WD(π) (where we regard α as a representation of the Weil–Deligne group via∼
the isomorphism Wpab −→ Q×
p of local class field theory).
N0
Proof. Write πfs:= E[T ] ⊗E[T + ] π N0 . (Here the subscript fs stands for “finite slope”.) It follows from the discussion of Remark 5.6.1 together with [35,Prop. 4.3.4] and its proof that the natural projection π → πN induces an isomorN0 ∼phism of T -representations πfs−→ πN ⊗ (| |−1p ⊗ | |p ). (This is a reformulationof Casselman’s theory of the canonical lifting of the Jacquet module [19, §4].)
Since π is absolutely irreducible, its Jacquet module πN is either zero, one, or
N0. Clearly OrdB (π) is identified
two-dimensional, and thus the same is true of πfs
N0with that subrepresentation of πfs on which T acts (possibly after an extensionof scalars) via unitary characters (i.e. characters with values in O× , rather thanmerely in E × ). In the case when πN is two-dimensional, if χ1 ⊗ χ2 is one character
appearing in πN , the other is equal to χ2 | |p ⊗ χ1 | |−1
p , and thus the representation
πN ⊗ (| |−1
⊗
|
|
)
can
containatmosta
one-dimensional
subspace on which T +ppacts via a unitary character. Thus in all cases OrdB (π) is at most one-dimensional.
If OrdB (π) = α ⊗ β is one-dimensional, then we obtain an embedding α| |p ⊗β| |−1p ,→ πN , and so indeed an embedding α ,→ WD(π), as claimed.

In the following lemma, we change our notation. Namely, we will let π denote a
cuspidal automorphic representation of GL2 (A) attached to a weight two cuspidal
newform defined over E, and let πf denote the finite part of π (i.e. the restrictedtensor product of the local components π` for all finite primes `), which is an admissible smooth representation of GL2 (Af ) which we can take to be defined over E∼[20, Prop. 3.2]. (Here we are implicitly using our chosen identification C −→ Qp , aswell as our embedding E ,→ Qp .) If K p is any compact open subgroup of GL2 (Apf ),pthen πfK is an admissible smooth representation of G to which the previous resultspapply. (Note that πfK embeds into H 1 (Kp )E , which contains the admissible Gp
invariant O-lattice H 1 (Kp )O , and thus πfK also contains an admissible G-invariant
O-lattice.)
5.6.8. Lemma. As in the preceding paragraph, let π be a cuspidal automorphic
representation of GL2 (A) attached to a weight two cuspidal newform defined overE, so that πf is an admissible smooth representation of GL2 (Af ) over E. If ρπ :GQ → GL2 (E) denotes the $-adic Galois representation attached to π, then for
any compact open subgroup K p of GL2 (Apf ), the action of GQp on the cokernel ofthe natural embeddingp ab,Sp
ρπ ⊗E OrdB (πfK ),→ ρπ ⊗E OrdB (πfK )factors through Gab
Qp . (We refer to Definition 3.6.2 for an explanation of the notaab,Stion (– ).)Proof. Let πfp denote the restricted tensor product of the local components π` (withp` 6= p), so that πf = πp ⊗E πfp , and hence πfK = πp ⊗ (πfp )Kp . Evidently, then, weLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q
p

∼

61

p

have that OrdB (πfK ) −→ OrdB (πp ) ⊗E (πfp )K , and soab,Sp ab,Sp∼
ρπ ⊗E OrdB (πfK )−→ ρπ ⊗E OrdB (πp )⊗E (πfp )K .
Thus, to prove the lemma, it suffices to prove that the action of GQp on the cokernelof the embedding
ab,S(5.6.9)ρπ ⊗E OrdB (πp ),→ ρπ ⊗E OrdB (πp )factors through Gab
Qp .
Lemma 5.6.7 shows that OrdB (πp ) is either zero or one-dimensional. If OrdB (πp )vanishes, then there is nothing to prove, and so we may suppose that OrdB (πp )is one-dimensional, with the T -action begin given by the character α ⊗ β. Notethat S then acts on OrdB (πp ) via α. It follows from the main result of [75] that,
if WD(ρπ |GQp ) denotes the Weil–Deligne representation attached to the potentiallysemi-stable Dieudonné module of ρπ |GQp , then WD(ρπ |GQp ) and WD(πp ) coincide
up to Frobenius semi-simplification, and Lemma 5.6.7 then implies that α is asubrepresentation of WD(ρπ |GQp ). Since α is a unitary (i.e. O× -valued) characterof Q×
p , we see that in fact ρπ |GQp is reducible, and contains α as a subrepresentation(now thinking of α as a character GQp → O× ). Thus α⊗E OrdB (πp ) is a subspace of
ab,Sρπ ⊗E OrdB (πp ), and hence the cokernel of (5.6.9) is a quotient of (ρπ /α) ⊗E
OrdB (πp ). Since ρπ /α is a one-dimensional representation of GQp , the GQp -action
on this space certainly factors through Gab
Qp .
5.6.10. Remark. One needn’t appeal to [75] to deduce × the reducibility of ρπ in the
Zp 0
preceding argument. Indeed, if we write Jr :=I , then when restricted
0 1 r


p 0
corresponds precisely to the action
to πpJr , the Hecke action of the matrix
0 1of the Up -operator on modular forms of level Γ1 (pr ) (together with some auxiliary
tame level); see e.g. [36, Prop. 4.4.2]. Thus OrdB (πp ) is non-zero if and only if the
newform attached to a twist of π is a ordinary at p, in the sense that it (or one ofits p-stabilizations, if its conductor is prime-to-p) has a p-adic unit Up -eigenvalue.13
The fact that ρπ is reducible locally at p is then well-known, and is due to Delignein the case when the conductor of π is prime-to-p [28], and to Wiles in general[85, Thm. 2] (building on earlier results of Mazur–Wiles [66, §8, Prop. 1]). Indeed,
Lemma 5.6.8 is essentially a reformulation of [85, Thm. 2] (taking into account thefact that we use cohomological conventions for ρπ and geometric conventions in ournormalization of the reciprocity maps of class field theory, while in [85] the opposite
conventions are used).
5.6.11. Theorem. For any n ≥ 0, the action of GQp on the cokernel of the embedding
11
1
1
b O,ρ,Σ
b O,ρ,Σ
b O,ρ,Σ
b O,ρ,Σ
/$n H)ab,S ,→ OrdB (H/$n H)
OrdB (Hfactors through Gab
Qp . (We refer to Definition 3.6.2 for an explanation of the notaab,Stion (– ).)13Thus the theory of ordinary parts [39] is not quite an abstraction of the theory of ordinary
modular forms, but rather of what are usually called nearly ordinary modular forms.

62

MATTHEW EMERTONn b1
b 1 (KΣ )O,ρ /$n H
b 1 (KΣ )O,ρ , where KΣ
b1
H
Proof. Since H000O,ρ,Σ /$ HO,ρ,Σ := lim−→
KΣ0runs over all allowable levels for ρ in GΣ0 , it follows from [39, Lem. 3.2.2] that there
is a natural isomorphism ∼11b 1 (KΣ )O,ρ −→b O,ρ,Σ
b O,ρ,Σ
b 1 (KΣ )O,ρ /$n H
OrdB (Hlim
OrdB H
/$n H
).
0
0
−→
KΣ
0

Thus, to prove the theorem, it suffices to prove that the action of GQp on thecokernel of the embeddingb 1 (KΣ )O,ρ ab,S ,→ OrdB H
b 1 (KΣ )O,ρb 1 (KΣ )O,ρ /$n H
b 1 (KΣ )O,ρ /$n H
OrdB H00
00
factors through KΣ0 , for each allowable level KΣ0 .
The smooth G-representation H 1 (KΣ0 )E,ρ contains H 1 (KΣ0 )O,ρ as an admissible
G-invariant O-lattice, and so Definition 5.6.2 applies to each of H 1 (KΣ0 )E,ρ andH 1 (KΣ0 )O,ρ .14 Lemma 5.3.8 implies that the natural maprrb 1 (KΣ )O,ρ Ir
b 1 (KΣ )O,ρ /$n H
H 1 (KΣ0 )IO,ρ
/$n H 1 (KΣ0 )IO,ρ
→ H
00
is an isomorphism for each r ≥ 0 (indeed, both spaces are naturally identified with
H 1 (Ir KΣ0 K0Σ )O/$n O,ρ ). Passing to the inductive limit over r, we find that thenatural map
N0
N0
n 1
n b1
0
b1
H 1 (KΣ0 )NO,ρ /$ H (KΣ0 )O,ρ → H (KΣ0 )O,ρ /$ H (KΣ0 )O,ρis surjective, and hence, by Lemma 5.6.3, that
OrdB H 1 (KΣ0 )O,ρ /$n OrdB H 1 (KΣ0 )O,ρ∼
b 1 (KΣ )O,ρ /$n H
b 1 (KΣ )O,ρ .
−→ OrdB H
00Consequently, it suffices to prove that GQp acts on the cokernel of the embedding
OrdB H 1 (KΣ0 )O,ρab,S,→ OrdB H 1 (KΣ0 )O,ρ

through Gab
Qp , or equivalently, that GQp acts on the cokernel of the embedding(5.6.12)

OrdB H 1 (KΣ0 )E,ρab,S,→ OrdB H 1 (KΣ0 )E,ρ

through Gab
Qp .
We may write H 1 (KΣ0 )E,ρ as the inductive limit of its finitely generated Gsubrepresentations V , and so it suffices for each such V to show that the GQp action on the cokernel of OrdB (V )ab,S ,→ OrdB (V ) factors through Gab
Qp . Now if
we extend scalars sufficiently, we may writeKΣV ∼= ⊕π ρπ ⊗ πf 0

Σ

K0 0

,

14The only reason to cut down to a fixed auxiliary level K
Σ0 is to be be able to apply thepreceding results, which, for the sake of simplicity, were developed in the context of an admissible
representation of G, rather than in the context of an admissible representation of GL2 (Af ).

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

63

where π ranges over a finite collection of cuspidal automorphic representationsof GL2 (A) corresponding to a cuspidal newforms of weight two.15 The required
property of the cokernel of (5.6.12) is now seen to follow from Lemma 5.6.8.

5.6.13. Corollary. The action of GQp on the cokernel of the embedding
11b O,ρ,Σ
b O,ρ,Σ
)ab,S ,→ OrdB (H)
OrdB (Hfactors through Gab
Qp .
Proof. This follows directly from the preceding theorem, by passing to the projective limit over n.
1. We give a representation-theoretic re5.7. Multiplicities of weights in Hk,ρ,Σ
formulation of the classical “mod p multiplicity one” results related to the appearance of Galois representations in the mod p (or more precisely, in our setting,
mod $) cohomology of modular curves.

5.7.1. Definition. Let V be a Serre weight, as in Definition 3.5.1. We say that V
1
is a global weight of ρ if HomGL2 (Zp ) (V, Hk,ρ,Σ) is non-zero. We let W gl (ρ) denotethe set of global Serre weights of ρ.
1is annihilated by
5.7.2. Remark. Since every element of the Tρ,Σ -module Hk,ρ,Σ
1).
some power of m, the same is true of the Tρ,Σ -module HomGL2 (Zp ) (V, Hk,ρ,Σ
1
1
Thus HomGL2 (Zp ) (V, Hk,ρ,Σ ) is non-zero if and only if HomGL2 (Zp ) (V, Hk,ρ,Σ [m]) =1
HomGL2 (Zp ) (V, Hk,ρ,Σ)[m] is non-zero.

We will frequently impose the following hypothesis on ρ:
5.7.3. Assumption. The representation ρ is non-scalar locally at p, i.e. the restriction ρ |GQp is not the direct sum of two copies of the same character.
5.7.4. Remark. It is known that W gl (ρ) = W (ρ) (here, to ease notation, we are
writing W (ρ) := W (ρ|GQp ) to denote the set of Serre weights attached to ρ|GQp ,as in Subsection 3.5); indeed, this is the weight part of Serre’s conjecture. (See[14, Thm. 3.15].) However, we will not need this result in this paper (other thanin the proof of Theorem 3.3.22); we will we only need the simpler result that
W gl (ρ) ⊂ W (ρ). For those ρ whose restriction to GQp satisfies Assumptions 3.3.1,
the results of this paper will then give a new proof that W gl (ρ) = W (ρ). (SeeRemark 6.2.15 below.)1[m]) is a
If V ∈ W gl (ρ), then Remark 5.7.2 shows that HomGL2 (Zp ) (V, Hk,ρ,Σ
non-zero k-vector space. It is equipped with commuting actions of H(V ) (seeRemark 3.5.3), GQ , and GΣ0 , and so is an H(V )[GQ × GΣ0 ]-module. We write11
socH(V ) HomGL2 (Zp ) (V, Hk,ρ,Σ
[m]) to denote the socle of HomGL2 (Zp ) (V, Hk,ρ,Σ[m])
as an H(V )-module. This is again an H(V )[GQ × GΣ0 ]-module.
15The reason for restricting attention from H 1 (K )
Σ0 E,ρ to its finitely generated subrepresentations is just to be sure that only finitely many π contribute to V ⊗E Qp , so that we may indeed
find a common finite extension of E over which they are all defined. This is a purely technicalissue, related to the fact that all our definitions have been made in the context of a finite extensionof Qp , rather than (say) in the context of Qp .

64

MATTHEW EMERTON

5.7.5. Definition. We say that a weight (Symr k 2 )∨ ⊗k dets in W gl (ρ) (where0 ≤ r ≤ p − 1 and 0 ≤ s ≤ p − 2) is good if either r < p − 1, or r = p − 1 and theweight dets does not lie in W gl (ρ).
5.7.6. Remark. Note that W gl (ρ) always contains a good weight. Indeed, if(Symr k 2 )∨ ⊗k dets is a weight in W gl (ρ) that is not good, then it must be thecase that r = p − 1 and that dets is also a weight in W gl (ρ); the weight dets is thengood.
5.7.7. Theorem.
(1) W gl (ρ) ⊂ W (ρ).
(2) If ρ satisfies Assumption 5.7.3, if V ∈ W gl (ρ) is a good weight (in thesense of Definition 5.7.5), and if we write socH(V ) m(V, ρ) to denote theH(V )-socle of the H(V ) module m(V, ρ) defined in Definition 3.5.4 (to ease
notation, we write m(V, ρ) rather then m(V, ρ |GQp )), then there is an isomorphism of H(V )[GQ ]-modules1FΣ0 socH(V ) Homk[GL2 (Zp )] (V, Hk,ρ,Σ
[m]) ∼= ρ ⊗k socH(V ) m(V, ρ).
(3) If ρ satisfies Assumption 5.7.3, then for any weight V ∈ W gl (ρ), the GΣ0 1[m]) is generic (in the sense of Defrepresentation Homk[GL2 (Zp )] (V, Hk,ρ,Σ
inition 4.1.3).
Proof. The proof of part 1 is standard, but for completeness, we recall it here (ina form adapted to the representation-theoretic view-point of this paper). At thesame time, we will prove that any simple H(V )-submodule of1
socH(V ) Homk[GL2 (Zp )] (V, Hk,ρ,Σ[m])also appears in socH(V ) m(V, ρ).
First of all, since the formation of W gl (ρ) and W (ρ) are compatible with twisting ρ, we need only consider the case when V ∈ W gl (ρ) is of the form (Symr k 2 )∨ .
For any allowable level KΣ0 for ρ, reduction modulo $ induces a map of Tρ,Σ [GQ ]modulesb 1 (KΣ )O,ρ(5.7.8) HomO[GL2 (Zp )] (Symr O2 )∨ , H
0→ Homk[GL2 (Zp )] V, H 1 (KΣ0 )k,ρ ,which, if KΣ0 is chosen to be sufficiently small, is surjective, by Proposition 5.3.15.
If we write Kf := GL2 (Zp )KΣ0 , and let Vr denote the locally constant sheaf on the
curve Y (Kf ) corresponding to the representation Symr E 2 of GL2 (Q), then there
are isomorphismsb 1 (KΣ )O
E ⊗O HomO[GL2 (Zp )] (Symr O2 )∨ , H0
∼b 1 (KΣ )E−→ HomE[GL2 (Zp )] (Symr E 2 )∨ , H
0∼−→ H 1 Y (Kf ), Vr E ,where the last isomorphism follows from [36, (4.3.4)]. Recall from Subsection 3.5
that H(V ) = k[T, Z, Z −1 ], which we regardas a quotient of the ring O[T, Z, Z −1 ]−1in the evident way. We make H 1 Y (Kf ), Vr a module over O[T, Z, Z] byhaving
p 0
T act via the classical Hecke operator Tp , and Z act via the matrix
. One
0 p

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

65

then verifies that this action of O[T, Z, Z −1 ] preservesb 1 (KΣ )O,ρ ,HomO[GL2 (Zp )] (Symr O2 )∨ , H
0and that the map (5.7.8) is a map of O[T, Z, Z −1 ]-modules.

Since, by part 2 of Lemma 5.3.8, the space Homk[GL2 (Zp )] V, H 1 (KΣ0 )k,ρ isfinite-dimensional, the same is true of Homk[GL2 (Zp )] V, H 1 (KΣ0 )k,ρ [m] . Thus, byextending scalars sufficiently, we may assume that the H(V )-socle of this latter
space is a direct sum of one-dimensional H(V )-submodules.
Fix a non-zero H(V )
eigenvector in Homk[GL2 (Zp )] V, H 1 (KΣ0 )k,ρ [m] , on which H(V ) acts via the systemof eigenvalues λ : H(V ) → k. Taking into account the surjectivity of (5.7.8), theDeligne–Serre lemma shows that, extending scalars further if necessary, we may
lift this eigenvector to an eigenvector for T(KΣ0 )[T, Z] in H 1 Y (Kf ), Vr E . Thislatter eigenvector corresponds to a weight r + 2 cuspidal eigenform f of level KΣ0defined over E, with Tp -eigenvalue ap that is congruent to λ(T ) mod $, and whose
associated Galois representation ρf lifts ρ. Since 2 ≤ r+2 ≤ p+1, the known results
relating the weight r + 2 and the eigenvalue ap to the structure of ρ := ρf mod $(see e.g. [33]) then imply that V ∈ W (ρ) (proving 1), and that λ appears in m(V, ρ).
We turn to proving part 2 of the theorem. Let V ∈ W gl (ρ), and hence, bywhat we have already proved, in W (ρ). Again, by twisting, we may assume thatV = (Symr k 2 )∨ for some 0 ≤ r ≤ p−1. Extending scalars, if necessary, we may also
assume that socH(V ) m(V, ρ) is spanned by H(V )-eigenvectors. Let λ : H(V ) → k
be a system of eigenvalues appearing in socH(V ) m(V, ρ). Assumption 5.7.3 implies
that the λ-eigenspace of m(V, ρ) is one-dimensional. We will show that there is an
isomorphism1[m])H(V )=λ ∼(5.7.9)FΣ0 Homk[GL2 (Zp )] (V, Hk,ρ,Σ= ρ.
Since we have shown that any irreducible constituent of1
socH(V ) Homk[GL2 (Zp )] (V, Hk,ρ,Σ[m])

is an irreducible constituent of socH(V ) m(V, ρ), this will serve to establish part 2of the theorem.
As in Definition 4.1.1, write


Y
Y
Y
a b
P0 := {|a ∈Z×,b∈Z` } ⊂
GL2 (Z` ) ⊂ GΣ0 .
`
0 1
`∈Σ0

`∈Σ0

`∈Σ0

To see that the isomorphism (5.7.9) holds, fix an admissible level KΣ0 for ρ containing P0 , and chosen small enough for Kf := GL2 (Zp )KΣ0 to be neat. Let V k denotethe locally constant sheaf on Y (Kf ) corresponding to the representation Symr k 2
of GL2 (Fp ), and then note that∼
(5.7.10)
Homk[GL2 (Zp )] (V, H 1 (KΣ0 )k,ρ ) −→ H 1 Y (Kf ), V r m(using the Hochschild–Serre spectral sequence as in the proof of Lemma 5.3.8 (2),and taking into account the fact that ρ is irreducible). As we already noted above,
under this isomorphism the action of H(V ) becomes identified with the classicalHecke action at p. It then follows from the classical mod p multiplicity one results
for modular curves (see [33] for example) together with the isomorphism (5.7.10)
that
H(V )=λ,U` =0 ∀`∈Σ0∼
Homk[GL2 (Zp )] V, H 1 (KΣ0 )k,ρ [m]
= ρ.

66

MATTHEW EMERTON

Passing to an inductive limit over all KΣ0 containing P0 , and recalling the definitionof FΣ0 (see Definition 4.1.1), we obtain the isomorphism (5.7.9).
Part 3 of the theorem follows from Ihara’s lemma. Indeed,11[m]) ,→ V ∨ ⊗k Hk,ρ,Σ[m],Homk[GL2 (Zp )] (V, Hk,ρ,Σ

and so it suffices to show that this latter representation is generic as a GΣ0 representation. If it is not, then it contains a finite-dimensional representationof GL2 (Q` ), for some ` ∈ Σ0 , contradicting Ihara’s lemma. This completes theproof of the theorem.
b1
6. Local-global compatibility and the structure of HO,ρ,ΣIn this section we first introduce a local-global compatibility conjecture, namely
Conjecture 6.1.6, which refines Conjecture 1.1.1. We also recall from [42] the (somewhat more technical) Conjecture 6.1.4. The remainder of the section is then devoted
to stating and proving our main results in the direction of these two conjectures.
The statements of the main results are presented in Subsection 6.2, and we explain
there how they follow from the more technically involved results that are the subject of the following subsections. Subsection 6.2 also contains the derivations ofTheorem 1.2.1 and Theorem 1.2.6 from these results.
Throughout this section we fix an absolutely irreducible modular Galois representation ρ : GQ → GL2 (k), whose restriction to GQp we furthermore assume
satisfies Assumption 3.3.1. We also fix a finite set of primes Σ0 , not containing p,
such that Σ := Σ0 ∪{p} contains all primes at which ρ is ramified. We freely employ
the notation related to ρ, Σ0 , and Σ that was introduced in the preceding section.
6.1. A refined local-global compatibility conjecture. Our goal in this subsection is to state a refinement Conjecture 1.1.1, and to establish some of its consequences. In order to state the conjecture, we will need to apply the local Langlands
correspondences, both at p and away from p, to ρmΣ . We begin with the local
correspondence at p.
By construction, ρm
Σ is a deformation of ρ over Tρ,Σ . Furthermore, it follows fromcrys
Corollary 5.4.6 that the restriction ρm(ρ |GQp ) of
Σ |GQp lies in the subgroupoid Def
Def(ρ |GQp ). Thus, if we let π denote the admissible smooth G-representation overk attached to ρ |GQp by Theorem 3.3.2, then Theorem 3.3.13 shows that there ismof π to an admissible G-representation on an orthonormalizablea deformation πΣTρ,Σ -module, uniquely determined, up to isomorphism, by the requirements that
∼
m
m
m
MF(πΣ
) −→ ρm
Σ |GQp and that πΣ admits det(ρΣ |GQp )ε as a central character.
We introduce the following notation, in analogy with the notation related to ρmΣthat was introduced in Definition 5.2.9.
mm
6.1.1. Definition. For any p ∈ Spec Tρ,Σ , we write π(p)◦ := πΣ/pπΣ, and π(p) :=◦m ∼κ(p) ⊗Tρ,Σ πΣ
κ(p)
⊗
π(p)
.
(Note
that
if
p
is
a
closedpointof Spec Tρ,Σ ,=Tρ,Σ /pthen by construction π(p) = B ρ(p) |GQp ; see Remark 3.3.19 above.)

We now consider the local correspondence away from p, related to which we havethe following result.
6.1.2. Theorem. The coadmissible smooth representation πΣ0 (ρmΣ ) of GΣ0 overTρ,Σ (as defined in Definition 4.4.2 above) exists. Furthermore, if p > 2 then there∼mis an isomorphism πΣ0 (ρmΣ )/$πΣ0 (ρΣ ) [m] −→ ⊗`∈Σ0 π(ρ|GQ ).
`LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

67

Proof. This is proved in [42]. In the case when ρ satisfies Assumption 5.7.3, the existence of πΣ0 (ρmΣ ) also follows from Corollary 6.4.13, together with Theorems 6.4.16
and 6.4.20, below.

6.1.3. Remark. I expect that the second statement of Theorem 6.1.2 holds also
when p = 2, but this is currently not known in general; we have only the weakerstatement that, for each ` ∈ Σ0 , there is a GΣ0 -equivariant embeddingmπ(ρ|GQ ) ,→ π` (ρmΣ )/$π` (ρΣ ) [m],`which is an isomorphism when ρ|GQ is non-scalar, and has cokernel of dimension`
at most one otherwise. Lassina Dembélé [29] has provided computational evidence
that this map is indeed an isomorphism even in those cases when ρ|GQ is scalar.
`

If p ∈ Spec Tρ,Σ [1/p] is a closed point, then by Theorem 4.4.1 (5) there is anembedding E ⊗O πΣ0 (ρmΣ )[p] ,→ ⊗`∈Σ0 π` (ρ(p) |GQ` ). In [42] we made the following
conjecture:
6.1.4. Conjecture. For any closed point p ∈ Spec Tρ,Σ [1/p], there is an isomorphism◦∼πΣ0 (ρmΣ )[p] = πΣ0 ρ(p) ,and hence also an isomorphismE ⊗O πΣ (ρm )[p] ∼= πΣ (ρ(p)) := ⊗`∈Σ π` (ρ(p) |G ) .
0

Σ

0

0

Q`

6.1.5. Remark. In order to prove Conjecture 6.1.4, it suffices to show that, ifΣng
0 ⊂ Σ0 denotes the set of primes at which ρ(p) is non-generic principal series,
and if χ : GQ → E × is a character chosen so that χ ⊗ ρ(p) is unramified at eachng
` ∈ Σng
0 , then χ ⊗ ρ(p) is promodular of level divisible only by primes in Σ0 \ Σ0 .
We now state the main conjecture of this section.
6.1.6. Conjecture. There is a GQ × G × GΣ0 -equivariant, Tρ,Σ -linear isomorphism
f

∼

m
m
b1
ρm
Σ ⊗Tρ,Σ πΣ ⊗Tρ,Σ πΣ0 (ρΣ ) −→ HO,ρ,Σ .

(6.1.7)f(Here ⊗ denotes the completed tensor product of Definition C.43.)
We will also state a slightly weaker variation of this conjecture, but before doing
so, we introduce some further notation.
6.1.8. Definition. If θ is a deformation of π over Tρ,Σ , then for any p ∈ Spec Tρ,Σ ,mthat was introduced in Defiwe write (in analogy with the notation related to πΣnition 6.1.1):θ(p)◦ := θ/pθ,
andθ(p) := κ(p) ⊗Tρ,Σ θ ∼= κ(p) ⊗Tρ,Σ /p θ(p)◦ .
6.1.9. Conjecture. There is a deformation θ of π over Tρ,Σ , such that for every
closed point p ∈ Spec Tρ,Σ [1/p] there is an isomorphism θ(p) ∼= π(p), and such that
there is a GQ × G × GΣ0 -equivariant, Tρ,Σ -linear isomorphism(6.1.10)
f

f

∼ b1
m
ρm
Σ ⊗Tρ,Σ θ ⊗Tρ,Σ πΣ0 (ρΣ ) −→ HO,ρ,Σ .

(Here ⊗ denotes the completed tensor product of Definition C.43.)

68

MATTHEW EMERTON

6.1.11. Remark. If ρ, and thus π (by [61, Lem. 2.1.2]), has only trivial endomorphisms, so that Def(π) is representable, then the condition that θ(p) ∼= π(p) for allm(since the closed points of Spec Tρ,Σ [1/p]p ∈ Spec Tρ,Σ [1/p] implies that θ ∼= πΣare Zariski dense in Spec Tρ,Σ ). Thus in this case Conjecture 6.1.9 is equivalent toConjecture 6.1.6.
In general, Conjecture 6.1.6 implies Conjecture 6.1.9 (as one sees by takingmθ = πΣ
). The reason for considering Conjecture 6.1.9 is that, on the one hand,
we will be able to establish this conjecture in some situations in which we cannot
establish the stronger Conjecture 6.1.6, and on the other, it is already strong enoughto imply the main results described in the introduction.
In order to relate Conjectures 6.1.4 and 6.1.9 to Conjecture 1.1.1, we establish aslight strengthening of [12, Prop. 3.2.3] and [38, Prop. 7.7.7]. Let ρ : GQ → GL2 (E)be a continuous, absolutely irreducible representation, unramified outside the finite
set of primes Σ. Let KΣ0 be a compact open subgroup of GΣ0 , and let p denotethe ideal in T(KΣ0 )[1/p] generated by the elements T` − trace ρ(Frob` ) for ` 6∈ Σ
(so p is either a closed point of Spec T(KΣ0 ), if ρ is promodular of tame level KΣ0 ,or else is the unit ideal).
6.1.12. Proposition. The inclusionb 1 (KΣ )E [p]) ⊂ HomE[G ] (ρ, H
b 1 (KΣ )E )HomE[GQ ] (ρ, H00Q
is an equality.
Proof. The map Frob` 7→ `S` (for ` 6∈ Σ) extends to a continuous character GQ,Σ →T(KΣ0 )× , where GQ,Σ denotes the Galois group of the maximal extension of Q in
Q unramified outside Σ.
b 1 (KΣ )E ). The T(KΣ )-action on H
b 1 (KΣ )E induces
Define U := HomE[GQ ] (ρ, H000a T(KΣ0 )-action on U . Evaluation gives a map(6.1.13)b 1 (KΣ )E ,ρ ⊗E U → H
0which is injective, since ρ is absolutely irreducible, and is T(KΣ0 )[GQ ]-linear, if
T(KΣ0 )[GQ ] acts on the tensor product through the action of GQ on the first factor
and T(KΣ0 ) on the second factor. Applying the Eichler–Shimura relations as in the
proof of [12, Prop. 3.2.3], we find that
(6.1.14) ρ(Frob` )(v) ⊗ T` − trace(ρ(Frob` )) u + v ⊗ (`S` − det(ρ(Frob` ))u = 0
for every v ∈ ρ, u ∈ U , and ` 6∈ Σ. If ` is such that ρ(Frob` ) does not act by ascalar, then by choosing v so that ρ(Frob` )(v) and v are linearly independent, weconclude thatT` − trace(ρ(Frob` )) = `S` − det(ρ(Frob` )) = 0
on U . Since the second of these quantities is the difference of two continuous
characters on GQ,Σ , and since the Frob` for which ρ(Frob` ) is not scalar generatea dense subgroup of GQ,Σ (by Cebotarev density, together with the fact that ρ
is absolutely irreducible, and so in particular not identically scalar), we conclude
that these two characters coincide identically on GQ,Σ , and thus in particular that`S` = det(ρ(Frob` )) on U for every ` 6∈ Σ. From (6.1.14) we conclude that T` =trace(ρ(Frob` )) on U for every ` 6∈ Σ, and thus that (6.1.13) has image lying in
b 1 (KΣ )E [p]. This proves the proposition.
H
0LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

69

6.1.15. Remark. Although we will not need it in what follows, we note that theb 1 ) 6= 0, then necessarily p ispreceding proposition shows that if HomE[GQ ] (ρ, HE,Σ
a proper ideal of T(KΣ0 )[1/p], and thus ρ is promodular. (This is a strengtheningof [38, Cor. 7.7.9].) In particular, Conjecture 1.1.1 implies that every odd irreducible continuous two-dimensional representation of GQ over E is promodular.
(Theorem 1.2.3 provides strong evidence in this direction.)
6.1.16. Corollary. For each closed point p ∈ Spec Tρ,Σ [1/p], each of the inclusions
1
1
1
b E,ρ,Σ
b E,ρ,Σ
b E,Σ
HomE[GQ ] (ρ(p), H[p]) ,→ HomE[GQ ] (ρ(p), H) ,→ HomE[GQ ] (ρ(p), H)

is an equality.
Proof. We may verify the claim of the equality after making an extension of scalars,
if necessary, and thus we may assume that κ(p) = E. The claim of the corollary
is then seen to follow from Proposition 6.1.12, by passing to the limit over allcompact open subgroups KΣ0 of GΣ0 , taking into account the fact that the inclusionb1
b1
HE,ρ,Σ [p] ⊂ HE,Σ [p] is an equality, since the ideal p is associated to the deformation
ρ(p) of ρ.
We now prove the following proposition, which relates Conjectures 6.1.4 and 6.1.9
to Conjecture 1.1.1.
6.1.17. Proposition. If Conjectures 6.1.4 and 6.1.9 hold for our fixed choice of ρ,and for every possible choice of Σ0 , then Conjecture 1.1.1 holds for every promodular
lift V of ρ.
Proof. Let V be a promodular lift of ρ over E, and let p denote the closed point ofSpec Tρ,Σ corresponding to V . Corollary 6.1.16 yields the equality(6.1.18)

1
1b E,Σ
b E,ρ,Σ
[p]) = HomE[GQ ] (V, H).
HomE[GQ ] (V, H

Tensoring with E over O, then passing to p-torsion parts, and taking into accountLemma 6.1.19 below, together with Conjecture 6.1.4, the isomorphism (6.1.10) gives
rise to an isomorphismO∼1b E,ρ,Σπ` (V ),
H[p] −→ V ⊗E B(V |GQp ) ⊗E`∈Σ0from which we deduce an isomorphism∼
1b E,ρ,Σ
HomE[GQ ] (V, H[p]) −→ B(V |GQp ) ⊗E

Oπ` (V ).

`∈Σ0

Combined with the equality (6.1.18) and passing to the inductive limit over all Σ0 ,this yields the isomorphism of Conjecture 1.1.1.

6.1.19. Lemma. If Conjecture 6.1.4 holds, if θ is an orthonormalizable admissible
representation of G over Tρ,Σ , and if p is a point of Spec Tρ,Σ [1/p], then there is
an isomorphism

f
∼
m
◦
◦
◦
ρm
Σ ⊗Tρ,Σ θ ⊗Tρ,Σ πΣ0 (ρΣ ) [p] −→ ρ(p) ⊗Tρ,Σ /p θ(p) ⊗Tρ,Σ /p πΣ0 (ρ(p) ).
Proof. By definition,
f

m
m H
ˆ
ρm
ρm
Σ ⊗Tρ,Σ θ ⊗Tρ,Σ πΣ0 (ρΣ ) = limΣ ⊗Tρ,Σ θ ⊗Tρ,Σ πΣ0 (ρΣ ) ,
−→
H

70

MATTHEW EMERTON

where H runs over the open subgroups of GΣ0 , and so it suffices to prove that

∼
m H
ˆ
ρm
[p] −→ ρ(p)◦ ⊗Tρ,Σ /p θ(p)◦ ⊗Tρ,Σ /p πΣ0 (ρ(p)◦ )H ,
Σ ⊗Tρ,Σ θ ⊗Tρ,Σ πΣ0 (ρΣ )for each open subgroup H. Since ρmΣ is free of rank two over Tρ,Σ , while θ isan orthonormalizable admissible G-representation over Tρ,Σ , the tensor productρmΣ ⊗Tρ,Σ θ is again an orthonormalizable admissible G-representation over Tρ,Σ ,and thus (taking into account our assumption that Conjecture 6.1.4 holds) therequired isomorphism is provided by Lemma 3.1.17, applied with M := Tρ,Σ /p,Hm
X := πΣ0 (ρmΣ ) , and π := ρΣ ⊗Tρ,Σ θ. (Note that we may and do omit the $-adic
completion on the right hand side of this isomorphism, since πΣ0 (ρ(p)◦ )H is a freeO-module of finite rank.)11[m] := lim
Hk,ρ,Σ
[m]. Conjecture 6.1.9 has the following implication
We write Hk,ρ−→
Σ0

1
for the structure of the mod $ cohomology space Hk,ρ[m].

6.1.20. Proposition. If Conjecture 6.1.9 holds for our fixed choice of ρ, and forevery possible choice of Σ0 , and if p > 2, then there is a GQ × GL2 (Af )-equivariant,k-linear, isomorphism
O
∼
0
1
[m],
ρ ⊗k π ⊗kπ(ρ |GQ ) −→ Hk,ρ`

`6=pwhere, for each ` 6= p, the representation π(ρ |GQ ) of GL2 (Q` ) is attached to ρ |GQ``
via the mod p local Langlands correspondence of Theorem 4.3.1.
Proof. Reducing the isomorphism (6.1.10) modulo $ and then passing to m-torsion
parts, and taking into account Lemma C.45 and the second statement of Theorem 6.1.2, we obtain an isomorphism∼1ρ ⊗k π ⊗k π Σ0 (ρ) −→ Hk,ρ,Σ[m].

Passing to the inductive limit over all finite sets Σ0 then induces the required
isomorphism.

6.1.21. Remark. I expect that the preceding proposition also holds when p = 2; see
Remark 6.1.3. From the weaker statement made in that remark relating π(ρ|GQ )`mand π` (ρmΣ )/$π` (ρΣ ) [m], for eachN` ∈ Σ0 , we may deduce a correspondingly1
weaker statement relating ρ ⊗k π ⊗k `6=p 0 π(ρ |GQ ) and Hk,ρ[m]. In particular, we`
see that the former always embeds into the latter.
6.1.22. Remark. In the context of Proposition 6.1.20, in the case when ρ |GQp is not
scalar, we see that Conjecture 6.1.9 has as a consequence various classical “mod p
multiplicity one results”; see Remark 1.2.8 above. On the other hand, in the casewhen ρ |GQp is scalar, the representation π is the direct sum of two copies of thesame parabolically induced representation (see Remark 3.3.3 (1)), and so we findthat Conjecture 6.1.9 implies a mod p multiplicity two result for such ρ. This isconsistent with [84, Cor. 4.4 and 4.5], which establish unconditionally a mod p
multiplicity ≥ 2 result for such ρ.
6.1.23. Remark. Suppose (for the duration of this remark) that the restriction to
GQp of the absolutely irreducible modular representation ρ does not satisfy Assumption 3.3.1. We nevertheless expect that there will be a GQ × GL2 (Af )-equivariant,

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

71

k-linear, isomorphismρ ⊗k π ⊗k

O

0

∼

1
,
π(ρ |GQ ) −→ Hk,ρ`

`6=pwhere, just as in the statement of Proposition 6.1.20, for each ` 6= p, the representation π(ρ |GQ ) of GL2 (Q` ) is attached to ρ |GQ via the mod p local Langlands``
correspondence of Theorem 4.3.1, and where π is some representation of G over kwhich depends only on ρ |GQp . However, in this case, we do not expect remark 3.3.4
to hold, i.e., we do not expect that the semi-simplification of π will match with
ρ |GQp via the semi-simple mod p local Langlands correspondence of [8].
As an example, we will describe the form that we expect π to take in the casewhen p ≥ 5,1 ∗
ρ |GQp ∼ χ ⊗0 ε
for some k-valued character χ of GQp , and ρ is indecomposable: Let 1 denotethe trivial character of G, and let St denote the Steinberg representation of G.
The space Ext1 (1, St) is two-dimensional, and so we may construct the universal
extension
0 → St → U → W → 0,
1∨
where W = Ext (1, St) is a two-dimensional k-vector space equipped with trivial
G-action. We then expect π to be a non-split extension
0 → (χ ◦ det) ⊗ (IndG
ε−1 ⊗ ε) → π → (χ ◦ det) ⊗ U → 0.
B
(Any two such non-split extensions are isomorphic.)
We expect a similar
to occur in the context of Conjecture 1.1.1,
 phenomenon
1 ∗when V |GQp ∼ χ ⊗. Namely, while in this case we expect a formula0 εof the type stated in the conjecture to hold, we don’t expect B(V |GQp ) to be therepresentation associated to V |GQp via the p-adic local Langlands correspondence
as described for the reducible representation V |GQP in [38], but rather to be aslightly longer representation (longer in the sense that it will have an additional
one-dimensional topological Jordan–Hölder factor). Thus, in such a case, we don’t
expect Conjecture 1.1.1 to be true in the precise form that it was stated in [38].
6.2. Main results. We begin with the following result, which gives some informab1
tion about the multiplicity spaces HomE[GQ ] (ρ, HE,ρ,Σ ), for promodular liftings ρof ρ.
6.2.1. Theorem. If p is a closed point of Spec Tρ,Σ [1/p], then there is a non-zerob1
continuous G-equivariant map B ρ(p) |GQp → HomE[GQ ] (ρ(p), H
E,ρ,Σ ).
Proof. Remark 6.4.3 below shows that there is a non-zero GQ × G-equivariant hob1
momorphism ρ(p) ⊗κ(p) π(p) → H
E,ρ,Σ [p]. Since B ρ(p) |GQp = π(p), the present
theorem follows.

Related to the preceding theorem, we have the following result, which extends[12, Thm. 5.7.2]. The latter result treats the case when p is a classical point
for which ρ(p) |GQp is reducible, and potentially crystalline with distinct Hodge–
Tate weights, but in fact its proof applies directly to the more general situationconsidered in Proposition 6.2.2. We present a slight variant of that proof here.

72

MATTHEW EMERTON

6.2.2. Proposition. If p is a closed point of Spec Tρ,Σ [1/p] for which ρ(p) |GQp isneither the direct sum of two characters, nor an extension of a character by itself,then any non-zero continuous κ(p)-linear, G-equivariant map
1
b E,ρ,Σ
B ρ(p) |GQp → HomE[GQ ] (ρ(p), H)
is necessarily a closed embedding.
Proof. If ρ(p) |GQp is irreducible, then the same is true of B ρ(p) |GQp , by Proposition 3.3.24, and the proposition is immediate. Thus we may assume that ρ(p) |GQp
is the non-split extension of two distinct characters, say an extensionof χ2 by χ1 .
If χ1 χ−1
=
6
ε,
then
Proposition
3.4.2
shows
that
B
ρ(p)
is
a
non-split
exten|GQp
2
Gχ ⊗χ1 ε by IndB
χ1 ⊗χ2 ε, these two representations being topologicallysion of IndG
B 2−1
irreducible, while if χ1 χ2 = ε, then Proposition 3.4.5 shows that B ρ(p) |GQp isa non-split extension of the topologically irreducible representation IndGχ ⊗ χ1 εB 2by a representation which is itself a non-split extension of a one-dimensional repb
resentation by the topologically irreducible representation (χ1 ◦ det) ⊗E St.
We claim that any κ(p)-linear, G-equivariant map
1
b E,ρ,Σ
)B ρ(p) |GQp → HomE[GQ ] (ρ(p), Hχ ⊗ χ1 ε ofwhich is not an embedding must factor through the quotient IndG
B 2

B ρ(p) |GQp . This is obvious in the first case. In the second case, it follows from theb 1 contains no one-dimensional G-invariant subspaces, by Lemma 5.1.3.
fact that H
E
The adjointness between induction and the functor of ordinary parts provides
an isomorphism(6.2.3)1
b E,ρ,Σ
Homκ(p)[G] (IndGχ ⊗ χ1 ε, HomE[GQ ] (ρ(p), H
)
B 2


∼
1
b E,ρ,Σ
))−→ Homκ(p)[T ] χ2 ⊗ χ1 ε, OrdB (HomE[GQ ] (ρ(p), H∼1b E,ρ,Σ
) .
−→ Homκ(p)[GQ ×T ] ρ(p) ⊗κ(p) (χ2 ⊗ χ1 ε), OrdB (HWe claim that the target of this isomorphism vanishes. Granting this, we find thatthe source also vanishes, and the proposition is proved.
To see the claimed vanishing, first recall that, by Corollary 5.6.13, the GQp ab,Sb1b1
action on OrdB (Hfactors through Gab
E,ρ,Σ )/OrdB (HE,ρ,Σ )Qp . On the other
hand, since ρ(p) |GQp is non-split (by assumption), the GQp -action on ρ(p) does notfactor through Gab
Qp . It follows that the image of any non-zero κ(p)[GQ × T ]-linearmap
1b E,ρ,Σ
φ : ρ(p) ⊗κ(p) (χ2 ⊗ χ1 ε) → OrdB (H)(which is necessarily injective, since ρ(p) is injective as a GQ -representation) mustab,Sb1
have non-trivial intersection with OrdB (H. On the other hand, since χ1 6=
E,ρ,Σ )ab,S
χ2 (again by assumption), we see that ρ(p) ⊗κ(p) (χ2 ⊗ χ1 ε)= 0. Thus the1ab,Sb
image of φ necessarily has trivial intersection with OrdB (HE,ρ,Σ ), and hence
any such φ in fact vanishes, as claimed.

6.2.4. Theorem. Conjecture 6.1.4 holds.
Proof. As noted in Remark 6.1.5, in proving Conjecture 6.1.4, we may replace theclosed point p by the closed point pχ for which ρ(pχ ) ∼= χ ⊗ ρ(p), chosen so that

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

73

ρ(pχ ) is unramified at each prime ` ∈ Σng
0 , and then (now writing p rather than pχ )we must prove that ρ(p) is promodular of level divisible only by primes in Σ0 \ Σng
0 .
We will remove the primes ` ∈ Σng
from
the
level
of
ρ(p)
one
at
a
time.
Thus
we
0
fix such an `, write Σ∗0 := Σ0 \{`}, and similarly write Σ∗ := Σ\{`} = Σ∗0 ∪{p}. Let
J` denote an Iwahori subgroup of GL2 (Z` ). (We use the letter J to avoid confusion
with I` , the inertia subgroup of GQ` .) Since ρ(p) is unramified non-generic principal
series at `, we see that ρ(p) is in fact promodular of tame level KΣ∗0 J` K0Σ , for somecompact open subgroup KΣ∗0 of GΣ∗0 .
Write KΣ0 := KΣ∗0 J` , an open subgroup of GΣ0 . Remark 6.4.3 below shows that
there is a non-zero GQ -equivariant map (which is necessarily an embedding, since
ρ(p) is irreducible)(6.2.5)b 1 (KΣ )E,ρ [p].
ρ(p) ,→ H
0

Our goal is to show that p is promodular of level KΣ∗0 , and we will do this byb 1 (KΣ∗ )E,ρ [p] is non-zero.
showing that H
0
Of course, we have to use the fact that ρ(p) |GQ` is unramified non-generic principal series. More precisely, we will use the following facts: ρ(p) is unramified locallyat `, so that ρ(p)(Frob` ) is defined, and the ratio of the eigenvalues of ρ(p)(Frob` )is equal to `±1 , so that in particular ρ(p)(Frob` ) is not a scalar.
To begin with, we define an extension T(KΣ0 )∗ρ of T(KΣ0 )ρ , by adjoining theoperator U` (whose definition was given in Definition 4.1.1). If f` ∈ GQ` denotes
a choice of lift of Frob` ∈ GQ` /I` , and if x2 + ax + b ∈ T(KΣ0 )ρ [x] denotes thecharacteristic polynomial of ρ(KΣ0 )(f` ), then U`2 + aU` + b = 0, and so T(KΣ0 )∗ρ
is finite over T(KΣ0 )ρ . Since also ρ(p) is an irreducible GQ -representation, we seethat we may find a prime p∗ of T∗ρ,Σ lying over p, such that (6.2.5) may be refined
to an embeddingb 1 (KΣ )E,ρ [p∗ ].
ρ(p) ,→ H
0

Since ρ(p) is unramified locally at `, in fact this embedding factors through anembedding(6.2.6)b 1 (KΣ )I` [p∗ ].
ρ(p) ,→ H
0 E,ρ

Below we will construct an exact sequence


I`
I`
b1
b 1 (KΣ )I`
b1(6.2.7)0→ H
0 E,ρ `−new → H (KΣ0 )E,ρ → H (KΣ0 )E,ρ `−old → 0of T(KΣ0 )∗ρ [GQ` × G]-modules, with the following properties: the action of Frob`∗b 1 (KΣ )I`
on H
0 E,ρ `−new is scalar, given by the element U` ∈ T(KΣ0 )ρ , and there is
an isomorphism of T(KΣ0 )ρ [GQ` × G]-modules

⊕2
∼
b1
b 1 (KΣ )I`∗(6.2.8)H.
0 E,ρ `−old −→ H (KΣ0 )E,ρ
b 1 (KΣ )I`
If (6.2.6) were to factor through H
0 E,ρ `−new , then we would find that
ρ(p)(Frob` ) is equal to a scalar, namely the image of U` modulo p∗ . On the other
hand, we have already observed that ρ(p)(Frob` ) is not a scalar. Thus (6.2.6)

b 1 I`
must not factor through H[p∗ ], consequently the composite of (6.2.6) with`−new
b 1 (KΣ )I` [p∗ ] → H
b 1 (KΣ )I`
the map H
0 E,ρ `−old [p∗] must be non-zero, and so in
0 E,ρ

b 1 (KΣ )I`particular H
0 E,ρ `−old [p∗] 6= 0. A consideration of the isomorphism (6.2.8)

74

MATTHEW EMERTON

b 1 (KΣ∗ )E,ρ [p] 6= 0, and thus we conclude that ρ(p) is promodular
then shows that H
0of tame level KΣ∗0 , as required.
It remains to construct the exact sequence (6.2.7). This essentially arises froma consideration of the nearby cycles spectral sequence for computing the p-adicétale cohomology of the modular curves X(Kp KΣ0 K0Σ ) over Spec Z` (for compact
open subgroups Kp of G). However, since we have to work with finite and integral
coefficients, we will find it simpler, and more concrete, to instead consider the
p-power torsion in the Néron models of the Jacobians of these curves.
Fix, for the moment, a compact open subgroup Kp of G chosen small enough∗to ensure that Kp KΣ∗0 K0Σ is neat, in the sense of Definition 5.3.7. For any
compact open subgroup Kf of GL2 (Af ), we let X(Kf ) denote the completionof the modular curve Y (Kf ); it is a projective smooth curve over Q, we write
J(Kf ) := Pic0 X(Kf ) , an abelian variety over Q, and we let J (Kf ) denote the∗Néron model of J(Kf ) over Z` . The modular curve X(Kp KΣ∗0 K0Σ ) has good reduction at `, while the modular curve X(Kp KΣ0 K0Σ ) admits a semi-stable model overSpec Z` , with special fibre equal to the union of two copies of the special fibre of∗∗X(Kp KΣ∗0 K0Σ ). Thus J (Kp KΣ∗0 K0Σ ) is an abelian scheme, while the special fibreof J (Kp KΣ0 K0Σ ) is an extension of a finite étale group scheme by a semi-abelianvariety. More precisely, we have an exact sequence(6.2.9)0 → J (Kp KΣ0 K0Σ )0/F` → J (Kp KΣ0 K0Σ )/F` → Φ → 0,

where the superscript 0 denotes the connected component of the identity, and Φ,
the group scheme of connected components of J (Kp KΣ0 K0Σ )/F` , is a finite étalegroup scheme over F` , as well as an exact sequence(6.2.10)

0 → T → J (Kp KΣ0 K0Σ )0/F`∗∗→ J (Kp KΣ∗0 K0Σ )/F` × J (Kp KΣ∗0 K0Σ )/F` → 0,
where T is a torus.
We now tensor with O over Zp , and then localize at the maximal ideal m ofT(KΣ0 ) corresponding to ρ. Since the Hecke action on Φ is Eisenstein [73, Cor.,∗
p. 140] (note that our assumption that Kp KΣ∗0 K0Σ is neat ensures that condition (i)of the corollary is satisfied), while m is associated to the absolutely irreducible Galois
representation ρ, we see that (6.2.9) gives rise to an isomorphism ∼O ⊗Zp J (KΣ0 K0Σ )0/F` [pr ] m −→ O ⊗Zp J (KΣ0 K0Σ )/F` [pr ] m ,and hence that (6.2.10) gives rise to a short exact sequence of T(KΣ0 )ρ∗ [GQ` /I` ]modules(6.2.11) 0 → O ⊗Zp T [pr ] m → O ⊗Zp J (Kp KΣ0 K0Σ )/F` [pr ] m∗∗→ O ⊗Zp J (Kp KΣ∗0 K0Σ )/F` [pr ] m × O ⊗Zp J (Kp KΣ∗0 K0Σ )/F` [pr ] m → 0.
It is known (see
 e.g. [72, Prop. 3.8]) that Frob` (the geometric Frobenius) acts on
O ⊗Zp T [pr ] m via `−1 U` .
Now for any smooth projective curve X over a field k, with separable closure k s ,∼1there is a natural isomorphism Hét
(X/ks , Z/pr Z) −→ Pic0 (X)[pr ](−1), where (−1)
denotes a twist by the inverse cyclotomic character. In particular, there is an∼
isomorphism H 1 X(KΣ0 K0Σ ), Z/pr Z) −→ J(KΣ0 K0Σ )[pr ](−1). Furthermore, under
this isomorphism, the I` -invariants on the left hand side are identified with theLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

75

subgroup J (KΣ0 K0Σ )/F` [pr ] of the right-hand side. Thus we may rewrite the inverse
cyclotomic twist of (6.2.11) as a short exact sequence of T(KΣ0 )∗ρ [GQ` /I` ]-modulesI 
0 → H 1 X(Kp KΣ0 K0Σ ), O/pr O m`

`−new

I
→ H X(Kp KΣ0 K0Σ ), O/pr O m`I 
→ H 1 X(Kp KΣ0 K0Σ ), O/pr O m`
1

→ 0,
`−old

with the properties that Frob` acts on the first non-trivial term via U` , and that
there is a T(KΣ0 )ρ [GQ` /I` ]-equivariant isomorphism

I 
 ⊕2
∼
H 1 X(Kp KΣ0 K0Σ ), O/pr O m`−→ H 1 X(Kp KΣ∗0 K0Σ∗ ), O/pr O m.
`−oldPassing to the inductive limit over Kp , and then to the projective limit over r,
and noting that, since we have localized at a non-Eisenstein maximal ideal, we may
replace the closed modular curves by the corresponding open ones without changing
the spaces we are computing, we obtain the desired short exact sequence (6.2.7). 
6.2.12. Remark. The preceding argument is a p-adic analogue of the argumentused to prove [72, Thm. 6.1], an argument sometimes known as “Mazur’s principle”.
Another approach to proving Conjecture 6.1.4 is via promodularity theorems. For
example, if p > 2, ρ |GQ(ζp ) is absolutely irreducible, and ρ is p-distinguished, thenConjecture 6.1.4 follows from Theorem 1.2.3. Indeed, that theorem shows that any
lift of ρ that is ramified at only finitely many primes, and is unramified at a prime` 6= p, is promodular of level prime-to-`, and Conjecture 6.1.4 follows from this, as
was noted in Remark 6.1.5.
6.2.13. Theorem. If ρ is p-distinguished (i.e. if the semi-simplification of ρ |GQp
is not the direct sum of two copies of the same character), then Conjecture 6.1.6
holds.
Proof. This follows from Corollary 6.4.13 and Theorems 6.4.9, 6.4.16, and 6.4.20
below.

Proof of Theorem 1.2.1. Let V be a promodular two-dimensional representationof GQ unramified away from a finite set of primes, as in the statement of thetheorem, and let ρ := V denote the reduction mod $ of V . By hypothesis, ρsatisfies the running assumptions of this section, and V corresponds to a closed
point p ∈ Spec Tρ,Σ [1/p] for some Σ containing the ramified primes for V as well
as p. Theorem 6.2.1 shows that there is a non-zero map(6.2.14)1
b E,ρ,Σ
B(V |GQp ) → HomE[GQ ] (V, H[p]).

If V |GQp is neither the direct sum of two characters, nor an extension of a characterby itself, then Proposition 6.2.2 shows that (6.2.14) is in fact an embedding. Arguing as in the proof of [38, Prop. 7.8.7], we then see that (6.2.14) gives rise to a
map
O
0
1
bE
B(V |G ) ⊗Eπ` (V ) → HomE[G ] (V, H
),
Qp

Q

`6∈Σ

76

MATTHEW EMERTON

which is an embedding if (6.2.14) is. This proves claim 1 of the theorem. Claim 2 of
the theorem follows directly from Theorems 6.2.4 and 6.2.13 and Proposition 6.1.17.
Proof of Theorem 1.2.6. This theorem follows from Theorem 6.2.13 and Proposition 6.1.20.

6.2.15. Remark. In the proof of Theorem 1.2.6 we used various mod p multiplicity
one theorems, but we did not use level-lowering at the prime p (i.e. [72, Thm. 6.1]
in the case when ` = p, and its generalizations), the existence of companion forms[48], or the more difficult part of [86, Thm. 2.1 (ii)] in which the mod p nebentypus
character has conductor prime to p. Furthermore, each of these results is easily
deduced from Theorem 1.2.6 (for those ρ satisfying the hypotheses of the theorem).
In particular, Theorem 1.2.6 gives a new proof that W gl (ρ) = W (ρ) for those ρ towhich it applies.
Of course this last-mentioned result is now well-understood, both in the context
of proving that “the weak Serre’s conjecture implies the strong Serre’s conjecture”,
and as a part of the much stronger result that any odd ρ is in fact modular [53, 54,
55, 60]. Nevertheless, it seems interesting to note that it can also be proved by themethods of the p-adic Langlands program.
6.3. The Tρ,Σ [GΣ0 ]-modules X(θ). In this subsection we begin our investigationof Conjectures 6.1.6 and 6.1.9. Throughout the subsection we fix a deformation θof π over Tρ,Σ , and use the notation of Definition 6.1.8. The deformation θ will
always be assumed to satisfy the following key hypothesis:
6.3.1. Assumption. If p ∈ Spec Tρ,Σ [1/p] is an allowable closed point , then there
is a κ(p)-linear G-equivariant isomorphism θ(p) ∼= π(p).
6.3.2. Remark. If EndG (π) = k, then Def(π) is representable, and so, since
the allowable closed points p ∈ Spec Tρ,Σ [1/p] are Zariski dense in Spec Tρ,Σ (bym. However, thisLemma 5.4.9), we see that Assumption 6.3.1 implies that θ ∼= πΣneed not be true in general (i.e. if Def(π) is not representable).
6.3.3. Lemma. For any closed point p ∈ Spec Tρ,Σ [1/p], there is an isomorphism
θ(p)ss ∼= π(p)ss .
Proof. Applying M F to θ, we obtain a deformation ψ of ρ over Spec Tρ,Σ . For eachpoint p ∈ Spec Tρ,Σ , we write ψ(p) := κ(p) ⊗Tρ,Σ ψ. Assumption 6.3.1 implies that
ψ(p) ∼= ρ(p) for each allowable closed point p. Lemma 5.4.9 shows that these points
are Zariski dense in Spec Tρ,Σ , and hence we conclude that the characters of ψ andρmΣ coincide on Spec Tρ,Σ . The lemma follows from this.
Our goal is now to introduce, and to begin the study of, a certain Tρ,Σ [GΣ0 ]module, denoted X(θ), the investigation of which is intimately connected to theinvestigation of Conjectures 1.1.1 and 6.1.6.
b1
6.3.4. Definition. Write X(θ) := HomTρ,Σ [GQ ×G]−cont (ρmΣ ⊗Tρ,Σ θ, HO,ρ,Σ ), themTρ,Σ -module of Tρ,Σ -linear, GQ × G-equivariant homomorphisms from ρmΣ ⊗Tρ,Σ πΣb1
to HO,ρ,Σ that are continuous, when the source is given its m-adic topology and thetarget is given its O-linear inductive limit topology. The module X(θ) is equippedin a natural way with a Tρ,Σ -linear action of GΣ0 (induced by the action of GΣ0b1
on H
O,ρ,Σ ).

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

77

6.3.5. Remark. Ideally, for the purposes of this paper we would like to be able tomrestrict our attention to the case when θ = πΣ. However, due to limitations in our
methods, we have to develop our results for more general θ (satisfying Assumption 6.3.1).
6.3.6. Remark. For any allowable level KΣ0 for ρ, Lemma 5.3.8 yields an identification(6.3.7)∼b1
X(θ)KΣ0 −→ HomTρ,Σ [GQ ×G]−cont (ρmΣ ⊗Tρ,Σ θ, H (KΣ0 )O,ρ ).

6.3.8. Remark. If p ∈ Spec Tρ,Σ [1/p] is a closed point, then from the definition ofX(θ) we obtain natural isomorphisms
∼1b O,S,ρ
(6.3.9)X(θ)[p] −→ HomT[GQ ×G] ρ(p)◦ ⊗O θ(p)◦ , H
[p] ,
and
(6.3.10)∼1b E,ρ,Σ
[p]).
E ⊗O X(θ)[p] −→ Homκ(p)[GQ ×G]−cont (ρ(p) ⊗κ(p) θ(p), H(We do not need to include a continuity condition in the space of homomorphisms
considered in (6.3.9) only because the source and target are both naturally endowedwith the $-adic topology, with respect to which the indicated homomorphisms,
being in particular O-linear, are automatically continuous.)
Also, there is an induced embedding(6.3.11)1[m])(X(θ)/$X(θ))[m] ,→ Homk[GQ ×G] (ρ ⊗k π, Hk,ρ,Σ
∼1−→ Homk[GQ ×G] (ρ ⊗k π, Hk,ρ,Σ
).

6.3.12. Theorem. The Tρ,Σ -module X(θ) is a cofinitely generated and coadmissible
smooth representation of GΣ0 over Tρ,Σ (in the sense of Definitions C.23 and C.32).
Proof. We beginSby showing that X(θ) is a smooth representation of GΣ0 , thatis, that X(θ) = KΣ X(θ)KΣ0 , where KΣ0 runs over all allowable levels for ρ in
0GΣ0 . Equivalently, taking into account the isomorphism (6.3.7), we have to showmb1that any continuous Tρ,Σ -linear, GQ × G-equivariant map φ : ρm
Σ ⊗Tρ πΣ → HO,ρ,Σ
1b (KΣ )O,ρ , for some allowable level KΣ .
factors through H00
It follows from Lemmas 5.3.6 and 5.3.8 that\1b 1 (KΣ )E,ρ H
b O,ρ,Σ
b 1 (KΣ )O,ρ = H
H00b1(the intersection taking place in H
O,ρ,Σ ), and thus it suffices to show that anymb1
continuous Tρ,Σ -linear, GQ × G-equivariant map φ : ρm
Σ ⊗Tρ πΣ → HE,ρ,Σ (where
the source is given its m-adic topology and the target is given its locally convex
b 1 (KΣ )E,ρ , for some KΣ . In fact we will
inductive limit topology) factors through H00b1show that this is true of any O-linear continuous map. Since H
E,ρ,Σ is the inductive
1blimit of the Banach spaces H (KΣ0 )E,ρ , with closed transition maps, any boundedb1
b1
subset of H
E,ρ,Σ is contained in some H (KΣ0 )E,ρ [76, Prop. 5.6]. Thus it suffices
to show that the image of φ is bounded. For this, let U be an open neighbourhoodb 1 (KΣ )E,ρ . Then φ−1 (U ) is a neighbourhood of 0 in ρm ⊗T π m , sinceof 0 in H
0
ρ
Σ
Σ

m
φ is continuous, and so contains ms ρmΣ ⊗Tρ πΣ for some s ≥ 0. In particular itmsmm
contains $s ρmΣ ⊗Tρ πΣ , and hence $ φ(ρΣ ⊗Tρ πΣ ) ⊂ U. Thus the image of φ isindeed bounded, as claimed.

78

MATTHEW EMERTON

We now turn to showing that X(θ) is cofinitely generated and coadmissible asa smooth representation of GΣ0 over Tρ,Σ . Again taking into account the isomorphism (6.3.7), we see that for each allowable level KΣ0 for ρ, the Tρ,Σ -module
X(θ)KΣ0 inherits the properties of being $-adically separated and complete, andb1
O-torsion free, directly from the corresponding properties of H
O,ρ,Σ . Furthermore,KΣ0
the continuity property that the elements of X(θ)
are assumed to satisfy implies
that the action map Tρ,Σ × X(θ)KΣ0 → X(θ)KΣ0 is continuous, when Tρ,Σ is given
is m-adic topology and X(θ) is given its $-adic topology. Thus, in order to provethat the smooth Tρ,Σ [GΣ0 ]-module X(θ) is cofinitely generated and coadmissible,
it suffices, by Lemma C.25 and Proposition C.33, to prove that the smooth k[GΣ0 ]module (X(θ)/$X(θ))[m] is finitely generated and admissible. Taking into account1) isthe embedding (6.3.11), it suffices in turn to show that Homk[GQ ×G] (ρ⊗k π, Hk,ρ,Σ
a finitely generated admissible smooth GΣ0 -representation over k. If we let W ⊂ πdenote a finite-dimensional k-vector subspace of π that generates π over G, andif we let Kp ⊂ G denote a compact open subgroup that fixes W pointwise, thenrestriction to ρ ⊗k W induces an embedding(6.3.13)

1
1
Homk[GQ ×G] (ρ ⊗k π, Hk,ρ,Σ
) ,→ ρ∨ ⊗k W ∨ ⊗k (Hk,ρ,Σ
)Kp .

It follows from Proposition 5.3.11 and Corollary 5.3.14 that the target of (6.3.13)
is a finitely generated admissible smooth representation of GΣ0 over k, and thus so
is its source.

6.3.14. Definition. If p ∈ Spec Tρ,Σ [1/p] is a classical closed point, then we letb1b1b1
M (p)E denote the closure in H[p] (or equivalently, in H
E,ρ,Σ ) of HE,ρ,Σ [p]l.alg .
T b 1 E,ρ,Σ
b1
Write M (p)O := M (p)E HO,ρ,Σ (the intersection taking place in H
E,ρ,Σ ).
Note that M (p)E and M (p)O are each invariant under GQ × G × GΣ0 , and that,b1
by construction, M (p)O is saturated in H
O,ρ,Σ .
6.3.15. Proposition. Let p ∈ Spec Tρ,Σ [1/p] be a closed point which is allowable inthe sense of Definition 5.4.7.
(1) There is a GQ × G × GΣ0 -equivariant isomorphism∼

M (p)E −→ ρ(p) ⊗E π(p) ⊗E πΣ0 (ρ(p)).
b1(2) The inclusion M (p)E ,→ H
E,ρ,Σ [p] induces an isomorphism∼
1b E,ρ,Σ
HomE[GQ ×G] (ρ(p) ⊗E π(p), M (p)E ) −→ HomE[GQ ×G] (ρ(p) ⊗E π(p), H[p]).

Proof. Let π
ep (ρ(p) |GQp ) be the locally algebraic representation attached to ρ(p) |GQp ,as in the statement of [38, Conj. 3.3.1 (7)]. Since ρ(p) |GQp is crystabelline, absolutely irreducible, and not exceptional, the E-Banach representation π(p) of G
associated to p is isomorphic to the universal unitary completion (in the sense of[37]) of π
ep (ρ(p) |GQp ) [3]. It then follows from [38, Thm. 7.4.2], together with themain results of [16] and [75], that there is an isomorphism∼

M (p)E −→ ρ(p) ⊗E π(p) ⊗E πΣ0 (ρ(p)),
proving 1.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

79

Again using the fact that π(p) is isomorphic to the universal unitary completionof the locally algebraic representation π
ep (ρ(p) |GQp ), we find that1b E,ρ,Σ
[p]
HomE[GQ ×G] ρ(p) ⊗E π(p), H1b E,ρ,Σ
= HomE[GQ ×G] ρ(p) ⊗E π
ep (ρ(p) |GQp ), H[p]1b E,ρ,Σ
[p]l.alg= HomE[GQ ×G] ρ(p) ⊗E π
ep (ρ(p) |GQp ), H
= HomE[GQ ×G] ρ(p) ⊗E π
ep (ρ(p) |GQp ), M (p)= HomE[GQ ×G] ρ(p) ⊗E π(p), M (p)E ,proving 2.

The preceding proposition, together with Assumption 6.3.1, allows us to rewrite
the isomorphism (6.3.10), for those p satisfying the hypotheses of the proposition,as an isomorphism∼(6.3.16)
E ⊗O X(θ) [p] −→ HomTρ,Σ [GQ ×G] ρ(p) ⊗E π(p), M (p)E .
6.3.17. Corollary. If p ∈ Spec Tρ,Σ [1/p] is a closed point which is allowable in thesense of Definition 5.4.7, then there is a GΣ0 -equivariant κ(p)-linear isomorphism(6.3.18)∼

E ⊗O X(θ)[p] −→ πΣ0 (ρ(p)),

and evaluation induces an isomorphism(6.3.19)∼ρ(p) ⊗E π(p) ⊗E (E ⊗O X(θ)[p]) −→ M (p)E .

Proof. Both claims follow from the isomorphism (6.3.16), together with the isomorphism of part 1 of Proposition 6.3.15.
The following proposition provides a refinement of Corollary 6.3.17, for certain
points p.
6.3.20. Proposition. Suppose that p ∈ Spec Tρ,Σ [1/p] is a closed point which is∼allowable in the sense of Definition 5.4.7. Furthermore, suppose that E −→ κ(p),and that ρ satisfies Assumption 5.7.3. Then the isomorphism (6.3.18) induces aGΣ0 -equivariant O-linear isomorphism(6.3.21)∼

X(θ)[p] −→ πΣ0 (ρ(p)◦ )(where πΣ0 (ρ(p)◦ ) is the admissible smooth representation of GΣ0 over O attachedto ρ(p)◦ by the local Langlands correspondence of Section 4).
∼∼

Proof. The assumption that E −→ κ(p) implies that O −→ Tρ,Σ /p. Thus X(θ)[p]is an O-lattice in the admissible smooth GΣ0 -representation E ⊗O X(θ)[p]. The
inclusion X(θ)[p] ⊂ X(θ) induces an embedding
X(θ)[p]/$X(θ)[p] ,→ (X(θ)/$X(θ))[m],and it follows from Corollary 6.3.24 below that X(θ)[p]/$X(θ)[p] is a generic GΣ0 representation, in the sense of Definition 4.1.3. The proposition follows from thisand Corollary 6.3.17, together with the characterization of πΣ0 (ρ(p)◦ ) provided by
Proposition 4.4.3.
1[m]) is6.3.22. Proposition. If ρ satisfies Assumption 5.7.3, then HomG (π, Hk,ρ,Σ
a generic GΣ0 -representation.

80

MATTHEW EMERTON

Proof. Suppose first that π is irreducible, and let V be a weight in socGL2 (Zp ) π(the GL2 (Zp )-socle of π). In the case when socGL2 (Zp ) contains a one-dimensional
representation, we furthermore choose V to be one-dimensional. Since π is irre1
ducible, V generates π over G. Thus the restriction map HomG (π, Hk,ρ,Σ
[m]) →
1
HomGL2 (Zp ) (V, Hk,ρ,Σ [m]) is injective. If the target is non-zero, then we find that
V belongs to the set of global weights W gl (ρ). It is furthermore good, and so the
proposition follows in this case from part 3 of Theorem 5.7.7.
Suppose next that ρ|GQp is an extension 0 → χ1 → ρ → χ2 → 0, and thatχ1 χ−16= ε. The representation π then sits in a short exact sequence 0 → π 1 →2G
π → π 2 → 0, where π 1 = IndB
χ1 ⊗ χ2 ε and π 2 = IndGχ ⊗ χ1 ε are irreducible
B 2
principle series. In this case we choose Vi to be a weight in socGL2 (Zp ) π i (in fact,
socGL2 (Zp ) π i is irreducible, so there is a unique choice for Vi ). There is an exact
sequence(6.3.23)1[m])
0 → HomG (π 2 , Hk,ρ,Σ
1
1→ HomG (π, Hk,ρ,Σ
[m]) → HomG (π 1 , Hk,ρ,Σ[m]),

and for each i = 1, 2, an embedding
1
1
HomG (π i , Hk,ρ,Σ
[m]), ,→ HomGL2 (Zp ) (Vi , Hk,ρ,Σ
[m]).
1[m]) is non-zero then Vi is a weightIf either of the spaces HomGL2 (Zp ) (Vi , Hk,ρ,Σ
glin W (ρ), which is automatically good. Part 3 of Theorem 5.7.7 then shows that1
HomGL2 (Zp ) (Vi , Hk,ρ,Σ
[m]) is a generic GΣ0 -representation. The same is thus true1[m]). A consideration of the exact seof its subrepresentation HomG (π i , Hk,ρ,Σ
1
quence (6.3.23) then shows that HomG (π, Hk,ρ,Σ
[m]) is generic.
Suppose, finally, that ρ|GQp is an extension 0 → χ → ρ → χε−1 → 0, for somecharacter χ. Assumption 3.3.1 guarantees that this extension is non-split. The
representation π then sits in a short exact sequence 0 → π 1 → π → π 2 → 0, whereπ 1 itself sits in a short exact sequence 0 → (χ ◦ det) ⊗k St → π 1 → χ ◦ det → 0,
while π 2 = IndG
χε−1 ⊗ χε. Now socGL2 (Zp ) π 2 contains a single weight which doesB1[m]) = 0. Thus restriction induces an
not lie in W (ρ), and hence HomG (π 2 , Hk,ρ,Σ
injection
1
1
HomG (π, Hk,ρ,Σ
[m]) ,→ HomG (π 1 , Hk,ρ,Σ[m]),

and it suffices to show that the latter GΣ0 -representation is generic.
If ρ|GQp is a trés ramifieé extension, then W (ρ) contains no one-dimensional
1[m]) = 0, and so if we let
representations. Consequently, HomG (χ ◦ det, Hk,ρ,Σ
V = socGL2 (Zp ) (χ ◦ det) ⊗ St , then restriction to V induces an injection11[m]) → HomGL2 (Zp ) (V, Hk,ρ,Σ[m]).
HomG (π 1 , Hk,ρ,Σ

If the target of this map is non-zero, then V is a weight in W gl (ρ), automaticallygood because ρ is trés ramifieé. Thus the proposition in this case again follows frompart 3 of Theorem 5.7.7.
If ρ|GQp is peu ramifieé, then we may find a GL2 (Zp )-equivariant splitting of thesurjection π 1 → χ ◦ det . If V denotes the image of this splitting, then V generates
π 1 over G, and so restriction to V induces an injection
1
1
HomG (π 1 , Hk,ρ,Σ
[m]) → HomGL2 (Zp ) (V, Hk,ρ,Σ[m]).

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

81

If the target of this map is non-zero, then V is a weight in W gl (ρ), and V is good,since it is one-dimensional. A last application of part 3 of Theorem 5.7.7 proves
the proposition in this case.

6.3.24. Corollary. If ρ satisfies Assumption 5.7.3, then the GΣ0 -representation(X(θ)/$X(θ))[m] is generic, in the sense of Definition 4.1.3.
Proof. The corollary follows from the existence of the embedding (6.3.11), together
with Proposition 6.3.22.

6.4. Evaluation maps. If Conjecture 6.1.6 holds, then the isomorphism (6.1.7)
gives rise to an embedding of πΣ0 (ρmΣ ) as a saturated, coadmissible Tρ,Σ [GΣ0 ]m
submodule of X(πΣ), and (6.1.7) can in turn be recovered by restricting the natural
evaluation map
f

m
m
b1
ρm
Σ ⊗Tρ,Σ πΣ ⊗Tρ,Σ X(πΣ ) → HO,ρ,Σ
f

m
m
to ρm
Σ ⊗Tρ,Σ πΣ ⊗Tρ,Σ πΣ0 (ρΣ ). Thus, in order to study Conjecture 6.1.6, it is naturalm), and the corresponding
to study the coadmissible Tρ,Σ [GΣ0 ]-submodules of X(πΣevaluation maps. In fact, as in the last section, we work in a more general context,
in which we fix a deformation θ of π over Tρ,Σ satisfying Assumption 6.3.1, andstudy submodules of X(θ) and the associated evaluation maps. This will allow us
to establish cases of the weaker conjecture 6.1.9, in situations where we are not able
to prove Conjecture 6.1.6 itself.

6.4.1. Definition. If Y is a coadmissible Tρ,Σ [GΣ0 ]-submodule of X(θ), then we
let evY denote the restriction of the natural evaluation mapfb1
ρmΣ ⊗Tρ,Σ θ ⊗Tρ,Σ X(θ) → HO,ρ,Σfto ρmΣ ⊗Tρ,Σ θ ⊗Tρ,Σ Y. (Note that, by Lemma C.48, the inclusion of Y into X(θ)
ffminduces an embedding ρmΣ ⊗Tρ,Σ θ ⊗Tρ,Σ Y ,→ ρΣ ⊗Tρ,Σ θ ⊗Tρ,Σ X(θ).) We introduce
further notation for various maps induced by evY .

(1) We denote byfb1
evE,Y : E ⊗O ρmΣ ⊗Tρ,Σ θ ⊗Tρ,Σ Y → HE,ρ,Σ
the map obtained by tensoring evY with E over O.
(2) If KΣ0 is an allowable level for ρ, then evY induces a map
KΣ0
b 1 (KΣ )O,ρ ,ˆ
ev(KΣ0 )Y : ρm→H
Σ ⊗Tρ,Σ θ ⊗Tρ,Σ Y
0

as well as a map
KΣ0
b 1 (KΣ )E,ρ .
ˆ
ev(KΣ0 )E,Y : E ⊗O ρm→HΣ ⊗Tρ,Σ θ ⊗Tρ,Σ Y
0(3) If p ∈ Spec Tρ,Σ [1/p] is a closed point, then evY induces a map
1
b E,ρ,Σ
ev(p)Y : ρ(p) ⊗κ(p) θ(p) ⊗κ(p) (E ⊗O Y [p]) → H[p].

(4) Reducing evY modulo $ and passing to m-torsion parts, and taking into
account Lemma C.45, we obtain a map
1ev(m)Y : ρ ⊗k π ⊗k (Y /$Y )[m] → Hk,ρ,Σ
[m]

82

MATTHEW EMERTON

as the compositeρ ⊗k π ⊗k (Y /$Y )[m] → ρ ⊗k π ⊗k (X(θ)/$X(θ))[m] ,→1
1
ρ ⊗k π ⊗k Homk[GQ ×G] (ρ ⊗k π, Hk,ρ,Σ
[m]) → Hk,ρ,Σ[m],where the first arrow is induced by the inclusion of Y in X(θ), the second
arrow is induced by the embedding (6.3.11), and the third arrow is thenatural evaluation map.
The following result gives an important characterization of faithful and saturated
coadmissible Tρ,Σ [GΣ0 ]-submodules of X(θ).
6.4.2. Proposition. If Y is a saturated coadmissible Tρ,Σ [GΣ0 ]-submodule of X(θ),
then the following conditions are equivalent:(1) Y is a faithful Tρ,Σ -module.
(2) For each allowable level KΣ0 ⊂ GΣ0 for ρ, the submodule of invariants
Y KΣ0 is a faithful T(KΣ0 )ρ -module.
(3) For each classical closed point p ∈ Spec Tρ,Σ [1/p] which is allowable in thesense of Definition 5.4.7, the inclusion E ⊗O Y [p] ⊂ E ⊗O X(θ)[p] is anequality.
(4) For each allowable level KΣ0 ⊂ GΣ0 for ρ, and for each classical closed pointp ∈ Spec T(KΣ0 )ρ [1/p] which is allowable in the sense of Definition 5.4.7,
the inclusion E ⊗O Y KΣ0 [p] ⊂ E ⊗O X(θ)KΣ0 [p] is an equality.
(5) If X(θ)ctf denotes the maximal Tρ,Σ -cotorsion free submodule of X(θ) (following Definition C.39), then X(θ)ctf ⊂ Y .
Proof. Note that if Y is saturated in X(θ), then Y is cofinitely generated overTρ,Σ [GΣ0 ], by Theorem 6.3.12 and Corollary C.34. It is clear that 2 implies 1, andthat 3 and 4 are equivalent. Corollary 6.3.17 shows that for any point p ∈ Spec Tρ,Σsatisfying the requirements of condition 3, the GΣ0 -representation E ⊗O X(θ)[p] isirreducible. Thus for any such point p, we have that E ⊗O Y [p] 6= 0 if and only if
E ⊗O Y [p] = E ⊗O X(θ)[p]. Proposition C.36, together with the facts that Tρ,Σ isreduced, and that the allowable closed points p ∈ Spec Tρ,Σ [1/p] are Zariski dense inSpec Tρ,Σ (by Lemma 5.4.9), then shows that 1 and 3 are equivalent. If condition 4
holds, then Corollary 6.3.17 shows that Y KΣ0 [p] 6= 0 for each allowable closed pointp ∈ Spec T(KΣ0 )ρ . Since these points are Zariski dense in Spec T(KΣ0 )ρ (again byLemma 5.4.9), and since T(KΣ0 )ρ is reduced, it then follows from Proposition C.22
that 4 implies 2.
It remains to show that 5 is equivalent to the first four conditions. This is aconsequence of the following
Claim: X(θ)ctf is the unique saturated, coadmissible Tρ,Σ [G]-submodule of X(θ)
which is both faithful and cotorsion free over Tρ,Σ .
Indeed, Proposition C.40 shows that Yctf satisfies condition 1 if Y does. The
claim then implies that Yctf = X(θ)ctf , and hence that X(θ)ctf ⊂ Y. Thus 1 implies 5. As for the converse, the claim implies that X(θ)ctf satisfies condition 1.
Hence, if X(θ)ctf ⊂ Y, we deduce that Y also satisfies condition 1. Thus 5 implies 1.
It remains to prove the claim. Since X(θ) evidently satisfies condition 4, it mustalso satisfy 1. Proposition C.40 then implies that X(θ)ctf also satisfies condition 1,
and so is indeed a faithful Tρ,Σ -cotorsion free Tρ,Σ [GΣ0 ]-submodule of X(θ). Now
if Y is any saturated, coadmissible Tρ,Σ [GΣ0 ]-submodule of X(θ) which is faithful

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

83

and cotorsion free over Tρ,Σ , then certainly Y ⊂ X(θ)ctf . Also, since we havealready shown that 1 implies 3, we see (taking into account Lemma 5.4.9) that theinclusion E ⊗O Y [p] ⊂ E ⊗O X(θ)ctf [p] is an equality for a Zariski dense set ofpoints p ∈ Spec Tρ,Σ . Proposition C.41 thus implies that Y = X(θ)ctf .

6.4.3. Remark. As we already noted in the proof of Proposition 6.4.2, X(θ) itself satisfies condition 4 of the proposition, and thus satisfies all the equivalentconditions of the proposition. In particular, if p ∈ Spec Tρ,Σ [1/p] is a closed
point, then Proposition C.36 shows that E ⊗O X(θ)[p] 6= 0. From the isomorm
phism (6.3.10), we conclude (taking, e.g., θ = πΣ, which certainly satisfies Assumption 6.3.1) that for any such p, there is a non-zero GQ × G-equivariant homob1
morphism ρ(p) ⊗κ(p) π(p) → H
E,ρ,Σ [p].
We give an alternative description of X(θ)ctf , after making a preliminary definition.
6.4.4. Definition.
We will say that an ideal I ⊂ Tρ,Σ is allowable if it is of theTnform I := i=1 pi , where each pi is an allowable closed point of Tρ,Σ [1/p], in thesense of Definition 5.4.7.
Clearly the intersection of any two allowable ideals of Tρ,Σ is again an allowableideal.
Thus the allowable ideals form a directed set, ordered by inclusion. ThusS
I allowable X(θ)[I] is a Tρ,Σ [GΣ0 ]-submodule of X(θ) (the union, as indicated, being
taken over all allowable ideals).
[
X(θ)[I]
6.4.5. Lemma. The closure (in the sense of Definition C.28) of
I allowablein X(θ) coincides with X(θ)ctf .
Proof. Let I be an allowable ideal of Tρ,Σ , and as in Definition 6.4.4, write I :=Tni=1 pi , where each pi is a classical closed point of Tρ,Σ [1/p] for which ρ(pi ) |GQp is
crystabelline and absolutely irreducible, and not exceptional. If Y is a saturated,coadmissible Tρ,Σ [GΣ0 ]-submodulePof X(θ), which is faithful as a Tρ,Σ -module,nthen Proposition
Pn 6.4.2 shows that i=1 X(θ)[pi ] ⊂ Y. Since X(θ)[I] is equal to thesaturation of i=1 X(θ)[pS i ] in X(θ), we see that in fact X(θ)[I] ⊂ Y. Thus, if we let
W denote the closure I allowable X(θ)[I] in X(θ), in the sense of Definition C.28,
then we find that W ⊂ Y . On the other hand, W is a saturated, coadmissibleTρ,Σ [GΣ0 ]-submodule of X(θ) (by Lemmas C.30 and C.31), which is faithful as aTρ,Σ -module (by Lemma 5.4.9), and so W is the minimal such submodule of X(θ).
In particular, W = Wctf , and so W is cotorsion free over Tρ,Σ . Since W satisfies
condition 1 of Proposition 6.4.2, it must also satisfy condition 5 of that proposition,and so X(θ)ctf ⊂ W. We conclude that W = X(θ)ctf , as claimed.
We will need the following strengthening of Lemma 6.3.3.
6.4.6. Proposition. Suppose that there exists a saturated coadmissible Tρ,Σ [GΣ0 ]submodule Y of X(θ), satisfying the equivalent conditions of Proposition 6.4.2, andfor which evY is injective. Then if p ∈ Spec Tρ,Σ [1/p] is a closed point for whichρ(p) is an extension (possibly trivial) of distinct characters, there is an isomorphism
θ(p) ∼= π(p).
Proof. Write ρ(p) as an extension 0 → χ1 → ρ(p) → χ2 → 0, with χ1 6= χ2 . Since−1
ρ|GQp satisfies Assumption 3.3.1, we see that χ1 χ−1, while if χ1 χ−1
2 6= ε2 = ε, then

84

MATTHEW EMERTON

p ≥ 5 and ρ|GQp is a twist of a non-split extension of ε−1 by 1. In particular, in thislatter case ρ|GQp , and so also π, has trivial endomorphisms, the deformation functorm(see Remark 6.3.2), and, in particular,Def(π) is representable, hence θ ∼
= πΣ
±1
∼
θ(p) = π(p). Thus we may assume that χ1 χ−1.
2 6= 1, ε∼Choose an allowable level KΣ0 for ρ so that T(KΣ0 )ρ −→ Tρ,Σ , and hence suchKΣ0that Y
is faithful as a Tρ,Σ -module (by Proposition 6.4.2 (2)). The evaluation map evY is assumed to be an embedding, and thus so is the evaluation mapev(KΣ0 )Y . This latter map in turn induces an embeddingb 1 (KΣ )O,ρ [pn ].
(θ/pn θ) ⊗T Y KΣ0 [pn ] ,→ H
ρm
Σ ⊗T
ρ,Σ

ρ,Σ

0

Since OrdB is a left-exact functor, Corollary 5.6.13 then implies that the action of
GQp on the cokernel of the embedding

KΣ n
n
ρm
[p ]
Σ ⊗Tρ,Σ OrdB (θ/p θ) ⊗Tρ,Σ /pn Yab,SKΣ [pn ]n θ) ⊗
n Y
ρm
⊗
Ord
(θ/p
T
B
/p
T
ρ,Σ
ρ,Σ
Σ
factors through Gab
Qp . The claim of the proposition now follows from Proposition 3.6.4.

We now begin our study of evaluation maps.
6.4.7. Theorem. Let Y be a saturated coadmissible Tρ,Σ [GΣ0 ]-submodule of X(θ)
satisfying the equivalent conditions of Proposition 6.4.2, and suppose furthermoremm), or that evY is injective. Then the map evY,E isand Y = X(πΣeither that θ = πΣsurjective (or equivalently, for each allowable level KΣ0 ⊂ GΣ0 , the map evY (KΣ0 )Eis surjective).
Proof. The equivalence of the two conditions is seen by alternately passing to KΣ0 invariants, and passing to the inductive limit over all KΣ0 . Suppose that they hold.
Lb1
We claim that the image of evY,E contains p∈C H
E,ρ,Σ [p]l.alg , where C is the set ofclosed points of p ∈ Spec T(KΣ0 )ρ,Σ [1/p] which are classical, and whose associatedb1
Galois representations are crystalline at p. Since H
E,ρ,Σ [p]l.alg is an irreducible
GQ × Gp × GΣ0 -representation, by [38, Thm. 7.4.2], it suffices in fact to show thatthe image of ev(p)Y contains a non-zero locally algebraic vector, for each p ∈ C.
If the Galois representation ρ(p) associated to p ∈ C is irreducible locally at p,
and is not exceptional at p (in the sense of Definition 3.3.18), then the isomorphism (6.3.19) of Corollary 6.3.17, together with the equality E ⊗O Y [p] = E ⊗O
X(θ)[p] given by part 3 of Proposition 6.4.2, shows directly that the image of ev(p)Y
b1
contains H
E,ρ,Σ [p]l.alg . Suppose next either that ρ(p) is irreducible locally at p, but
is exceptional at p (note that, as a special case of the Tate conjecture, this is in fact
conjectured never to occur; see e.g. [21]), or else that ρ(p) is reducible, but nonsplit. Note that in the latter case, since p is classical, the two characters of whichρ(p) is an extension must be distinct. The hypothesis of the theorem, together with
Proposition 6.4.6, shows that θ(p) ∼= π(p). Theorem 3.3.21 then implies that θ(p)contains locally algebraic vectors, and Proposition C.36 implies that E ⊗O Y [p] 6= 0.
Taking into account Proposition 6.2.2 (which does apply, since, as already noted,
in the case when ρ(p) is an extension of two characters, the characters are distinct),
we conclude that the image of ev(p)Y contains a non-zero locally algebraic vector.
Suppose now that ρ(p) is the direct sum of two characters. TheLrepresentation
θ(p) ∼= π(p) may then be decomposed as a direct sum θ(p) ∼= π1 π2 , where π1

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

85

is equal to the universal unitary completion (in the sense of [37]) of the locallyalgebraic representation π
ep (ρ(p) |GQp ) attached to ρ(p) |GQp as in the statement of[38, Conj. 3.3.1 (7)]. (See the discussion of Subsection 3.4.) If evY is an injection,then the same is true of the induced evaluation map ev(p)Y . Since Proposition C.36
implies that E ⊗O Y [p] 6= 0, we see again in this case that the image of ev(p)Ycontains a non-zero locally algebraic vector. Finally, we consider the case whenm
Y = X(πΣ). Since π1 is the universal unitary completion of π
ep (ρ(p) |GQp ), we
see from [38, Thm. 7.4.2] that there is a non-zero GQ × G-equivariant embeddingb1
ρ(p)⊗E π1 ,→ H
E,ρ,Σ [p]. Composing with the projection ρ(p)⊗E π(p) → ρ(p)⊗E π1 ,m
we obtain an element of E ⊗O X(πΣ)[p] whose image contains a non-zero locallyalgebraic vector. Thus the image of ev(p)X(πΣm ) contains a non-zero locally algebraic
vector in this case as well.
Lb1
b1
Since, by Corollary 5.4.5, the direct sum p∈C H
E,ρ,Σ [p]l.alg is dense in HE , web 1 . On the other hand, it follows
have proved that the image of evY,E is dense in H
E
KΣ0
ˆfrom Theorem 6.3.12, together with Lemma 3.1.16, that ρmΣ ⊗Tρ,Σ θ ⊗Tρ,Σ (X(θ))
is a $-adically admissible representation of G over Tρ,Σ for any allowable levelb 1 (KΣ )E,ρ,Σ , by Lemma 5.3.5. It follows from PropoKΣ0 . The same is true of H
0sition 3.1.3 that the image of ev(KΣ0 )E,Y is closed. Passing to the limit over allKΣ0 , we find that the image of evE,Y is closed. This proves the theorem.

6.4.8. Remark. As remarked in footnote 9, in the case when p > 2, Paškūnas [68]has extended the results of [3] to cover the exceptional cases. Thus in this case
it is possible to avoid the appeal to Theorem 3.3.21 in the proof of the preceding
theorem. (Note that in the case when ρ(p) is reducible it is easy to show directlythat π(p) (and hence θ(p)) contains non-zero locally algebraic vectors, whether ornot ρ(p) is split).
On the other hand, it is possible to replace the application of the density results ofSubsection 5.4 in the proof of Theorem 6.4.7 by instead making a wholesale appeal
to Theorem 3.3.21. Indeed, if p is a classical closed point of Spec T(KΣ0 )ρ,Σ [1/p]corresponding to a modular form of weight two, then Theorem 3.3.21, together with
Remark 3.3.23, shows that π(p) contains non-zero smooth vectors, and so, arguing
just as in the proof given above, one finds that the image of evY,E contains all of theb1b1
smooth vectors of H
E,ρ,Σ . These are dense in HE,ρ,Σ by construction, and hence
the argument can again be concluded via an appeal to Proposition 3.1.3.
6.4.9. Theorem. Let Y be a coadmissible Tρ,Σ [GΣ0 ]-submodule of X(θ). The following are equivalent:(1) The map evY is an isomorphism.
(2) Y is a faithful Tρ,Σ -module, and the map evY (m) is injective.
If Y satisfies these equivalent conditions, then furthermore:(3) Y = X(θ)ctf .
b1
Proof. Suppose first that evY is an isomorphism. Since HO,ρ,Σ is a faithful Tρ,Σ module, we see that Y must be a faithful Tρ,Σ -module. Reducing the map evY
modulo $, and then passing to the m-torsion in the source and target, we find thatevY (m) is again an isomorphism, and so in particular, is injective. Thus 1 implies 2.
Conversely, if evY (m) is injective, then by Lemma C.46, we see that evY is
injective, with saturated image. Lemma C.52 then implies that Y must be saturatedin X(θ). If Y is furthermore faithful as a Tρ,Σ -module, then Theorem 6.4.7 shows

86

MATTHEW EMERTON

that evE,Y is surjective. It follows that evY is in fact an isomorphism, and so 2
implies 1.
Suppose now that evY satisfies conditions 1 and 2. As we have already noted, it
follows that Y is saturated in X(θ), and so condition 5 of Proposition 6.4.2 showsthat
X(θ)ctf ⊂ Y.

(6.4.10)Since X(θ)ctf is saturated in X(θ), and so in Y , we conclude that the induced map
X(θ)ctf /$X(θ)ctf [m] → (Y /$Y )[m]is an embedding, and thus that evX(θ)ctf (m) is injective, since evY (m) is. The
equivalence of 1 and 2 of the present proposition, applied now to X(θ)ctf , thenshows that evX(θ)ctf is an isomorphism. Thus the inclusion (6.4.10) induces anisomorphism
f

f

∼

m
m
m
m
⊗Tρ,Σ X(θ)ctf −→ πΣ⊗Tρ,Σ Y.
⊗Tρ,Σ πΣπΣ⊗Tρ,Σ πΣIt follows from Lemma C.51 that the inclusion (6.4.10) is an equality, i.e. that part 3
holds.

6.4.11. Theorem. Suppose that ρ satisfies Assumption 5.7.3. If evX(θ)ctf is an
isomorphism, then there is an isomorphism X(θ)ctf ∼= πΣ0 (ρmΣ ).
Proof. We will let Π denote the set of closed points p ∈ Spec Tρ,Σ [1/p] that areallowable in the sense of Definition 5.4.7, and will show that X(θ)ctf and Π together
satisfy the conditions of Theorem 4.4.1. That proposition will then show that∼
X(θ)ctf −→ πΣ0 (ρ).
Since X(θ)ctf is saturated in X(θ), there is a natural embedding
X(θ)ctf /$X(θ)ctf ,→ X(θ)/$X(θ),
and hence also an embedding
X(θ)ctf /$X(θ)ctf [m] ,→ X(θ)/$X(θ) [m].
A consideration of the embedding(6.3.11), together with Proposition 6.3.22, thenshows that X(θ)ctf /$X(θ)ctf [m] is a generic representation of GΣ0 .
Our assumption on evX(θ)ctf , together with the equivalence of conditions 1 and 2
in Theorem 6.4.9, shows that the map evX(θ)ctf (m) induces an embedding1ρ ⊗k π ⊗k X(θ)ctf /$X(θ)ctf [m] ,→ Hk,ρ,Σ[m].
(6.4.12)Choose V to be a good weight in W gl (ρ) (in the sense of Definition 5.7.5). It
follows from part 1 of Theorem 5.7.7 that V lies in W (ρ). Lemma 3.5.5 gives anisomorphism∼∼m(V, ρ) −→ HomGL2 (Zp ) (V, π) −→ HomG (c−IndG
GL2 (Zp ) V, π).
Thus, applying the functor FΣ0 socH(V ) HomGL2 (Zp ) (V, –) to the embedding (6.4.12)
yields an embedding ρ ⊗k socH(V ) m(V, ρ) ⊗k FΣ0 X(θ)ctf /$X(θ)ctf [m]1,→ FΣ0 socH(V ) HomGL2 (Zp ) (V, Hk,ρ,Σ[m])∼−→ ρ ⊗k socH(V ) m(V, ρ),LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

87

where
by part 2 of Theorem 5.7.7. We conclude that the isomorphism is provided
FΣ0 X(θ)ctf /$X(θ)ctf [m] is at most one-dimensional. Combined with the resultof the preceding paragraph, this shows that condition 1 of Theorem 4.4.1 holds.
Since each point of Π is a classical closed point, the associated local representations ρ(p) |GQ` (` ∈ Σ0 ) are always generic [51, p. 354]. Corollary 6.3.17 andLemma 6.4.5 together show that for each p ∈ Π, we have a GΣ0 -equivariant isomor∼
phism (and so in particular a GΣ0 -equivariant map)
P πΣ0 (ρ(p)) −→ E ⊗O X(θ)ctf [p].
Lemma 6.4.5 also shows that the saturation of p X(θ)ctf [p] (where p ranges over
all elements of Π) is dense in X(θ)ctf . Thus conditions 2 (a), (b), and (c) of Theorem 4.4.1 all hold. As already noted, this completes the proof of the theorem. 
6.4.13. Corollary. If there is an isomorphism of Tρ,Σ [GQ × G × GΣ0 ]-modules
f

∼

m
b1
ρm
Σ ⊗Tρ,Σ θ ⊗Tρ,Σ πΣ0 (ρΣ ) −→ HO,ρ,Σ ,(6.4.14)then X(θ)ctf ∼= πΣ0 (ρmΣ ), and the evaluation map evX(θ)ctf is an isomorphism. Conversely, if evX(θ)ctf is an isomorphism, and if ρ satisfies Assumption 5.7.3, thenX(θ)ctf ∼= πΣ0 (ρmΣ ), and there is an isomorphism of the form (6.4.14).
Proof. If there is an isomorphism of the form (6.4.14) then it gives rise to anembedding πΣ0 (ρmΣ ) ,→ X(θ), such that, letting Y denote the image of this map,the evaluation map evY is an isomorphism. Theorem 6.4.9 then shows that Y =X(θ)ctf .
Conversely, if evX(θ)ctf is an isomorphism, and if ρ satisfies Assumption 5.7.3,
then Theorem 6.4.11 shows that X(θ)ctf ∼= πΣ0 (ρmΣ ), and so evX(θ)ctf gives rise to
an isomorphism of the form (6.4.14).
The preceding results are most easily applied in the case when ρ |GQp has only
trivial endomorphisms (in which case, as we noted in Remark 6.3.2, we necessarily
have that θ ∼
= ρm
Σ ).
We first prove a lemma.
6.4.15. Lemma. Suppose that π 1 and π 2 are smooth representations of G overk, that U is a k-vector space, and that f : π 1 ⊗k U → π 2 is a G-equivariant klinear map (the G-action on the source being defined via its action on the first
factor in the tensor product). If furthermore π 1 is admissible and soc(π 1 ) (theG-socle of π 1 ) is multiplicity free, and if for every non-zero element u ∈ U , the∼map π 1 −→ π 1 ⊗k (k · u) → π 2 induced by f is an embedding, then f itself is anembedding.
Ls
Proof. Write soc(π 1 ) =i=1 π 1,i , where the π 1,i are pair-wise non-isomorphic
irreducible admissible smooth G-representations. We have the evident isomorphismsM∼∼
π 1,i ⊗k U −→ soc(π 1 ) ⊗k U −→ soc(π 1 ⊗k U ).
i=1

If f has a non-zero kernel, then this kernel has a non-zero socle, and so has non-zero
intersection with π 1,i ⊗k U for at least one value of i (since the π 1,i are pair-wise nonisomorphic), and so contains π 1,i ⊗k (k · u) for some non-zero u ∈ U . Consequently,
the map π 1 ⊗k (k · u) → π 2 is not injective, contradicting our assumption.

We now prove our main theorem in the case when ρ|GQp has only trivial endomorphisms.

88

MATTHEW EMERTON

6.4.16. Theorem. If ρ |GQp has only trivial endomorphisms, then the evaluation
map evX(πΣm ) is an isomorphism, and hence Conjecture 6.1.6 holds.
Proof. If we show that evX(πΣm ) is an isomorphism, then part 3 of Theorem 6.4.9
and Corollary 6.4.13 together imply that Conjecture 6.1.6 holds. To prove thatevX(πΣm ) is an isomorphism, it suffices, by 2 of Theorem 6.4.9, to prove that the map
ev(m)X(πΣm ) is injective. For this, it suffices in turn to prove that the evaluationmap
11
ρ ⊗k π ⊗k Homk[GQ ×G] (ρ ⊗k π, Hk,ρ,Σ
[m]) → Hk,ρ,Σ[m](6.4.17)is injective. Since ρ is absolutely irreducible, the evaluation map
11[m]) → Hk,ρ,Σ[m]ρ ⊗k Homk[GQ ] (ρ, Hk,ρ,Σ

is injective, and so to prove the injectivity of (6.4.17), it suffices in turn to provethat the induced map(6.4.18)11[m]) → Homk[GQ ] (ρ, Hk,ρ,Σ[m])
π ⊗k Homk[GQ ×G] (ρ ⊗k π, Hk,ρ,Σ

is injective.
By Lemma 6.4.15, to prove the injectivity of (6.4.18), it suffices to prove thatany non-zero G-equivariant map
1[m])
π → Homk[GQ ] (ρ, Hk,ρ,Σ

is injective. Since
∼

1
1
1
1
Homk[GQ ] (ρ, Hk,ρ,Σ
[m]) ,→ ρ∨ ⊗k Hk,ρ,Σ
−→ Hk,ρ,Σ
[m] ⊕ Hk,ρ,Σ[m](the latter G-equivariant isomorphism being obtained by choosing a k-basis for ρ),
we see that it in fact suffices to show that any non-zero G-equivariant map(6.4.19)1
π → Hk,ρ,Σ[m]

is injective. If ρ|GQp , and hence π, is irreducible, this is clear. If not, then write
ρ|GQp as an extension
0 → χ1 → ρ → χ2 → 0.
Since ρ is assumed to have only trivial endomorphisms, we have χ1 6= χ2 , and thisextension is non-split. By Assumption 3.3.1, we also have χ1 χ2−1 6= ε−1 . Thusif χ1 χ−1
6= ε, then π is an extension of IndGχ ⊗ χ1 ε by the non-isomorphic
2
B 2
G
representation IndB χ1 ⊗ χ2 ε, with each of these principal series representations
being irreducible. The GL2 (Zp )-socle of the first of these principal series consistsof a single Serre weight, which does not line in W (ρ). Taking into account part 1 of1χ ⊗ χ1 ε into Hk,ρ,Σ
Theorem 5.7.7, we find that there is no embedding of IndG[m],
B 2
and thus that indeed any non-zero map of the form (6.4.19) is injective.
If χ1 χ2−1 = ε, then, writing χ := χ1 , we find that π is a non-split extension ofG
IndB
χε−1 ⊗ χε (which is irreducible) by a representation π 1 which itself sits in a
non-split short exact sequence
0 → (χ ◦ det) ⊗k St → π 1 → χ ◦ det → 0.
The unique Serre weight in the GL2 (Zp )-socle of IndG
χε−1 ⊗ χε does not lie in
B
W (ρ), and so a non-zero map of the form (6.4.19) cannot contain all of π 1 in its
kernel. On the other hand, if it contains just (χ ◦ det) ⊗k St in its kernel, then we
1
find that Hk,ρ,Σ
contains a one-dimensional G-invariant subrepresentation, which

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

89

is impossible, by Ihara’s Lemma. Thus any non-zero map of the form (6.4.19) mustin fact be injective. This completes the argument.
When ρ|GQp is the sum of two distinct characters, we can establish a slightly
weaker result.
6.4.20. Theorem. If ρ |GQp is isomorphic to the direct sum of distinct characters,then Conjecture 6.1.9 holds.
The somewhat lengthy proof of the theorem is presented in the following subsection.
6.5. Proof
Throughout this subsection we suppose that
L of Theorem 6.4.20.
±1ρ = χ1 χ2 , with χ1 χ−1
=
6
1,
ε
.
(The
arguments typically apply more generally,2
but we have no need for them other than in this case.) We introduce the following
hypothesis on a point p ∈ Spec Tρ,Σ [1/p]:
6.5.1. Assumption. The closed point p is allowable, in the sense of Definition 5.4.7,∼∼
and E −→ κ(p) (or, equivalently, O −→ Tρ,Σ /p).
The following result lies at the heart of our argument.
6.5.2. Proposition. If p ∈ Spec Tρ,Σ [1/p] satisfies Assumption 6.5.1, then there is
an isomorphism of O[GΣ0 × G]-modules∼

ρ(p)◦ ⊗O π(p)◦ −→ FΣ0 (M (p)O )(where we refer to Definition 6.3.14 for the definition of M (p)O ).
Proof. Applying the functor FΣ0 to the isomorphism of part 1 of Proposition 6.3.15,
we obtain an isomorphism(6.5.3)∼FΣ0 (M (p)E ) −→ ρ(p) ⊗E π(p).

Thus FΣ0 (M (p)O ) is a bounded open GQ ×G-invariant lattice in ρ(p)⊗E π(p). Sinceρ is absolutely irreducible, any such lattice is of the form
ρ(p)◦ ⊗O π ◦ ,for some bounded open G-invariant lattice π ◦ in π(p). In particular, we may write∼FΣ0 (M (p)O ) −→ ρ(p)◦ ⊗O π ◦ ,∼for some lattice π ◦ . Our goal is to show that π ◦ −→ π(p)◦ .
b1
Since M (p)O is saturated in H
O,ρ,Σ , we see that the induced embedding∼
1b O,ρ,Σ
ρ(p)◦ ⊗O π ◦ −→ FΣ0 (M (p)O )) ,→ FΣ0 (H)

is saturated, and hence that reduction modulo $n induces an embedding
11b O,ρ,Σ
b O,ρ,Σ
/$n H),(ρ(p)◦ /$n ρ(p)) ⊗O/$n (π ◦ /$n π ◦ ) ,→ FΣ (H0

for any n ≥ 0. Taking ordinary parts, we obtain an embedding11b O,ρ,Σ
b O,ρ,Σ
(ρ(p)◦ /$n ρ(p)) ⊗O/$n OrdB (π ◦ /$n π ◦ ) ,→ OrdB FΣ0 (H
/$n H
)

∼
1
1
b O,ρ,Σ
b O,ρ,Σ
/$n H) .
−→ FΣ0 OrdB (HFrom Theorem 5.6.11, we conclude that the action of GQp on the quotient(ρ(p)◦ /$n ρ(p)) ⊗O/$n OrdB (π ◦ /$n π ◦ )(ρ(p)◦ /$n ρ(p)) ⊗O/$n OrdB (π ◦ /$n π ◦ )ab,S

90

MATTHEW EMERTON

factors through Gab
Qp . The theorem is now seen to follow from Proposition 3.6.3. 
Before stating our next results, we introduce some notation.
00
6.5.4. Definition. We write Tρ,Σ:= HomO−cont (Tρ,Σ) to denote the space of Olinear, m-adically continuous maps from Tρ,Σ to O.

6.5.5.
Tn Definition. Suppose that I is an allowable ideal in Tρ,Σ , and write I =p are distinct allowable closed points of Spec Tρ,Σ [1/p]. Therei=1 pi , where theLnib1
is an embedding i=1 M (pi )E ,→ H
E,ρ,Σ0 . Since∼

E ⊗O (Tρ,Σ /I) −→n
Y

E ⊗O (Tρ,Σ /pi ),i=1

we see that the image of this embedding may be identified with the closure ofb1
b1
the space H
E,ρ,Σ [I]l.alg of locally algebraic elements of HE,ρ,Σ that are annihilated
by I. Generalizing the notation of Definition 6.3.14, we denote the image of thisT b1embedding by M (I)E , and write M (I)O := M (I)E H
O,ρ,Σ .
Following Proposition 5.5.3, we make the factorization∼ b1
ρmΣ ⊗Tρ,Σ UΣ −→ HO,ρ,Σ ,b1where UΣ := HomTρ,Σ [GQ ] (ρmΣ , HO,ρ,Σ ) is a G × GΣ0 -representation. This factorization induces a corresponding factorization ∼mρmΣ ⊗Tρ,Σ HomTρ,Σ [GQ ] ρΣ , M (I)O −→ M (I)O .
6.5.6. Definition. If I ⊂ Tρ,Σ is an allowable ideal, then we write


 ∼

−→ HomTρ,Σ [GQ ] ρm,
WI := FΣ0 HomTρ,Σ [GQ ] ρm
Σ , M (I)O
Σ , FΣ0 M (I)Oand
θI := HomTρ,Σ (Tρ,Σ /I)0 , WI )(where, following the notation introduced in Subsection 3.7, we write (Tρ,Σ /I)0 :=HomO (Tρ,Σ /I, O)), so that there is a natural evaluation map(6.5.7)(Tρ,Σ /I)0 ⊗Tρ,Σ θI → WI .

Both WI and θI are naturally G-representations over Tρ,Σ /I, and (6.5.7) is a mapof (Tρ,Σ /I)[G]-modules.
6.5.8. Lemma. If J ⊂ I is an inclusion of allowable ideals in Tρ,Σ , then there is a∼natural isomorphism WJ −→ WI [J].
Proof. Evidently the inclusion M (J)O ⊂ M (I)O induces an isomorphism∼

M (J)O −→ M (I)O [J].
The lemma follows directly from this and from the definition of WI and WJ in
terms of M (I)O and M (J)O .

We immediately deduce the following corollary from Proposition 6.5.2.
6.5.9. Corollary. If p ∈ Spec Tρ,Σ [1/p] satisfies Assumption 6.5.1, then there is aG-equivariant isomorphism Wp ∼= π(p)◦ .

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

91

Our goal is now to prove a partial generalization of the preceding corollary fora general allowable ideal. In the course of the argument it will be necessary to
make an extension of scalars (so that we may put ourselves in a situation where
the prime ideals under consideration satisfy Assumption 6.5.1), and so we begin byintroducing some notation, and proving a lemma, related to such an extension.
Suppose that E 0 is a finite extension of E, let O0 be the integral closure of O in E 0 ,and let k 0 be the residue field of O0 (a finite extension of k). Write ρ0 := k 0 ⊗k ρ.
b1
b10 0 , Hb1
b 1 0 0 , and Tρ0 ,Σ in analogy with H
We define HE,ρ,Σ , HO,ρ,Σ , and Tρ,Σ .
E ,ρ ,Σ
O ,ρ ,Σ
m0
mm
m0
We furthermore define ρm0Σ and πΣ in analogy with ρΣ and πΣ . Thus ρΣ denotesm0
the universal promodular deformation of ρ0 over Tρ0 ,Σ , while πΣ
is defined so that
∼
m0
m0
m0
MF(πΣ
) −→ ρm0
Σ |GQp , with πΣ |GQp admitting the central character det(ρΣ |GQp )ε.
∼∼b 1 0 0 , O0 ⊗O H
b1
b1
−→ H
−→
There are natural isomorphisms E 0 ⊗E H
E,ρ,ΣE ,ρ ,Σ

O,ρ,Σ

∼
∼
m0
0
m ∼
m0
b 1 0 0 , O0 ⊗O Tρ,Σ −→HTρ0 ,Σ , O0 ⊗O ρmΣ −→ ρΣ , and O ⊗O πΣ −→ πΣ .
O ,ρ ,Σ

6.5.10. Lemma. If I is an allowable ideal (in the sense of Definition 6.4.4) in Tρ,Σ ,then I 0 := O0 ⊗O I is an allowable ideal in Tρ0 ,Σ .
Tn
Proof. Write I = i=1 pn as in Definition 6.4.4. If we tensor the embeddingnYTρ,Σ /I ,→κ(pi )

i=1

with O0 over O, we obtain an embedding
O0 ⊗O (Tρ,Σ /I) ,→nY

E 0 ⊗E κ(pi ).

i=1

Since E 0 and each κ(pi ) are fields of characteristic zero, the target of this embedding
is reduced. Thus so is the source. From the isomorphism∼Tρ0 ,Σ /I 0 −→ O0 ⊗O (Tρ,Σ /I),Tr
we conclude that I 0 is a radical ideal, say I 0 = j=1 qj . Each qj contains one ofthe pi , and since Tρ0 ,Σ is a finite extension of Tρ,Σ , the corresponding residue field
extension κ(qj )/κ(pi ) is finite. Thus qj is a classical closed point of Tρ0 ,Σ , and there∼
is an isomorphism ρ(qj ) −→ κ(qj )⊗κ(pi ) ρ(pi ). (If p0i corresponds to a cuspidal Hecke
eigenform defined over κ(pi ), then qj corresponds to the same Hecke eigenform, but
regarded as being defined over κ(qj ).) Thus I 0 is allowable.

We now state and prove our generalization of Corollary 6.5.9.
6.5.11. Proposition. If I is an allowable ideal, then θI is a deformation of π overTρ,Σ /I, and the evaluation map (6.5.7) is an isomorphism.
Tn
Proof. Write I = i=1 pi , as in Definition 6.4.4. Choose E 0 to be a finite Galois
extension of E, sufficiently large so that each of the residue fields κ(pi ) admits an00embedding into E 0 . Lemma 6.5.10
ideal
Tr shows that I := O ⊗O I is an allowable0of Tρ0 ,Σ , and if we write I = j=1 qj , then by virtue of our choice of E 0 , each of
the residue fields κ(qj ) is equal to E 0 .
∼Clearly there are natural isomorphisms WI 0 −→ O0 ⊗O WI and θI 0 := O0 ⊗O θI ,and the evaluation map(6.5.12)(Tρ0 ,Σ00 /I 0 )0 ⊗Tρ0 ,Σ0 θI 0 → WI 0 .

92

MATTHEW EMERTON

is obtained from the map (6.5.7) by tensoring with O0 over O. Thus it suffices toprove the proposition with E, ρ, I, etc., replaced by E 0 , ρ0 , I 0 , etc., and hence,
replacing E by E 0 , ρ by ρ0 , I by I 0 , and so on, it is no loss of generality to assume
from the beginning that for each pi appearing in the decomposition of I, the residue
field κ(pi ) is equal to E, and we do so from now on.
We will now prove the statement of the theorem (under the additional assumption that κ(pi ) = E for each i = 1, . . . , n) by induction on n. The base case n = 1
Tn−1
follows from Corollary 6.5.9. Suppose now that n > 1, and write J := i=1 pi ,p := pn . By induction, θJ is a deformation of π over Tρ,Σ /J, and the evaluationmap (Tρ,Σ /J)0 ⊗Tρ,Σ θJ → WJ is an isomorphism.
∼∼
Lemma 6.5.8 provides isomorphisms WI [J] −→ WJ and WI [p] −→ Wp , andso it follows from the induction hypothesis, Theorem 5.7.7, Lemma 3.5.5 (2), andProposition 3.7.11 that WI := (Tρ,Σ /I)0 ⊗Tρ,Σ X for some deformation X of π∼
over Tρ,Σ /I. It then follows from Lemma 3.7.10 that X −→ θI , showing that
θI is a deformation of π over Tρ,Σ /I, and that the evaluation map (6.5.7) is an
isomorphism.
We note one particular corollary of the preceding result.
6.5.13. Corollary. If J ⊂ I is an inclusion of allowable ideals, then there is a∼natural isomorphism θI /JθI −→ θJ .
Proof. Proposition 6.5.11 shows that the natural evaluation maps provide isomorphisms(6.5.14)∼(Tρ,Σ /I)0 ⊗Tρ,Σ θI −→ WI

and(6.5.15)∼(Tρ,Σ /J)0 ⊗Tρ,Σ θJ −→ WJ .

Now consider the composite of the sequence of isomorphisms:
∼∼(Tρ,Σ /J)0 ⊗Tρ,Σ θI /JθI −→ (Tρ,Σ /J)0 ⊗Tρ,Σ θI −→ (Tρ,Σ /I)0 ⊗Tρ,Σ θI [J]
∼

∼

∼

0
−→ WI [J] −→ WJ −→ Tρ,Σ[J] ⊗Tρ,Σ θJ ,of which the first is evident, the second is obtained by applying part 3 of Lemma B.6
with A taken to be Tρ,Σ /I, M taken to be Tρ,Σ /J, and N taken to be (Tρ,Σ /I)0 ,the third is (6.5.14), the fourth is obtained from Lemma 6.5.8, and the fifth is
the inverse of (6.5.15). Applying HomTρ,Σ (T0ρ,Σ [J], –) to the source and target of
this composite isomorphism, and taking into account Lemma 3.7.10, we obtain the∼required isomorphism of deformations θI /JθI −→ θJ .
We are now ready to give the:
Proof of Theorem 6.4.20. Corollary 6.5.13 allows us to form the projective limitθ := limθI , the projective limit being taken over all allowable ideals I of Tρ,Σ , to
←−
I

∼

∼

obtain a deformation θ of π over Tρ,Σ −→ lim
Tρ,Σ /I. By construction, θ/Iθ −→ θI←−Ifor any allowable ideal I ⊂ Tρ,Σ . In particular, if p is an allowable prime ideal, we
compute that
 ∼∼∼θ(p) −→ E ⊗O θp −→ FΣ0 HomGQ (ρ(p), M (I)E ) −→ π(p),LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

93

the final isomorphism following from part 1 of Proposition 6.3.15. Thus θ satisfies Assumption 6.3.1. Our goal is to prove that evX(θ)ctf is an isomorphism; theclaims of the theorem will follow from this, together with Corollary 6.4.13 andProposition 6.4.6.
Proposition 6.4.2 shows that X(θ)ctf is a faithful Tρ,Σ -module, and so by Theorem 6.4.9, it suffices to provethat ev(m)X(θ)ctf is injective. Since, by Lemma 6.4.5,
S
X(θ)ctf is the closure of I X(θ)[I], with I ranging over the allowable ideals in Tρ,Σ ,∼ S
we see that X(θ)/$X(θ) [m] −→ I X(θ)[I]/$X(θ)[I] [m], and thus it suffices
to prove that the evaluation map
1ρ ⊗k π ⊗k X(θ)[I]/$X(θ)[I])[m] → Hk,ρis injective, for each allowable ideal I. By Lemma 4.1.4 and Corollary 6.3.24, forthis it suffices in turn to show that1)ρ ⊗k π ⊗k FΣ0 X(θ)[I]/$X(θ)[I] [m] → FΣ0 (Hk,ρ
is injective, which itself will follow in turn if we show that, for each I, the evaluationmap
b1(6.5.16)ρmΣ ⊗Tρ,Σ θ ⊗Tρ,Σ FΣ0 X(θ)[I] → FΣ0 (HO,ρ,Σ )is injective, with saturated image.
Now(6.5.17)b1
X(θ)[I] = HomTρ,Σ [GQ ×G] (ρmΣ ⊗Tρ,Σ θ, HO,ρ,Σ )[I]∼b1−→ HomTρ,Σ [GQ ×G] (ρmΣ ⊗Tρ,Σ θ/Iθ, HO,ρ,Σ ).

If we write I =Tn

i=1 pi , where the pi are distinct allowable prime ideals, then since1b O,ρ,Σ
,M (I)O := ⊕ni=1 M (pi )E ∩ H
we conclude from the isomorphisms (6.3.16) (applied to p = p1 , . . . , pn ) and (6.5.17)
that we have an isomorphism∼X(θ)[I] −→ HomTρ,Σ [GQ ×G] ρmΣ ⊗Tρ,Σ θ/Iθ, M (I)O ,which in turn induces an isomorphism


 ∼
.
(6.5.18)
FΣ0 X(θ)[I] −→ HomTρ,Σ [GQ ×G] ρmΣ ⊗Tρ,Σ θ/Iθ, FΣ0 M (I)Ob1
Since by construction M (I)O is a saturated O-submodule of H
O,ρ,Σ , we see that
1b
FΣ0 M (I)O is saturated in FΣ0 (HO,ρ,Σ ). Hence, in order to verify that (6.5.16) is
injective with saturated image, it suffices to verify that the evaluation map
(6.5.19)ρmΣ ⊗Tρ,Σ θ/Iθ ⊗Tρ,Σ FΣ0 X(θ)[I] → FΣ0 M (I)O ,which arises from the description of FΣ0 X(θ)[I] provided by (6.5.18), is injective,
with saturated image.
 ∼Now FΣ0 M (I)O −→ ρmΣ ⊗TΣ WI , and so we may rewrite (6.5.18) as an isomorphism ∼(6.5.20)
FΣ0 X(θ)[I] −→ HomTρ,Σ [G] (θ/Iθ, WI ).
This in turn gives rise to an evaluation map(6.5.21)θ/Iθ ⊗Tρ,Σ FΣ0 X(θ)[I] → WI ,

94

MATTHEW EMERTON

from which (6.5.19) is obtained by tensoring with ρmΣ over Tρ,Σ . It thus suffices toshow that (6.5.21) is injective, with saturated image.
∼As already noted, there is an isomorphism θ/Iθ −→ θI , while Proposition 6.5.11∼
0gives an isomorphism (Tρ,Σ /I) ⊗TΣ,ρ θI −→ WI . Thus we may rewrite the isomorphism (6.5.20) as ∼FΣ0 X(θ)[I] −→ HomTρ,Σ [G] θI , (Tρ,Σ /I)0 ⊗TΣ,ρ θI ,and hence rewrite the evaluation map (6.5.21) as the evaluation map
 ∼(6.5.22)
θI ⊗Tρ,Σ HomTρ,Σ [G] θI , (Tρ,Σ /I)0 ⊗Tρ,Σ θI −→ (Tρ,Σ /I)0 ⊗Tρ,Σ θI .
It follows from Lemma 3.1.19 that the natural map(Tρ,Σ /I)0 → HomTρ,Σ [G] θI , (Tρ,Σ /I)0 ⊗Tρ,Σ θIis in fact an isomorphism, and hence that (6.5.22) is also an isomorphism. In
particular it is injective with saturated image, and so we are done.

7. Applications
7.1. Applications to the Fontaine–Mazur conjecture. In this subsection, we
prove the following theorem, which is a rephrasing of part 2 of Corollary 1.2.2.
7.1.1. Theorem. Let ρ : GQ → GL2 (E) be a pro-modular lift of the absolutely
irreducible representation ρ : GQ → GL2 (k), unramified outside of the finite set ofprimes S ∪ {p}, and suppose that


1 ∗
;
(1) ρ |GQp 6∼ χ ⊗0 ε(2) If ρ |GQp is isomorphic to the direct sum of two characters, then these characters have non-isomorphic reductions modulo $;(3) ρ |GQp is de Rham with distinct Hodge–Tate weights.
Then ρ is a twist of the Galois representation attached to a classical cuspidal eigenform of weight k ≥ 2.
Proof. It follows from Theorem 1.2.1 (1) that there is a G-equivariant embedding
1bE
B(ρ) ,→ HomE[GQ ] (ρ, H).

b1
By Theorem 3.3.21, the space B(ρ)l.alg is non-zero, and thus HomE[GQ ] (ρ, H
E,l.alg )is non-zero. The theorem is thus seen to follow from [38, Thm. 7.4.2].

7.2. Applications to a conjecture of Kisin. In this subsection, we prove thefollowing theorem, which is a rephrasing of part 1 of Corollary 1.2.2.
7.2.1. Theorem. Let ρ : GQ → GL2 (E) be a pro-modular lift of the absolutely
irreducible representation ρ : GQ → GL2 (k), unramified outside of the finite set ofprimes S ∪ {p}, and suppose that1 ∗(1) ρ |GQp 6∼ χ ⊗;0 ε(2) ρ |GQp is trianguline.
Then ρ is a twist of the Galois representation attached to an overconvergent p-adic
cuspidal eigenform of finite slope.

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

95

Proof. We begin by recalling some constructions from [38, §7.5]. For any tameb 1 (K p )E )an is an essentiallylevel K p , the locally analytic Jacquet module JB (Hadmissible locally analytic representation of T , corresponding to a rigid analyticb of locally analytic characters of T . Since JB
coherent sheaf E(K p ) on the space Tpb 1 (K p )E )an , and so on E(K p ),is a functor, the Hecke algebra T(K ) acts on JB (Hand generates a coherent sheaf of endomorphisms of E(K p ), which we denote byA(K p ). Suppose now that (in the notation of Subsection 5.2), the tame level K p isof the form KΣ0 K0Σ , where KΣ0 is a compact open subgroup of GΣ0 . We then writeE(KΣ0 ) := E(K p ) and A(KΣ0 ) := A(K p ). If KΣ0 is chosen to be allowable for ρ,
b 1 (KΣ )E,ρ is a direct summand of Hb 1 (KΣ )E , and so JB (Hb 1 (KΣ )E,ρ )anthen H000
b 1 (K p )E )an . We write E(KΣ )ρ and A(KΣ )ρ for theis a direct summand of JB (H00associated direct summands of E(KΣ0 ) and A(KΣ0 ).
Let p ∈ Spec Tρ,Σ denote the kernel of the map Tρ,Σ → E corresponding to ρ.
Theorem 1.2.1 (1) shows that there is a non-zero G-equivariant map
b 1 (KΣ )E,ρ [p]),B(ρ) → HomE[GQ ] (ρ, H
0
for a sufficiently small choice of KΣ0 . Passing to locally analytic Jacquet modules
for B, we obtain a T -equivariant mapb 1 (KΣ )E,ρ [p])an .
JB B(ρ)an → HomE[GQ ] ρ, JB (H0Since ρ is trianguline, the locally analytic vectors in each topological Jordan–Hölder
factor of B(ρ) have a non-vanishing Jacquet
 module (see [38, §§5.2 and 6]), and so(by left-exactness of the functor JB (–)an ) this map is again non-zero. Thus weconclude thatb 1 (KΣ )E,ρ )an [p] = JB (Hb 1 (KΣ )E,ρ [p])an 6= 0,
JB (H
0
0
and hence that the system of Hecke eigenvalues attached to ρ appears in the space
Spec A(K p )ρ . It then follows from [38, Thm. 7.5.8] that ρ is a twist of the Galois
representation attached to a finite slope overconvergent p-adic cuspidal eigenform.
We note that the proof of [38, Thm. 7.5.8] is only sketched, and so the above
argument may rightly be regarded as incomplete. However, we can avoid the appeal
to [38, Thm. 7.5.8] as follows: As is explained in the beginning of the proof of [38,Thm. 7.5.8], to conclude that every point of Spec A(KΣ0 )ρ (and so in particular, theGalois representation ρ) arises as the twist of a point corresponding to a finite slope
overconvergent p-adic cuspidal eigenform, it suffices to show that Spec A(KΣ0 )ρ is
equidimensional of dimension two. For this, it suffices in turn to show that the
support of E(KΣ0 )ρ is equidimensional of dimension two, and that E(KΣ0 )ρ has noembedded primes as a module over its support, i.e. that the support of every nonzero section of E(KΣ0 )ρ has dimension two. Since E(KΣ0 )ρ is the coherent sheaf onb associated to JB (Hb 1 (KΣ )E,ρ )an , this follows from Corollary 5.3.19 and [35,T0Prop. 4.2.36]. (In fact, as stated, this last result shows merely that the support ofE(KΣ0 )ρ is of dimension two. But in fact the considerations in its proof suffice toestablish the stronger result that each non-zero element of E(KΣ0 )ρ has support ofdimension two.)
7.3. Promodularity. In this subsection we present the:96

MATTHEW EMERTON

Proof of Theorem 1.2.3. As we noted prior to stating this theorem, its proof is asynthesis of results and methods due to Böckle, Diamond–Flach–Guo, and Khare–
Wintenberger and Kisin. (In particular, the p-adic Langlands correspondence plays
no role in its proof!)
As in the statement of the theorem, let V be a continuous odd irreducible representation of GQ over E, whose reduction V satisfies the given conditions 1 and 2.
Serre’s conjecture (proved by Khare–Wintenberger and Kisin [53, 54, 55, 60]) implies that V is modular, and thus we may apply the results (and more generally
the methods) of Böckle [4] to study the deformation space of V , parametrizing
deformations of V unramified outside Σ.
In the case when V |GQp is absolutely reducible, Theorem 1.2.3 follows from [4,Cor. 3.8]. (Note that our condition 2 ensures that the condition “µ` 6= χ` ” of that
result is satisfied.) Thus we may assume that V |GQp is absolutely irreducible.
If V |GQp is furthermore finite flat (up to a twist), then Theorem 1.2.3 again
follows from [4, Cor. 3.8]. In this (the finite flat) case, the proof of [4, Cor. 3.8]
depends on two ingredients: the main theorem of Wiles [86] and Taylor–Wiles [83],as strengthened by Diamond [30], which identifies the universal deformation ring of
V parametrizing deformations that are finite flat locally at p with an appropriate(weight two) Hecke algebra, and the result of Ramakrishna [70] showing that theuniversal deformation ring of V |GQp parametrizing finite flat deformations of fixed
determinant is formally smooth of relative dimension one over O.
Both of these ingredients have since been extended to the case of an arbitrary V
for which V |GQp is absolutely irreducible, by Diamond, Flach, and Guo [31]. Moreprecisely, twisting V if necessary, we may assume that V is in the image of the
Fontaine–Laffaille functor [45], i.e. arises from a two-dimensional Fontaine–Laffaillemodule with coefficients in k, having filtration indices 0 and k − 1, with 2 ≤ k ≤p. In [31] it is proved that the universal deformation ring of V , parametrizing
deformations over objects A of Art(O) that locally at p arise from a rank two
Fontaine–Laffaille module with coefficients in A having filtration indices 0 and k −1, may be identified with an appropriate (weight k − 1) Hecke algebra (see [31,Thm. 3.6] and its proof; note that although this result includes the hypothesis k < p,the proof is in fact valid in the case k = p also). Furthermore, the corresponding
universal deformation ring of V |GQp is shown to be formally smooth of relative
dimension one over O [31, Cor. 2.3]. With these two ingredients at hand, theargument proving [4, Cor. 3.8] in the finite flat case extends to handle the caseof arbitrary V for which V |GQp is absolutely irreducible, which in turn suffices toprove Theorem 1.2.3 for such V .

7.4. Locally algebraic vectors. In this subsection we present the:Completion of the proof of Theorem 3.3.22. Colmez has proved the theorem inthe case when VE is crystabelline or the twist of a semi-stable representation [25,Thm. VI.6.50]. Thus we assume that VE is potentially crystalline but not crystabelline. Let Dpcrys (VE ) denote the potentially crystalline Dieudonné module of VE ,and let a > b denote the Hodge–Tate weights of VE .
Colmez has shown [25, Thm. VI.6.42] that B(VE )l.alg depends only on Dpcrys (VE )as a (ϕ, N, GQp )-module, together with the pair of weights a, b (i.e. it is independent

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

97of the Hodge filtration on DdR (VE )). Equivalently, it depends only on the Weil–
Deligne representation WD(VE ) attached to the (ϕ, N, GQp )-module Dpcrys (VE ),together with the pair of weights a and b. Equivalently again, it depends only on
the admissible smooth GL2 (Qp )-representation πp (VE ) attached to WD(VE ) via theclassical local Langlands correspondence, together with the pair of weight a and b.
Thus in order to prove the theorem, it suffices to construct a single two-dimensional
continuous representation VE0 of GQp over E, with Hodge–Tate weight a and b, suchthat πp (VE0 ) = πp (VE ), and such that∼

B(VE0 )l.alg −→ Syma−b−1 E 2 ⊗E detb ⊗E πp (VE ).
Indeed, it suffices to do this after making an extension of scalars from E to somefinite extension E 0 , since the formation of locally algebraic vectors is compatible
with the extension of scalars. We may also replace VE by an unramified twist, since
the formation of locally algebraic vectors is compatible with such twisting.
We now set about constructing such a VE0 . Note that our assumption that VE
is potentially crystalline, but not crystabelline, implies (indeed, is equivalent torequiring) that πp (VE ) be cuspidal. Let σ be the minimal type of πp (VE ) [49],2and let σ0 be a GL2 (Zp )-invariant OE -lattice in σ. Write τ0 := Syma−b−1 OE⊗OEbdet ⊗σ0 (thought of as a GL2 (Zp )-representation) and τ := E ⊗OE τ0 , and let W
be a Jordan–Hölder constituent of the reduction τ0 /$τ0 .
Choose a modular ρ such that ρ|GQp is irreducible and such that W ∈ W (ρ),
i.e. W is a Serre weight of ρ. (It is easy to find such a ρ, e.g. as the reduction of the induction of a Grössen-character of the appropriate weight, for somequadratic imaginary extension of Q in which p is inert.) Let Σ be the set ofprimes at which ρ is ramified, together with p, and let K(Σ0 ) be an allowablelevel for ρ, chosen so that GL2 (Zp )K(Σ0 )K0Σ is neat. We use the notation of Section 5; thus we let m denote the maximal ideal in T(KΣ0 ) corresponding to ρ.
Since W is a Serre weight of ρ, the weight part of Serre’s conjecture shows that
there is a GL2 (Zp )-equivariant embedding W ,→ H 1 (KΣ0 )k,ρ,Σ [m]. It then follows from Proposition 5.3.15 that we may lift this to a GL2 (Zp )-equivariant map
b 1 (KΣ )O ,ρ,Σ , which in turn gives rise to a GL2 (Zp )-equivariant map
τ0 → H0E
b 1 (KΣ )E,ρ,Σ , which is necessarily an embedding, since τ is irreducibleτ → H
0b 1 (KΣ )E,ρ,Σ )as a GL2 (Zp )-representation over E. The space HomE[GL2 (Zp )] (τ, H0
is finite-dimensional over E, and is a T(KΣ0 )ρ -module, and hence we may findb 1 (KΣ )E,ρ,Σ )[p] is non-zero,p ∈ Spec T(KΣ0 )ρ [1/p] such that HomE[GL2 (Zp )] (τ, H
0or, in other words, such that there is an embedding(7.4.1)b 1 (KΣ )E,ρ,Σ [p].
τ ,→ H0b 1 (KΣ )E,ρ,Σ [p] contains non-zero locally Syma−b−1 E 2 ⊗E detb -algebraic
Since H0vectors (e.g. those in the image of the embedding (7.4.1)), we see that p is a classicalpoint of Spec T(KΣ0 ), corresponding (up to a cyclotomic twist) to a classical Hecke
eigenform defined over E 0 := κ(p), a finite extension of E. The local factor at pof the corresponding automorphic representation π then has σ as a type, and somust coincide with E 0 ⊗E πp (VE ), up to an unramified twist. To ease notation,
we replace E with E 0 and VE with an appropriate unramified twist (which, as wasnoted above, we may do without changing the problem), and thus assume that
κ(p) = E and that the local factor at p is equal to πp (VE ).

98

MATTHEW EMERTON

If we let N denote the conductor of π, and let

 Y
a b
p
K1 (N ) := {∈GL2 (Z` ) | c ≡ 0 mod N, d ≡ 1 mod N },
c d
`6=p

then it follows from [38, Thm. 7.4.2] that there is a GQ × GL2 (Qp )-equivariant
isomorphism∼b 1 K p (N )[p]l.alg −→ ρ(p) ⊗E Syma−b−1 E 2 ⊗E detb ⊗E πp (VE ).
H
1

E,ρ,Σ

On the other hand, Theorem 1.2.1 shows that there is a GQ × GL2 (Qp )-equivariant
isomorphism ∼b 1 K p (N )[p].
ρ(p) ⊗E B ρ(p)|GQp −→ H
1
E,ρ,Σ
Hence


∼ b1
K1p (N ) E,ρ,Σ [p]l.alg
ρ(p) ⊗E B ρ(p)|GQp l.alg −→ H∼−→ ρ(p) ⊗E Syma−b−1 E 2 ⊗E detb ⊗E πp (VE ).
As remarked above, this serves to complete the proof of the theorem.



7.4.2. Remark. We don’t need the full strength of Theorem 1.2.1 in order to prove
the preceding result. Indeed, it suffices to have a GQ × GL2 (Qp )-equivariant map
b 1 K p (N )ρ(p) ⊗E B ρ(p)|GQp → H
[p]
1
E,ρ,Σ
(which will then necessarily be an embedding, by Proposition 3.3.24), the existenceof which is guaranteed by Remark 6.4.3. Such an embedding induces a corresponding embeddingb 1 K p (N )ρ(p) ⊗E B ρ(p)|GQp l.alg ,→ H[p]l.alg1E,ρ,Σ
∼−→ ρ(p) ⊗E Syma−b−1 E 2 ⊗E detb ⊗E πp (VE ).
Since the target of this embedding is irreducible, and the source is non-zero [25,Thm. VI.6.18], it is then necessarily an isomorphism.
Appendix A. Banach modules
Let A be a complete Noetherian local ring with maximal ideal m, and let p ∈Spec A be a prime ideal of A. We will define the notion of a Banach module over Ap ,after first making some preliminary definitions.
A.1. Definition. If V is an Ap -module, then we say that two A-submodules W1 andW2 of V are commensurable if there exists a ∈ A\p such that aW1 ⊂ W2 ⊂ a−1 W1 .
One immediately checks that commensurability is an equivalence relation on theset of A-submodules of any given Ap -module.
A.2. Definition. If V is an Ap -module, we say that an A-submodule W of V is an
A-lattice in V if W generates V as an Ap -module, or, equivalently, if the natural
map Ap ⊗A W → V is an isomorphism.
A.3. Definition. A Banach module over Ap consists of an Ap -module V , together
with the choice of a non-empty commensurability class of m-adically complete andseparated A-lattices in V . (Note that we are not suggesting that any lattice commensurable to a given m-adically complete and separated A-lattice in V is itself

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

99

necessarily complete and separated. Rather, we are considering commensurability
as a relation on the set of complete and separated A-lattices in V .)If V is a Banach module over Ap , and if W is a member of the chosen commensurability class of m-adically complete and separated A-lattices in V , then we will
sometimes refer to W as a choice of unit ball in V .
A.4. Remark. If A = O and p is the zero ideal, so that Ap = E, then the notionof Banach module over E reduces to the standard notion of a Banach space over E.
(Here, and throughout the paper, by a Banach space over E we mean a complete
topological vector space whose topology can be defined by a norm; we do not regard
a Banach space as having a fixed choice of norm defining its topology, and thus its
unit ball is well-defined only up to commensurability.)
Appendix B. Orthonormalizable and pro-free modules
Let A be a complete Noetherian local ring.
B.1. Definition. We say that an A-module V is orthonormalizable if V is madically complete and separated, and if the quotient V /mi V is free over A/mi , foreach i ≥ 0.
We begin by showing that an A-module V is orthonormalizable if and only if
its isomorphic to the m-adic completion of a free A-module, or equivalently, if andonly if its admits an orthonormal basis in a sense analogous to that used in thetheory of p-adic Banach spaces.
B.2. Lemma. Let V be an orthonormalizable A-module, let {ei }i∈I be a basis of
V /mV, and let {ei }i∈I denote some choice of lifts of these basis elements to elementsof V . If F denotes the A-submodule of V generated by {ei }i∈I , then:(1) F is freely generated by {ei }i∈I , i.e. F is a free A-module with {ei }i∈I as
a basis;(2) For each r ≥ 0, the natural map(B.3)

F/mr F → V /mr V
is an isomorphism. In particular, if Fb denotes the m-adic completion of F ,then the natural map Fb → V is an isomorphism.

Proof. Evidently, V = F + mV. Iterating this, we find that V = F + mr V for any
r ≥ 0. Thus the map (B.3) is surjective. This proves a part
 ofP2.
Now consider the map ı : AI → V defined by ıI (ai ) = i∈I ai ei . We intend
to show that ı is an embedding. If r > 0, then reducing ı mod mr yields a map
ır : (A/mr )I → V /mr V between free A/mr -modules, which when reduced mod m
becomes an embedding. One easily concludes that ır is itself an embedding. Sincer > 0 was arbitrary, we find that ı is an embedding, and thus that the image F of
ı is free. This proves 1.
Finally, we note that the preceding argument shows that the map F/mr F →V /mr V is injective for any r > 0. This completes the proof of 2.

B.4. Definition. In the context of Lemma B.2, we refer to {ei }i∈I as an orthonormal basis for the orthonormalizable A-module V . (The terminology is motivatedby the obvious analogy with the usual notion of an orthonormal basis of a p-adic
Banach space.)

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MATTHEW EMERTON

B.5. Remark. If F is the free A-module generated by a basis {ei }i∈I , then them-adic completion Fb of F admits the following concrete description:
Fb ∼= {(ai )i∈I ∈ AI | ai → 0 m-adically },where we write ai → 0 m-adically to denote that for any r ≥ 0, there exists a finite
subset J ⊂ I such ai ∈ mr if i ∈ I \ J. Consequently, if {ei }i∈I is an orthonormal
basis for an orthonormalizable A-module V , then V admits the same concrete
description.
The following lemmas establish some elementary facts regarding orthonormalizable modules.
B.6. Lemma. Let V be an orthonormalizable A-module.
(1) V is flat.
(2) If M is a finitely generated A-module, then M ⊗A V is m-adically complete.
(3) If M and N are A-modules, with M being finitely generated, then the natural map
HomA (M, N ) ⊗A V → HomA (M, N ⊗A V )is an isomorphism.
(4) If B is a finite local A-algebra, than B ⊗A V is an orthonormalizable Bmodule.
Proof. Fix an orthonormal basis {ei } for V , and if M is any finitely generated
A-module, define
F (M ) := {(mi )i∈I ∈ M I | mi → 0 m-adically }.
The formation of F (M ) is functorial on the category of finitely generated Amodules, and it follows from the Artin–Rees lemma that it is an exact functor.
∼∼
Since F (A) −→ V (see Remark B.5), we conclude that F (M ) −→ M ⊗A V for
any finitely generated A-module M . (Apply each of the right exact functors F and– ⊗A V to a presentation As → Ar → M → 0 for M .) Thus – ⊗A V is exact
on the category of finitely generated A-modules, and hence on the category of all
A-modules. (Since tensor products commute with inductive limits, any A-moduleis the inductive limit of finitely generated A-submodules, and the formation of inductive limits is exact.) Claim 1 follows. Obviously F (M ) is m-adically complete,and so 2 also follows.
We turn to proving 3, and so fix A-modules M and N , with M finitely generated.
If we note that the functors HomA (M, –) and – ⊗A V are compatible with theformation of inductive limits (the former since M is finitely generated over A,
and the latter as a general property of tensor products), we see (writing N as an
inductive limit of finitely generated A-modules) that we may in addition assume
that N is finitely generated over A. Since HomA (M, N ) ⊗A V and N ⊗A V areboth m-adically complete, by 2, we see that the natural map HomA (M, N ) ⊗A V →HomA (M, N ⊗A V ) may be computed as the projective limit over r of the maps

HomA (M, N )/mr HomA (M, N ) ⊗A/mr (V /mr V )→ HomA M/mr M, (N/mr N ) ⊗A/mr (V /mr V ) .

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

101

We may factor each of these maps in the form

HomA (M, N )/mr HomA (M, N ) ⊗A/mr (V /mr V )→ HomA (M/mr M, N/mr N ) ⊗A/mr (V /mr V )→ HomA M/mr M, (N/mr N ) ⊗A/mr (V /mr V ) .
The second map in this factorization is evidently an isomorphism, since V /mr V is
free over A/mr for each r > 0, and the first of these maps yields an isomorphism
after passing to the projective limit over r, since by the Artin–Rees lemma, themap of projective systems{HomA (M, N )/mr HomA (M, N )}r → {HomA (M/mr M, N/mr N )}ris an equivalence, in the sense that in both its kernel and cokernel, the compositionof sufficiently many consecutive transition maps vanishes. This proves 3.
To prove 4, we note that, if n denotes the maximal ideal of B, then the m-adic
and n-adic topologies on B coincide. Thus to show that B ⊗A V is orthonormal, it
suffices to show that (B ⊗A V )/mr (B ⊗A V ) is free over B/mr B for each r > 0. This∼
follows from the isomorphism (B ⊗A V )/mr (B ⊗A V ) −→ (B/mr B)⊗A/mr (V /mr V ),rr
and the fact that V /m V is free over A/m by assumption.

B.7. Lemma. Let U and V be orthonormalizable A-modules. If M is any finitely
generated A-module, then the natural map
M ⊗A HomA (U, V ) → HomA (U, M ⊗A V )is an isomorphism.
Proof. Choose an orthonormal basis of U , and let F ⊂ U be the free A-submodule
spanned by this orthonormal basis, so that U is the m-adic completion of F . Since
V and M ⊗A V are m-adically complete (the latter by part 2 of Lemma B.6),∼∼
there are isomorphisms HomA (U, V ) −→ HomA (F, V ) and HomA (U, M ⊗A V ) −→HomA (F, M ⊗A V ). Thus it suffices to prove that
M ⊗A HomA (F, V ) → HomA (F, M ⊗A V )is an isomorphism.
Applying the exact functor HomA (F, –) to the tensor product with V over A ofa presentation As → Ar → M → 0 of M yields an exact sequence
As ⊗A HomA (F, V ) → Ar ⊗A HomA (F, V ) → HomA (F, M ⊗A V ) → 0.
On the other hand, tensoring this presentation of M with HomA (F, V ) over A yields
an exact sequence
As ⊗A HomA (F, V ) → Ar ⊗A HomA (F, V ) → M ⊗A HomA (F, V ) → 0.
Comparing the two exact sequences yields the desired isomorphism.



B.8. Lemma. If A is flat overT O, and if V is a non-zero orthonormalizable Amodule, then A = (E ⊗O A) EndA (V ), where the intersection takes place inEndE⊗O A (E ⊗O V ).
Proof. Choose an orthonormal basis for V , and let F be the free A-submodule F of
V spanned by thisTorthonormal basis, so that V is the m-adic completionof F . IfTa/$i ∈ (E ⊗O A) EndA (V ), with a ∈ A and i ≥ 0, then aF ⊂ F $i V = $i F ,102

MATTHEW EMERTON

and hence (since F is non-zero and free over A) we see that in fact a/$i ∈ A. The
lemma follows.

The next lemma establishes a certain faithfulness result for completed tensor
products with orthonormalizable modules.
B.9. Lemma. Let M be an A-module which is $-adically separated, and with theproperty that any element of M/$M is annihilated by some power of m. If theˆ A V vanishes for some orthonormalizable$-adically completed tensor product M ⊗A-module V , then either V = 0 or M = 0.
ˆ A V = 0 and that V 6= 0, and deduce that M = 0. Since
Proof. We will suppose M ⊗M is $-adically separated, by assumption, it suffices to show that M/$M = 0. If
we consider the composite∼

ˆ A V )/$(M ⊗ˆ A V ) → (M ⊗A V )/$(M ⊗A V ) −→ (M/$M ) ⊗A V,(M ⊗the first arrow being the natural surjection, we deduce that (M/$M ) ⊗A V = 0.
∼
Also by assumption M/$M −→ lim(M/$M )[mi ], and so (since tensor product−→icommute with inductive limits) we have an isomorphism∼

0 = (M/$M ) ⊗A V −→ lim(M/$M )[mi ] ⊗A V.
−→
i

Since V is A-flat, by Lemma B.6, the transition maps in the indicated inductive
system are injective, and so we conclude that (M/$M )[mi ] ⊗A V = 0, for eachi ≥ 0. There is an evident isomorphism∼(M/$M )[mi ] ⊗A V −→ (M/$M )[mi ] ⊗A/mi (V /mi V ),and thus (M/$M )[mi ] ⊗A/mi (V /mi V ) = 0, for each i ≥ 0. Since V is a non-zeroorthonormalizable A-module, its quotient V /mi V is a non-zero free A/mi -module.
We conclude that (M/$M )[mi ] = 0 for each i ≥ 0, and passing to the inductive
limit over i, that M/$M = 0, as required.
Finally, we give a “dual” description of orthonormalizable A-modules.
B.10. Definition. We say that a profinite A-module M is pro-free if it is topologically isomorphic to a direct product of copies of A, each factor being equipped withits m-adic topology.
B.11. Proposition.

(1) If V is an orthonormalizable A-module, then∼

HomA (V, A)/mi HomA (V, A) −→ HomA/mi (V /mi V, A/mi ),for each i > 0.
(2) If for any orthonormalizable A-module V we equip HomA (V, A) with the
topology of point-wise convergence, with respect to the m-adic topology on
A, then the functor V 7→ HomA (V, A) induces an equivalence of categories
between the category of orthonormalizable A-modules and the category of
pro-free profinite A-modules.
Proof. Part 1 follows from Lemma B.7, applied with M = A/mi and U = A.
Part 2 is easily verified by the reader. A quasi-inverse functor is provided by
M 7→ HomA−cont (M, A), where the target is the space of continuous A-linear maps
from M to A (where A is given its m-adic topology).
LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

103

Appendix C. Coadmissible representations
Throughout this appendix we let Γ denote a topological group which contains
a profinite open subgroup whose pro-order if prime-to-p. The example we have inmind is Γ = GΣ0 , where Σ0 is a finite set of primes
distinct from p, and, as in theQbody of the paper, we have written GΣ0 := `∈Σ GL2 (Q` ). However, in view ofpossible future applications, it seems sensible to write this appendix in the naturallevel of generality with respect to which its results hold.
Our goal in this appendix is to introduce, for any object A of Comp(O), thenotion of a coadmissible smooth representation of Γ over A, and to describe some
simple results and constructions pertaining to such representations.
Coadmissible smooth representations are dual to admissible smooth representations, in an appropriate sense, and so, in order to define them, it is necessary to
first define a class of A-modules that are dual to finitely generated O-torsion freeA-modules.
C.1. Definition. We say that an A-module X is cofinitely generated if it satisfies
the following conditions:(1) X is $-adically complete and separated.
(2) X is O-torsion free.
(3) The action map A × X → X induced by the A-module structure on X iscontinuous, when A is given its m-adic topology, and X is given its $-adic
topology.
(4) (X/$X)[m] is finite-dimensional over k.
C.2. Remark.
(1) Condition 3 of Definition C.1 is equivalent to the requirement that for each i > 0, and each element x ∈ X/$i X, there is a j > 0such that mj annihilates x.
(2) From the fact that A/mj has finite length, for any j > 0, one easily infers that condition 4 of the Definition C.1 is equivalent to the apparently
stronger condition that for every i, j > 0, the O-module (X/$i X)[mj ] is offinite length.
As already intimated, an important fact about cofinitely generated A-modules
is that they are dual (in an appropriate sense) to finitely generated O-torsion freeA-modules. In order to prove this, and to exploit it to prove further results about
cofinitely generated A-modules, we must recall some facts about duality for $adically complete and separated, O-torsion free modules.
If X is any O-torsion free, $-adically complete and separated O-module, thenits O-dual HomO (X, O) has a natural profinite topology, namely the topology ofpointwise convergence. If we write M := HomO (X, O), then we have the following
natural isomorphisms:
(C.3)

∼

M/$i M −→ HomO (X/$i X, O/$i O),

for each i > 0. Each of the spaces M/$i M is again equipped with a natural
profinite topology of pointwise convergence, and there is a topological isomorphism
(C.4)

∼

M −→ lim
M/$i M.
←−i

The functor X 7→ HomO (X, O) induces an anti-equivalence of categories between
the category of O-torsion free, $-adically complete and separated O-modules and104

MATTHEW EMERTON

the category of O-torsion free, profinite linear-topological16 O-modules, with aquasi-inverse functor given by M 7→ HomO−cont (M, O), where HomO−cont (M, O)
denotes the space of continuous O-linear maps from M to O. (This is proved in
the course of proving [77, Thm. 1.2].)
Suppose now that X satisfies conditions 1, 2, and 3 of Definition C.1. The Aaction on X induces a transposed A-action on M := HomO (X, O), making M a
profinite linear-topological A-module. The reader may easily verify that the functor
X 7→ HomO (X, O) induces an anti-equivalence of categories between the categoryof A-modules satisfying conditions 1, 2, and 3 of Definition C.1 and the categoryof O-torsion free profinite linear-topological A-modules; of course, a quasi-inverse
functor is again provided by M 7→ HomO−cont (X, O).
We now give the promised description of cofinitely generated A-modules in termsof their duals.
C.5. Proposition. If X is an A-module satisfying conditions 1, 2, and 3 of Definition C.1, then X is cofinitely generated over A if and only if HomO (X, O) is
finitely generated over A. Furthermore, the functor X 7→ HomO (X, O) induces
an anti-equivalence between the category of cofinitely generated A-modules, and thecategory of O-torsion free finitely generated A-modules, with a quasi-inverse functor being provided by M 7→ HomO−cont (M, O) (where M is equipped with its m-adic
topology).
Proof. Note that any profinite linear-topological A-module that is finitely generated
over A is necessarily equipped with its m-adic topology. Thus, in light of the above
discussion, it suffices to show that (X/$X)[m] is finite-dimensional over k if andonly if M := HomO (X, O) is finitely generated over A.
Since any profinite linear-topological A-module is necessarily m-adically complete, we see that M is finitely generated over A if and only if M/mM is finitedimensional over k. The isomorphism(C.3) (in the case i = 1) induces an isomorphism between HomO (X/$X)[m], k and M/mM. Thus (X/$X)[m] is indeed
finite-dimensional over k if and only if M/mM is finite-dimensional over k. The
proposition follows.

C.6. Definition. As usual, we say that a submodule Y of an O-torsion free Omodule X is saturated if the quotient X/Y is O-torsion free. If X is an O-module,
and Y is an O-submodule of X, then we let satX (Y ) denote the saturation of Y in
X, i.e. satX (Y ) := {x ∈ X | $i x ∈ Y for some i ≥ 0}.
We note the following simple lemma.
C.7. Lemma. If X is a $-adically complete and torsion free O-module, and if
Y is a saturated O-submodule of X, then the $-adic closure of Y in X is again
saturated.
Proof. Let Y denote the $-adic closure of Y in X, and suppose that $i x ∈ Y forsome x ∈ X. For any j ≥ i, there exists xj ∈ X such that $i x − $j xj ∈ Y. Since
Y is saturated in X, we infer that x − $j−i xj ∈ Y. Letting j → ∞, we find thatx ∈ Y , and thus Y is saturated in X, as claimed.
The following results record some additional properties of cofinitely generated
A-modules, and of maps between them.
16I.e. having a neighbourhood basis of the origin consisting of open O-submodules.

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C.8. Proposition. If φ : Y → X is an A-linear
map between cofinitely generated
A-modules, then both φ(Y ) and satX φ(Y ) are cofinitely
generated A-modules, asare the kernel of φ and the quotient X/satX φ(Y ) (which is the maximal O-torsion
free quotient of the cokernel of φ). Furthermore, the O-torsion submodule of thecokernel of φ is of bounded exponent.
Proof. We begin by noting that morphisms in the category of A-modules satisfying conditions 1, 2, and 3 of Definition C.1 admit kernels, cokernels, images,
and coimages. If φ : Y → X is a morphism in this category, then the categorical kernel of φ coincides with the usual module-theoretic kernel of φ, and thecategorical coimage of φ coincides with the usual module-theoretic image.
The
categorical image of φ coincides with the $-adic closure of satX φ(Y ) in X, andthe categorical cokernel of φ coincides with the quotient of X by the categoricalimage of φ. Similarly, the category of O-torsion free, profinite linear-topologicalA-modules admits kernels, cokernels, images, and coimages (as it must, in light of
the anti-equivalence described above), which may be described in the same manner,namely: if φ : M → N is a morphism in this category, then its categorical kernel
coincides with its usual module-theoretic kernel, its categorical coimage coincides
with its usual module-theoretic image (equipped with the quotient
topology), its

categorical image coincides with the closure in N of satN φ(M ) , and its categoricalcokernel coincides with the quotient of N by its categorical image. Of course, theanti-equivalence X 7→ HomO (X, O) interchanges categorical kernels and cokernels,
and categorical images and coimages.
Suppose now that φ : M → N is a morphism of finitely generated A-modules.
The Artin–Rees Lemma shows that satN φ(M ) is closed in N , and so coincides
with the categorical image of φ. Furthermore, the quotient satN φ(M ) /φ(M ),which is torsion as an O-module, is finitely generated over A, and so of finite
exponent. Thus $i satN φ(M ) ⊂ φ(M ) ⊂ satN φ(M ) for some i ≥ 0. BothsatN φ(M ) and φ(M ) (which is the categorical coimage of φ) are finitely generated
over A, being submodules of N . The kernel of φ is also finitely
generated over A,being a submodule of M , as is the quotient N/satN φ(M ) (which is equal to thecategorical cokernel of φ).
Applying the anti-equivalence of categories of Proposition C.5, we find that if
φ : Y → X is an A-linear morphism of cofinitely generated A-modules, then the
categorical coimage of φ, the categorical image of φ, the categorical kernel of φ, andthe categorical cokernel of φ are all cofinitely generated over A; that is, φ(Y ), the$-adic closure in X of satX φ(Y ) , the kernel of φ, and the quotient of X by the$-adic closure in X of satX φ(Y ) are all cofinitely generated over
 A. Furthermore,
if we write Z to denote the $-adic closure in X of satX φ(Y ) , then we find thati$ Z ⊂ φ(Y ) ⊂ Z for some
 i ≥ 0. This shows that in fact Z ⊂ satX φ(Y ) , andthus that Z = satX φ(Y ) , i.e. that satX φ(Y ) is already $-adically closed
 in X,and is cofinitely generated over A. Furthermore, we see that satX φ(Y ) /φ(Y ) isof bounded exponent. This proves the proposition.

C.9. Proposition. If X is a cofinitely generated A-module, and if Y is an Asubmodule of X, then the following are equivalent:(1) Y is a cofinitely generated A-module.
(2) Y is $-adically closed in X, and the O-torsion submodule of the quotient
X/Y is of bounded exponent.

106

MATTHEW EMERTON(3) Y is $-adically closed in X, and the $-adic topology on X induces the$-adic topology on Y .
Proof. The reader may easily check that if X is any O-torsion free O-module,
and if Y is a submodule of X, then the $-adic topology on X induces the $-adic
topology on Y if and only if the torsion submodule of X/Y is of bounded
Texponent.
Indeed, T
the following conditions are equivalent (for some i ≥ 0): $i+1 X Y ⊂ $Y ;$i+j X Y ⊂ $j Y for all j > 0; any O-torsion element of X/Y is annihilated
by $i . In particular, we conclude that conditions 2 and 3 of the proposition areequivalent.
Suppose now that condition 1 holds. Applying Proposition C.8 to the inclusion
Y ⊂ X, we find that X/Y has bounded exponent. The observation of the preceding
paragraph shows that the $-adic topology on X induces the $-adic topology on Y .
Since Y is $-adically complete (being cofinitely generated over A), we see that Y
is $-adically closed in X. Thus condition 2 (and so also condition 3) holds.
Suppose conversely that conditions 2 and 3 hold. Since satX (Y )/Y is equal to
the O-torsion submodule of X/Y, we see that
(C.10)$i satX (Y ) ⊂ Y ⊂ satX (Y )for some i ≥ 0. From this we conclude that satX (Y ) is also $-adically closed
in X, and of course, since satX (Y ) is saturated in X, by construction, the $adic topology on X induces the $-adic topology on satX (Y ). Thus satX (Y )satisfies conditions
1, 2, and 3 of Definition C.1. Furthermore, the natural map
satX (Y ) /$ satX (Y ) [m] ,→ (X/$X)[m] is injective, and thus the source of thismap is finite-dimensional over k, since the target is so. This shows that satX (Y ) is
cofinitely generated over A.
Now condition 3 shows that Y is $-adically complete, and thus Y also satisfies
conditions 1, 2, and 3 of Definition C.1. From (C.10), we obtain an embedding∼

HomO (Y, O) ,→ HomO ($i satX (Y ), O) −→ HomO (satX (Y ), O).
Since satX (Y ) is cofinitely generated over A, Proposition C.5 shows that the targetof this embedding is finitely generated over A, and thus so is the source. Anotherapplication of the same proposition then shows that Y is cofinitely generated over A,
i.e. that condition 1 holds.

C.11. Proposition. If M is a finitely generated A-module and X is a cofinitely
generated A-module, then HomA (M, X) is a cofinitely generated A-module. Furthermore, there is a natural isomorphism∼M ⊗A HomO (X, O) / M ⊗A HomO (X, O) [$∞ ] −→ HomO HomA (M, X), O .
Proof. The fact that HomA (M, X) is cofinitely generated over A can be verified
directly from the definition. Indeed, the fact that HomA (M, X) is $-adically complete and separated and O-torsion free follows directly from the corresponding
properties of X. The continuity condition on the A-action is also easily verified.
Finally, there is a natural embedding

HomA (M, X)/$ HomA (M, X) [m] ,→ Homk (M/mM, (X/$X)[m]).
Since M is finitely generated over A, the quotient M/mM is finite-dimensional overk, while (X/$X)[m] is finite-dimensional over k by assumption. The target of thisembedding is thus finite-dimensional over k, and hence so is the source.

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We now give another proof of the cofinite generation of Hom A (M, X), which alsoestablishes the claimed description of HomO HomA (M, X), O . To this end, choose
a finite presentation As → Ar → M → 0 of M . Applying the functor HomA (–, X),
we obtain an exact sequence
0 → HomA (M, X) → X r → X s .
It now follows from Proposition C.8 that HomA (M, X) is cofinitely generated, since
it is the kernel of an A-linear map between cofinitely generated A-modules. On theother hand, tensoring this presentation of M with HomO (X, O) over A yields anexact sequence
HomO (X, O)s → HomO (X, O)r → M ⊗A HomO (X, O) → 0,
and the same proposition (or, more precisely, its proof), shows that there is an
isomorphism between HomO HomA (M, X), O and the quotient of HomO (X, O)rby the saturation of the image of HomO (X, O)s , or, more canoncically, that thenatural map
M ⊗A HomO (X, O) → HomO HomA (M, X), O
induces an isomorphism∼M ⊗A HomO (X, O) / M ⊗A HomO (X, O) [$∞ ] −→ HomO HomA (M, X), O ,as claimed.



C.12. Lemma. If A → B is a finite morphism between objects of Comp(O), and if
X is a B-module, then X is cofinitely generated as a B-module if and only if X is
cofinitely generated as an A-module.
Proof. This is easily verified directly, and also follows from the fact that the Bmodule HomO (X, O) is finitely generated over B if and only if its finitely generated
over A.

C.13. Proposition. Let A → B be a finite morphism between objects of Comp(O),and let X be a cofinitely generated A-module.
(1) HomA (B, X) is a cofinitely generated B-module.
(2) If X is faithful as an A-module, and if B is reduced, then HomA (B, X) is
faithful as a B-module.
Proof. Proposition C.11 shows that HomA (B, X) is cofinitely generated as an Amodule, and Lemma C.12 then shows that it is also cofinitely generated as a Bmodule. This proves (1).
To prove (2), note that if M = HomO (X, O) and N = HomO HomA (B, X), O ,∼then by Proposition C.5, there is an isomorphism X −→ HomO−cont (M, O) (resp.
∼
HomA (B, X) −→ HomO−cont (N, O)), and hence X (resp. HomA (B, X)) is faithfulas an A-module (resp. B-module) if and only if the same is true of M (resp. N ).
Also, by Proposition C.11, there is an isomorphism∼(B ⊗A M )/(B ⊗A M )[$∞ ] −→ N.
Thus in order to establish (2), it suffices to show that if M is a faithful A-module,
then (B ⊗A M )/(B ⊗A M )[$∞ ] is a faithful B-module. Since B is reduced, it
suffices for this to show this quotient has non-zero localization at each minimal
prime of B. Since B is flat over O, the localization of B at each of its minimal
primes is torsion free as an O-module, and thus the localization of this quotient at

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MATTHEW EMERTON

such a prime coincides with the localization of B ⊗A M itself at that prime. Since B
is finite over A, and since M is faithful as an A-module, this localization is indeed
non-zero, and (2) follows.

Since A is a complete local ring, its localization A[1/p] is a Jacobson ring. A
point p ∈ Spec A[1/p] ⊂ Spec A is closed if and only if A/p is a finite O-algebra,
or equivalently, if and only if κ(p) (= A/p[1/p], the fraction field of A/p) is a finite
extension of E.
C.14. Lemma. Let X be a cofinitely generated A-module. If p ∈ Spec A[1/p] is a
closed point, then E ⊗O X[p] and κ(p) ⊗A HomO (X, O) are dual finite-dimensional
κ(p)-vector spaces.
∼

Proof. There is a natural isomorphism X[p] −→ HomA (A/p, X), and so Proposition C.13 (1) shows that X[p] is cofinitely generated over A/p. The equivalenceof categories of Proposition C.5 (applied with A replaced by A/p) implies that∼
X[p] −→ HomO−cont (M, O) for some finite generated torsion free A/p-module M .
However, the ring A/p is finite over O, and hence M is in fact finitely generated
as an O-module. The same is thus true of HomO−cont (M, O), and hence of X[p].
This immediately implies that E ⊗O X[p] is finite-dimensional over E, and so also
over κ(p).
Proposition C.11 also provides an isomorphism∼(A/p) ⊗A HomO (X, O)/ (A/p) ⊗A HomO (X, O) [$∞ ] −→ HomO (X[p], O),
and hence an isomorphism∼

κ(p) ⊗A HomO (X, O) −→ HomE (E ⊗O X[p], E).
Linear algebra provides a κ(p)-linear isomorphism
∼
HomE (E ⊗O X[p], E) −→ Homκ(p) E ⊗O X[p], κ(p) ,and so we see that E ⊗O X[p] and κ(p)⊗A HomO (X, O) are dual κ(p)-vector spaces,
as claimed.

We now prove some results regarding the completed tensor products of orthonormalizable and cofinitely generated A-modules.
C.15. Lemma. Let be f : Y → X be a A-linear map of cofinitely generated Amodules, and let V be an orthonormalizable A-module.
(1) If f is an embedding, then the induced map
ˆ AY → V ⊗ˆ AXV⊗of $-adically completed tensor products is again an embedding.
(2) If f furthermore has saturated image, then the same is true of the inducedmapˆ AY → V ⊗ˆ A X.
V⊗
Proof. Suppose given an embedding f : Y ,→ X, with cokernel C. We then have ashort exact sequence
0 → Y → X → C → 0,
which upon tensoring by V over A yields a short exact sequence(C.16)

0 → V ⊗A Y → V ⊗A X → V ⊗A C → 0,

LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

109

since since V is A-flat, by Lemma B.6. Proposition C.9 shows that O-torsion
submodule of C has bounded exponent, say $i for some i ≥ 0, and thus (again
using the fact that V is flat over A) the O-torsion submodule of V ⊗A C has thesame bounded exponent. Thus passing to $-adic completions in (C.16) preserves
exactness, i.e. we obtain a short exact sequence(C.17)

ˆ AY → V ⊗ˆ AX → V ⊗ˆ A C → 0.
0→V⊗In particular, the second arrow is injective, and so part 1 is proved.
If C is O-flat (i.e. if we can take i = 0), then the preceding discussion showsˆ A C. A
that the same is true of V ⊗A C, and hence of its $-adic completion V ⊗consideration of (C.17) then shows that the second arrow is not only injective, but
has saturated image, proving 2.
C.18. Lemma. Let be f : Y → X be a A-linear map of cofinitely generated Amodules, and let V be a non-zero orthonormalizable A-module.
ˆ A X of $-adically completed tensor prodˆ AY → V ⊗(1) If the induced map V ⊗ucts is an embedding (with saturated image), then f is an embedding (withsaturated image).
ˆ AY → V ⊗ˆ A X of $-adically completed tensor prod(2) If the induced map V ⊗ucts is an isomorphism, then f is an isomorphism.
Proof. Consider the short exact sequence
0→K→Y →X→C→0(so K, resp. C, is the kernel, resp. cokernel, of f ). Tensoring with the A-flat (byLemma B.6) module V , we obtain an exact sequence(C.19)

0 → V ⊗A K → V ⊗A Y → V ⊗A X → V ⊗A C → 0.

Now K, Y, and X are O-torsion free, while Proposition C.9 shows that the O-torsion
submodule of C has bounded exponent, and hence the same is true of the O-torsion
submodule of V ⊗A C (since V is A-flat). Thus $-adically completing (C.19) yields
an exact sequence(C.20)

ˆ AK → V ⊗ˆ AY → V ⊗ˆ AX → V ⊗ˆ A C → 0.
0→V⊗

Proposition C.9 shows that K is cofinitely generated over A, and so in particular,
any element of K/$K is annihilated by some power of m. The same is true of theelements of C/$C (since this is a quotient of X/$X, and X is cofinitely generated
ˆ A K (resp. V ⊗ˆ A C) vanishes ifover A). Consequently Lemma B.9 shows that V ⊗ˆand only if K (resp. C) vanishes, and also that V ⊗A C is O-torsion free if and only
if C is. These facts, taken together with the exact sequence (C.20), serve to prove
the lemma.

We now introduce the notion of cosupport for cofinitely generated A-modules.
C.21. Definition. If X is a cofinitely generated A-module, then we define the cosupport of X (which is a subset of Spec A) to be the support of its dual HomO (X, O)(which is a finitely generated A-module).
Thus if I denotes the annihilator ideal in A of X, so that I is simultaneously theannihilator ideal in A of the finitely generated A-module HomO (X, O), then thecosupport of X is equal to the closed subset Spec A/I of Spec A.

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MATTHEW EMERTON

C.22. Proposition. If X is a cofinitely generated A-module, then a closed pointp ∈ Spec A[1/p] is contained in the cosupport of X if and only if X[p] 6= 0, or
equivalently, if and only if E ⊗O X[p] 6= 0.
Proof. By definition, p lies in the cosupport of X if and only if p lies in the supportof HomO (X, O). Since this latter module is finitely generated over A, Nakayama’s
lemma shows that p lies in its support if and only if κ(p)⊗A HomO (X, O) is non-zero.
The proposition then follows from Lemma C.14.

We now introduce the class of coadmissible smooth Γ-representations over A.
C.23. Definition. We say that a smooth Γ-representation on an O-torsion free Amodule X is coadmissible if for each open subgroup KΣ0 of Γ, the space of invariants
X KΣ0 is a cofinitely generated A-module.
C.24. Definition. If X is a smooth representation of Γ on an O-torsion free Ae to denote the smooth contragredient to X, i.e.
module, then we write X
e := {φ ∈ Hom(X, O) | φ is fixed by some compact open subgroup of Γ}.
X(Here Γ acts on HomO (X, O) via the contragredient action.)
C.25. Lemma. Let X be a smooth representation of Γ over A, that satisfies thefollowing conditions:(1) X is O-torsion free.
(2) For each (or equivalently, for a cofinal collection of) open subgroups H of Γ,
the space of invariants X H is $-adically complete and separated.
(3) The action map A × X → X induced by the A-module structure on X iscontinuous, when A is given its m-adic topology, and X is given its $-adic
topology.
Then the smooth Γ-representation X is coadmissible if and only if (X/$X)[m](which is a smooth representation of Γ over k) is admissible.
Proof. Let H be a open subgroup of Γ. Conditions 1, 2, and 3 of the lemma
imply that X H satisfies conditions 1, 2, and 3 of Definition C.1. Now choose H
to be a compact open subgroup of Γ, and so small that its pro-order is prime-to-p.
Reduction modulo m commutes with the formation of H-invariants, and thus we
see that X H satisfies condition 4 of Definition C.1 for every such H if and only if(X/$X)[m] is admissible.
e induces an anti-equivalence of categories beC.26. Lemma. The functor X 7→ X
tween the category of coadmissible smooth representations of Γ on O-torsion freeA-modules, and the category of admissible smooth representations of Γ on O-torsion
free A-modules.
Proof. If X is a coadmissible smooth representation of Γ over A, and if H is acompact open subgroup of Γ, chosen so that the pro-order of H is prime-to-p, then∼e H := HomO (X, O)H −→X
HomO (X H , O).
e H is finitely generated over A, and hence X
e isThus Proposition C.5 shows that X
an admissible smooth representation of Γ over A. The claimed anti-equivalence of
categories is easily verified by the reader, taking into account the anti-equivalenceof categories of Proposition C.5.


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C.27. Lemma. If X is a coadmissible smooth A[Γ]-module and Y is a coadmissible
A[Γ]-submodule of X, then satX (Y ) is again a coadmissible A[Γ]-module.
Proof. If H is any compact open subgroup of Γ, then satX (Y H ) = satX (Y )H . The
present lemma is thus seen to follow from Proposition C.8.

C.28. Definition. If X is a coadmissible smooth A[Γ]-module, we say that an
A[Γ]-submodule Y of X is closed if Y H is $-adically closed in X H for any compact
open subgroup (or equivalently, for a cofinal collection of compact open subgroups)
H of Γ. If Y is any A[Γ]-submodule of X, then we define the closure of Y tobe the intersection of all closed A[Γ]-submodules of X (in the preceding sense)
containing Y . (More concretely, the closure of Y is equal to the union, over allcompact open subgroups H of Γ, of the $-adic closure of Y H in X H .)C.29. Remark. If X is a coadmissible smooth A[Γ]-module, then there is a nat∼ural isomorphism lim
X H −→ X, where H runs over all (or equivalently, a cofinal−→
H

collection of) compact open subgroups of Γ. Equipping each X H with its $-adic
topology, we may then endow X with the resulting O-linear inductive limit topology.17 One may then prove that an O-submodule Y of X is then closed in X, withrespect to this topology, if and only if Y H is $-adically closed in X H for each H.
This gives some justification for the preceding definition.
C.30. Lemma. If X is a coadmissible smooth A[Γ]-module, and if Y is a saturated
A[Γ]-submodule of X, then the closure of Y in X is again saturated.
H

Proof. If we let Y denote the closure of Y in X, then it suffices to show that YH
is saturated in X H , for each compact open subgroup H of Γ. Since Y coincides
with the $-adic closure of Y H in X H , and since the assumption that Y is saturated
in X implies that Y H is saturated in X H , this follows from Lemma C.7.

C.31. Lemma. Let X be a coadmissible smooth A[Γ]-module. If Y is a saturated
A[Γ]-submodule Y of X, then Y is coadmissible if and only if Y is closed in X, inthe sense of Definition C.28.
Proof. This follows directly from Proposition C.9.



C.32. Definition. A coadmissible smooth A[Γ]-module X is called cofinitely gene is finitely generated over A[Γ].
erated over A[Γ] if X
C.33. Proposition. A coadmissible smooth A[Γ]-module X is cofinitely generated
over A[Γ] if and only if (X/$X)[m] is finitely generated over k[Γ].
Proof. Recall that an admissible smooth representation W of Γ over k is finitely
generated if and only if it is of finite length. Since W is of finite length if and
f , we conclude that W
only if the same is true of its smooth contragredient Wf
is finitely generated if and only if W is. In particular, since (X/$X)[m] is thee X,e we see that (X/$X)[m] is finitely generated oversmooth contragredient to X/meek[Γ] if and only if X/mX is. The proposition thus follows from the fact that thee is finitely generated over A[Γ] if and only
admissible smooth A[Γ]-representation X
e X
e is finitely generated over k[Γ] [43].
if X/m
17I.e. we take as a basis of neighbourhoods of the origin in X those O-submodules Y such thatY H is $-adically open in X H for each H.

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MATTHEW EMERTON

C.34. Corollary. If X is a coadmissible smooth A[Γ]-module that is cofinitely generated over A[Γ], and if Y is a saturated coadmissible A[Γ]-submodule of X, thenY is again cofinitely generated over A[Γ].
Proof. The assumption that Y is saturated in X implies that the natural map
Y /$Y → X/$X is injective. The corollary then follows from the preceding
proposition, and the fact that a subrepresentation of a finitely generated admissible smooth Γ-representation over k is again finitely generated. (Indeed, as was
already noted in the proof of the preceding result, finitely generated admissible
smooth Γ-representations over k are necessarily of finite length.)C.35. Lemma. If X is a coadmissible smooth Γ-representation over A, and if p ∈Spec A[1/p] is a closed point, then E ⊗O X[p] is naturally isomorphic to the smoothe over κ(p).
contragredient of the admissible smooth Γ-representation κ(p) ⊗A X
Proof. As already noted in the proof of Lemma C.26, if H is a compact open
subgroup of Γ of pro-order prime-to-p, then∼e H := HomO (X, O)H −→X
HomO (X H , O).
Lemma C.14 then yields the second of the pairs of isomorphisms∼∼e H −→e H −→
κ(p) ⊗A X
κ(p) ⊗ X
Homκ(p) E ⊗O X H [p], κ(p) .
(The first follows from the fact that H has pro-order prime-to-p.) Letting H rune and E ⊗O
over a cofinal set of compact open subgroups of Γ, we find that κ(p)⊗A X
X[p] are indeed mutually dual admissible smooth representations of Γ over κ(p). 
C.36. Proposition. If the ring A is reduced, and if X is a coadmissible smooth
Γ-representation over A which is cofinitely generated over A[Γ], then the following
are equivalent:(1) X[p] 6= 0, or equivalently, E ⊗O X[p] 6= 0, for every closed point p ∈Spec A[1/p].
(2) X[p] 6= 0, or equivalently, E ⊗O X[p] 6= 0, for a Zariski dense subset ofclosed points p ∈ Spec A[1/p].
(3) X is a faithful A-module.
e in A,
Proof. Clearly the annihilator of X in A coincides with the annihilator of X
eand Lemma C.35 shows that E ⊗O X[p] 6= 0 if and only if κ(p) ⊗A X 6= 0, for any
e 6= 0 for
closed point p ∈ Spec A[1/p]. Thus it suffices to verify that (10 ) κ(p) ⊗A X
e 6= 0 for a Zariski
every closed point p ∈ Spec A[1/p] if and only if (20 ) κ(p) ⊗A X
0 e
dense subset of closed points p ∈ Spec A[1/p] if and only if (3 ) X is faithful as an
A-module.
e is finitely generated over A[Γ], and thus is generated overBy assumption, X
HeA[Γ] by X , for some sufficiently small compact open subgroup H of G. ShrinkingeH
H if necessary, we may assume that the pro-order of H is prime-to-p. Since X
e H 6= 0 for every
is finitely generated, the following are equivalent: (100 ) κ(p) ⊗A X
00
Heclosed point p ∈ Spec A[1/p]; (2 ) κ(p) ⊗A X 6= 0 for a Zariski dense subset ofe H is faithful as an A-module.
closed points p ∈ Spec A[1/p]; (300 ) X
e 6= 0 if and only if κ(p) ⊗A X
e H 6= 0. Indeed,
We now note that κ(p) ⊗A X
e is generated over κ(p)[Γ] by κ(p) ⊗A X
e H , and so the only if direction
κ(p) ⊗A X
Hee since p doesn’t divide
is clear. On the other hand, X is a direct summand of X,

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113

e H is a direct summand of κ(p) ⊗A X.
ethe pro-order of H, and thus κ(p) ⊗A X
000This establishes the if direction. Consequently (1 ) and (1 ) are equivalent, as aree H generates Xe over A[Γ], conditions (30 ) and (300 ) are also(20 ) and (200 ). Since X
equivalent. This completes the proof of the proposition.

C.37. Definition. Suppose that A is reduced. We say that a coadmissible smoothe is torsion free as an A-module (i.e. if multiA[Γ]-module X is cotorsion free if X
eplication by any non-zero divisor of A induces an injection on X).
C.38. Proposition. If A is reduced, and if X is a coadmissible smooth A[Γ]-module,
then X contains a unique maximal cotorsion free coadmissible A[Γ]-submodule,
which is furthermore saturated in X.
etor denote the maximal A-torsion submodule of X,e and let X
etf :=
Proof. Let Xeeee
X/Xtor denote the maximal A-torsion free quotient of X. Clearly Xtor is an A[Γ]e and is saturated in X.
e Thus X
etf is an A[Γ]-module, and is Osubmodule of X,
e is, and corresponds under the anti-equivalence
torsion free. It is admissible, since X
eof Lemma C.26 to the sort-after submodule of X.
C.39. Definition. If A is reduced, and if X is a coadmissible smooth A[Γ]-module,
then we let Xctf denote the maximal cotorsion free coadmissible A[Γ]-submoduleof X whose existence is guaranteed by the preceding proposition.
C.40. Proposition. If A is reduced, and if X is a cofinitely generated coadmissible
smooth Γ-representation over A, which is faithful as an A-module, then Xctf is alsofaithful as an A-module.
e it suffices to show that if X
e is a finitely generated
Proof. Passing from X to X,
etf (the maximal
smooth A[Γ]-module, which is faithful as an A-module, then XeA-torsion free quotient of X) is again faithful as an A-module. If we choose ae H generates Xe over A[Γ], then we findcompact open subgroup H of Γ such that X
HHe must be a faithful A-module. Since X
e is finitely generated over A, thisthat Xe H )tf is again a faithful A-module. The inclusion X
eH ⊂ Xe induces
implies that (XHeeean embedding (X )tf ,→ Xtf . Thus Xtf contains a faithful A-submodule, and so isitself faithful as an A-module.

C.41. Proposition. If A is reduced, if X is a coadmissible cotorsion free smoothA[Γ]-module, and if Y is a saturated coadmissible A[Γ]-submodule of X with theproperty that Y [p] = X[p] (or equivalently, since Y is saturated in X, that E ⊗O
Y [p] = E ⊗O X[p]) for a Zariski dense set of closed points p ∈ Spec A[1/p], thenY = X.
Proof. If H is a compact open subgroup of Γ whose pro-order is prime-to-p, thenY H is a saturated cofinitely generated A-submodule of the cofinitely generated
A-module X H , and thus the induced map of finitely generated A-modules(C.42)

HomO (X H , O) → HomO (Y H , O)is surjective. By assumption Y H [p] = X H [p] for a Zariski dense set of closed points
p ∈ Spec A[1/p], and so from Lemma C.14, we conclude that the induced surjection
κ(p) ⊗A HomO (X H , O) → κ(p) ⊗A HomO (Y H , O)

114

MATTHEW EMERTON

is an isomorphism for a Zariski dense set of closed points p ∈ Spec A[1/p]. Thus thekernel of the surjection (C.42) is torsion. The A-submodule X H is a direct summandof X, since the pro-order of H is prime-to-p, and consequently HomO (X H , O)e Thus HomO (X H , O) is also torsion free as an Ais a direct summand of X.
module, and hence so are all of its submodules. In particular, the kernel of (C.42)
is simultaneously torsion and torsion free as an A-module, and so must vanish.
Consequently, the surjection (C.42) is in fact an isomorphism, and thus Y H = X H .
Taking H to be arbitrarily small, we conclude that in fact Y = X, as claimed. 
C.43. Definition. If V is any A-module, and if X is a coadmissible smooth A[Γ]module, then we writefˆ AX H ,V ⊗A X := lim V ⊗−→H

ˆ A X H denotes thewhere H runs over the collection of open subgroups of Γ, and V ⊗H$-adic completion of the tensor product V ⊗A X .
f

The formation of V ⊗A X is evidently functorial in both V and X, and the Γfaction on X induces a corresponding Γ-action on V ⊗A X. In the remainder of thisappendix, we prove some simple lemmas involving this functor.
C.44. Lemma. If V is an orthonormalizable A-module, X is a coadmissible A[Γ]f
ˆ A X H → V ⊗A X
module, and H is an open subgroup of Γ, then the natural map V ⊗is an embedding with saturated image.
Proof. Let H 0 ⊂ H be an inclusion of open subgroups. Since X H is evidently
0saturated in X, it is saturated in X H , and hence, by part 2 of Lemma C.15, the0ˆ AX H → V ⊗ˆ A X H is injective, with saturated image. Passing to
induced map V ⊗the inductive limit over all H 0 , the lemma follows.

C.45. Lemma. If V is an orthonormalizable A-module and X is a coadmissible
representation of Γ over A, then there is a natural isomorphism
ff∼
V /mV ⊗k (X/$X)[m] −→ (V ⊗A X)/$(V ⊗A X) [m].
f

ˆ H , where H runs over the open subgroups of Γ,
Proof. Since V ⊗A X = lim
V ⊗X
−→
H

we see that
f

f

∼

∼

(V ⊗A X)/$(V ⊗A X) −→ lim(V ⊗A X H )/$V ⊗A X H ) −→ V /$V ⊗A/$A X/$X−→
H

(the last isomorphism holding since tensor produces commute with inductive limits).
We now apply the functor (–)[m], which is isomorphic to the functor HomA/$A (k, –),to the composite of these isomorphisms, to obtain the first in the sequence ofisomorphisms
ff∼(V ⊗A X)/$(V ⊗A X) [m] −→ (V /$V ⊗A/$A X/$X)[m]∼∼−→ V /$V ⊗A/$A (X/$X)[m] −→ V /m ⊗k (X/$X)[m];the second isomorphism is given by part 3 of Lemma B.6, which applies, since
V /$V is orthonormalizable over A/$A, by part 4 of the same lemma, while thethird isomorphism is evident. The composite of this sequence of isomorphisms
yields the required isomorphism.
LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

115

C.46. Lemma. If V is an orthonormalizable A-module, X is a coadmissible representation of Γ over A, and W is an O-flat A-module, then an A-linear mapf

V ⊗A X → W is injective, with saturated image, if and only if the map V /mV ⊗k(X/$X)[m] → (W/$W )[m] obtained by reducing it modulo $ and passing to minvariants (and then applying the isomorphism of Lemma C.45) is injective.
f

Proof. A map V ⊗A X → W is injective, with saturated image, if and only if theinduced mapff(V ⊗A X)/$(V ⊗A X) → W/$W

(C.47)

is injective. Now, as was noted in the proof of the preceding lemma, there is anatural isomorphismf∼f(V /$V ) ⊗A (X/$X) −→ (V ⊗A X)/$(V ⊗A X).
Since each element of (X/$X) is annihilated by some power of m, the same is thustrue of each element of the source of this isomorphism, and thus also of each elementof its target. An easy argument then shows that (C.47) is injective if and only if
the induced map on m-torsion submodules is injective. This proves the lemma. 
C.48. Lemma. Let Y ,→ X be an A-linear, Γ-equivariant embedding of coadmissible
A[Γ]-modules. If V is an orthonormalizable A-module, then the induced mapff

V ⊗A Y → V ⊗A X

(C.49)
is again an embedding.

∼∼

Proof. Write Y −→ lim
Y H , X −→ lim
X H , where H runs over the open subgroups
−→
−→
H

H

ˆ AY → V ⊗ˆ A X, whichof Γ. The map (C.49) is the inductive limit of the maps V ⊗are injective, by Lemma C.15. Thus (C.49) is also injective, as claimed.

C.50. Lemma. Let H be a compact open subgroup of Γ whose pro-order is primeto-p. If V is a $-adically complete A-module and if X is a coadmissible smoothfˆ H → (V ⊗X)H is an isomorphism.
A[Γ]-module, then the induced map V ⊗X∼

Proof. We may write X −→ lim
X N , where N runs over normal open subgroups of
−→
N

f

∼

ˆ A X N . Passing to the H-invariants, we find thatH, and that V ⊗A X −→ lim
V⊗
−→
N

f

H

(V ⊗A X)∼∼N

as required.

∼ˆ A X N )H −→ lim V ⊗ˆ A X H −→ V ⊗ˆ AX H ,−→ lim
(V ⊗
−→
−→
N



C.51. Lemma. Let Y → X be an A-linear, Γ-equivariant map between coadmissible
smooth A[Γ]-modules, and let V be an orthonormalizable A-module. If the inducedff

map V ⊗A Y → V ⊗A X is an isomorphism, then the same is true of the given map
Y → X.
Proof. Let H be a compact open subgroup of Γ whose pro-order is prime-to-p.
ˆ AY H → V ⊗ˆ A X H is an isomorphism. Part 2
Lemma C.50 shows that the map V ⊗Hof Lemma C.18 then shows that the map Y → X H is an isomorphism. Passing∼
to the inductive limit over all H, we find that Y −→ X, as claimed.
116

MATTHEW EMERTON

C.52. Lemma. Let V be an orthonormalizable A-module, let Y ,→ X be an embedding of coadmissible A[Γ]-modules, let W be an A-module, and suppose give anfA-linear map V ⊗A X → W. If the compositeff

V ⊗A Y → V ⊗A X → W
has saturated image, then the embedding Y ,→ X has saturated image.
Proof. It suffices to show that the induced embedding Y H ,→ X H has a saturated
image, for each open subgroup H of Γ. Now Lemma C.44 shows that the naturalfˆ A Y H → V ⊗A Y is an embedding, with a saturated image. Thus so is themap V ⊗
compositefˆ AY H → V ⊗ˆ A X H → V ⊗A X → W,V⊗and hence so is the first of these maps. Part 1 of Lemma C.18 then implies thatthe embedding Y H ,→ X H has a saturated image, as required.

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Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston,
IL 60208
E-mail address: emerton@math.northwestern.edu