Draft: August 19, 2009
A LOCAL-GLOBAL COMPATIBILITY CONJECTURE IN THE
p-ADIC LANGLANDS PROGRAMME FOR GL2/Q
MATTHEW EMERTON

To John Coates, for his 60th birthday

Contents
1. Introduction
2. Classical Langlands for GL2/Q
3. The local p-adic Langlands conjecture for GL2/Qp
4. Refinements and trianguline local Galois representations
5. Some invariants of admissible unitary GL2 (Qp )-representations
6. The local correspondence for trianguline representations
7. Local-global compatibility
References

1
9
14
20
29
44
61
92

1. Introduction
1.1. The local-global compatibility conjecture. Fix a prime p, as well as ab p ) (referred to as afinite extension E of Qp . If K p is an open subgroup of GL2 (Z
b 1 (K p )E , equipped“tame level”), then one can define a certain E-Banach space H
with an action of GQ × GL2 (Qp ), by taking the inductive limit of the étale cohomology with coefficients in E of the modular curves of arbitrary p-power leveland of tame level K p , and then completing with respect to the norm induced by
the OE -submodule of integral cohomology classes. Passing to the locally convex
inductive limit over all tame levels K p , we obtain a complete locally convex topob 1 equipped with a representation of GQ × GL2 (Af ) that
logical E-vector space H
E
is (so to speak) “smooth in the prime-to-p-directions, but unitary Banach in thep-adic direction”. (See Subsection 7.2 for the precise definitions of these various
topological vector spaces.) The object of this note is to explain a conjecture on theb 1 . This conjecture
multiplicities with which certain GQ -representations appear in H
E
is in some sense the most optimistic possible, in light of what is already known, or
believed, to be true.
In order to state the conjecture, we must first admit the truth of a “local padic Langlands conjecture for GL2 ”. The idea that such a conjecture should (oreven could) exist is due largely to Breuil, and has been extensively developed bothby him and others. In what follows, we will take as given the most optimistic
version of this conjecture, namely that to any continuous representation of GQp
on a two dimensional E-vector space V there is associated in a natural manner an
The author was supported in part by NSF grant DMS-0401545.
1

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MATTHEW EMERTON

admissible unitary Banach space representation B(V ) of GL2 (Qp ) over E. Granting
the existence of this local correspondence, we may now state our conjecture.
1.1.1. Conjecture (Local-global compatibility). If V is an arbitrary odd irreducible continuous two dimensional representation of GQ over E that is unramifiedb 1 ) (which is naturally
outside of a finite set of primes, then the space HomGQ (V, H
E
a GL2 (Af )-representation) decomposes as a restricted tensor productO∼0 m
b 1E ) −→HomGQ (V, H
B(V |Dp ) ⊗π` (V ),
`6=p

where B(V |Dp ) is the admissible unitary Banach space representation of GL2 (Qp )associated to V |Dp via the local p-adic Langlands correspondence, while for eachprime ` 6= p, π`m (V ) is the admissible smooth representation of GL2 (Q` ) associatedto V |D` via the modified classical local Langlands correspondence discussed in 2.1.1
below.1
At least for those V that are attached to classical modular forms, this conjectureis due to Breuil [13]. One of our goals will be to provide some justification forextending his conjecture to more general V .
Of course, for Conjecture 1.1.1 to have a precise meaning, one has to make thelocal p-adic Langlands correspondence precise. Thus a second goal of our paper is
to discuss various results and conjectures related to the local correspondence. So
far this correspondence has been specified for two dimensional GQp -representations
that are trianguline in the sense of [27] (with some caveats in the case of reducible
representations). This is work primarily of Breuil and Colmez. In Section 6 we willrecall the definition of this trianguline correspondence in detail.
Currently, very little seems to have been proved regarding the existence of thelocal correspondence for those two dimensional GQp -representations V that are nottrianguline.2
Nevertheless, it is possible to formulate a minimal set of requirements that the
local p-adic correspondence should satisfy, and we do this in Conjecture 3.3.1 below.
Our set of requirements is by no means definitive (and it certainly does not specify
the correspondence uniquely; cf. Remark 3.3.6 below). However, it does serve toformalize some of the ideas and expectations about the relationship between p-adic
Galois representations and p-adic representations of GL2 (Qp ) that have appearedin the literature in the last several years.3 Of all the conditions that we incorporate
into our conjecture, perhaps the most novel is a condition (namely condition (8)of Conjecture 3.3.1) that relates locally analytic Jacquet modules on the GL2 (Qp )side (as defined in [32]) to a certain notion of refinement on the GQp side (related
1The representation π m (V ) coincides with the representation π (V ) attached to V`|D` 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).
2Breuil and Strauch (in unpublished work) have proposed a candidate for B(V ) in (at least
some of) those cases when V is potentially semi-stable but not trianguline. However, as far as I
am aware, little is known about their proposed B(V ). For example, it does not seem to be known
whether or not it is non-zero.
3A caveat: There is an important connection between the G
Qp and GL2 (Qp ) worlds, discovered
by Colmez, which we have not attempted to incorporate into our conjecture; see Remark 3.3.7
below.

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

3

to the one introduced in [45] for crystalline representations, and also closely related
to the theory of two dimensional trianguline representations developed in [27]).
Having formulated Conjecture 3.3.1, we are immediately confronted with the
problem of proving that it is satisfied by the correspondence as it has been defined in
the trianguline case. A good part of the paper is devoted to discussing this problem
either directly or indirectly (although we do not solve it – see Subsection 6.6 for a
discussion of what parts of the problem remain open). We focus in particular on
describing the relationship between the trianguline correspondence and the theoryof Jacquet modules. For this we rely heavily on the constructions and results of[27, 32, 35].
When Conjectures 1.1.1 and 3.3.1 are combined, they provide an appealing
framework within which to consider a number of important conjectures relating
p-adic Galois representations and modular forms. For example, we will prove thefollowing result.
1.1.2. Proposition. Assume that the local p-adic Langlands correspondence exists,and satisfies Conjecture 3.3.1. Then Conjecture 1.1.1 has the following consequences. (As usual we let V denote an odd irreducible continuous two dimensional
representation of GQ over E.)(1)? The representation V |Dp is potentially semi-stable with distinct Hodge-Tateweights if and only if V is the twist by a power of the p-adic cyclotomic
character of a Galois representation attached to a classical cuspidal Hecke
eigenform of weight k ≥ 2.
(2)? The representation V |Dp is trianguline if and only if V is a twist of a
Galois representation attached to a p-adic overconvergent cuspidal Hecke
eigenform of finite slope.
The ? are to indicate that each assertion is contingent on Conjectures 1.1.1
and 3.3.1. The first statement is a conjecture of Fontaine and Mazur [37, Conj. 3c].
The second statement implies [41, Conj. 11.8], but is in fact stronger; it also implies
the equality Xfs◦ = Xfs discussed in note (2) on p. 450 of that reference.
Proposition 1.1.2 is proved in Subsection 7.9 below. (More precisely, it is areformulation of parts (3)? and (4)? of Proposition 7.9.1.) The proof of (2)? depends
on a result which may be of interest in its own right. If K p is a fixed tame level,b 1 (K p )E one canthen applying the theory of locally analytic Jacquet modules to H
2b of characters of (Q×construct a certain coherent sheaf of algebras on the space Tp) ,p
whose relative spectrum Spec A(K ) was shown in [33, §4.4] to contain the reduced
eigensurface of tame level K p . The question of whether or not this inclusion isan equality was left open in that reference. Theorem 7.5.8 below shows that thisinclusion is (essentially) an equality. As the proof of this theorem is rather technical,
we only sketch it here. The details will appear elsewhere.
We now state some partial results in the direction of Conjecture 1.1.1 related tothe case when V is an irreducible two dimensional GQ -representation associated toan overconvergent p-adic eigenform of finite slope. (In this case V |Dp is trianguline –see [41] and also Thm. 7.6.1 below. Thus B(V |Dp ) is defined, and Conjecture 1.1.1
has a precise meaning.) The following theorem incorporates results of Berger,
Breuil, and Colmez, as well as of the author.
1.1.3. Theorem. Let V be an irreducible continuous GQ -representation attached toan overconvergent eigenform f over E of finite slope. Assume furthermore that:

4

MATTHEW EMERTON(a) If V |Dp is irreducible and potentially semi-stable (in which case f is necessarily classical of weight k ≥ 2, by [41, Thm. 6.6]) then it is Frobenius
semi-simple.
(b) If V |Dp is reducible then it is potentially crystalline, and f is classical ofweight k ≥ 2.
The following results then hold:(1) If V |Dp is indecomposable and f is classical then there is an isomorphism
O

∼
0 m
b 1E ) .
π` (V ) −→ HomGL2 (Qp ) B(V |Dp ), HomGQ (V, H`6=p(2) In general, if f has tame level K p , then there is a GL2 (Qp )-equivariantb 1 )K p .
embedding B(V |Dp ) → HomGQ (V, H
E
The next theorem provides something of a converse to the previous one.
1.1.4. Theorem. Let V be as in Conjecture 1.1.1, and suppose furthermore that
V is not a twist of a representation having finite image. Let W be an irreducible
trianguline continuous two dimensional representation of GQp . If there is a nonb 1 ), then V is azero continuous GL2 (Qp )-equivariant map B(W ) → HomGQ (V, H
E
twist of a representation attached to an overconvergent eigenform of finite slope,and V |Dp ∼= W.
These theorems follow from the results proved in Subsection 7.10 below.
Let us close this introductory discussion by remarking that one could hope fora still stronger conjecture than Conjecture 1.1.1, for while this conjecture predicts
the multiplicities with which all odd irreducible two dimensional Galois represenb 1 , it does not explain how they are “glued together” inside
tations appear inside H
E
1
b . By contrast, the classical Langlands correspondence provides a complete deH
Escription of the parabolic cohomology of modular curves as a representation of
GQ × GL2 (Af ) (see Theorem 2.5.1 below). In a forthcoming paper we will formub1
late a strengthening of Conjecture 1.1.1 that gives an analogous description of HE1b
(or more precisely, of the localization of H E at the maximal ideal in an appropriate Hecke algebra corresponding to some fixed absolutely irreducible residual two
dimensional GQ -representation) as a GQ × GL2 (Af )-module.
1.2. Contents and arrangement of the paper. In Section 2 we briefly recall theclassical Langlands correspondence for GL2/Q in both the local and global contexts.
Given that our ultimate subject is Conjecture 1.1.1, we put a particular emphasis on
the realization of the global correspondence in the cohomology of modular curves.
Sections 3, 4, 5, and 6 are devoted to discussing various aspects of the local
p-adic Langlands conjecture.
In Section 3, after presenting some initial motivation and recalling the notionof admissible unitary Banach space representations of GL2 (Qp ), we present ourconjecture regarding the local p-adic Langlands correspondence (Conjecture 3.3.1).
In Section 4 we introduce the concept of a refinement of a two dimensional
GQp -representation, and establish some basic properties on the existence and classification of refinements. (For this we rely heavily on the work of Colmez [27].) A
fundamental point is that a two dimensional GQp -representation admits a refinement if and only if it is trianguline. Furthermore, as we explain, the language ofLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

5

refinements gives a convenient way to express the classification (due to Colmez) of
two dimensional trianguline representations.
In Section 5 we recall what little is known about the classification of topologicallyirreducible admissible unitary GL2 (Qp )-representations. One basic technique forstudying such a representation B is to pass to the subspace Ban of locally analytic
vectors, which is dense in B, and is a strongly admissible locally analytic GL2 (Qp )representation. Unfortunately, not much is known about the classification of such
representations in general. However, one is much closer to having a classificationof those B for which either Blalg (the space of locally algebraic vectors in B) or
JP(Qp ) (Ban ) (the Jacquet module of Ban with respect to the Borel subgroup of
upper triangular matrices in GL2 (Qp )) is non-zero, as we explain in some detail.
The discussion relies heavily on work of Colmez [26, 27] and of the author [32, 35].
As we remarked above, the local correspondence V 7→ B(V ) has been definedfor most trianguline two dimensional GQp -representations V . More precisely, it
has been defined for irreducible trianguline V by Breuil and Colmez [12, 26], andfor Frobenius semi-simple potentially crystalline reducible V by Breuil and theauthor [6, 14]. In Section 6 we recall the definition of the correspondence in thesevarious cases, and also discuss the expected structure of B(V ) in the remaining
reducible cases. We discuss the extent to which this explicit correspondence is
known to satisfy the conditions of Conjecture 3.3.1. We also state a conjecture on
the structure of the space of locally analytic vectors B(V )an in B(V ) (generalizing
a conjecture of Breuil in the potentially crystalline case).
With our discussion of the local p-adic Langlands conjecture completed, in Section 7 we return to the subject of Conjecture 1.1.1. We begin by describing in
b 1 (K p )E and Hb 1 introduced above. After an aside
more detail the completions H
E
on various notions and results related to systems of Hecke eigenvalues, we disb 1 (K p )E,an .
b 1 )lalg , and of the Jacquet module JP(Q ) H
cuss the structure of (HEp
In particular, we state and sketch the proof of Theorem
7.5.8, which establishes
b 1 (K p )E,an and the eigensurface of tame
the precise relationship between JP(Qp ) H
level K p . After another aside, in which we reformulate some of the results of [41]in the language of refinements, we turn to our main topic: the local-global compatibility conjecture. We state the conjecture in various forms, and deduce someof its consequences (perhaps the most interesting of which are the two stated in
Proposition 1.1.2). Finally, we state and prove some results (summarized in Theorems 1.1.3 and 1.1.4 above) that provide some small amount of evidence for the
conjecture.
1.3. Notation, terminology, and conventions. We fix an algebraic closure Q of
Q, as well as an algebraic closure Q` of Q` for each prime `. We also fix embeddingsı` : Q ,→ Q` for each `, and an embedding ı∞ : Q ,→ C. As usual, we write
GQ := Gal(Q/Q) and GQ` := Gal(Q` /Q` ). The embedding ı` induces an embedding
GQ` ,→ GQ for each `; we sometimes denote the image by D` (the decompositiongroup at `).
For each prime `, the residue field of the integral closure of Z` in Q` is an algebraic
closure F` of the finite field F` . There is a natural surjection GQ` → Gal(F` /F` ),with kernel equal to the inertia subgroup I` ; the target of this surjection is procyclic,
topologically generated by the Frobenius Frob` . For each prime ` we let W` denotethe Weil group at `; i.e. the subgroup of GQ` consisting of elements which act on F`
via an integral power of Frobenius. We topologize W` in the usual way, by declaring

6

MATTHEW EMERTON

the inertia subgroup (endowed with the topology it inherits as a closed subgroup∼abof GQ` ) to be open. The local Artin map induces an isomorphism Q×` −→ W` ,×
which we normalize by identifying the uniformizer ` in Q` with a lift of Frob−1`(i.e. geometric Frobenius).
b denote the profinite completion of Z, and as usual we write Af :=We let Z
b for the ring of finite adèles over Q. More generally, if Σ is any finite set of
Q ⊗Z Zb Σ to denote the prime-to-Σ profinite completion of Z, andprimes then we write ZbΣAΣf := Q ⊗Z Z to denote the prime-to-Σ finite adèles. If Σ = {`} for a single primeb ` := Zb {`} and A` := A{`} . The embeddings Q → Af and Z
b → Af`, we write Zf∼fb × −→ A× , and hence an isomorphism
induce an isomorphism Q>0 × Z
f(1)∼
b × −→
Z
Q>0 \A×f .

The global Artin map (which we normalize to be compatible with our choice of
normalization of the local Artin maps) induces an isomorphism
(2)

∼

ab
Q>0 \A×f −→ GQ .

As usual, we will use the isomorphisms (1) and (2) to identify characters ψ of GQabb × . If ψ is unramified outside of a finite set of primes Σ, thenwith characters of Zb × factors through the projection Z
b× → Zb × /(Zb Σ )× ,the corresponding character of Zb × /(Zb Σ )× . Let us point out that if ψ
and thus we will regard ψ as a character of Zabis a character of GQ unramified outside of Σ, if ` is a prime not in Σ, and if ψ`denotes the restriction of ψ to W`ab , identified with a character of Q×` via the local
Artin map, then we have the formula ψ(`) = ψ` (`−1 ) (where on the left hand sideb × /(Zb Σ )× and on the right hand side as an element` is regarded as an element of Z×
of Q` ).
×The prime p will be fixed throughout the paper. We let | |: Q×
p → Qp denotethe p-adic absolute value, normalized by | p |= p−1 . (We will occasionally use thesame notation to denote the `-adic absolute value on Q×` , for some prime ` 6= p,
normalized so that | ` |= `−1 . We will always make it clear when we are doing×
this.) For any integer w, we write simply z w to denote the character Q×
p → Qpw
defined by z 7→ z .
We let ε : GQ → Z×p denote the p-adic cyclotomic character. We use the same
symbol ε to denote its restriction to GQp . (It will be clear from the context whether
we are referring to the global or local character.) Global class field theory identifies
b × given by projection onto the pth factor: Z
b × → Z× .
ε with the character of Zp
We fix a finite extension E of Qp contained in Qp . We let OE denote the ring ofintegers in E, $ the uniformizer of OE , and F := OE /$ the residue field of OE .
A continuous character η : Wp → E × extends to a continuous character of×
GQp if and only if η is unitary, i.e. if and only if η takes values in OE. If thisextension exists, it is unique, and we denote it by the same symbol η. If α ∈ E × ,×
we let ur(α) : Q×denote the character that maps p to α and is trivial
p → E
×on Zp . Regarded as a character of Wp , the character ur(α) is unramified (hence
the notation) and maps a geometric Frobenius element to α. This character isunitary, and hence extends to a continuous E-valued character of GQp , precisely
×
when α ∈ OE
. Assuming this condition holds, then according to the convention justsignalled, we use the same notation ur(α) to denote the extended character. The

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

7

same convention allows us to regard the local cyclotomic character ε as a unitarycharacter of Q×
p . We then have the following useful formula: ε =| | z.
For any commutative ring R with 1 we let GL2 (R) denote (as usual) the groupof invertible 2 × 2 matrices over R, let P(R) denote the subgroup of upper triangular matrices,Tlet P(R) denote the subgroup of lower triangular matrices, and let
There is a canonT(R) := P(R) P(R) denote the subgroup of diagonalmatrices.

ical identification T(R) = (R× )2 (given by identifying

a
0

0
d

with the ordered pairof units (a, d)).
If B is an E-Banach space equipped with a continuous action of GL2 (Qp ), thenwe let Blalg (resp. Ban ) denote the GL2 (Qp )-subspace of locally algebraic (resp.
locally analytic) vectors in B (as defined in [31]; see also [51] in the case of locallyanalytic vectors). Recall that the space Ban may be equipped in a natural way withan inductive limit topology, making it a barrelled locally convex E-space equippedwith a locally analytic action of GL2 (Qp ). (See [31], but note that in that reference
Ban is denoted by Bla .) The topology on Ban is finer than the topology inducedon it as a subspace of B. In [51] another topology on Ban is considered, which
realizes it as a closed subspace of the space of locally analytic B-valued functions
on GL2 (Qp ). When B is admissible unitary, which is the main case of interest to us,
these two topologies coincide [14, Rem. A.1.1], and [51, Thm. 7.1] shows that Ban
is a strongly admissible locally analytic representation of GL2 (Qp ) (in the sense of[49]), which is dense as a subset of B.
b denote the rigid analytic space over E that classifies the continuous
We let T(equivalently, locally analytic) characters of T(Qp ). For any finite extension E 0 ofb 0 ) of E 0 -valued points of Tb is thus canonically identified with
Qp , the space T(Ethe space of continuous characters T(Qp ) → E 0× . Via the canonical identification2b
T = (Q×p ) we may regard an element χ ∈ T(E) as an ordered pair (χ1 , χ2 ) of××
characters χi : Qp → E ; we will then write χ = χ1 ⊗ χ2 . We may equally well× 2
identify the pair (χ1 , χ2 ) with a representation Q×
p → T(E) = (E ) , and hence
via local class field theory with a representation Wp → T(E). In summary, we havea natural identification Tb(E) = Homcont (Wp , T(E)).
We will have cause to consider several kinds of parabolically induced representab
tions attached to characters χ1 ⊗ χ2 ∈ T(E).
For any such character we may form
GL2 (Qp )the continuous induction IndP(Q ) χ1 ⊗χ2 cont and the locally analytic inductionpGL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 an . Each is defined as a space of functions f : GL2 (Qp ) → E
psatisfying the condition(3)f (pg) = χ1 (a)χ2 (d)f (g)
for all g ∈ GL2 (Qp ) and p = a∗ d0 ∈ P(Qp ). The GL2 (Qp )-action is via
right translation. For the continuous (resp. locally analytic) induction, the function f is required to be continuous (resp. locally analytic). Since the projection
GL2 (Zp ) → P(Qp )\GL2 (Qp ) is surjective, restricting functions to GL2 (Zp ) embeds the continuous (resp. locally analytic) induction as a closed subspace of thespace of continuous (resp. locally analytic) functions on GL2 (Zp ). Thus it is naturally an admissible Banach space representation of GL2 (Qp ), in the sense of [50](resp. a strongly admissible locally analytic representation of GL2 (Qp ), in the senseof [49]). The locally analytic induction is naturally identified (as a topological

8

MATTHEW EMERTON

GL2 (Qp )-representation) with the space of locally analytic vectors in the continuous induction (as follows from [31, Prop. 3.5.11]).
If χ1 χ−1
has non-negative integral Hodge-Tate weight, then we may also form2GL (Q )
the locally algebraic induction IndP(Q2 ) p χ1 ⊗ χ2 lalg , defined to consist of thosepfunctions f satisfying condition (3), and such that χ−1
2 ◦ det (g)f (g) is a locallyalgebraic function on GL2 (Qp ). If χ2 itself is of integral Hodge-Tate weight, thenthe locally algebraic induction is naturally identified with the subspace of locallyalgebraic vectors in the continuous induction. In general (i.e. if χ2 is not of integral
Hodge-Tate weight), then it consists of the subspace of locally SL2 -algebraic vectors
in the continuous induction. (Hopefully this slight inconsistency in our use of theterminology “locally algebraic” will not cause confusion.)
Finally, if χ1 and χ2 are smooth characters of T(Q` ) for some prime `, with valuesGL (Q )
in any field, then we will write IndP(Q2 ) ` χ1 ⊗ χ2 sm to denote the corresponding`bsmooth induction. (In the case when ` = p and χ1 ⊗ χ2 ∈ T(E)with both χ1 andχ2 being smooth characters, smooth induction is a special case of locally algebraic
induction.)
If we are given a family π` of GL2 (Q` ) vector spaces indexed by the primes `lying outside some finite set of primes Σ, such that for almost all ` the representavector e` , unique up to multiplication
tion π` contains a non-zero GL2 (Z` )-invariant
N0
by a scalar, then we will denote byπ`6∈Σ ` the restricted tensor product of therepresentations π` with respect
N to the vectors e` (as discussed in [40, §9], for example, where the notation is e` π` .). The restricted tensor product is naturally a
GL2 (AΣf )-representation, which up to isomorphism is independent of the choice ofthe e` .
We let 1 denote the trivial character (of any group; in any particular usage, thegroup in question will be clear from the context).
If M is a Zp -module, or a sheaf of Zp -modules, and A is a Zp -algebra, then wesometimes write MA := A ⊗Zp M .
We assume that the reader is familiar with the basic aspects of the p-adic Hodge
theory of GQp -representations, including Hodge-Sen-Tate weights, the period ringsof Fontaine and the related conditions on a representation of being Hodge-Tate, deRham, (potentially) semi-stable, or (potentially) crystalline. We use the standard
notation for these period rings, and for the associated Dieudonné modules. (Wealways form Dieudonné modules in the covariant sense – i.e. by tensoring with thecorresponding period ring and then passing to GQp -invariants.) A useful survey ofthis material is given in [4]. (We caution the reader that in this reference, HodgeSen-Tate weights are referred to as generalized Hodge-Tate weights.)

1.4. Acknowledgments. It is a pleasure to thank Christophe Breuil for all theconversation and correspondence in which he has explained various aspects of his“p-adic Langlands philosophy” to me, as well as for his thoughtful remarks onsome earlier drafts of the present work. I would also like to thank Kevin Buzzard,Frank Calegari, Gaetan Chenevier, Toby Gee, Michael Harris, Haruzo Hida, Mark
Kisin, Robert Pollack, Peter Schneider, and Eric Urban for helpful discussions andremarks related to various aspects of this note.

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

9

2. Classical Langlands for GL2/Q
In this section we briefly describe the classical Langlands correspondence for GL2
over Q.
2.1. The classical local Langlands correspondence. For any prime `, the local Langlands correspondence for GL2 (Q` ) (whose existence was established byTunnell [61], building on earlier results of Jacquet-Langlands [40] and Langlands[43]) establishes a certain bijection between the set of isomorphism classes of irreducible admissible smooth representations of GL2 (Q` ) on Qp -vector spaces, and theset of isomorphism classes of two dimensional Frobenius semi-simple Weil-Deligne
representations WD` → GL2 (Qp ) over Qp (as defined in [29, §8] or [58, §4]).
In fact there are various choices of correspondence, depending on the desired
normalization. The so-called unitary correspondence is uniquely determined by therequirement that the local L- and ε-factors attached to a pair of corresponding isomorphism classes should coincide. However, in our discussion we will always employ
the so-called Tate normalization (as described in [29]). If σ is a two dimensional
Frobenius semi-simple representation of WD` over Qp , then we let π` (σ) denotethe irreducible admissible smooth representation of GL2 (Q` ) associated to σ via
the Tate normalized local Langlands correspondence. One advantage of the Tatenormalization is that if the isomorphism class of the Weil-Deligne representation
σ is in fact defined over a finite extension E of Qp , then the same is true of theisomorphism class of the GL2 (Q` )-representation π` (σ).
L
For definiteness, we recall the definition of π` (σ) in the case when σ ∼= χ1 χ2∼ ab (the isomorphism being provided by the local
for two characters χi of Q×` = W`
Artin reciprocity map). If χ1 χ−1
6=| |±1 (where | | denotes the absolute value
2
×character of Q` ), thenGL (Q )π` (σ) := IndP(Q2 ) ` χ1 ⊗ χ2 | | sm ,`while if χ1 χ−1
2 = || (resp. | |−1 ), then π` (σ) := χ1 ◦ det (resp. χ2 ◦ det). (As werecall in more detail below, in these latter two cases, the smooth induction of χ1 ⊗χ2
is not irreducible.) The collection of representations of GL2 (Q` ) attached to such
σ is known as the principal series. In particular, the collection of representationsπ` (σ) attached to those σ for which χ1 and χ2 are both unramified is known as
the unramified principal series. Its members can be characterized intrinsically as
those irreducible admissible smooth GL2 (Q` )-representations that contain a nonzero GL2 (Z` )-fixed vector.
For later use, it will be convenient to recall some additional terminology used in
the classification of irreducible admissible smooth GL2 (Q` )-representations. NoteGL (Q )
that the parabolic induction IndP(Q2 ) ` 1⊗1 sm evidently contains the trivial char`GL (Q )
acter of GL2 (Q` ) as a subrepresentation. The quotient of IndP(Q2 ) ` 1 ⊗ 1 sm by`this one dimensional subrepresentation is an irreducible representation of GL2 (Q` )called the Steinberg representation [19, §8], which we will denote by St. The twistsof St are known collectively as the special representations of GL2 (Q` ). They correspond under the local Langlands correspondence to those Weil-Deligne representations that are reducible but not semi-simple.
The irreducible admissible smooth representations of GL2 (Q` ) that are neither
principal series nor special are called cuspidal. They are characterized intrinsically10

MATTHEW EMERTON

by the property that their matrix coefficients are compactly supported modulo thecentre of GL2 (Q` ). They correspond under the local Langlands correspondence tothe irreducible Weil-Deligne representations.
2.1.1. A modified local Langlands correspondence. In what follows we will need aslightly modified version of the local Langlands correspondence. Recall that an
irreducible admissible smooth representation π of GL2 (Q` ) over Qp is called genericif π admits a Whittaker model, or equivalently, if it is infinite-dimensional (seeThm. 2.14 of [40], together with the discussion on p. 62 of this reference). Otherwise
we say that π is non-generic. The only non-generic irreducible admissible smooth×
representations are those of the form χ ◦ det, where χ is a Qp -valued character of
Q×` [40, Prop. 2.7].
If σ is a two dimensional Frobenius semi-simple representation of WD` for whichπ` (σ) is generic, then we define π`m (σ) := π` (σ). The only σ for which π` (σ) is
non-generic are those of the form
χ | |−1 0
,
σ∼
=
0
χ
where χ is some character of Q×` (uniquely determined by σ), and | | denotes theabsolute value character of Q×` ; as we recalled above, the usual Tate normalized
local Langlands correspondence is defined on such σ by π` (σ) := χ ◦ det. For suchGL (Q )
σ, we define π`m (σ) := IndP(Q2 ) ` χ | |−1 ⊗χ | | sm , which is a reducible but`
indecomposable infinite dimensional representation of GL2 (Q` ). It is a non-split
extension of the character χ ◦ det by the twist St ⊗ χ [19, §8].
2.1.2. From local Galois representations to GL2 (Q` )-representations. We finish thissubsection by recalling the manner in which the local Langlands correspodence may
be used to attach admissible representations of the groups GL2 (Q` ) to two dimensional Galois representations. Let V be a continuous two dimensional representation of GQ` defined over Qp . If ` is a prime distinct from p, then we may attach a
Weil-Deligne representationσ(V ) : WD` → GL2 (Qp )to V via a recipe of Deligne [29, §8] (see also [58, Thm. 4.2.1]). We may furthermore Frobenius semi-simplify σ(V ) (see [29, 8.6]) to obtain a Frobenius semi-simple
Weil-Deligne representation σ ss (V ). We then write π` (V ) := π` (σ ss (V )), and alsoπ`m (V ) := π`m (σ ss (V )). If ` = p and V is furthermore potentially semi-stable, thenwe may attach a Weil-Deligne representationσ(V ) : WDp → GL2 (Qp )to V via a recipe of Fontaine [36, §2.3.7]. (See also [59], which gives a succinct description of the construction for representations having coefficients in Qp .) Again,
we may Frobenius semi-simplify to obtain a Frobenius semi-simple Weil-Deligne
representation σ ss (V ), and we write πp (V ) := πp (σ ss (V )), and also πpm (V ) :=πpm (σ ss (V )).
If V is a two-dimensional continuous GQ` -representation over Qp for which π` (V )is defined (i.e. V is arbitrary if ` 6= p, and V is potentially semi-stable if ` = p),then we say that V is generic if π` (V ) is generic, or equivalently, if π` (V ) = π`m (V ).
Otherwise, we will say that V is non-generic.

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

11

2.1.3. Remark. If V is defined over the finite extension E of Qp , then the isomorphism class of the Weil-Deligne representation σ ss (V ) is defined over E, and thus
so is the isomorphism class of each of πp (V ) and πpm (V ) (since we have used theTate normalization for the local Langlands correspondence). It follows that each ofthese representations may be defined over E. (See [20, Prop. 3.2] for the analogous
statement in the case of GLn (F ), for any finite extension F of Q` , and any n ≥ 1.)
2.2. Representations of GL2 (Af ) attached to two dimensional global Galois representations. One may use the local Langlands correspondence to attach
representations of adèlic groups to global Galois representations. Suppose that V isa two dimensional Qp -vector space equipped with a continuous action of GQ that isunramified away from a finite number of primes. For each prime ` distinct from p,
the local Langlands correspondence recalled in the preceding subsection gives riseto an irreducible admissible smooth representation π` (V ) := π` (V |D` ) of GL2 (Q` ),defined over Qp . Since V is unramified at almost all primes `, the representationπ` (V ) lies in the unramified principal series for almost all `, and so we may form
the restricted tensor productO0
π p (V ) :=π` (V ),
`6=p

which is an irreducible admissible smooth representation of GL2 (Apf ) defined overQp .
We also introduce notation for the modified version of this construction, in which
we use the modified local Langlands correspondence described in 2.1.1. Namely, wedefineO0 mπ` (V ).
π m, p (V ) :=`6=p

This is an admissible smooth GL2 (Apf )-representation.
If V |Dp is furthermore potentially semi-stable, then we may form the irreducible
admissible smooth representation πp (V ) := πp (V |Dp ) of GL2 (Qp ). We may thendefineO0π` (V )
π(V ) := πp (V ) ⊗ π p (V ) =`(the restricted tensor product now running over all primes). This is an irreducible
admissible smooth representation of GL2 (Af ). (We could also define a modified
representation π m (V ), but we will not have need of this.)
2.2.1. Remark. Remark 2.1.3 shows that if V is defined over the finite extension
E of Qp , then so are π p (V ), π m, p (V ), and (if V |Dp is potentially semi-stable) π(V ).
2.3. Galois representations attached to newforms. If f is a cuspidal newformof weight k ≥ 1 and conductor N defined over Qp , then associated to f is a twodimensional irreducible continuous representation Vf of the absolute Galois group
GQ (constructed by Shimura [54] when k = 2, by Deligne [28] when k > 2, andby Deligne-Serre [30] when k = 1). This representation is characterized by thefollowing condition: it is unramified outside of the primes dividing N p, and foreach ` - N p, the geometric Frobenius at ` has the characteristic polynomial(4)X 2 − a` X + χ(`)`k−1 ;12

MATTHEW EMERTON

here a` denotes the ` th Hecke eigenvalue of f , and χ : (Z/N )× → Q×p denotes thenebentypus character of f . As we now recall, one can formulate a more precise
relationship between the Galois representation Vf and the newform f , which takes
into account the local properties of each of them at all primes (including those
dividing N p), in terms of their associated GL2 (Af )-representations.
On the one hand, there is a natural action of the group GL2 (Af ) on the space of
cuspforms over Qp of weight k and arbitrary level which makes the space of modular
forms an admissible smooth representation of GL2 (Af ). Under this action, thenewform f generates an irreducible representation π(f ).4 On the other hand, Vf
is unramified away from a finite number of primes, and is known to be potentiallysemi-stable at p, and so we may associate to Vf an irreducible admissible smooth
representation π(Vf ) of GL2 (Af ) via the procedure of the preceding Subsection.
∼

2.3.1. Theorem. There is a GL2 (Af )-equivariant isomorphism π(f ) −→ π(Vf ).
This theorem expresses the compatibility of the local and global Langlands correspondence. As already intimated, it provides a significant strengthening of theformula (4) for the characteristic polynomials of Frobenius acting on Vf . (One recovers that formula from the isomorphism of the theorem by considering the action
just of those groups GL2 (Q` ) for ` - N p.)
Both the construction of the representations Vf and the proof of Theorem 2.3.1
rely on an analysis of the étale cohomology of modular curves. We briefly recall
some of the details in the following subsections.
b
2.4. Cohomology of modular curves. If Kf is an open subgroup of GL2 (Z)(which we think of as a “level”), then we let Y (Kf ) denote the open modular curve
over Q that classifies elliptic curves with a Kf -level structure (i.e. elliptic curves∼
E equipped with a Kf -orbit of isomorphisms (Q/Z)2 −→ Etor := limE[n]). This−→Tn
curve is a fine moduli space provided that the intersection SL2 (Z) Kf is torsion
free. Under this assumption on Kf , let pr denote the projection to Y (Kf ) fromthe universal elliptic curve over Y (Kf ), and define V3 = R1 pr∗ Zp . For any integerk ≥ 2, set Vk := Symk−2 V3 ; this is a free of rank k − 1 étale local system of
Zp -modules on Y (Kf ).
If Kf0 ⊂ Kf is an inclusion of levels, then there is an induced surjection Y (Kf0 ) →Y (Kf ) (“pass to the underlying Kf -level structure”), and the pullback under thismap of the local system Vk on the target is naturally isomorphic to the corresponding local system Vk on the source (which justifies our omission of the level Kf fromthe notation for Vk ).
There is a natural right action of GL2 (Af ) on the directed system of curves
Y (Kf ), and the local systems (Vk )Qp are equivariant with respect to this action.
This action, and the stated equivariance, are most easily seen on the level of complex
points. The Riemann surface of C-valued points of Y (Kf ) is equal to
Y (Kf )(C) := GL2 (Q)\ C r R × GL2 (Af ) /Kf(where GL2 (Q) acts on CrR via linear fractional transformations, and on GL2 (Af )via left multiplication, while Kf acts trivially on C r R, and on GL2 (Af ) via right
4To be precise about our choice of normalization, if π (f ) denotes the unitary GL (A )u2f

representation attached to f , then π(f ) := πu (f )⊗ | |(2−k)/2 .

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

13

multiplication). For each integer k ≥ 2 the pullback of Vk to Y (Kf )(C) is a local
system in the complex topology admitting the description

Vk/Y (Kf )(C) := GL2 (Q)\ C r R × GL2 (Af ) × Symk−2 Z2p /Kf(where the double quotient is taken with respect to the action map

GL2 (Q) × C r R × GL2 (Af ) × Symk−2 Z2p × Kf → C r R × GL2 (Af ) × Symk−2 Z2p
defined via γ ×(τ, g, v)×k 7→ (γτ, γgk, kp−1 v); here kp denotes the pth component of
the element k ∈ Kf ). The right action of GL2 (Af ) on the complex curves Y (Kf )(C)and on the sheaves (Vk/Y (Kf )(C) )Qp is then defined via(τ, g) × g 0 7→ (τ, gg 0 )and(τ, g, v) × g 0 7→ (τ, gg 0 , (gp0 )−1 v)
for g 0 ∈ GL2 (Af ) (where gp0 denotes the pth component of g 0 ). We refer to [42, §3]
for a detailed discussion of the GL2 (Af )-action on the curves Y (Kf ) and the local
systems (Vk )Qp over Q that underlies the action just described on the C-points.
For any integer n, and any choice of ∗ ∈ {∅, c, par}, we let H∗n denote étale
cohomology computed with respect to the support condition ∗: i.e. either no support condition, compact supports, or parabolic cohomology (that is, the image ofcompactly supported cohomology in cohomology). Now fix an integer k ≥ 2, anddefineH∗1 (Vk ) := lim
H∗1 (Y (Kf )/Q , Vk ).
−→KfThis space is equipped with a natural continuous action of GQ := Gal(Q/Q), andhence so is the base-change H∗1 (Vk )Qp . The right action of GL2 (Af ) on the direct
system of curves Y (Kf ), and the equivariance of the local systems (Vk )Qp under
this action, induces a left action of GL2 (Af ) on H∗1 (Vk )Qp that commutes with the
GQ -action on this space.
2.5. Cohomological realization of the classical Langlands correspondence.
Let us fix a weight k ≥ 2. Eichler-Shimura theory, which describes the space
1
Hpar(Vk ) (with its GL2 (Af )-action) in terms of the space of cuspforms of weightk (see [42, §2] for an account in representation-theoretic language), shows that1
Hpar
(Vk )Qp is semi-simple as a GL2 (Af )-representation, that its simple direct summands are precisely isomorphic to the representations π(f ) (as f runs over all cuspidal newforms of the given weight k), and that each representation π(f ) appears
1
with multiplicity two in Hpar
(Vk )Qp . Thus we may write
M
∼
1
Hpar
(Vk )Qp −→Mf ⊗Qp π(f ),f1
where Mf = HomGL2 (Af ) (π(f ), Hpar
(Vk )) is a uniquely determined continuous twodimensional representation of GQ over Qp .
Since each multiplicity space Mf is a Galois subrepresentation of the étale cohomology of a curve with coefficients in the local system (Vk )Qp of geometric origin,
it is unramified away from a finite number of primes, and is potentially semi-stable
at p. Thus we may define the GL2 (Af )-representation π(Mf ) using the procedure

14

MATTHEW EMERTON

described in Subsection 2.1. The key result concerning the multiplicity spaces Mfthen states that for each f there is a GL2 (Af )-equivariant isomorphism(5)∼π(Mf ) −→ π(f ).

By considering the local factors of these two representations just at primes ` - N p,
one finds that Mf is a model for Vf , the Galois representation associated to f .
(Indeed, this is the approach taken by Deligne to constructing the representation
Vf .) The isomorphism (5) then implies Theorem 2.3.1 (for f of weight k ≥ 2). The
following theorem summarizes the situation.
2.5.1. Theorem. For any given k ≥ 2, there is a GQ × GL2 (Af )-equivariant isomorphism
M
∼
1
Hpar
(Vk )Qp −→Vf ⊗Qp π(Vf ),fwhere f runs over all cuspidal newforms defined over Qp of weight k.
The isomorphism (5) was established in general by Carayol [18] (for the local
factors at the primes ` 6= p) and Saito [47] (for the local factors at p), building on thework of many people, including Eichler, Shimura, Igusa, Deligne, and Langlands.
∼
Note that since Vf is determined up to isomorphism by π(Vf ) −→ π(f ), we seethat the Vf are non-isomorphic for distinct f . Combining the conjecture of Fontaine
and Mazur on geometric Galois representations [37, Conj. 1] and Langlands’ conjecture relating automorphic forms and motives [20], one expects that (for fixed k)
the representations Vf range over all irreducible continuous two dimensional p-adic
representations of GQ that are unramified outside of a finite number of primes, andthat are potentially semi-stable at p with Hodge-Tate weights (1 − k, 0) (cf. [37,Conj. 3c]). Thus Theorem 2.5.1 and the Fontaine-Mazur conjecture together imply
the following conjecture, on which Conjecture 1.1.1 is modelled.5
2.5.2. Conjecture. If V is an irreducible continuous GQ -representation over Qp
that is semi-stable at p with Hodge-Tate weights (1−k, 0) for some k ≥ 2, and is un1
ramified outside of a finite set of primes, then the space HomGQ (V, Hpar
(Vk )) (whichis naturally a GL2 (Af )-representation) is isomorphic to the irreducible smooth
GL2 (Af )-representation π(V ) defined in Subsection 2.2 above.
3. The local p-adic Langlands conjecture for GL2/Qp
In this section we recall some ideas of Breuil and others related to a possible
local p-adic Langlands conjecture.
3.1. Motivation. Let ` be a prime, and let V be a two dimensional continuous
representation of the decomposition group GQ` over Qp . As was recalled in Subsection 2.1, applying either the recipe of Deligne if ` 6= p, or the recipe of Fontaine if` = p and V is potentially semi-stable, we may attach a Frobenius semi-simple WeilDeligne representation σ ss (V ) : WD` → GL2 (Qp ) to V , which in turn corresponds
via local Langlands to an admissible smooth representation π` (V ). The representation σ ss (V ) is in turn determined up to isomorphism by π` (V ) := π` (σ ss (V )).
In the case when ` 6= p, Deligne’s procedure for constructing σ(V ) from V isreversible, and so if the original representation V was itself Frobenius semi-simple
5Note that in this conjecture we don’t explicitly require that V should be odd, since this should
follow automatically from the other assumptions on V .

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

15(as is conjectured to be the case when V arises as the restriction to GQ` of a global
Galois representation attached to a cuspidal newform – this is a particular case of aconjecture of Tate [57]) then since W` is dense in GQ` , we see that V is determinedby V |W` . Altogether, we see that if V is Frobenius semi-simple, then it is determined
up to isomorphism by the associated GL2 (Q` )-representation π` (V ).
On the other hand, if ` = p and V is potentially semi-stable, then the construction of σ(V ) from V involves passing to the potentially semi-stable Dieudonnémodule of V (which is an admissible filtered (ϕ, N, GQp )-module), and then forgetting the Hodge filtration. In general, one can equip a given (ϕ, N, GQp )-modulewith a filtration so as to make it an admissible filtered (ϕ, N, GQp )-module in manydifferent ways. Thus in this case V is typically not determined by π` (V ).
Breuil has conjectured that there should be a “local p-adic Langlands conjecture”, which attaches to V a certain representation B(V ) of GL2 (Qp ) on a p-adic
Banach space. This representation B(V ) should determine V up to isomorphism.
Breuil originally limited his conjecture to the case when V is potentially semi-stable;
however, in light of developments in the field (in particular, the results of Colmez[27]), it seems reasonable to conjecture the existence of such a correspondence forarbitrary continuous two dimensional representations of GQp . Although there is so
far no precise formulation of the conjecture (even in the case when V is potentiallysemi-stable), one can formulate a list of at least some of the properties that itshould be required to satisfy. We will do this below, after first recalling some basic
definitions regarding Banach space representations of GL2 (Qp ).
3.2. Admissible unitary Banach representations of GL2 (Qp ). Let B be an
E-Banach space equipped with a continuous action of GL2 (Qp ).
3.2.1. Definition. We call B admissible unitary if there exists a norm that determines the topology of B, such that the unit ball L ⊂ B of with respect to this norm
is GL2 (Qp )-invariant, and such that the induced GL2 (Qp )-action on L/$L is anadmissible smooth representation of GL2 (Qp ) over OE /$. (Recall that $ denotes
a uniformizer in OE .)The notion of admissibility introduced here is due to Schneider and Teitelbaum[50]. We refer the reader to that reference for a number of equivalent characterizations of admissibility.
3.2.2. Definition. Let B be an admissible unitary Banach space of GL2 (Qp ) andlet L be a choice of GL2 (Qp )-invariant unit ball in B. If the admissible smooth
GL2 (Qp )-representation L/$L is of finite length, then we let B denote the semisimplification of L/$L as a GL2 (Qp )-representation. (Note that if the hypothesis
holds for one choice of L then it holds for any such choice, and the semi-simple admissible smooth GL2 (Qp )-representation B is then well-defined up to isomorphism
independently of the choice of L.)
3.3. Statement of the conjecture. As we recalled above, in the classical local
Langlands correspondence, the matching between isomorphism classes of Frobenius
semi-simple two dimensional Weil-Deligne representations at p and isomorphismclasses of irreducible admissible smooth GL2 (Qp ) representations (with respect tothe unitary normalization) is uniquely determined by the requirement that the
local L- and ε-factors attached to a pair of matched objects should coincide. No
such precise formulation of a local p-adic Langlands conjecture has yet been given.

16

MATTHEW EMERTON

However, following the ideas of Breuil, a consensus seems to have emerged that astatement of the following type should be true.
3.3.1. Conjecture. There is a local p-adic Langlands correspondence that associates
to each continuous two dimensional representation V of GQp over E a corresponding
non-zero admissible unitary Banach space representation B(V ) of GL2 (Qp ) over E,
well-defined up to GL2 (Qp )-equivariant topological isomorphism. This correspondence satisfies the following properties:(1) If V and V 0 are continuous two dimensional representations of GQp over E,then V and V 0 are isomorphic if and only B(V ) and B(V 0 ) are GL2 (Qp )equivariantly topologically isomorphic.
(2) If V has determinant χ, then B(V ) has central character χε.
(3) For any continuous character χ : GQp → E × , there is an isomorphism∼

B(V ⊗ χ) −→ B(V ) ⊗ (χ ◦ det)
(where on the right hand side we regard χ as a character of Q×
p via local
class field theory).
(4) The association of B(V ) to V is compatible with extending scalars to any
finite extension of E.
(5) The representation B(V ) satisfies the hypothesis of Definition 3.2.2, andB(V ) is associated to the semi-simplification V of the reduction mod $ ofV via the local mod $ Langlands correspondence of [10, Déf. 1.1].6(6) If V is irreducible, then B(V ) is topologically irreducible. In general, if V ss
denotes the semi-simplification of V , then the representations B(V ) andB(V ss ) have isomorphic topological semi-simplifications.
(7) The two dimensional representation V is potentially semi-stable, with distinct Hodge-Tate weights, if and only if the subspace B(V )lalg of locallyalgebraic vectors in B(V ) is non-zero. Furthermore, if these conditions
hold, then (letting w1 < w2 denote the Hodge-Tate weights of V ) the subrepresentation B(V )lalg is isomorphic to the locally algebraic representationπep (V ) := πpm (V ) ⊗ Symw2 −w1 −1 E 2 ⊗ detw1 +1(where πpm (V ) is the admissible smooth representation of GL2 (Qp ) attached
to V via the modified classical local Langlands correspondence, as explainedin 2.1.2; note that by Remark 2.1.3 this representation is defined over E).
(8) The set of refinements of V is related to the structure of JP(Qp ) (B(V )an ) inb
the following manner: for any character η ⊗ ψ ∈ T(E)there is an equalityof dimensions
dim Ref η⊗ψ (V ) = dim Expη|

|⊗ψε|

|−1

(B(V )an ).

Here JP(Qp ) denotes the locally analytic Jacquet module functor of [32] (see−1also Subsection 5.2 below), while Ref η⊗ψ (V ) and Expη| |⊗ψε| | (B(V )an )are both projective spaces over E, whose definitions can be found in Definitions 4.1.7 and 5.2.3 below respectively.
The preceding conjecture merits several remarks.
6We normalize this mod $ correspondence so that it satisfies the analogue of condition (2)above.

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

17

3.3.2. Remark. Conditions (1), (2), (3), and (4) are standard requirements of any
Langlands-type conjecture. (The appearance of the factor ε in condition (2) reflects
our particular choice of normalization.) By imposing condition (5), expressing
the compatibility of the conjectured correspondence with reduction modulo $, wehave incorporated one of Breuil’s original motivations for introducing a local p-adic
Langlands conjecture.
3.3.3. Remark. Condition (6) reflects one way in which the conjectured local padic Langlands correspondence is quite different from the classical correspondence,in that the representations B(V ) will in some instances be topologically reducible.
In fact the analysis of the reducible case that we make in Section 6 below showsthat for any correspondence satisfying Conjecture 3.3.1, the representation B(V )is irreducible (resp. indecomposable) if and only if V is irreducible (resp. indecomposable).
3.3.4. Remark. Condition (7) relates the conjectural p-adic correspondence to theclassical Langlands correspondence. Note that for a potentially semi-stable representation V with distinct Hodge-Tate weights, the locally algebraic representationπep (V ) encodes the Hodge-Tate weights of V , together with the Weil-Deligne representation σ(V ) attached to V by Fontaine’s recipe – or equivalently, the (ϕ, N, GQp )module underlying Dpst (V ). 7 Breuil’s idea is that the Banach space B(V ) (at least
when V is irreducible) should be regarded as a completion of π
ep (V ) with respect
to a certain GL2 (Qp )-invariant norm (depending on V ), and that the extra data of
this norm should actually determine the Hodge filtration on Dpst (V ).
In some instances the representation π
ep (V ) admits only one non-zero unitarycompletion up to isomorphism. If this is the case, and if Conjecture 3.3.1 holds,
then (taking into account the preceding discussion together with condition (1) of
Conjecture 3.3.1) we see that V would have to be uniquely determined by the(ϕ, N, GQp )-module that underlies Dpst (V ) (i.e. the Hodge filtration would have to
be uniquely determined, up to isomorphism, by the weak admissibility condition).
This latter condition holds precisely when V is irreducible and potentially crystalline and σ(V ) is abelian (equivalently, πp (V ) is principal series), and indeed forsuch V , Breuil has conjectured that π
ep (V ) does admit a unique non-zero unitarycompletion (up to isomorphism); see Conjecture 5.1.5 below. (This conjecture has
been proved by Berger and Breuil [8] for Frobenius semi-simple V , as we recall in
Theorem 5.1.6 below.)If πp (V ) is special or cuspidal then V is not determined by the (ϕ, N, GQp )module underlying Dpst (V ), and so Conjecture 3.3.1 implies that π
ep (V ) should
admit a whole family of admissible unitary completions, corresponding to the different possible ways of putting a Hodge filtration on the (ϕ, N, GQp )-module underlying Dpst (V ) so as to make it a weakly admissible filtered (ϕ, N, GQp )-module.
Such a family has been constructed in the case when πp (V ) is special, i.e. when
V is potentially semi-stable but not potentially crystalline. (See Theorem 5.1.13
below.)
7Note that, in contrast to the `-adic setting, no information is lost about the ϕ-action on
Dpst (V ) in the process of Frobenius semi-simplifying when one passes from V to πp (V ). Indeed,in those cases when σ(V ) contains two copies of the same character, the weak admissibility of
Dpst (V ) implies that the ϕ-action on Dpst (V ) is necessarily not semi-simple; cf. the proof of [23,Thm. 3.1].

18

MATTHEW EMERTON

3.3.5. Remark. As the discussions of Sections 5 and 6 below will make clear,
Condition (8) is an analogue in the p-adic setting of the requirement in the classicalLanglands correspondence that reducible Weil-Deligne representations be matchedwith subrepresentations of parabolically induced representations of GL2 (Qp ).
3.3.6. Remark. The conditions of Conjecture 3.3.1 are not enough to uniquely
specify the local p-adic correspondence. As we have explained in Remark 3.3.4,
these conditions imply that if V is potentially semi-stable and πp (V ) is special
or cuspidal, then π
ep (V ) should admit a family of admissible unitary completions
corresponding to the various potentially semi-stable representations V 0 for whichπep (V 0 ) = π
ep (V ). However, there is no specification in our conjecture of the particular way that the representations V 0 and the completions of π
ep (V ) (assuming thatthey do exist) should be matched.
Nevertheless, we expect that there will be a “natural” way to match each of therepresentations V 0 under consideration with a certain specific completion of π
ep (V ).
(For example, in the case when πp (V ) is special, the various representations V 0 areclassified by their L-invariants, and Breuil has defined a family of completions ofπ
ep (V ) that also depend on a parameter L – see [12] and also 5.1.7 below.) One
would like to find a condition (or conditions) to add to Conjecture 3.3.1 that would
uniquely determine the correspondence (analogous to the condition on L-factorsand ε-factors in the classical correspondence). Recent work of Colmez suggests one
possible such condition (see the following remark).
3.3.7. Remark. When V is irreducible and trianguline the correspondence V 7→
B(V ) has been specified – “by hand”, as it were – by Breuil and Colmez [12, 27].
It has also been specified for most reducible potentially crystalline V by Breuil andthe author [6, 14]. We will recall the definitions of the representations B(V ) in thecases where they have been defined in Section 6 below, and also consider the extent
to which this explicit correspondence satisfies the conditions of Conjecture 3.3.1.
When V is irreducible, trianguline, and not potentially crystalline, Colmez has
shown that there is in fact an intrinsic relationship between V and the representation B(V ) that is associated to V ; namely, one can recover B(V ) as a P(Qp )representation directly from the overconvergent (ϕ, Γ)-module D† (V ) attached to V .
Building on Colmez’s methods, Berger and Breuil have done the same for those Vthat are crystalline and Frobenius semi-simple. It seems likely that some suitable formulation of this relationship between V and B(V ) will be a key conditionin the ultimate statement of the local p-adic correspondence. However, we have
not wanted to try to predict the general form that this relationship should take,and for this reason have not incorporated Colmez’s results into the statement ofConjecture 3.3.1.
3.3.8. Remark. One point emphasized by Breuil in his initial postulation of a local
p-adic correspondence is that the structure of the GL2 (Qp )-representation on thespace of locally analytic vectors B(V )an should reflect the p-adic Hodge theory of
the Galois representation V (see [12, §1.3]). This requirement does not appear tohave been formalized as of yet, and so we have not incorporated it as one of theconditions in Conjecture 3.3.1. However, in Subsection 6.7 we will give a conjecturaldescription of the space B(V )an in the case when V is trianguline. Comparing thisconjecture with the work of Colmez, Berger, and Breuil discussed in the preceding
remark, which relates B(V ) to D† (V ), suggests that there may be an analogous

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

19

relationship between B(V )an and Drig (V ) (see [3, 27] for a definition of the latter
object). Establishing such a relationship could be one approach to formalizing
Breuil’s proposed link between B(V )an and the p-adic Hodge theory of V .
I should note, though, that there is one condition relating B(V )an to the p-adic
Hodge theory of V that can be formulated precisely. Namely, it should be thecase that B(V )an admits a generalized infinitesimal character, and this character(suitably normalized) should match with the Hodge-Sen-Tate weights of V . (Iwould like to thank Michael Harris for pointing this out to me.) Our conjecture on
the structure of B(V )an for trianguline V is compatible with this requirement.
3.3.9. Remark. As in the classical case, one test of the correctness of the proposed
local correspondence will be its compatibility with the global situation. Thus there
is an interplay between the local Conjecture 3.3.1 and the local-global compatibility
Conjecture 1.1.1. The various results proved in Subsections 7.8, 7.9, and 7.10
below in the direction of Conjecture 1.1.1 may all be regarded as evidence for thecorrectness of the formulation of the local conjecture.
3.3.10. Remark. In this remark we discuss the possible “surjectivity” of the local
p-adic correspondence. We begin with the following definition (which extends tothe characteristic zero situation terminology introduced in [2, §7] by Barthel andLivné for mod $ representations of GL2 (Qp )).
3.3.11. Definition. Let B be a topologically irreducible admissible unitary representation of GL2 (Qp ) over E. We say that B is ordinary if it is a subquotient of
GL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 cont for some pair of unitary E-valued characters χ1 and χ2
pof Q×
p . Otherwise, we say that B is supersingular.
The ordinary representations are precisely the ones appearing in the list of Corollary 5.3.6 below, together with those of the form η ◦ det for unitary characters ηof Q×p (as follows from Proposition 5.3.4 below).
Condition (8) of Conjecture 3.3.1, when combined with the results of Subsection 4.4 and with Proposition 5.2.1 and Lemma 5.3.3 (2) below, implies that if V
is irreducible then B(V ) is supersingular. This observation naturally suggests thefollowing question.
3.3.12. Question. If B is a supersingular topologically irreducible admissible unitary GL2 (Qp )-representation, then is B isomorphic to a representation of the form
B(V ) for some irreducible continuous two dimensional GQp -representation V ?
Given the current state of our knowledge, it seems reasonable to hope that thisquestion has an affirmative answer. Certainly, a large number of the conjecturesthat we make in Sections 5 and 6 below are premised on the expectation that various
known phenomena in the context of GQp -representations will be mirrored by corresponding phenomena in the context of admissible unitary GL2 (Qp )-representations.
Finally, we note that the preceding discussion shows that the ordinary representations are never of the form B(V ) for an irreducible GQp -representation V . In fact
we will see in the discussion of Section 6 that they appear as constituents of therepresentations B(V ) attached to reducible V . This parallels the case of the local
mod $ Langlands correspondence defined in [10, Déf. 1.1].

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MATTHEW EMERTON

4. Refinements and trianguline local Galois representations
In this section we first introduce a notion of refinement of a continuous twodimensional representation V of GQp that generalizes the one introduced by Mazur
for crystalline representations in [45]. This notion is closely related to that oftrianguline representations: indeed, one can define the trianguline continuous twodimensional representation of GQp to be those that admit a refinement. We nextclassify the possible refinements of a trianguline two dimensional representation,
before finally discussing the classification (due to Colmez [27]) of trianguline twodimensional representations themselves.
4.1. Refinements. Let V be a continuous two dimensional representation of GQp
over E.
4.1.1. Definition. A refinement of V is a triple R = (η, α, r), where:(1) η is a continuous character GQp → E × such that V ⊗ η −1 has at least one
Hodge-Sen-Tate weight equal to 0;(2) α ∈ E × ;+(3) r is a non-zero GQp -equivariant E-linear map V ∨ ⊗ η → (E ⊗Qp Bcrys)ϕ=α .
+
Note that if (η, α, r) is a refinement of V , then in particular (E⊗Qp Bcrys)ϕ=α 6= 0,
and so α ∈ OE r {0}. Note also that we may regard the morphism r as a non-zero+element of Dcrys(V ⊗ η −1 )ϕ=α .

4.1.2. Definition. We say that a pair of refinements R = (η, α, r) and R0 =×(η 0 , α0 , r0 ) of V are equivalent if there exists β ∈ OE
and 0 6= x ∈ (E ⊗Zp W (Fp ))ϕ=β00such that r = x · r (and hence such that η = ηur(β −1 ) and α0 = αβ).
Recall that T (E) = (E × )2 (regarded as the diagonal torus of GL2 (E)).
4.1.3. Definition. If R = (η, α, r) is a refinement of V , we define the associated∼abelian Weil group representation to be the map σ(R) : Wpab −→ Q×
p → T (E)defined via the pair of characters (ηur(α), (det V )η −1 ur(α)−1 ).
By construction the determinant of σ(R) coincides with the restriction to Wpabof the determinant of V . Also, σ(R) depends only on R up to equivalence. In fact
we have the following more precise result.
L
4.1.4. Lemma. Suppose that V is not of the form η η for some continuous Evalued character η of GQp . If R and R0 are two refinements of V , then σ(R) = σ(R0 )if and only if R and R0 are equivalent.
Proof. We have already observed that σ(R) = σ(R0 ) if R and R0 are equivalent.
We now prove the converse. Write R = (η, α, r) and R0 = (η 0 , α0 , r0 ). By definition,
σ(R) = σ(R0 ) if and only if ηur(α) = η 0 ur(α0 ). If the latter holds, then (since η and×
η 0 are Galois characters, and hence unitary) we see that β := α0 /α ∈ OE
. Fix 0 6=
ϕ=βx ∈ (E ⊗Zp W (Fp )). Replacing R by the equivalent refinement (ηur(β −1 ), αβ, x·r), we see that we may assume that η = η 0 and α = α0 . If r and r0 are not+linearly dependent over E, then we find that Dcrys(V ⊗η −1 )ϕ=α is two dimensional,−1
and thus that V ⊗ η is a crystalline representation such that Dcrys (V ⊗ η −1 ) =+Dcrys(V ⊗ η −1 )ϕ=α . It follows directly that V ⊗ η −1 is isomorphic to the direct
sum of two copies of some crystalline E-valued character of GQp , and thus that V isisomorphic to the direct sum of two copies of some continuous character of GQp . LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

21

The exceptional case of the preceding lemma is easily analyzed by hand:L
4.1.5. Lemma. If V = η η for some continuous representation η : GQp → E × ,then any equivalence class of refinements of V contains a unique refinementof theLform (η, 1, r) where r is a non-zero homomorphism V ⊗ η −1 = E
E → E =+(E ⊗Qp Bcrys)ϕ=1 . Conversely, any such r gives rise to a refinement (η, 1, r) of V .
Thus the equivalence classes of refinements of V are indexed by the points of P1 (E).
Proof. If (η 0 , α0 , r0 ) is a refinement of V , then η 0 η −1 has Hodge-Sen-Tate weightzero, and also admits an embedding into E ⊗Qp Bcrys . Thus it must be unramified.
Thus (η 0 , α0 , r0 ) is equivalent to a refinement of the form (η, 1, r), which is thenclearly unique in its equivalence class. The remainder of the lemma is clear.

4.1.6. Definition. We say that a continuous two dimensional GQp -representation
V is trianguline if it admits at least one refinement.
The notion of a trianguline GQp -representation (of arbitrary finite dimension)
was introduced by Colmez in [27, Déf. 5.1] (cf. [27, Prop. 5.3] for a proof of theequivalence between the definition given here in the two dimensional case and thedefinition given by Colmez).
4.1.7. Definition. Let V be a continuous two dimensional representation of GQp
over E.
(1) Let Ref(V ) denote the set of equivalence classes of refinements.
(2) For any σ ∈ Homcont (Wp , T(E)), set Ref σ (V ) := {[R] ∈ Ref(V ) | σ(R) =σ}. (Here [R] denotes an equivalence class of refinements, with representative R.)σ
If we fix σ,
the exceptional
Lthen Ref (V ) is either empty or a point, except incase V = η η considered in Lemma 4.1.5, in which case Ref η⊗η (V ) ∼= P1 (E).
σThus we regard Ref (V ) as a projective space over E of dimension −1, 0, or 1.

4.1.8. Twisting refinements. Let ψ : GQp → E × be a continuous character. If
R = (η, α, r) is a refinement of V , then R ⊗ ψ := (η ⊗ ψ, α, r) is a refinement ofV ⊗ ψ. The passage from R to R ⊗ ψ evidently induces a bijection between theset of (equivalence classes of) refinements of V and of V ⊗ ψ. In particular, V istrianguline if and only if V ⊗ ψ is. The formation of Weil group representations isclearly compatible with twisting: i.e. σ(R ⊗ ψ) = σ(R) ⊗ ψ.
4.2. Classifying refinements: the non-potentially semi-stable case. We begin by recalling the following proposition, due to Colmez.
4.2.1. Proposition. Let V be a two dimensional continuous representation of GQp
over E, and suppose that η1 and η2 are two continuous characters of GQp over E
such that Dcrys (V ⊗ η1−1 ) and Dcrys (V ⊗ η2−1 ) are both non-zero. If the weights of
η1 and η2 do not differ by an integer, then V is the direct sum of two continuous
characters.
Proof. This a restatement of [27, Prop. 5.10].



The following corollary of the preceding proposition will provide the key to classifying the possible refinements of two dimensional GQp -representations.

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MATTHEW EMERTON

4.2.2. Corollary. If V is a trianguline two dimensional continuous representationof GQp over E that is not a twist of a representation of Hodge-Tate type, and does
not split as the direct sum of two continuous characters of GQp , then V admits aunique equivalence class of refinements.
Proof. Since V is trianguline, it admits a refinement R = (η, α, r). Replacing V
by V ⊗ η −1 , we may assume that V has a Hodge-Sen-Tate weight of zero, but is
not Hodge-Tate, and that η = 1, so that r is a non-zero GQp -equivariant morphism+r : V ∨ → (E ⊗Qp Bcrys)ϕ=α . If (0, w) are the Hodge-Sen-Tate weights of V , thensince V is not Hodge-Tate, either w = 0 or w 6∈ Z.
Now suppose that R0 = (η 0 , α0 , r0 ) is another refinement of V . Since V has
Hodge-Sen-Tate weights (0, w), we see that the weight of η 0 is either 0 or w. As(1, α, r) and (η 0 , α0 , r0 ) are each refinements of V , we also see that Dcrys (V ) andDcrys (V ⊗ (η 0 )−1 ) are both non-zero. Recalling that w is either zero or non-integral,and that by assumption V does not split as the direct sum of two characters, we
infer from Proposition 4.2.1 that η 0 must in fact be of weight zero.
+ GQp =η 0−1
If we choose 0 6= x ∈ (E ⊗Qp BdR), then x · r0 induces a GQp -equivariant+∨
map V → E ⊗Qp BdR . Thus both r and x · r0 induce elements of DdR (V ). Since
V is assumed to not be Hodge-Tate, it is in particular not de Rham, and thus x · r0
must be a scalar multiple of r. Consequently η 0 must be a crystalline characterof weight zero, which is to say, an unramified character. Thus we see that R0 is
equivalent to R.

4.2.3. Definition. We say that a refinement R = (η, α, r) of V is ultracritical ifθ++the composite r : V ∨ ⊗ η → (E ⊗Qp Bcrys)ϕ=α ⊂ E ⊗Qp Bcrys→ Cp vanishes. (Theterminology is motivated by a comparison of these refinements with the “critical”
refinements introduced in Definition 4.4.7 below; see in particular Definition 4.5.2
and Lemma 4.5.3.)

Clearly the property of being ultracritical depends only on the equivalence classof the refinement R. If R is ultracritical, then V ∨ ⊗ η is necessarily Hodge-Tate,with weights (0, w) for some w > 0. One may then define another refinement
R(−w) := (η · ε−w , αp−w , t−w · r) of V (here t has its usual meaning as an element+of E ⊗Qp Bcrys), whose associated Weil group representation is defined via thecharacters (ηur(α)z −w , det(V )η −1 ur(α)−1 z w ).
4.2.4. Proposition. If V is Hodge-Tate and trianguline, but not potentially semistable, then V admits a unique ultracritical refinement R, and any refinement of V
is equivalent either to R or to R(−w).
Proof. Since V is assumed to be Hodge-Tate but not potentially semi-stable, its
Hodge-Sen-Tate weights are distinct integers, say w1 < w2 . If R = (η, α, r) is any
refinement of V , then since V ⊗ η −1 has zero as a Hodge-Sen-Tate weight, we seethat η is Hodge-Tate of weight either w1 or w2 , hence potentially semi-stable, andthus V ⊗η −1 is also Hodge-Tate but not potentially semi-stable – and so also not deRham [3]. Thus DdR (V ⊗ η −1 ) is at most one dimensional over E. The argumentof [41, Thm. 6.9] shows that in fact DdR (V ⊗ η −1 ) is precisely one dimensional overE, and is supported in filtration degree either 0 (if η is of weight w1 ) or w2 − w1
(if η is of weight w2 ). In particular, if Dcrys (V ⊗ η −1 ) is non-zero, then it is one
dimensional, and supported in filtration degree either 0 or w2 − w1 . Furthermore, ifη and η 0 are two characters of weights w, w0 ∈ {w1 , w2 }, and if Dcrys (V ⊗ η −1 ) 6= 0,

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

23

then Dcrys (V ⊗ η 0−1 ) 6= 0 if and only if η 0 η −1 is crystalline, and hence the product0of an unramified character and εw −w .
Altogether, we see that if V is trianguline, then for each wi (i = 1, 2) we mayfind a character ηi of weight wi , uniquely determined up to multiplication by an+unramified character, such that Dcrys(V ⊗ ηi−1 ) is non-zero (and hence one dimensional over E), and is supported in filtration degree 0 (respectively w2 − w1 ) if i = 1(resp. i = 2), and such that η2 = εw2 −w1 η1 . Thus if we let αi denote the eigenvalue+of ϕ on Dcrys(V ⊗ ηi−1 ), and let ri denote a non-zero element of this space, then we
obtain refinements Ri := (ηi , αi , ri ) of V such that any refinement of V is equivalent
to one of the Ri . Note that R2 is ultracritical, and that R1 = R2 (−(w2 − w1 )).
This proves the lemma.

4.3. Classifying refinements: the potentially semi-stable case. We now consider the case of potentially semi-stable representations.
4.3.1. Proposition. If V is a potentially semi-stable continuous two dimensional
representation of GQp over E, then V is trianguline if and only if the Weil-Deligne
representation σ(V ) contains a one dimensional subrepresentation that can be defined over E. If V is furthermore indecomposable, then the refinements of V are in
bijection with such one dimensional Weil-Deligne subrepresentations of σ(V ).
Proof. Replacing V by a twist if necessary, we may assume that V has Hodge-Tateweights 0 and w ≤ 0. Thus if R = (η, α, r) is any refinement of V , the Hodge-Tateweight of η is either 0 or w. Suppose that w < 0, and that the Hodge-Tate weight ofη equals w. Then the Hodge-Tate weights of V ∨ ⊗ η are w and 0, and the existenceof the non-zero homomorphism r : V ∨ ⊗ η → (E ⊗Qp Bcrys )+ shows that V ∨ ⊗ η isan extension of a character of Hodge-Tate weight 0 by a character of Hodge-Tateweight w. Since V ∨ ⊗ η is potentially semi-stable by assumption, this extension
must split, and thus V ∨ ⊗ η, and so also V itself, must be a direct sum of two
characters. In this case σ(V ) is also a direct sum of two characters, each definable
over E, and V is certainly trianguline. (See Proposition 4.4.5 for a detailed analysisof this case.)
Thus we may suppose for the rest of the proof that V is indecomposable, in whichcase if R = (η, α, r) is any refinement of V , the character η is of Hodge-Tate weightzero. The homomorphism r then corresponds to a non-zero element of Dcrys (V ⊗
η −1 )ϕ=α , or equivalently, a non-zero element of σ(V )Wp =ηur(α),N =0 . Conversely,
any such non-zero element gives rise to a refinement (since our assumption that V
is indecomposable with Hodge-Tate weights w ≤ 0 implies that+
Dcrys
(V ⊗ η −1 )ϕ=α = Dcrys (V ⊗ η −1 )ϕ=α

for any η of Hodge-Tate weight zero and any eigenvalue α). The proposition follows.

4.3.2. Remark. If V is potentially semi-stable, but not potentially crystalline, andindecomposable, then from the proposition we see that V is trianguline, and that
V admits a unique refinement R, up to equivalence. If w1 < w2 are the Hodge-Tateweights of V , then the reader can easily compute that σ(R) = (η, η | |−1 z w1 −w2 )for some character η of Q×p.
4.3.3. Remark. If V is potentially crystalline, trianguline, and indecomposable,
then V admits two equivalence classes of refinements if V is Frobenius semi-simple,

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MATTHEW EMERTON

and one equivalence class otherwise. Let w1 ≤ w2 be the Hodge-Tate weightsof V . In the Frobenius semi-simple case, if R1 and R2 are two inequivalent refinements of V , and if σ(R1 ) = (η, ψ), then the reader can easily check thatσ(R2 ) = (ψz w2 −w1 , ηz w1 −w2 ). If V is not Frobenius semi-simple, and if R is arefinement of V , then σ(R) = (η, ηz w1 −w2 ) for some character η of Q×p.
4.4. Classifying refinements: the reducible case.
4.4.1. Lemma. Let R = (η, α, r) be a refinement of V . Consider the following
conditions:×(1) One has α ∈ OE;
(2) For some (equivalently, any) refinement R0 = (η 0 , α0 , r0 ) equivalent to R,×one has α0 ∈ OE;
(3) The equivalence class of R contains a refinement of the form R0 = (η 0 , 1, r0 ),which is then necessarily unique up to scaling r0 by an element of E × ;(4) The image of r is one dimensional, and R is not ultracritical;(5) For some (equivalently, any) refinement R0 = (η 0 , α0 , r0 ) equivalent to R,
the image of r0 is one dimensional, and R0 is not ultracritical;(6) V is reducible.
Then (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4) ⇐⇒ (5) =⇒ (6).
Proof. It is clear from the definitions that items (1), (2), and (3) are equivalent,+
and also that items (4) and (5) are equivalent. Since (E ⊗Qp Bcrys)ϕ=1 = E is one
dimensional, we see that (3) implies (5). Also, if the image of r is one dimensional,
then V must be reducible, and so (5) implies (6). This image is then a crystalline+
character of GQp which embeds into (E ⊗Qp Bcrys)ϕ=α , and so is of non-negative
Hodge-Tate weight. If R is not ultracritical, then it must in fact be of Hodge-Tate×weight zero, and hence unramified. Thus α ∈ OE
, and we see that (4) implies (1).

4.4.2. Definition. We say that a refinement R = (η, α, r) of V is ordinary if itsatisfies the equivalent conditions (1), (2), (3), (4), and (5) of the preceding lemma.
The preceding lemma shows that any refinement equivalent to an ordinary refinement of V is also ordinary, and that V admits an ordinary refinement only if V
is reducible. More precisely:
4.4.3. Lemma. If R is an ordinary refinement of V , and if σ(R) = (η, ψ) is theassociated Weil group representation, then each of η and ψ is a unitary characterof Q×p (and thus extends uniquely to a continuous character of GQp ), and V is anextension of ψ by η.
Proof. Consider the unique refinement of the form (η, 1, r) in the equivalence classof R. The map r then realizes E (with trivial Galois action) as a quotient ofV ∨ ⊗ η, and thus η embeds as a one dimensional subrepresentation of V . Sinceψ = det(V )η −1 by definition, the lemma follows.

We now prove a series of results that completely classify the refinements ofreducible two dimensional continuous representations V of GQp . We begin with thecase when V is reducible but indecomposable.
4.4.4. Proposition. If V may be written as a non-split extension
0 → η → V → ψ → 0,

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

25

where η and ψ are two continuous characters of GQp , then V admits a uniqueequivalence class of ordinary refinements (and so in particular is trianguline). Furthermore, if we we let w denote the weight of ψη −1 , then:(1) If w 6∈ Z (so that V is not of Hodge-Tate type up to twist), then V admits just one equivalence class of refinements, namely the class of ordinaryrefinements;(2) If w ∈ Z>0 (so that V is of Hodge-Tate type, but not potentially semi-stable,
up to a twist), then V admits two equivalence classes of refinements, namely
the class of ordinary refinements, and the class of ultracritical refinements;(3) If w ∈ Z<0 (so that V is potentially semi-stable up to a twist), then:(a) If V is furthermore potentially crystalline up to a twist, then V
admits two equivalence classes of refinements, namely the class of ordinaryrefinements, and one class of non-ordinary refinements;(b) If V is not potentially crystalline up to a twist, then V admits
a unique equivalence class of refinements, namely the ordinary equivalenceclass;(4) If w = 0 (so that either V is unramified and Frobenius non-semi-simple, up
to a twist, or else V is not of Hodge-Tate type up to a twist), then V admits
a unique equivalence class of refinements, namely the ordinary equivalenceclass.
Proof. The fact that V admits a unique equivalence class of ordinary refinements
follows immediately from Lemma 4.4.3 and our assumption on V , and claims (1)and (2) are direct consequences of Propositions 4.2.2 and 4.2.4 respectively.
To prove claim (3), we may replace V by a twist if necessary, and so assume that
V is potentially semi-stable. The claim then follows from Proposition 4.3.1.
The non-Hodge-Tate case of claim (4) again follows from Proposition 4.2.2. Finally, if V is a twist of an unramified but non-semi-simple representation, then,replacing V by a twist, if necessary, we may assume that in fact V is an unramified non-split extension 0 → η → V → η → 0. (The two characters appearing in
V are necessarily equal, since V is unramified but non-split.) Now suppose thatR = (η 0 , α, r) is a refinement of V . Since V has both Hodge-Tate weights equal tozero, the same must be true of η 0 . Similarly, since η is a crystalline character, thesame must be true of η 0 . Thus η 0 is in fact unramified, and so we may replace R
by an equivalent refinement R0 of the form R0 = (1, α0 , r0 ). The map r0 is a GQp 0+
equivariant non-zero homomorphism V ∨ → (E ⊗Qp Bcrys)ϕ=α . Since V (and soV ∨ ) is not semi-simple, the map r cannot be an embedding. Thus R0 is necessarily
an ordinary refinement of V , as claimed.

We now consider the case when V is the direct sum of distinct characters.
L
4.4.5. Proposition. Suppose that V = η1 η2 for a pair of distinct characters η1 6=
η2 of GQp . Then V admits exactly two equivalence classes of ordinary refinements.
Furthermore:(1) If the Hodge-Sen-Tate weight of η1 η2−1 is either zero or non-integral, thenV admits exactly two equivalence classes of refinements, namely the twoordinary equivalence classes;(2) If the Hodge-Sen-Tate weight of η1 η2−1 is a non-zero integer, then V admits
three equivalence classes of refinements, namely the two ordinary equivalence classes and a unique equivalence class of ultracritical refinements.

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MATTHEW EMERTON

Proof. Lemma 4.4.3 shows that V admits exactly two equivalence classes of ordinaryrefinements. Suppose now that R = (η, α, r) is a non-ordinary refinement
L of V .
Then r is a non-zero GQp -equivariant morphism r : V ∨ ⊗ η = ηη1−1 ηη2−1 →+(E ⊗Qp Bcrys)φ=α . Since η1 6= η2 , we see that r must vanish on one of the direct
summands in its source (and on exactly one, since it is non-zero). Since R is nonordinary, Lemma 4.4.1 (4) implies that it must be ultracritical. In particular η1 η2−1
must be of non-zero integral Hodge-Sen-Tate weight. Interchanging η1 and η2 if−1necessary, we may in fact assume that η2 ηL
has Hodge-Tate weight w > 0. We
1now show (under this assumption) that η1 η2 does in fact admit an ultracriticalrefinement, and that this refinement is unique up to equivalence.
If (η, α, r) is an ultracritical refinement, then we see that ηη1−1 (resp. ηη2−1 ) must
be of weight w (resp. weight 0), and that r must induce an embedding ηη1−1 →+E ⊗Qp Bcrys(and vanish on ηη2−1 ). Thus η is a character with the property thatηη1−1 is crystalline of weight w. Clearly such an η exists, and any two differ bymultiplication byLan unramified character. Thus we can construct an ultracriticalrefinement of η1 η2 , and any two such are equivalent.

4.4.6. Remark. If V is the direct sum of two copies of the same character of GQp ,then the refinements of V are classified by Lemma 4.1.5.
We conclude this section by introducing some terminology.
4.4.7. Definition. If V is a reducible two dimensional continuous representation of
GQp over E, then we say that a refinement R of V is critical if it is not ordinary.
(The terminology “critical” is used in analogy with the terminology “critical slope”
as it is employed in the theory of modular forms.8)
Any refinement equivalent to a critical refinement is again critical. The preceding
results show that a reducible representation V admits at most one equivalence classof critical refinements. In order for V to admit such a class, V should be HodgeTate with distinct Hodge-Tate weights up to a twist, and either not potentiallysemi-stable up to a twist, or else potentially crystalline up to a twist.
4.5. The classification of trianguline representations. Colmez has classified
the trianguline representations [27, Thm. 0.5]. We will recall his result here, butwill rephrase it in the language of refinements. Note that if R is any refinement of atrianguline two dimensional representation V , and if we write σ(R) = (η, ψ), thenηψ is unitary, while | η(p) |≤ 1 (and hence in fact | η(p) |< 1 unless both η and ψ
are unitary). The cited result of Colmez implies in particular that any continuous
representation σ ∈ Homcont (Wp , T(E)) satisfying these conditions is of the formσ(R) for some refinement R of some trianguline two dimensional representation
V . Furthermore, it allows us to classify the representations V that give rise to aparticular σ.
We begin with some definitions.
4.5.1. Definition. Let Homcont (Wp , T(E))+ denote the set of maps σ = (η, ψ) ∈Homcont (Wp , T(E)) such that ηψ is unitary and | η(p) |≤ 1.
8A classical finite slope p-stablized Hecke eigenform of weight k ≥ 2 over Q is said to be ofpcritical slope if its slope is equal to k − 1 (its largest possible value). Any such form is the “eviltwin” of an ordinary p-stabilized eigenform; see for example the discussion of [14, §4.1], as well as
the proof of Theorem 7.6.1 (3) below.

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

27

4.5.2. Definition.
(1) We say that σ = (η, ψ) ∈ Homcont (Wp , T(E))+ is unitary if η and ψ are unitary.
(2) We say that σ = (η, ψ) ∈ Homcont (Wp , T(E))+ is critical if ηψ −1 is ofintegral Hodge-Tate weight w > 0, and if σ(−w) := (ηz −w , ψz w ) again liesin Homcont (Wp , T(E))+ , and is furthermore unitary.
(3) We say that σ = (η, ψ) ∈ Homcont (Wp , T(E))+ is ultracritical if ηψ −1 is ofintegral Hodge-Tate weight w > 0, and if σ(−w) := (ηz −w , ψz w ) again liesin Homcont (Wp , T(E))+ , and is not unitary.
This terminology is almost compatible with the corresponding terminology forrefinements.
4.5.3. Lemma. Let V be a trianguline continuous two dimensional representationof GQp over E, and R a refinement of V .
(1) σ(R) is ultracritical if and only if V is irreducible and R is ultracritical;(2) σ(R) is critical if and only if V is reducible and R is a critical refinementof V .
Proof. The reader can check this by going through the classification of all possible
refinements given in the preceding sections.

4.5.4. Theorem. Let σ = (η, ψ) ∈ Homcont (Wp , T(E))+ , and suppose that σ isneither unitary nor critical. Then there is a trianguline two dimensional continuous
representation V of GQp over E admitting a refinement R such that σ(R) = σ, andany such V is irreducible. Furthermore:(1) If ηψ −1 is not of the form εz n for some n > 0, then V is unique up toisomorphism.
(2) If ηψ −1 = εz n for some n > 0, then the set of isomorphism classes ofsuch V can be indexed by the elements L ∈ P1 (E) in such a way that, if
L ∈ E, then the members of the corresponding isomorphism class are twistsof a semi-stable representation of Hodge-Tate weights (0, −1 − n) with Linvariant equal to L, while if L = ∞, then the members of the corresponding
isomorphism class are twists of a non-generic crystalline representation of
Hodge-Tate weights (0, −1 − n).
Proof. The results of Subsection 4.4 show that the Weil group representation attached to any refinement of a reducible representation is either unitary or critical.
Thus if V exists it must be irreducible. Suppose first that σ is not ultracritical. Fix
L = ∞ if σ does not satisfy the condition of (2); otherwise let L be any element of
P1 (E). The pair s = (σ, L) is then an element of the space Sirr considered in [27],and the corresponding GQp -representation V (s) (see [27, p. 5]) admits a refinement
R with σ(R) = σ (see the proof of [27, Prop. 5.3]). Conversely, it follows from [27,Prop. 5.3] that if V admits a refinement R with σ(R) = σ, then V ∼= V (s) for somes ∈ Sirr lying over σ. This proves the theorem for non-ultracritical σ.
If σ is ultracritical, then applying the theorem to σ(−w) we obtain a representation V admitting a refinement R such that σ(R) = σ(−w). It follows that V
is Hodge-Tate up to twist, but not potentially semi-stable up to twist, and thus
by Proposition 4.2.4 we see that V admits an ultracritical refinement R0 such thatσ(R0 ) = σ. This completes the proof of the theorem in general.

The case when σ = (η, ψ) ∈ Homcont (Wp , T(E))+ is either unitary or critical isstraightforward. When σ is unitary, regarding η and ψ as Galois characters, we have

28

MATTHEW EMERTON

L
seen that the direct sum V = η ψ admits a refinement R such that σ(R)L = σ.
ψz w
If σ is critical, and w is the weight of ηψ −1 , then we have seen that ηz −w
admits a critical (in fact ultracritical) refinement R such that σ(R) = σ.
The following results describe the possible indecomposable V admitting such arefinement R for which σ(R) = σ. (Note that Lemmas 4.4.1 and 4.5.3 imply thatany such V is reducible.)
4.5.5. Proposition. If σ = (η, ψ) is a unitary element of Homcont (Wp , T(E))+ ,then there exists an indecomposable trianguline two dimensional continuous representation V of GQp over E admitting a refinement R such that σ(R) = σ. Furthermore:(1) If ηψ −1 6= 1, ε, then V is unique up to isomorphism.
(2) If ηψ −1 = ε, then the set of isomorphism classes of such V can be indexedby the elements L ∈ P1 (E) in such a way that, if L ∈ E, then the membersof the corresponding isomorphism class are twists of a semi-stable representation of Hodge-Tate weights (0, −1) with L-invariant equal to L, while if
L = ∞, then the members of the corresponding isomorphism class are twistsof a non-generic crystalline representation of Hodge-Tate weights (0, −1).
(3) If η = ψ, then the set of isomorphism classes of such V can be indexed by
the elements of P1 (E). There is a unique (up to isomorphism) such V thatis Hodge-Tate up to twist, which is then in fact crystalline (indeed, evenunramified) up to twist.
∼

Proof. There is an isomorphism Ext1GQp (ψ, η) −→ H 1 (GQp , ηψ −1 ) (where Ext1 iscomputed in the category of continuous GQp -representations, and H 1 indicatescontinuous Galois cohomology). Using Tate local duality and the local Euler characteristic formula (for example) one finds that H 1 (GQp , ηψ −1 ) is one-dimensional
unless ηψ −1 = 1 or ε; in these latter two cases it is two-dimensional. It follows thatin the situation of (1), any two non-trivial extensions of ψ by η are isomorphic as
GQp -representations. This proves (1).
The elements of H 1 (GQp , ε), and hence the corresponding extensions of 1 by ε,are easily computed via Kummer theory. We recall the results of this well-known
computation: one finds that any extension of 1 by ε is semi-stable, and that thenon-trivial extensions are classified (up to isomorphism as GQp -representations) by
their L-invariants (with L = ∞ corresponding to the unique isomorphism class of
crystalline extensions). This proves (2).
The element of H 1 (GQp , 1), and hence the corresponding extensions of 1 by itself,
are easily computed via local class field theory. Since this computation is also wellknown, we again content ourselves with recalling the outcome: one finds that (upto isomorphism as GQp -representations) there is a unique non-trivial extension of
1 by itself which is Hodge-Tate, namely the non-trivial unramified extension. This
proves (3).

4.5.6. Proposition. If σ = (η, ψ) is a critical element of Homcont (Wp , T(E))+ ,then there exists exactly two isomorphism classes of indecomposable trianguline twodimensional continuous representations V of GQp over E admitting a refinement R
such that σ(R) = σ. There is a unique such isomorphism class of V whose members
are furthermore potentially semi-stable up to a twist (and the members of this classare then in fact potentially crystalline, up to a twist).

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

29

Proof. Applying Proposition 4.5.5 to σ 0 := (ψz w , ηz −w ), we may find V that is
potentially crystalline up to a twist and admits a critical refinement R with σ(R) =σ. Furthermore we see that V is unique up to isomorphism. (Note that when σ 0
falls into case (2) of that result, only the representation corresponding to L = ∞admits a critical refinement.)
Applying Proposition 4.5.5 to σ(−w) := (ηz −w , ψz w ) yields a representation V
that admits an ultracritical refinement R with σ(R) = σ. Again we see that V
is unique up to isomorphism among the indecomposable representations admitting
such a refinement that are not potentially semi-stable up to a twist.

4.5.7. Terminology. We say that a trianguline two dimensional continuous GQp representation “admits an L-invariant” if and only if it is a twist of either a semistable, but non-crystalline, representation, or of an indecomposable non-genericcrystalline representation.
The trianguline representations that admit an L-invariant are precisely the representations that are classified by Theorem 4.5.4 (2) and Proposition 4.5.5 (2).
5. Some invariants of admissible unitary GL2 (Qp )-representations
The problem of classifying topologically irreducible admissible unitary GL2 (Qp )representations seems currently unapproachable in general, but there are two situations in which we can make some progress: namely if the admissible unitary
representation B being classified satisfies one of the following conditions:(6) The subspace Blalg of locally algebraic vectors in B is non-zero;(7) The Jacquet module JP(Qp ) (Ban ) is non-zero.
In this section we discuss some results and conjectures towards the classification of
topologically irreducible admissible unitary representations satisfying one of these
conditions.
We first discuss the problem of classifying such representations B satisfying (6).
The non-trivial results in this case are due to Colmez [26] and Berger and Breuil[8], and all progress has been motivated by Breuil’s initial postulation of the local
p-adic Langlands correspondence in the potentially semi-stable case [12]. After abrief review of some properties of the Jacquet module functor JP(Qp ) , we then turnto the problem of classifying those B satisfying (7). In addition to the theory of theJacquet module, our discussion relies heavily on the results of Colmez describing
the universal unitary completions of certain locally analytic induced representations[27].
While the two problems that we discuss are in principle independent of one another, since B can satisfy either one of conditions (6) or (7) without satisfying theother, all the results obtained so far related to the classification of representations
satisfying (6) apply only to representations that also satisfy (7). Nevertheless, evenin the cases where no progress has been made, the local p-adic Langlands correspondence suggests the form that the classification should take. (See Conjecture 5.1.19
below.) On the other hand, the representations that satisfy (7) but not (6) can
be completely classified, a result that can also be viewed as a manifestation of thelocal p-adic Langlands correspondence. (See Remark 5.3.13.)

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MATTHEW EMERTON

5.1. Admissible unitary representations of GL2 (Qp ) containing locally algebraic vectors. Suppose that B is a topologically irreducible admissible unitary
GL2 (Qp )-representation over E for which Blalg 6= 0. Then there exists an irreducible
admissible smooth representation U of GL2 (Qp ) over E, an irreducible algebraic
representation W of GL2 (Qp ) over E, and a GL2 (Qp )-equivariant embedding
π := U ⊗E W → B.
In particular, π (when regarded as a topological GL2 (Qp ) representation by being equipped with is finest locally convex topology) admits a non-zero universal
unitary completion πb (in the sense of [34]), and πb maps continuously and GL2 (Qp )equivariantly into B with dense image (since B is topologically irreducible).
5.1.1. Question. In the above situation, is the map πb → B necessarily surjective?
It seems reasonable to hope that the answer to this question is “yes”. (For example, this is the case if πb is admissible. 9 Proposition 5.1.10 below gives an example
where this is again the case, although π
b is not admissible.) Assuming that this isso, we see that B is a topologically irreducible admissible quotient of πb, and thatthe problem of classifying those B for which Blalg 6= 0 reduces to the following two
problems: (a) classify those irreducible admissible locally algebraic representations
π of GL2 (Qp ) for which πb 6= 0; (b) for such π, classify the topologically irreducible
admissible quotients of πb. Problem (a) is close to being solved in its entirety, andsubstantial progress has been made on problem (b). In order to discuss the situation in more detail, as above we write π = U ⊗E W where U is an irreduciblesmooth representation of GL2 (Qp ) and W is an irreducible algebraic representation of GL2 (Qp ). We may then write W = Symw2 −w1 −1 E 2 ⊗ detw1 +1 for someuniquely determined integers w1 < w2 .
We first note the following lemma.
5.1.2. Lemma. If B is a finite dimensional irreducible unitary representation of
SL2 (Qp ) then B is the trivial representation of SL2 (Qp ).
Proof. Any finite dimensional continuous representation of SL2 (Qp ) is necessarily
locally analytic [52, LG §5.9, Thm. 2], and hence is locally algebraic, since it is finite
dimensional. It follows from [31, Prop. 4.2.8] that we may factor B as a tensor product B ∼= U ⊗ W where U is a finite dimensional irreducible smooth representationof SL2 (Qp ) and W is an irreducible algebraic representation of SL2 (Qp ). It is wellknown that U is then necessarily trivial (see for example [40, Prop. 2.7]), and thus
we see that the irreducible algebraic representation W admits an SL2 (Qp )-invariant
norm. But this implies that W is also trivial, whence the lemma.
This lemma has the following immediate corollary.
9This is a consequence of the following result: Theorem. If φ : U → V is a G-equivariant
continuous morphism between two Banach space representations over E of the compact p-adic
locally analytic group G, and if U is admissible, then φ has closed image. This is well-known if V
is assumed to be admissible (cf. [31, Prop. 6.2.9]) but in fact holds true without that assumption,
as Breuil observed in the course of proving [12, Prop. 4.4.4 (v)]. Here is a sketch of the proof:
Replacing V with the closure of the image of φ, we may assume that φ has dense image, sothat the topological dual map φ0 : V 0 → U 0 is injective. Since U 0 is finitely generated overE[[G]] := E ⊗OE OE [[G]] (by our assumption that U is admissible), the same is true of V 0 ,since OE [[G]], and hence E[[G]], is Noetherian [44, V.2.2.4]. Thus V is also admissible (cf. [31,Def. 6.2.1]), and so φ has closed image, as claimed.

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

31

5.1.3. Corollary. The only finite dimensional irreducible unitary representationsof GL2 (Qp ) are the characters χ ◦ det (where χ is a unitary character of Q×p ).
From this corollary we see that in the above discussion we may restrict ourattention to the case when π (or equivalently U ) is infinite dimensional. Thus weassume this from now on. We also assume that π has a unitary central character,since this is one obvious necessary condition for πb to be non-zero.
5.1.4. The principal series case. Suppose that U is principal series (and infinite
GL (Q )
dimensional), so that we may write U = (IndP(Q2 ) p χ1 ⊗ χ2 )sm with χ1 χ−1
6=2
p1, | |−2 . The assumption that U has unitary central character corresponds to thecondition| χ1 (p)χ2 (p)pw1 +w2 +1 | = 1.

(8)

In this case, the theory of Jacquet modules provides an additional necessary condition for πb to be non-zero, namely| pw2 χ1 (p) | , | pw2 +1 χ2 (p) | ≤ 1(9)(see Lemma 5.2.4 below).
The following conjecture is due to Breuil. (See Remark 3.3.4 for motivation.)
5.1.5. Conjecture. If conditions (8) and (9) hold then πb is non-zero, and is a
topologically irreducible admissible unitary GL2 (Qp )-representation.
In particular, in this case we expect that the only topologically irreducible admissible unitary quotient of πb is πb itself.
5.1.6. Theorem. Conjecture 5.1.5 is true provided that χ1 6= χ2 | |−1 .
Proof. We may describe π as the locally algebraic induction
GL (Q )GL (Q )
π = (IndP(Q2 ) p χ1 z w2 ⊗ χ2 z w1 +1 )lalg ∼= (IndP(Q2 ) p χ2 ε−1 z w2 +1 ⊗ χ1 εz w1 )lalg ,ppwhere the isomorphism is provided by the theory of intertwining operators. If
either | pw2 χ1 (p) | = 1 or | pw2 +1 χ2 (p) | = 1, then π is equal to the locally algebraic
induction of a unitary character, in which case it is straightforward to check thatπb is equal to the corresponding continuous induction, and that this continuous
induction is topologically irreducible and admissible unitary. (See [14, Prop. 2.2.1],
and also the first paragraph of the proof of Proposition 5.3.4 below.) On the other
hand, if neither of these conditions hold, and if χ1 6= χ2 | |−1 , then the claim of
the theorem follows from the main result of [8].

5.1.7. The special case. Let us suppose now that U is special, say U = χ ◦ det ⊗St
for some character χ. The condition that π have a unitary central character thenbecomes(10)| χ(p)p(w1 +w2 +1)/2 | = 1.

5.1.8. Theorem. Suppose that condition (10) holds.
(1) If furthermore W is one dimensional (i.e. w2 = w1 +1), then πb is non-zero,
topologically irreducible, and admissible unitary.
b is non-zero, is not(2) If W is not one dimensional (i.e. w2 > w1 + 1) then π
topologically irreducible, and is not admissible unitary.

32

MATTHEW EMERTON

Proof. In case (1) we see that π is the twist of St by a unitary character, and theclaim of the theorem follows from the particular case π = St, which is proved in[12, Lem. 4.5.1].
In case (2), the fact that πb is non-zero is due to Teitelbaum [60] (resp. GrosseKlönne [38]) when W is odd (resp. even) dimensional. The fact that πb is notadmissible is proved in [12, Prop. 4.6.5]. The fact that πb is not topologically irreducible follows from the explicit construction of the series of non-zero topologicallyirreducible admissible unitary quotients described below.
In the situation of case (2) of the preceding Theorem, Breuil has described a
specific series of quotients of πb, depending on a parameter L ∈ P1 (E), which heconjectured, and Colmez has proved, to be non-zero, topologically irreducible, andadmissible unitary. We briefly recall their construction.
First note that it is no loss of generality to replace π by a unitary twist (since
twisting by a unitary character commutes with passage to the universal unitarycompletion). Thus we may assume that w2 = 0, and so write w = w1 < −1. In the√cases when w is even we furthermore extend E if necessary so that p ∈ E. Then,further twisting π if necessary, we may reduce to the case∨
π = St⊗ | det |(1+w)/2 ⊗ Sym−1−w E 2 .
Given L ∈ P1 (E), choose (u, v) ∈ E 2 \ {(0, 0)} so that u/v = L, and define thetwo dimensional representation σ(L) of P(Qp ) via


a !
a
1 u ordp ( ) + v logp ( )
a 0 
:=
σ(L)
d
d ,
c d
0
1
where ordp denotes the p-adic valuation (normalized so that ordp (p) = 1) andlogp denotes the Iwasawa p-adic logarithm (so logp (p) = 0). Note that up toisomorphism, σ(L) depends only on the value of u/v = L.
Let χw denote the characterχw :=| |(1+w)/2 ⊗ | |(−1+w)/2 z w ε =| |(1+w)/2 ⊗ | |(1+w)/2 z 1+wof P(Qp ), and consider the locally analytic induction
GL (Q )
(11)
IndP(Q2 ) p σ(L) ⊗ χw an .
p

Since σ(L) is an extension of the trivial character of P(Qp ) by itself, we see that (11)sits in a short exact sequence(12)GL (Q )GL (Q )
0 −→ IndP(Q2 ) p | |(1+w)/2 ⊗ | |(1+w)/2 z 1+w an −→ IndP(Q2 ) p σ(L) ⊗ χw anpp
pr
GL2 (Qp )(1+w)/2(1+w)/2 1+w−→ IndP(Q ) | |
⊗| |
z
−→ 0.
an
p

GL (Q )The representation IndP(Q2 ) p | |(1+w)/2 ⊗ | |(1+w)/2 z 1+w an contains
p
∨
(13)
| det |(1+w)/2 ⊗ Sym−1−w E 2
as a subrepresentation. Following [12], we defineΣ(1 − w, L) =pr−1 (| det |(1+w)/2 ⊗ Sym−1−w E 2

∨)/ | det |(1+w)/2 ⊗ Sym−1−w E 2∨LOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

33(i.e the quotient by the copy of (13) sitting in the first term of (12) of the preimage
under the projection pr of the copy of (13) sitting in the third term); then Σ(1−w, L)
is a locally analytic representation of GL2 (Qp ) that sits in an exact sequence
(14)
∨

GL (Q )
0 → IndP(Q2 ) p | |(1+w)/2 ⊗ | |(1+w)/2 z 1+w an / | det |(1+w)/2 ⊗ Symw−1 E 2p∨→ Σ(1 − w, L) →| det |(1+w)/2 ⊗ Sym−1−w E 2 → 0.
(Our notation follows that of [12], possibly up to a unitary twist.) Since σ(L) is a
non-split extension of the trivial character by itself, one easily sees that the exact
sequence containing Σ(1 − w, L) is also non-split.
The representation Σ(1 − w, L) has the topological Jordan-Hölder series∨(15) 0 ⊂ π = St⊗ | det |(1+w)/2 ⊗ Sym−1−w E 2∨GL (Q )
= IndP(Q2 ) p | |(1+w)/2 ⊗ | |(1+w)/2 z 1+w lalg / | det |(1+w)/2 ⊗ Sym−1−w E 2p∨GL (Q )
⊂ IndP(Q2 ) p | |(1+w)/2 ⊗ | |(1+w)/2 z 1+w an / | det |(1+w)/2 ⊗ Sym−1−w E 2
p⊂ Σ(1 − w, L),related to which we have the following easy lemma.
5.1.9. Lemma. Each of the inclusions after the first in (15) is non-split, and hasa cokernel whose universal unitary completion vanishes.
Proof. That the second inclusion is non-split is standard. Its cokernel is isomorphictoGL (Q )
IndP(Q2 ) p | |(1+w)/2 z w ⊗ | |(1+w)/2 z an ,pwhose universal unitary completion vanishes by Proposition 5.2.1 and Lemma 5.2.4
below.
We observed above that the short exact sequence (14) is non-split, which implies that the third inclusion is non-split. Its cokernel is isomorphic to | |(1+w)/2
∨⊗ Sym−1−w E 2 , which is finite dimensional but not a character (since w < −1),and so has vanishing universal unitary completion by Corollary 5.1.3.

It follows from the preceding lemma that the maps of universal unitary completions induced by the second and third inclusions have dense image. In fact we havethe following less elementary proposition.
5.1.10. Proposition. The second inclusion of (15) induces an isomorphism of universal unitary completions, while the third inclusion induces a surjection of universal unitary completions.
Proof. The first claim follows from [34, Prop. 2.5]. (See also Proposition 5.3.8
below.) The third claim is proved in [13, Cor. 3.3.4] in the case L =6 ∞. In the casewhen L = ∞, it is easily checked (see the remarks below).

5.1.11. Definition. For each L ∈ P1 (E), let B(1 − w, L) denote the universal
unitary completion of Σ(1 − w, L).
The preceding proposition shows that B(1 − w, L) is a quotient of πb.

34

MATTHEW EMERTON

5.1.12. Remark. The completion B(1 − w, L) may be described explicitly as asuitable quotient of a certain space of functions on Qp of class C (−1−w)/2 ; see [13,Cor. 3.3.4] for the case L =6 ∞.
5.1.13. Theorem. Each of the GL2 (Qp )-representations B(1 − w, L) is non-zero,
topologically irreducible, and admissible unitary. Furthermore, the representations
corresponding to distinct values of L are non-isomorphic.
Proof. In the case when L 6= ∞, the statement of the theorem is due to Colmez[26]. In the case when L = ∞, it follows from the results of Berger and Breuil [8],taking into account the isomorphism (16) below.
Thus the spaces B(1 − w, L) yield a series of topologically irreducible admissible
unitary quotients of πb indexed by the elements L ∈ P1 (E). This suggests thefollowing question.
5.1.14. Question. Do the spaces B(1 − w, L) (for L ∈ P1 (E)) exhaust the topologically irreducible admissible unitary quotients of πb?
One can at least say that no other such quotients have been described in theliterature to date.
5.1.15. Remark. The case when L = ∞ is a little different from the case when L
is finite, since the representation σ(∞) is smooth. In fact, one can check that there
is an isomorphism∼GL (Q )(16)
Σ(1 − w, ∞) −→ IndP(Q2 ) p | |(−1+w)/2 ⊗ | |(3+w)/2 z 1+w an .
pThus in this case the space of locally algebraic vectors in Σ(1 − w, ∞) is larger thanπ; namely one has an isomorphism∨ ∼GL (Q )
IndP(Q2 ) p | |−1 ⊗ | | sm ⊗ | det |(1+w)/2 ⊗ Sym−1−w E 2 −→ Σ(1 − w, ∞)lalg .
p5.1.16. Remark. In the case when w is odd and ≥ −p, Breuil and Mézard haveconstructed a certain torsion free sheaf of OE -modules on the p-adic upper halfplane whose space of global sections is the unit ball in the topological dual ofB(1 − w, L) [15]. They have used this description of B(1 − w, L) to compute themod $ representation B(1 − w, L).
5.1.17. The cuspidal case. We now assume that π = U ⊗ W with U irreducible
and cuspidal and W irreducible algebraic. We continue to suppose that the central
character of π is unitary. In this context we have the following easy proposition.
5.1.18. Proposition. The universal unitary completion πb is non-zero, and is notadmissible.
Proof. The non-vanishing of πb follows directly from the fact that the matrix coefficients of U are compactly supported modulo the centre (together with the assumption that the central character of π is unitary); see [34, Prop. 1.18]. It is then easily
checked (by using the construction of U as a compact induction, for example) thatin fact π
b is not admissible.
In this case nothing general seems to be known about the possible topologicallyirreducible admissible unitary quotients of πb. One can use global considerations
to show that π does admit some non-zero admissible unitary completions, which

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

35

one presumes (but cannot prove as of yet) are topologically irreducible quotientsof πb. (See Remark 7.7.16 below, and recall that we expect Question 5.1.1 to havea positive answer in general.)Motivated by the local p-adic Langlands conjecture, Breuil has made the following conjecture (cf. Remark 3.3.4 above and Remark 6.1.9 below).
5.1.19. Conjecture. The universal unitary completion πb admits a family of nonzero, mutually non-isomorphic, topologically irreducible, admissible unitary quotients, indexed by the elements of P1 (E).
An optimist might guess that in fact this conjectural series of quotients exhausts
all topologically irreducible, admissible unitary quotients of πb.
5.2. Jacquet modules. Recall that if U is any topological E-vector space ofcompact type equipped with a locally analytic GL2 (Qp )-representation, then theJacquet module JP(Qp ) (U ) is a certain locally analytic representation of T(Qp ) overE functorially associated to U [32]. We do not recall the definition here (see thediscussion following [32, Prop. 0.3]), but do recall that JP(Qp ) is additive and left
exact. In the remainder of this subsection we describe the additional properties ofthe functor JP(Qp ) that we will require.
The following result describes the Jacquet modules of induced representations.
5.2.1. Proposition. Fix χ1 ⊗ χ2 ∈ Tb(Qp ).
w−1(1) If χ1 χ2−1 is of integral Hodge-Tate weight w ≥ 0, and if χ1 χ−1,
2 6= z | |thenMGL (Q )
JP(Qp ) (IndP(Q2 ) p χ1 ⊗ χ2 )lalg ∼(χ2 z w ⊗ χ1 z −w ).
= (χ1 | | ⊗χ2 | |−1 )
p

−1
w
−1
(2) If χ1 χ−1,
2 is of integral Hodge-Tate weight w ≥ 0, and if χ1 χ2 = z | |GL2 (Qp )then JP(Qp ) (IndP(Q ) χ1 ⊗ χ2 )lalg is a non-split extension of the charp

acter χ1 | | ⊗χ2 | |−1 = χ2 z w ⊗ χ1 z −w by itself.
(3) If χ1 χ−1
2 is of integral Hodge-Tate weight w ≥ 0 then the inclusion
GL (Q )GL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 lalg ⊂ IndP(Q2 ) p χ1 ⊗ χ2 anpp

induces an isomorphism ∼GL (Q )GL (Q )
JP(Qp ) (IndP(Q2 ) p χ1 ⊗ χ2 )lalg −→ JP(Qp ) (IndP(Q2 ) p χ1 ⊗ χ2 )an .
pp(4) If χ1 χ−1
2 is not of non-negative integral Hodge-Tate weight thenGL (Q )
JP(Qp ) (IndP(Q2 ) p χ1 ⊗ χ2 )an ∼= χ1 | | ⊗χ2 | |−1 .
pProof. Claims (1) and (2) follow from [32, Prop. 4.3.6] and the well-known structureof the Jacquet modules of smooth induced representations of GL2 (Qp ) (cf. [19, Lem.
7.1.1 (a)]). The proofs of claims (3) and (4) (in a much more general context) willappear in [35] (and are in any case straightforward computations working from thedefinition of JP(Qp ) ).

The following corollary is an immediate consequence of the preceding proposition.

36

MATTHEW EMERTON

b
5.2.2. Corollary. For every χ1 ⊗ χ2 ∈ T(E)there is an embedding of T(E)representationsGL (Q )
χ1 | | ⊗χ2 | |−1 → JP(Qp ) (IndP(Q2 ) p χ1 ⊗ χ2 )an ,puniquely determined up to multiplication by a non-zero scalar.
5.2.3. Definition. Fix a compact type locally analytic GL2 (Qp )-representation U
defined over E.
(1) Define Exp(U ) to be the set of one dimensional T(Qp )-invariant subspacesof JP(Qp ) (U ).
b(2) For any line L ∈ Exp(U ), write χ(L) ∈ T(E)to denote the character via
which T(Qp ) acts on L.
b(3) For any character χ ∈ T(E),write Expχ (U ) := {L ∈ Exp(U ) | χ(L) = χ}.
b
If we fix a character χ ∈ T(E),then Expχ (U ) has the structure of a projective
space (possibly of infinite dimension, or possibly of dimension −1, i.e. empty),
χ
namely it is the projectivization of the χ-eigenspace JP(Q(U ) in JP(Qp ) (U ). Thusp)b
Exp(U ) is a disjoint union of projective spaces, indexed by the characters χ ∈ T(E).
b
Recall the identification T(E) = Homcont (Wp , T(E)) discussed in Subsection 1.3.
bb
We let T(E)+ denote the subset of T(E) that corresponds to Homcont (Wp , T(E))+(as defined in Definition 4.5.1) under this identification, and we will use this idenbtification to apply the terminology of Definition 4.5.2 to elements of T(E)+.
The following result imposes a limitation on the exponents that can appear inJP(Qp ) (U ) in the cases of interest to us.
5.2.4. Lemma. If B is an E-Banach space equipped with a unitary representationχof GL2 (Qp ) for which Ban is of compact type, and if JP(Q
(Ban ) 6= 0 for some
p)
−1
b
b
character χ = χ1 ⊗ χ2 ∈ T(E), then χ1 | | ⊗χ2 | | ∈ T(E)+ .
Proof. This is a consequence of [32, Lem. 4.4.2].

The following theorem provides a kind of converse to Corollary 5.2.2.
5.2.5. Theorem. Let B be an E-Banach space equipped with a unitary representation of GL2 (Qp ) for which Ban is of compact type, let L ∈ Exp(Ban ), and write
χ(L) = χ1 ⊗ χ2 . Suppose furthermore that the character χ1 | |−1 ⊗χ2 | | ε−1 isneither critical nor ultracritical in the sense of Definition 4.5.2. Then there is acontinuous GL2 (Qp )-equivariant map
GL (Q )
(17)
IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | an → Ban ,puniquely determined up to multiplication by a non-zero scalar, so that the inclusion
L ⊂ JP(Qp ) (Ban ) is obtained by applying the functor JP(Qp ) to (17) and composing
the resulting map with the inclusionGL (Q )
χ1 ⊗ χ2 ⊂ JP(Qp ) (IndP(Q2 ) p χ1 | |−1 ⊗χ2 | |)anp

provided by Corollary 5.2.2.
GL (Q )
Proof. As in [34, §2], we identify IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | an with a space ofplocally analytic E-valued functions on Qp satisfying a certain regularity conditionGL (Q )GL (Q )
at infinity. We let IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )sm (resp. IndP(Q2 ) p χ1 | |−1ppLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q37⊗χ2 | | (Qp )an ) denote the closed P(Qp )-invariant (resp. (gl2 , P(Qp ))-invariant)GL (Q )
subspace of IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | an consisting of locally constant (resp.
p

locally analytic) functions that are compactly supported on Qp . (Here gl2 denotes
the Lie algebra of GL2 (Qp ).) If χ1 χ2−1 is furthermore of integral Hodge-Tate weightGL (Q )w ≥ 0, then we let IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )lalg denote the (gl2 , P(Qp ))p
GL (Q )
invariant subspace of IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )an consisting of functionspthat are locally polynomial of degree ≤ w.

GL (Q )
The action of gl2 on IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )an induces a naturalp(gl2 , P(Qp ))-equivariant map(18)
GL (Q )
Ugl2 ⊗Up IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )smp
GL (Q )
→ IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )anp(where p denotes the Lie algebra of P(Qp ) and Ux indicates the universal enveloping
algebra of the Lie algebra x), which is an isomorphism if χ1 χ−1
2 is not of non-negative
integral Hodge-Tate weight, and which induces a surjection
GL (Q )
(19) Ugl2 ⊗Up IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )smp
GL (Q )
→ IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )lalgpif χ1 χ−1
2 is of non-negative integral Hodge-Tate weight.
The defining adjointness property of JP(Qp ) [32, Thm. 3.5.6] yields a natural
isomorphismGL (Q )
(20) Hom(gl2 ,P(Qp )) (Ugl2 ⊗Up IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )sm , Ban )p∼−→ HomT(Qp ) (χ1 ⊗ χ2 , JP(Qp ) (Ban )).
If χ1 χ−1
2 is not of non-negative integral Hodge-Tate weight, then upon combining
the isomorphism (18) with [14, Prop. 2.1.4] (which is a special case of the main
result of [35]) we find that passing to Jacquet modules yields an isomorphismGL (Q )
(21) HomGL2 (Qp )) ( IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | an , Ban )p∼−→ HomT(Qp ) (χ1 ⊗ χ2 , JP(Qp ) (Ban )),proving the theorem in this case.
If χ1 χ−1
is of non-negative integral Hodge-Tate weight, then by assumption2
χ1 | |−1 ⊗χ2 | | ε−1 is neither critical nor ultacritical. Taking this together withthe fact that B is unitary, it follows from [32, Thm. 4.4.5] that the canonical lift
to Ban of any element in JP(Qp ) (Ban ) (see [32, 0.9]) is locally SL2 -algebraic. It
follows directly from this, and from the construction of the map (20), that any map
in the source of (20) necessarily factors through the surjection (19), yielding an
isomorphismGL (Q )
(22) Hom(gl2 ,P(Qp )) ( IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )lalg , Ban )p∼−→ HomT(Qp ) (χ1 ⊗ χ2 , JP(Qp ) (Ban )).

38

MATTHEW EMERTON

The same argument that proves [34, Prop. 2.5] (which is a variation of the classicalargument of Amice-Vélu [1] and Vishik [62], and relies again on the fact that χ1 | |−1⊗χ2 | | ε−1 is neither critical nor ultacritical) shows that the restriction map
GL (Q )
HomGL2 (Qp )) ( IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | an , Ban )p
GL (Q )→ Hom(gl2 ,P(Qp )) ( IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )lalg , Ban )pis an isomorphism. Composing this isomorphism with the isomorphism (22), we
find that in this case passing to Jacquet modules again yields an isomorphism (21).
This completes the proof of the theorem.
The following supplement to Theorem 5.2.5 treats the case when Exp(B(V )an )contains a line L such that χ(L) is critical or ultracritical.
5.2.6. Proposition. Let B be an E-Banach space equipped with a unitary representation of GL2 (Qp ) for which Ban is of compact type, let L ∈ Exp(Ban ), and write
χ(L) = χ1 ⊗ χ2 . Suppose that χ1 χ−1
2 is of integral Hodge-Tate weight w ≥ 0.
−1
−1(1) If χ1 | | ⊗χ2 | | ε is critical, then either there exists L0 ∈ Exp(Ban )such that χ(L0 ) equals one of χ1 z −1−w ⊗χ2 z 1+w or χ2 | | z w ⊗χ1 | |−1 z −w ,or else χ1 = χ2 and B contains a copy of the one dimensional GL2 (Qp )invariant representation χ1 ◦ det.
(2) If χ1 | |−1 ⊗χ2 | | ε−1 is ultracritical, thendim Expχ1 ⊗χ2 (B(V )an ) ≤ dim Expχ1 z−1−w⊗χ2 z 1+w

(B(V )an ).

0

In particular, there exists L ∈ Exp(Ban ) such that χ(L0 ) equals χ1 z −1−w ⊗χ2 z 1+w .
Proof. We suppose that we are in either case (1) or case (2), that is, that χ1 | |−1⊗χ2 | | ε−1 is either critical or ultracritical. We will use the terminology andresults recalled in the proof of the preceding theorem. Let v ∈ Ban be the canonicallift of a non-zero element in L. Suppose first that v is locally SL2 -algebraic. Then
under the adjunction isomorphism (20), the inclusion
L → JP(Qp ) (Ban )(23)arises from a (gl2 , P(Qp ))-equivariant map
GL (Q )
IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )lalg → Ban .
p

One can show without difficulty that the restriction map
GL (Q )
HomGL2 (Qp )) ( IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | lalg , Ban )p
GL (Q )→ Hom(gl2 ,P(Qp )) ( IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | (Qp )lalg , Ban )pis an isomorphism (for any character χ1 ⊗ χ2 such that χ1 χ−1
2 is of integral HodgeTate weight, and any continuous GL2 (Qp )-Banach space representation B; this isa variant of [14, Lem. A.2.4]). Thus the map (23) is obtained by applying JP(Qp )to a non-zero continuous GL2 (Qp )-equivariant map
GL (Q )
(24)
IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | lalg → Ban .
p

If this map is injective, then it induces an injection on Jacquet modules, and Proposition 5.2.1 now shows that we obtain a copy of the character χ2 | | z w ⊗χ1 | |−1 z −w

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

39

in JP(Qp ) (B), and so Lemma 5.2.4 implies that χ2 z w ⊗ χ1 z −w , and hence also
−1b
χ2 z w ⊗ χ1 z −w ε−1 , lies in T(E)⊗χ2 | | ε−1 is assumed to be+ . Since χ1 | |−w −1either critical or ultracritical, so that χ1 z ε ⊗ χ2 z w (= χ1 | |−1 z −1−w ⊗ χ2 | |−w −1bε−1 z 1+w ) also lies in in T(E)ε ⊗ χ2 z w is uni+ , we conclude that in fact χ1 z
−1−1tary, and thus that χ1 | | ⊗χ2 | | ε is critical. Hence we are necessarily incase (1).
The source of (24) is irreducible (and consequently this map is necessarily injective) unless χ1 χ2−1 | |−1 =| |±1 z w . Again using the fact that χ1 | |−1 ⊗χ2 | |ε−1 = χ1 | |−1 ⊗χ2 z −1 is either critical or ultracritical, we see that the sourceof (24) is irreducible unless w = 0 and χ1 = χ2 . Thus if (24) is not injective, then
χ1 = χ2 and (24) must factor through the quotient χ1 ◦ det of its source, yielding
a copy of this one dimensional GL2 (Qp )-representation inside B. Clearly χ1 mustthen be unitary, and so we must again be in case (1). This completes our discussion
in the case when v is locally SL2 -algebraic.
Suppose now that thecanonicallift v of a non-zero element of L is not locallySL2 -algebraic. If X− := 01 00 ∈ gl2 , then [32, Prop. 4.4.4] shows that (X− )1+w v
−1−w

χ1 z
is the canonical lift of a non-zero element of JP(Qp)⊗χ2 z 1+w

(Ban ). In particular,χ1 ⊗χ2
this completes the proof of case (1). Since in case (2) no element of JP(Qhasp)
a locally SL2 -algebraic canonical lift (as we have seen above), we see that theχ1 ⊗χ2
action of (X− )1+w on canonical lifts in fact induces an injection JP(Q
(Ban ) →
p)
−1−w

1+w

χ1 z
⊗χ2 z
JP(Q
(Ban ). This yields the claimed inequality in case (2), and completesp)
the proof of the proposition.


5.3. Admissible unitary representations of GL2 (Qp ) with non-zero Jacquet
modules. In this subsection we consider the classification of those topologicallyirreducible admissible unitary GL2 (Qp )-representations B for which JP(Qp ) (Ban ) 6=
0. We begin with the following proposition.
5.3.1. Proposition. If B is an admissible unitary GL2 (Qp )-representation suchthat JP(Qp ) (Ban ) 6= 0, then we may find a finite extension E 0 of E such thatb 0 ) such that
Exp(E 0 ⊗E Ban ) 6= ∅ (or equivalently, there is a character χ ∈ T(E
χ
0
JP(Q
(E
⊗
B
)
=
6
0).
E
an
p)
Proof. Since B is admissible unitary, the locally analytic representation Ban is
admissible, and so JP(Qp ) (Ban ) is an essentially admissible T(Qp )-representationb (as explained[32, Thm. 0.5], corresponding to a rigid analytic coherent sheaf F on Tin [31, §6.4] – the global sections of F are naturally dual to JP(Qp ) (Ban )). Since
JP(Qp ) (Ban ) 6= 0, the coherent sheaf F is non-zero, and so has a non-empty support.
b 0 ) (for some finite extension E 0 of E) is a point in the support of F, thenIf χ ∈ T(Eχ
the fibre Fχ 6= 0. This fibre is naturally dual to JP(Q(E 0 ⊗E Ban ), and so we findp)that this latter space is also non-zero.
In light of the preceding result, we restrict our attention from now on to topologically irreducible admissible unitary B such that Exp(Ban ) 6= ∅. From Theorem 5.2.5 we see that the first step in classifying such B will be to describe theuniversal unitary completions of locally analytic induced representations.
We consider first the induction of a unitary character, and begin by introducing
some additional notation.

40

MATTHEW EMERTONb denote the universal unitary completion of the Steinberg
5.3.2. Definition. Let St
representation St.
b with its natural GL2 (Qp )-action, is5.3.3. Lemma.
(1) The Banach space St,
topologically irreducible and admissible unitary.
b induces an isomorphism(2) The inclusion St ⊂ St∼

b an ).
| | ⊗ | |−1 = JP(Qp ) (St) −→ JP(Qp ) (StProof. Part (1) was already observed in Theorem 5.1.8 (1). To prove claim (2),begin with the short exact sequenceGL (Q )
0 → 1 → IndP(Q2 ) p 1 ⊗ 1 sm → St → 0.
p

Passing to universal unitary completions yields the short exact sequence of admissible unitary representationsGL (Q )b →0
(25)
0 → 1 → IndP(Q2 ) p 1 ⊗ 1 cont → Stp(cf. the proof of the following proposition). Passing to locally analytic vectors(which is an exact functor on admissible unitary representations [51, Thm. 7.1])
yields the short exact sequenceGL (Q )b an → 0.
0 → 1 → IndP(Q2 ) p 1 ⊗ 1 an → Stp

Passing to Jacquet modules (and taking into account Proposition 5.2.1) yields anexact sequence of T(Qp )-representationsMb an ).
0→1→1| | ⊗ | |−1 → JP(Qp ) (St
Although JP(Qp ) is not a right exact functor in general, it is not difficult to checkthat in this case the preceding exact sequence is exact on the right as well, proving
claim (2).
The next proposition gives a precise description of the universal unitary completion of the locally analytic induction of a unitary character.
b5.3.4. Proposition. Let χ = χ1 ⊗ χ2 ∈ T(E)be a unitary character.
(1) The continuous injectionGL (Q )GL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 an → IndP(Q2 ) p χ1 ⊗ χ2 contpp

identifies the latter representation with the universal unitary completion of
the former.
GL (Q )
(2) The GL2 (Qp )-representation IndP(Q2 ) p χ1 ⊗χ2 cont is admissible unitary.
pGL (Q )
(3) If χ1 6= χ2 then IndP(Q2 ) p χ1 ⊗ χ2 cont is topologically irreducible.
pGL (Q )
(4) If χ1 = χ2 then IndP(Q2 ) p χ1 ⊗ χ2 cont sits in a non-split exact sequencep

0 → χ1 ◦ det →GL (Q )
b → 0.
IndP(Q2 ) p χ1 ⊗ χ2 cont → χ1 ◦ det ⊗St
pProof. Claim (1) follows directly from the fact that the character χ is unitary; it
is a straightforward generalization of the standard fact in p-adic analysis that the
space of locally analytic E-valued functions on P1 (Qp ) is dense in the space ofcontinuous E-valued functions with respect to the sup norm (which is the special

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

41

case of claim (1) when χ is the trivial character). We note that when χ1 ⊗ χ−1
2 is ofnon-negative integral Hodge-Tate weight, the same reasoning (locally polynomialGL (Q )
functions are dense in continuous functions) shows that IndP(Q2 ) p χ1 ⊗χ2 cont mayp
be regarded as the universal unitary completion of the locally algebraic induction
GL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 lalg .
p
It was already observed in Subsection 1.3 that the continuous induction is admissible. This proves claim (2).
Suppose now that χ1 6= χ2 . If χ1 ⊗ χ−1
2 is of non-negative integral Hodge-Tateweight then claim (3) is proved in [14, Prop. 2.2.1]. If not, then the same proofGL (Q )
works, but is even simpler, since IndP(Q2 ) p χ1 ⊗ χ2 an is topologically irreducible.
p

If χ1 = χ2 , then claim (4) results from twisting the short exact sequence (25) byχ1 = χ2 .

5.3.5. Remark. Parts (2), (3), and (4) of this proposition have also been observedby Schneider and Teitelbaum.
5.3.6. Corollary. If B is a topologically irreducible admissible unitary Banach space−1
representation of GL2 (Qp ) such that Expχ1 | |⊗χ2 | | (Ban ) 6= ∅ for some unitarybcharacter χ1 ⊗ χ2 ∈ T(E)then either:∼GL2 (Qp )(1) B −→ IndP(Q ) χ1 ⊗ χ2 cont (if χ1 6= χ2 ); orp∼b (if χ1 = χ2 ).
(2) B −→ χ1 ⊗ St
Proof. It follows from Theorem 5.2.5 and Proposition 5.3.4 (1) that there is a nonzero continuous GL2 (Qp )-equivariant map
GL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 cont → B.
p

Since the source and target of this map are admissible unitary (the source by
Proposition 5.3.4 (2) and the target by assumption), we see that it has closed
image [31, Prop. 6.2.9], and hence is surjective. The Corollary then follows from
Proposition 5.3.4 (3) and (4).
The next result, which treats the universal unitary completions of the locallyanalytic inductions of non-unitary characters, lies significantly deeper than Proposition 5.3.4.
b
5.3.7. Theorem. Let χ = χ1 ⊗ χ2 ∈ T(E)+ be such that the following conditions
hold:(1) χ1 ⊗ χ2 is not unitary;(2) The ratio χ1 χ−1
2 does not have non-negative integral Hodge-Tate weight.

GL (Q )
Then the universal unitary completion of IndP(Q2 ) p χ1 ⊗ χ2 an is a non-zero,p

topologically irreducible, admissible unitary GL2 (Qp )-representation.
Proof. Arguing as in the proof of [8, Thm. 4.3.1] (cf. the proof of [14, Thm. 2.2.2]),
one sees that the universal unitary completion in question may be identified withthe space B(s) of [27], taking s to be the character χ1 ⊗ χ2 ε. The theorem is thus
a restatement of [27, Thm. 0.12].
42

MATTHEW EMERTON−1b
5.3.8. Proposition. Let χ1 ⊗ χ2 ∈ T(E)+ be such that χ1 χ2 is of integral Hodge−1
is neither critical nor ultracritical, then the
Tate weight w ≥ 0. If χ1 ⊗ χ2 εGL (Q )GL (Q )
inclusion IndP(Q2 ) p χ1 ⊗χ2 lalg ⊂ IndP(Q2 ) p χ1 ⊗χ2 an induces an isomorphismpp
on universal unitary completions.

Proof. This is proved in [34, Prop. 2.5], and independently in [8]. (Although the−wcase χ1 χ−1=| |−2 is not explicitly considered in [34], the proof as written
2 zapplies to it perfectly well.)w
5.3.9. Remark. If χ1 χ−1
2 = z for some w > 0 then the surjection∨GL (Q )GL (Q )
IndP(Q2 ) p χ1 ⊗ χ2 lalg → IndP(Q2 ) p χ1 ⊗ χ2 lalg /χ1 ⊗ Symw E 2pp

induces an isomorphism on universal unitary completions, since Corollary
5.1.3
∨implies that the universal unitary completion of | |−w/2 ⊗ Symw E 2 vanishes.
−1b
5.3.10. Definition. If χ1 ⊗ χ2 ∈ T(E)is neither unitary,+ is such that χ1 ⊗ χ2 εcritical, nor ultracritical, then we write B(χ1 ⊗ χ2 ) to denote the universal unitaryGL (Q )
completion of IndP(Q2 ) p χ1 ⊗ χ2 an .
p

5.3.11. Remark. In the context of the preceding definition, if we write r :=ordp (χ1 (p)), then the completion B(χ1 ⊗ χ2 ) may be described explicitly as a suitable quotient of a certain space of functions on Qp of class C r . (See [13, §3.3] and[8, Thm. 4.3.1]; as we remarked in the proof of Theorem 5.3.7, the second citedresult extends to apply to any of the completions B(χ1 ⊗ χ2 ) under consideration.)
5.3.12. Remark. If, in the context of the preceding definition, the character χ1 ⊗χ2satisfies condition (2) of Theorem 5.3.7, then that theorem shows that B(χ1 ⊗ χ2 )is topologically irreducible and admissible unitary. Suppose on the other hand−wthat χ1 χ−1is of integral Hodge-Tate weight w ≥ 0. If χ1 χ−1
is furthermore
2
2 z
−1
−2
6= 1, | | , | | , then Proposition 5.3.8 shows that B(χ1 ⊗χ2 ) may be identified withGL (Q )
the unitary universal completion of IndP(Q2 ) p χ1 ⊗χ2 lalg , which is a unitary twistpof a representation to which Theorem 5.1.6 applies. Thus that Theorem shows thatin this case B(χ1 ⊗ χ2 ) is again topologically irreducible and admissible unitary. (IfConjecture 5.1.5 holds in full generality, then we see that the same is true without
the requirement that χ1 χ2−1 z −w 6=| |−1 .) If χ1 χ2−1 z −w =| |−2 , then Remark 5.1.15
shows that B(χ1 ⊗ χ2 ) is a unitary twist of B(2 − w, ∞), and so by Theorem 5.1.13−w
is also topologically irreducible and admissible unitary. Finally, if χ1 χ−1=1
2 z(in which case necessarily w > 0, since χ1 ⊗ χ2 is non-unitary by assumption) thenTheorem 5.1.8, Proposition 5.3.8 and Remark 5.3.9 together show that B(χ1 ⊗ χ2 )is neither topologically irreducible nor admissible unitary.
−1bis neither
5.3.13. Remark. For characters χ1 ⊗ χ2 ∈ T(E)+ such that χ1 ⊗ χ2 εunitary, critical, nor ultracritical, we have seen that the representation B(χ1 ⊗ χ2 )is (or should be) admissible unitary and topologically irreducible precisely when
χ1 ⊗χ2 ε−1 corresponds to a unique trianguline two dimensional GQp -representation
with respect to the classification scheme of Theorem 4.5.4. This can be viewed as
evidence in favour of a “p-adic local Langlands philosophy”.

5.3.14. Corollary. Let B be a topologically irreducible admissible unitary Banach−1
space representation of GL2 (Qp ) such that Expχ1 | |⊗χ2 | | (Ban ) 6= ∅ for somebcharacter χ1 ⊗ χ2 ∈ T(E)satisfying the following conditions:

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

43(1) χ1 ⊗ χ2 ε−1 is neither unitary, critical, nor ultracritical;−w(2) χ1 χ−1
6= 1, | |−1 for any integer w ≥ 0.
2 z
Then B is isomorphic to B(χ1 ⊗ χ2 ).
b
Proof. Note that by Lemma 5.2.4 we have χ1 ⊗ χ2 ∈ T(E)+ . The result now
follows by an argument similar to that used to prove Corollary 5.3.6, applying
Remark 5.3.12 in place of Proposition 5.3.4.

If Conjecture 5.1.5 were known to be true in full generality then the preceding corollary would hold with condition (2) replaced by the weaker condition thatw
χ1 χ−1
2 6= z for any w > 0.
5.3.15. Remark. Suppose that B is a topologically irreducible admissible unitary
Banach space representation of GL2 (Qp ) that is not one dimensional (and hence−1
is infinite dimensional by Corollary 5.1.3) such that Expχ1 | |⊗χ2 | | (Ban ) 6= ∅ for−1ba character χ1 ⊗ χ2 ∈ T(E)is critical (resp. ultracritical).
+ for which χ1 ⊗ χ2 εPart (1) (resp. part (2)) of Proposition 5.2.6 then shows that there is also a unitarycharacter (resp. a character satisfying the conditions of Theorem 5.3.7) ψ1 ⊗ ψ2 ∈b
T(E)for which Expψ1 ⊗ψ2 (Ban ) 6= ∅; the representation B is then classified by
Corollary 5.3.6 (resp. Corollary 5.3.14).
b
5.3.16. Lemma. If χ1 ⊗χ2 ε−1 ∈ T(E)
+ is neither unitary, critical, nor ultracritical,−1
if χ1 χ2 is of integral Hodge-Tate weight w ≥ 0, and if χ1 χ2−1 z −w 6= 1, | |−2 , thenthere is an GL2 (Qp )-equivariant topological isomorphism B(χ1 ⊗ χ2 ) ∼= B(χ2 | |−1w−w
z , χ1 | | z ).
Proof. Proposition 5.3.8 shows that B(χ1 ⊗ χ2 ) (resp. B(χ2 | |−1 z w , χ1 | | z −w ))GL (Q )
may be identified with the universal unitary completion of IndP(Q2 ) p χ1 ⊗ χ2 lalgpGL (Q )
(resp. IndP(Q2 ) p χ2 | |−1 z w ⊗ χ1 | | z −w lalg ). The theory of intertwiningpoperators for smooth parabolic induction yields an isomorphism between these two
locally algebraic inductions.

We close this section by considering one more universal unitary completion, thistime in a critical case.
5.3.17. Definition. Let B(2, ∞) denote the universal unitary completion of theGL (Q )
smooth induction IndP(Q2 ) p | |−1 ⊗ | | sm .
p(The notation will be explained in Subsection 6.5 below.)
5.3.18. Lemma. If B is a Banach space equipped with a unitary GL2 (Qp )-action,GL (Q )
then any continuous GL2 (Qp )-equivariant injection IndP(Q2 ) p | |−1 ⊗ | | sm → B
pextends uniquely to a continuous GL2 (Qp )-invariant injection B(2, ∞) → B.
Proof. By the very definition of the universal unitary completion, any continuous
GL (Q )
GL2 (Qp )-equivariant map φ : IndP(Q2 ) p | |−1 ⊗ | | sm → B extends uniquelyp
to a continuous GL2 (Qp )-invariant map φe : B(2, ∞) → B. Taking L = ∞ in thediscussion of Subsection 6.5 below shows that the representation B(2, ∞) sits in a
non-split short exact sequenceb → B(2, ∞) → 1 → 0.
0 → St

44

MATTHEW EMERTON

b is topologically irreducible, by Lemma 5.3.3 (1), we see that if φe is not
Since Stb vanishes. But this implies that the restriction of
injective, then its restriction to St
GL (Q )φ to the subrepresentation St of IndP(Q2 ) p | |−1 ⊗ | | sm vanishes, and so φ isp
not injective either.

6. The local correspondence for trianguline representations
In this section we will describe the explicit correspondence V 7→ B(V ) in thetrianguline case that was discussed in the introduction and in Remark 3.3.7. We
will see that in fact the conditions of Conjecture 3.3.1 serve to almost entirely determine this correspondence. For example, among the irreducible trianguline V ,the representations that are potentially semi-stable, but not potentially crystalline,
up to a twist, are the only ones for which B(V ) is not completly determined by
Conjecture 3.3.1. (This reflects the fact that these are the only irreducible trianguline representations whose classification requires Hodge-theoretic, in addition to
Weil-Deligne, invariants; cf. Remark 3.3.4 and Theorem 4.5.4.)
For reducible V the situation is more complicated. If V is the direct sum of two
characters then we can specify a corresponding GL2 (Qp )-representation B(V ), andthis representation is completely determined by Conjeture 3.3.1. However if V isindecomposable then we are not able to specify a candidate for B(V ) in general(although we can predict what structure it should have), due mainly to our lack ofknowledge about the structure of extensions in the category of admissible unitary
representations.
Although the case of reducible V may seem somewhat marginal in relation to
the problem of constructing the local p-adic correspondence in general, we havefound it useful to study this case carefully for two reasons: firstly, the analysis in
this case illustrates how the conditions of Conjecture 3.3.1 can be used to narrow
down the possibilities for the associated GL2 (Qp )-representation B(V ). Secondly,
the extensions that one expects to find on the level of Banach spaces in the case of
reducible indecomposable V provide a model for the extensions that are conjectured
to appear after passing to the subspace of locally analytic vectors in the case of
irreducible V (as we explain in Subsection 6.7 below).
I would like to close this introductory passage by mentioning that the discussionof Subsection 6.5 below has been strongly influenced by remarks and suggestionsof Breuil.
6.1. V is irreducible. If V is an irreducible continuous two dimensional trianguline representation of GQp , and if R is a refinement of V , then R is neither
ordinary nor critical. Furthermore, we may and do choose R so that it is not ultracritical. Write σ(R) = (η, ψ). According to condition (8) of Conjecture 3.3.1,
the admissible unitary GL2 (Qp )-representation associated to B(V ) should satisfy−1Expη| |⊗ψε| | (B(V )an ) 6= ∅. Suppose first that ηψ −1 6= | | z w for any w > 1.
Assuming that Conjecture 5.1.5 holds if necessary, we see that it follows from
Corollary 5.3.14 that B(V ) must be isomorphic to B(η ⊗ ψε). This motivates us
to make the following definitions.
6.1.1. Definition. If R is a non-ultracritical refinement of an irreducible trianguline
representation, then we define B(R) := B(η ⊗ ψε).

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

45

Note that the definition of B(R) depends only on R up to equivalence (since thisis true of σ(R)).
6.1.2. Definition. Let V be an irreducible continuous two dimensional trianguline
GQp -representation admitting a non-ultracritical refinement R such that, writing
σ(R) = (η, ψ), we have ηψ −1 6= | | z w for any w > 1. Then we define B(V ) := B(R).
6.1.3. Remark. If V admits two inequivalent refinements R1 and R2 both satisfying the conditions of the preceding definition, then V is necessarily a twistof a Frobenius semi-simple potentially crystalline representation, and if we writeσ(R1 ) = (η, ψ), and let w > 0 denote the Hodge-Tate weight of ηψ −1 , thenσ(R2 ) = (ψz w , ηz −w ) (see Remark 4.3.3). Lemma 5.3.16 yields an isomorphism
B(R1 ) ∼= B(R2 ), and so we see that B(V ) is well-defined, independent of the choiceof R.
6.1.4. Remark. If in the context of Definition 6.1.2 the representation V is potentially semi-stable, then ηψ −1 has Hodge-Tate weight w > 0 (since V is irreducible).
Thus Proposition 5.3.8 shows that B(V ) can also be defined as the universal unitarycompletion of the locally algebraic representation π
ep (V ). (See Conjecture 3.3.1 (7)for the definition of π
ep (V ).) Thus the definition of B(V ) given here coincides withthat given by Breuil in [12, §1.3] (perhaps up to a unitary twist, reflecting a difference between our choice of normalization of the correspondence and his).
6.1.5. Remark. If in the context of Definition 6.1.2 the representation V is notpotentially semi-stable up to a twist, so that ηψ −1 is not of positive integral HodgeTate weight, then the definition of B(V ) given here coincides with that given by
Colmez [27] (again up to a possible unitary twist, reflecting a difference in normalizations). As was noted in the proof of Theorem 5.3.7, the proof of this follows thesame lines as the proof of [8, Thm. 4.3.1].
The only irreducible trianguline V that do not admit a refinement satisfying theconditions of Definition 6.1.2 are those that are twists of semi-stable non-crystalline√
representations. If V is of this form, then adjoining p to E if necessary in thecase when w2 − w1 is even, and replacing V by a suitable twist (and taking into
account Conjecture 3.3.1 (3) and (4)) we may assume that V is semi-stable with
Hodge-Tate weights w < −1 and 0, and that the eigenvalues of ϕ on Dst (V ) areequal to p(−1−w)/2 and p(1−w)/2 . The representation V then admits a refinement
R that is unique up to equivalence, and σ(R) = (| |(1+w)/2 , | |(−1+w)/2 z w ).
Proposition 5.3.8 and Remark 5.3.9 show that B(R) may be identified with the∨universal unitary completion of π
ep (V ) = St⊗ | |(1+w)/2 ⊗ Sym−1−w E 2 . The
representation π
ep determines Dst (V ) as a (ϕ, N )-module, but it does not determine
the Hodge filtration on Dst (V ). This filtration depends on an additional parameter L ∈ E (the L-invariant of E – in defining which we follow the conventionof [26, 0.2]) that measures the relative position of the Hodge filtration on Dst (V )and the filtration induced by the kernel of N . The semi-stable representation V
is thus classified by the locally algebraic representation π
ep (V ) together with theinvariant L.
6.1.6. Definition. If V is as in the preceding discussion, with L-invariant equal toL, then we define B(V ) to be the corresponding quotient B(1 − w, L) of B(R), as
defined in Definition 5.1.11.

46

MATTHEW EMERTON

6.1.7. Remark. If V is a twist of a non-generic crystalline representation, then√adjoining p to E if necessary in the case when w2 − w1 is even, and replacing V
by a suitable twist, we may assume that V is crystalline with Hodge-Tate weights
w < −1 and 0, and that the eigenvalues of ϕ on Dcrys (V ) are equal to p(−1−w)/2 andp(1−w)/2 . The representation V admits two equivalence classes of refinements. We
may choose representatives R and R0 so that σ(R) = (| |(1+w)/2 , | |(−1+w)/2 z w ) andσ(R0 ) = (| |(−1+w)/2 , | |(1+w)/2 z w ). Definition 6.1.2 applies to V , and stipulates
that B(V ) := B(R0 ). On the other hand, Remark 5.1.15 shows that B(V ) is also
isomorphic to the quotient B(1 − w, ∞) of B(R). Thus we may treat all the cases
in which V admits an L-invariant in a uniform fashion.
6.1.8. Remark. If V is any irreducible trianguline representation admitting a refinement R with σ(R) = (η, ψ), and if we write r := ordp (η(p)), then B(V ) may
be described explicitly as a suitable quotient of a certain space of functions on Qp
of class C r ; cf. Remarks 5.1.12 and 5.3.11. The proofs of the results of Colmez
and of Berger and Breuil recalled in Remark 3.3.7, and in Theorems 5.1.6, 5.1.13,
and 5.3.7, rely heavily on this explicit description of B(V ).
6.1.9. Remark. Let V be a potentially semi-stable continuous two dimensional
GQp -representation for which πp (V ) is cuspidal. (Equivalently, V is potentiallycrystalline, but becomes crystalline only over a non-abelian extension of V .) Although V is not trianguline, this case is quite similar to the case in which πp (V )is special. As in that case, V is not uniquely determined by the associated locally algebraic representation π
ep (V ). Indeed, while this representation determines
the (ϕ, GQp )-module underlying Dpcrys (V ), as well as the Hodge numbers of thefiltration on Dpcrys (V ), there is a P1 (E) worth of possible choices for the Hodge
filtration itself on DdR (V ). This provides the motivation for Conjecture 5.1.19:
one hopes to find a series of topologically irreducible admissible unitary quotientsof the universal unitary completion of π
ep (V ) parameterized by the same set P1 (E),and then to define B(V ) to be that quotient of πb indexed by the point of P1 (E)corresponding to the Hodge filtration on DdR (V ).
L
6.2. V is the direct sum of two characters. Suppose that V = η ψ fortwo continuous E-valued characters η and ψ of GQp . If η 6= ψ, then V admits
two inequivalent ordinary refinements R1 and R2 such that σ(R1 ) = (η, ψ) andσ(R2 ) = (ψ, η), and so Conjecture 3.3.1 (8) stipulates that we have an embeddingM(26)
η | | ⊗ ψε | |−1
ψ | | ⊗ ηε | |−1 → JP(Qp ) (B(V )an ).
If η = ψ, then Lemma 4.1.5 shows that we have Ref η⊗η (V ) ∼= P1 (E). Thus Conjecture 3.3.1 (8) stipulates that−1Expη| |⊗ηε| | (B(V )an ) ∼= P1 (E).

Thus we must again have an embedding of the form (26). Theorem 5.2.5 andProposition 5.3.4 (1) show that the embedding (26) must be induced by a continuous
GL2 (Qp )-equivariant morphismMGL (Q )GL (Q )
IndP(Q2 ) p ψ ⊗ ηε cont → B(V ).
(27)
IndP(Q2 ) p η ⊗ ψε cont
p

p

−1

±1

6.2.1. Lemma. Suppose that ηψ 6= ε . If B(V ) satisfies conditions (5) and (8)of Conjecture 3.3.1, then (27) is necessarily an isomorphism.

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

47

L
Proof. The reduction modulo $ of V is equal to η ψ (the sum of the reductions
modulo $ of the characters η and ψ). Under the modulo $ local Langlands correspondence, this representation is matched with the semi-simplification of the direct
sum
 M

GL (Q )GL (Q )
IndP(Q2 ) p η ⊗ ψε sm
IndP(Q2 ) p ψ ⊗ ηε sm .
ppThis direct sum is also the reduction modulo $ of the source of (27). Thus Conjecture 3.3.1 (5) implies that the reductions modulo $ of the source and target of (27)have isomorphic semi-simplifications.
Since ηψ −1 6= ε±1 the two direct summands in the source of (27) are irreducible,
and so (27) must be an embedding. The observation of the preceding paragraphimplies that it is in fact an isomorphism.
We now consider the case when ηψ −1 = ε±1 . Interchanging η and ψ if necessary,
we may assume that in fact ηψ −1 = ε.
6.2.2. Lemma. If ηψ −1 = ε, and if B(V ) satisfies conditions (3), (5), (7), and (8)of Conjecture 3.3.1, then there is an isomorphismM∼
GL (Q )η ◦ det ⊗B(2, ∞)IndP(Q2 ) p ηε−1 ⊗ ηε cont −→ B(V ).
p

Proof. Twisting V by η −1 (and taking into account Conjecture
3.3.1 (3)), we seeLthat we may in fact assume that η = 1, so that V = 1 ε−1 . Conjecture 3.3.1 (7)then stipulates that we should have an isomorphism∼GL (Q )
(28)
IndP(Q2 ) p | |−1 ⊗ | | sm −→ B(V )lalg .
p

As a consequence we find that St is a subrepresentation of B(V ), and so (27) cannot
be an isomorphism. Also, by Lemma 5.3.18, the isomorphism (28) induces a closed
embedding(29)

B(2, ∞) → B(V ).

Since (27) is not an isomorphism, Proposition 5.3.4 (3) and (4) show that (27)factors through an embeddingMGL (Q )b
IndP(Q2 ) p ε−1 ⊗ ε cont → B(V ).
(30)
St
p

Considering reductions modulo $ as in the proof of the preceding lemma, we seebthat the cokernel of this embedding must be one dimensional. Thus the copy of St
appearing in B(V ) by virtue of the embedding (29) must coincide with the copy
b appearing in B(V ) by virtue of the embedding (30), and we see that B(V ) isof St
isomorphic to the direct sum of the sources of these two embeddings, amalgamatedb The lemma follows.
along their common subrepresentation St.
In light of the preceding results, we make the following definition.
L
6.2.3. Definition.
(1) If V = η ψ and ηψ −1 6= ε±1 then we defineMGL (Q )GL (Q )
B(V ) := IndP(Q2 ) p η ⊗ ψε cont
IndP(Q2 ) p ψ ⊗ ηε cont .
p
pL(2) If V = η ψ and ηψ −1 = ε then we defineMGL (Q )
B(V ) := η ◦ det ⊗B(2, ∞)IndP(Q2 ) p ηε−1 ⊗ ηε cont .
p

Note that this definition is an obvious extension of the one given in [14] in thecase when η and ψ are Hodge-Tate.

48

MATTHEW EMERTON

6.3. V is reducible and indecomposable – first case. Write V as a non-split
extension
0 → η → V → ψ → 0,

(31)
−1

±1

and suppose that ηψ 6= ε . Combining Definition 6.2.3 (1) with condition (6)of Conjecture 3.3.1 we see the topological Jordan-Hölder factors B(V ) should be
GL (Q )GL (Q )
IndP(Q2 ) p η ⊗ ψε cont and IndP(Q2 ) p ψ ⊗ ηε cont . (Note that both of these repreppsentations are topologically irreducible, by Proposition 5.3.4 (3).) Since V admits
an ordinary refinement R such that σ(R) = (η, ψ), condition (8) of Conjecture 3.3.1−1
requires that Expη| |⊗ψε| | (B(V )an ) 6= ∅, and thus by Theorem 5.2.5 and Proposition 5.3.4 (1) there should be an embeddingGL (Q )
IndP(Q2 ) p η ⊗ ψε cont → B(V ).
p

Altogether we see that if B(V ) is to satisfy the conditions of Conjecture 3.3.1 thenit must sit in a non-split short exact sequenceGL (Q )GL (Q )
(32)
0 → IndP(Q2 ) p η ⊗ ψε cont → B(V ) → IndP(Q2 ) p ψ ⊗ ηε cont → 0.
pp(This sequence must be non-split, by condition (1) of Conjecture 3.3.1, since byassumption V is not the direct sum of η and ψ.)
The preceding considerations, together with Proposition 4.5.5, motivate the following conjecture.
6.3.1. Conjecture. Let η and ψ be a pair of unitary characters of Q×p.
(1) If ηψ −1 6= 1, ε±1 then the space of extensionsGL (Q )GL (Q )
Ext1 ( IndP(Q2 ) p ψ ⊗ ηε cont , IndP(Q2 ) p η ⊗ ψε cont ),ppcomputed in the category of admissible unitary GL2 (Qp )-representations
over E, is one dimensional.
(2) For any unitary character η of Q×
p , the spaces of extensions


GL
(Q
)
GL (Q )
Ext1 ( IndP(Q2 ) p η ⊗ ηε cont , IndP(Q2 ) p η ⊗ ηε cont ),ppcomputed in the category of admissible unitary GL2 (Qp )-representations
over E that have central character η 2 ε, is two dimensional.
6.3.2. Remark. In case (1) of the preceding conjecture, any element of the Ext1
under consideration will automatically have central character ηψε, since the rep
GL (Q )GL (Q )
resentations IndP(Q2 ) p η ⊗ ψε cont and IndP(Q2 ) p ψ ⊗ ηε cont are topologicallyppirreducible and non-isomorphic. Thus there is not need to explicitly impose thecondition that the extensions under consideration admit a central character.
If the preceding conjecture is correct, then for a non-split extension (31) withη and ψ satisfying the condition of Conjecture 6.3.1 (1) we would define B(V ) to
be any non-zero element of the corresponding Ext1 (any two such elements being
topologically isomorphic as GL2 (Qp )-representations, by the conjecture). If on theother hand η = ψ, then set of isomorphism classes of GQp -representations that sit
in a non-split extension (31) is parameterized by the points of P1 (E), and we expect
to be able to define the correspondence V 7→ B(V ) so as to match these various isomorphism classes with the corresponding P1 (E)-worth of topological isomorphismclasses of non-zero elements of the Ext1 considered in Conjecture 6.3.1 (2).

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

49

If V is potentially crystalline and generic, up to a twist, and if η 6= ψ (whichputs us in case (1) of Conjecture 6.3.1), then Breuil has constructed an extension of
GL (Q )
the desired type by taking the universal unitary completion of IndP(Q2 ) p ψz w ⊗pηεz −w an (where w > 0 is the Hodge-Tate weight of ηψ −1 ) [6, 14]. (Note that thecharacter ψz w ⊗ ηz −w is critical.) Anticipating that Conjecture 6.3.1 (1) is correct,
we follow these references in making the following definition.
6.3.3. Definition. If 0 → η → V → ψ → 0 is a non-split extension that is potentially crystalline and generic, up to a twist, and if furthermore η 6= ψ, then we
GL (Q )
define B(V ) to be the universal unitary completion of IndP(Q2 ) p ψz w ⊗ ηεz −w an .
p

6.4. V is reducible and indecomposable – second case. Suppose that V is a
non-split extension of the form 0 → ψε−1 → V → ψ → 0. Twisting V by ψ −1 (andtaking into account Conjecture 3.3.1 (3)), we may assume that V is in fact of theform
0 → ε−1 → V → 1 → 0.

(33)

The representation V is then unique up to isomorphism, and is Hodge-Tate but notpotentially semi-stable. Definition 6.2.3 (2) and Conjecture 3.3.1 (6) show that theb 1, and IndGL2 (Qp ) ε−1 ⊗topological Jordan-Hölder factors of B(V ) should be St,
P(Qp )
ε cont . By Conjecture 3.3.1 (7) we should have B(V )lalg = 0, and thus neither 1 norb can be subobjects of B(V ). Consequently there must be an inclusionSt
GL (Q )
(34)
IndP(Q2 ) p ε−1 ⊗ ε cont → B(V ).
p

By Conjecture 3.3.1 (8), together with Proposition 4.4.4 (2), we expect thatM

JP(Qp ) B(V )an =| | ε−1 ⊗ ε | |−1
1 ⊗ 1.
If we let B denote the cokernel of (34), then the topological Jordan-Hölder factors
b and 1, and (taking into account Proposition 5.2.1 (4) and the factof B must be St
that JP(Qp ) is left exact) there is an injection1 ⊗ 1 → JP(Qp ) (Ban ).
These facts and Proposition 5.2.6 (or better, its proof) together imply that B
GL (Q )
contains a copy either of 1 or else of IndP(Q2 ) p | |−1 ⊗ | | sm . In the latterpcase we deduce from Lemma 5.3.18 and our knowledge of the topological JordanHölder factors of B that B is isomorphic to B(2, ∞). One can show that there areGL (Q )
no non-trivial extensions of 1 by IndP(Q2 ) p ε−1 ⊗ ε cont , and so since 1 is not apsubrepresentation of B(V ), we conclude that 1 cannot be a subrepresentation of
B either. Thus indeed we must have B ∼= B(2, ∞), and so B(V ) must sit in a
non-split short exact sequence(35)GL (Q )
0 → IndP(Q2 ) p ε−1 ⊗ ε cont → B(V ) → B(2, ∞) → 0.
pThis suggests the following conjecture.
6.4.1. Conjecture. The space of extensionsGL (Q )
Ext1 (B(2, ∞), IndP(Q2 ) p ε−1 ⊗ ε cont ),p

50

MATTHEW EMERTON

computed in the category of admissible unitary GL2 (Qp )-representations over E, is
one dimensional.
6.4.2. Remark. As there are no non-zero continuous GL2 (Qp )-equivariant homoGL (Q )
morphisms from B(2, ∞) to IndP(Q2 ) p ε−1 ⊗ε cont , any element of the Ext1 underpconsideration necessarily has trivial central character.
If this conjecture were true, we would then define B(V ) to be the extension
whose existence it predicts.
6.5. V is reducible and indecomposable – third case. Suppose now that V
is a non-split extension of the form 0 → η → V → ηε−1 → 0. Twisting V by η −1(and recalling condition (3) of Conjecture 3.3.1), we see that it suffices to consider
non-split extensions of the form
0 → 1 → V → ε−1 → 0.

(36)

Such an extension is either semi-stable (but not crystalline), and classified by thevalue of its L-invariant, or else is non-generic crystalline (and uniquely determined
up to isomorphism) – the case L = ∞. As in the preceding case, Definition 6.2.3 (2)together with condition (6) of Conjecture 3.3.1 shows that the topological Jordanb 1, and IndGL2 (Qp ) ε−1 ⊗ ε.
Hölder factors of B(V ) should be St,
cont
P(Q )
p

In the case when L = ∞, so that V is non-generic crystalline, Conjecture 3.3.1 (7)together with Lemma 5.3.18 shows that there must be an injection B(2, ∞) →B(V ), and thus that B(V ) must be a non-split extensionGL (Q )
(37)
0 → B(2, ∞) → B(V ) → IndP(Q2 ) p ε−1 ⊗ ε cont .
p(Non-split because of Conjecture 3.3.1 (1) and the fact that V is not the direct sumof 1 and ε−1 .)GL (Q )
6.5.1. Lemma. The universal unitary completion of IndP(Q2 ) p | |−1 ⊗ | | an ispa non-split extension of the form (37).
Proof. The natural injectionGL (Q )GL (Q )
IndP(Q2 ) p | |−1 ⊗ | | sm → IndP(Q2 ) p | |−1 ⊗ | | anppinduces an injection of B(2, ∞) into this completion We leave it to the reader toGL (Q )
identify the cokernel as IndP(Q2 ) p ε−1 ⊗ ε cont .
p

6.5.2. Conjecture. The space of extensionsGL (Q )
Ext1 ( IndP(Q2 ) p ε−1 ⊗ ε cont , B(2, ∞)),
pcomputed in the category of admissible unitary GL2 (Qp )-representations over E, is
one dimensional.
6.5.3. Remark. Since there no non-zero continuous GL2 (Qp )-equivariant homoGL (Q )
morphisms from IndP(Q2 ) p ε−1 ⊗ε cont to B(2, ∞), any element of the Ext1 underpconsideration necessarily has trivial central character.
Presuming that the conjecture is correct, we follow [14, Rem. 2.3.1] in making
the following definition.

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

51

6.5.4. Definition. If V is the non-split crystalline extension of the form (36) thenGL (Q )
we define B(V ) to be the universal unitary completion of IndP(Q2 ) p | |−1 ⊗ | | an .
p

Now suppose that V is a semi-stable, non-crystalline, extension of the form (36).
∼
In this case Conjecture 3.3.1 (7) requires us to have an isomorphism B(V )lalg −→ St.
b → B(V ), whose cokernel has 1 andThus we have must have an injection St
GL2 (Qp ) −1
IndP(Q ) ε ⊗ ε cont as topological Jordan-Hölder factors. For such V a candip

date for B(V ) has not yet been specified in the literature. However, we will statea conjecture regarding its structure.
We begin by following [12], and constructing a family of non-split extensions
b depending on a parameter L ∈ P1 (E). Recall theof the trivial character by St
representation σ(L) of P(Qp ) constructed in 5.1.7 above. This is evidently a unitary
representation of P(Qp ) which extends the trivial character of P(Qp ) by itself, and soGL (Q )
the continuous induction IndP(Q2 ) p σ(L) cont is an admissible unitary GL2 (Qp )prepresentation, which sits in a short exact sequenceGL (Q )GL (Q )
(38) 0 −→ IndP(Q2 ) p 1 ⊗ 1 cont −→ IndP(Q2 ) p σ(L) contpppr
GL (Q )−→ IndP(Q2 ) p 1 ⊗ 1 cont −→ 0.
p

Recall that the trivial character 1 of GL2 (Qp ) is a subrepresentation of the continuGL (Q )
ous induction IndP(Q2 ) p 1 ⊗ 1 cont . Following [12], we define B(2, L) = pr−1 (1)/1p(i.e the quotient by the copy of 1 sitting in the first term of (38) of the preimage
under the projection pr of the copy of 1 sitting in the third term); then B(2, L) isan admissible unitary representation of GL2 (Qp ) that sits in an exact sequenceb → B(2, L) → 1 → 0.
0 → St(The notation follows that used in [12].) Since σ(L) is a non-split extension of
the trivial character by itself, one easily sees that the exact sequence containing
B(2, L) is also non-split. In the case L = ∞, the reader can check that B(2, ∞)is topologically isomorphic to the representation defined by Definition 5.3.17 above(which explains our choice of notation in that definition).
We also remark that if L 6= ∞, then the evident inclusion St → B(2, L)lalg is an
isomorphism (since the function logp , which intervenes in the definition of σ(L), is
not locally algebraic).
For any extension of the form (36) we then expect the representation B(V ) to
sit in a non-split short exact sequence of admissible unitary representationsGL (Q )
(39)
0 → B(2, L) → B(V ) → IndP(Q2 ) p ε−1 ⊗ ε cont −→ 0,pwhere L is the L-invariant of V . Nothing is known about the existence or classification of such extensions except in the case of L = ∞, where such an extension
is constructed in Lemma 6.5.1 above. Nevertheless, we might hope that the obvious generalization of Conjecture 6.5.2, with ∞ replaced by an arbitrary element
L ∈ P1 (E), holds.
We close this section with the following Lemma, which bears on the question ofwhether the representation B(V ), if it were to be defined as an extension of theform (39), would satisfy conditions (7) and (8) of Conjecture 3.3.1.

52

MATTHEW EMERTON

6.5.5. Lemma. If L ∈ E then the evident inclusion St → B(2, L) induces isomor∼∼
phisms St −→ B(2, L)lalg and | | ⊗ | |−1 = JP(Qp ) (St) −→ JP(Qp ) (B(2, L)an ).
Proof. Both assertions are easily checked by the reader, taking into account theb in place of B(2, L) as well as the fact thattruth of the analogous assertions for St
the function logp (which intervenes in the definition of σ(L), and thus of B(2, L))
is not locally algebraic.

6.6. Compatibility with Conjecture 3.3.1. If V is a continuous two dimensional trianguline representation of GQp over E, then the discussion of the preceding subsections yields a precise definition of B(V ) provided V satisfies one of thefollowing (mutually exclusive) conditions:(1) V is irreducible (Definitions 6.1.2 and 6.1.6);(2) V is a direct sum of two characters (Definition 6.2.3);(3) V is reducible, indecomposable, and is a twist of a Frobenius semi-simple
potentially crystalline representation (Definitions 6.3.3 and 6.5.4).
In the remainder of this subsection we discuss the extent to which this explicitly
defined correspondence in the trianguline case satisfies the conditions of Conjecture 3.3.1. We first note that conditions (2) through (8) of that conjecture refer
only to properties of B(V ) for some particular V , while condition (1) involves comparing the representations B(V ) for all V . Thus in any verification of condition (1),we will have limit the scope of that statement to those V for which B(V ) has actually been defined. We also note that conditions (2), (3), and (4) are built into theconstruction of the correspondence in the trianguline case, and so there is no needto discuss them further.
We consider first the reducible case. To avoid irritating circumlocutions, we willassume that the correspondence V 7→ B(V ) has actually been constructed for all
reducible V (i.e. that the various extensions whose existence has been conjectured
in Subsections 6.2 through 6.5 have been actually shown to exist). In those cases
when B(V ) has actually been constructed (i.e. if V is a direct sum of two characters,
or is Frobenius semi-simple potentially crystalline up to a twist), our remarks will
apply unconditionally; in the remaining case, they will apply as soon as B(V ) has
been constructed, provided that it is an extension of the form conjectured above.
It is clear from the constructions that if V and W are two reducible continuous two dimensional GQp -representations then B(V ) is topologically isomorphic to
B(W ) as a GL2 (Qp )-representation if and only if V and W are isomorphic GQp representations. (In the case when V and W both admit an L-invariant, use thefact that the representations B(2, L) are non-isomorphic for distinct values of L,
since this is true of the representations σ(L) that are used in their construction.)
Condition (5) is satisfied in this case (essentially by the definition of B(V ) and of
the mod $ correspondence), as are the remaining conditions (6) (by construction),(7) (by a direct computation of locally algebraic vectors), and (8) (by a directcomputation of Jacquet modules).
We now turn to the case of irreducible V .
6.6.1. Theorem.
(1) If V is an irreducible trianguline two dimensional continuous GQp -representation, and if V is not a twist of a potentially crystalline Frobenius non-semi-simple representation, then B(V ) is infinite dimensional and topologically irreducible, and satisfies Schur’s lemma: any
GL2 (Qp )-equivariant continuous endomorphism of B(V ) is scalar.

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

53(2) If V and W are trianguline two dimensional continuous representations of
GQp , each satisfying the condition of part (1), then B(V ) is topologicallyisomorphic to B(W ) as a GL2 (Qp )-representation if and only if V and W
are isomorphic as GQp -representations.
Proof. Given the definition of B(V ) in the various cases, it follows from Theorems 5.1.6, 5.1.13, and 5.3.7 that B(V ) is non-zero and topologically irreducible.
Note that since B(V ) is a non-zero quotient of the universal unitary completion of
an irreducible infinite dimensional locally analytic representation, it must in fact
be infinite dimensional. The claim regarding Schur’s lemma is [8, Prop. 3.4.5].
(Actually, this reference only treats the case of Frobenius semi-simple potentiallycrystalline V , but the argument applies equally well to the situations considered in[26, 27].) This proves part (1), while part (2) is a restatement of [27, Thm. 0.11]. Part (1) of the preceding result verifies condition (6) of Conjecture 3.3.1 forthose representations to which it applies. Part (2) shows that the correspondence
V 7→ B(V ) distinguishes non-isomorphic irreducible trianguline representations
that satisfy the condition of part (1).
6.6.2. Remark. It is expected that an appropriate variation of the methods used in
the proof of the preceding result should apply to establish the remaining unproved
cases of Conjecture 5.1.5. If this expectation were to be realized, then part (1) of
the preceding theorem would apply without exception to all irreducible trianguline
two dimensional representations. These same methods are similarly expected toallow one to extend part (2) to cover the outstanding cases. (This is implicit in thestatement of [27, Thm. 0.11]; cf. [27, Rem. 0.13].)We also note the following lemma. Note that it does not require that V satisfy
the conditions of part (1) of the preceding theorem.
6.6.3. Lemma. Let V be an irreducible trianguline representation, let R be a nonultracritical refinement of V , and write σ(R) = (η, ψ). Let B be an E-Banach spaceequipped with an admissible unitary GL2 (Qp )-action. If B(V ) → B is a non-zero
η|

|⊗ψε|

continuous GL2 (Qp )-equivariant morphism, then JP(Qp )|−1(Ban ) 6= 0.

Proof. Let w denote the Hodge-Sen-Tate weight of ηψ −1 ε−1 . We begin by defining
a locally analytic GL2 (Qp )-representation U attached to the pair (η, ψ), namely
GL2 (Qp ) IndP(Qp ) η ⊗ ψεan if w 6∈ Z≥0 ;GL (Q )U=
IndP(Q2 ) p η ⊗ ψε lalg if w ∈ Z≥0 and ηψ −1 ε−1 6= z w ;

p

∨

η ◦ det ⊗St ⊗ Symw E 2 if w ∈ Z>0 and ηψ −1 ε−1 = z w .
(Since V is irreducible, the character η⊗ψε−1 is not unitary, and so falls into exactly
one of the three cases.) In each case we see from the definition of B(V ) that there isa continuous injection U → B(V ) with dense image, while Proposition 5.2.1 showsthat there is an inclusion(40)

η | | ⊗ψε | |−1 ⊂ JP(Qp ) (U ).

Except in the case when w ∈ Z≥0 and ηψ −1 ε−1 =| |−2 z w , the representation U
is furthermore topologically irreducible. Thus the non-zero map B(V ) → B induces
an injection U → Ban , and hence passing to Jacquet modules yields an injection

54

MATTHEW EMERTON

JP(Qp ) (U ) → JP(Qp ) (Ban ). Composing this injection with the inclusion (40) gives
the lemma.
Suppose finally that we are in the exceptional case where ηψ −1 ε−1 =| |−2 z w ,and that the map U → Ban is not an injection. It then∨ induces an injection of thefinite dimensional quotient (η | |) ◦ det ⊗ Symw E 2 of U into Ban . However theJacquet module of this quotient is precisely η | | ⊗ψε | |−1 . Thus the lemma
follows in this case also.
In light of Remark 6.6.2, for the remainder of this subsection we will write as if
Theorem 6.6.1 (or equivalently Conjecture 5.1.5) held true in full generality. The
cautious reader may add the necessary caveats.
6.6.4. Lemma. Let V be an irreducible trianguline representation and let χ1 ⊗ χ2 ∈b
T(E)+ satisfy one of the following conditions:(1) χ1 | |−1 ⊗χ2 | | ε−1 is unitary;(2) χ1 | |−1 ⊗χ2 | | ε−1 is critical.
Then Expχ1 ⊗χ2 (B(V )an ) = ∅.
Proof. Let R be a non-ultracritical refinement of V , and write σ(R) = (η, ψ). Since
V is irreducible, η ⊗ψ is neither unitary, critical, nor ultracritical (by Lemma 4.5.3).
By construction, together with Theorem 6.6.1, the representation B(V ) is a nonGL (Q )
zero quotient of the universal unitary completion of IndP(Q2 ) p η ⊗ ψε an . Takingpinto account the isomorphism (21) established in the course of the proof of Theoη| |⊗ψε| |−1

rem 5.2.5, we see that JP(Qp )(B(V )an ) 6= 0.
Theorem 6.6.1 (1) shows that B(V ) is topologically irreducible and infinite dimensional, and thus it cannot contain any one dimensional subrepresentations.
Proposition 5.2.6 thus implies that if Expχ1 ⊗χ2 (B(V )an ) 6= ∅ for a character χ1 ⊗χ2satisfying condition (2), then there is also a character χ01 ⊗χ02 satisfying condition (1)00for which Expχ1 ⊗χ2 (B(V )an ) 6= ∅. Thus we may assume that χ1 ⊗ χ2 satisfies condition (1). Corollary 5.3.6 then implies that B(V ) may be identified with one of therepresentations appearing in the statement of that corollary. But one checks (again
using Proposition 5.2.1) that none of these contains a copy of η | | ⊗ψε | |−1 in its
Jacquet module (since η ⊗ ψ is neither unitary nor critical). This contradicts theconclusion of the first paragraph, and the lemma follows.

The preceding lemma, when combined with the computation of the Jacquet modules of the representations B(V ) attached to reducible V , implies that if V is reducible and W is irreducible trianguline, then B(V ) and B(W ) are not isomorphic.
Combined with Theorem 6.6.1 (2), it shows that the correspondence V 7→ B(V ) for
trianguline representations satisfies Conjecture 3.3.1 (1), if one restricts the rangeof that condition to trianguline representations.
It remains to discuss conditions (5), (7), and (8) in the case of irreducible trianguline V . Condition (5) has been verified by Berger [5] (improving on earlier resultsof various authors [7, 9, 10, 15]). We now turn to condition (7). Write B(V )exp.lalg
to denote the expected subspace of locally algebraic vectors in B(V ), according
to Conjecture 3.3.1; that is B(V )exp.lalg = 0 if V is not potentially semi-stable
ep (V ) if V is potentiallywith distinct Hodge-Tate weights, and B(V )exp.lalg = π
semi-stable with distinct Hodge-Tate weights. It is clear from the construction of

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

55

B(V ), and the fact that it is non-zero (by Theorem 6.6.1), that there is an injection B(V )exp.lalg ⊂ B(V )lalg . However it doesn’t seem to be known whether or
not this inclusion is an equality. For example, if U (resp. W ) is a special or cuspidal irreducible admissible smooth representation (resp. an irreducible algebraic
representation) of GL2 (Qp ), and if the central characters of π := U ⊗ W and ofB(V ) coincide, then it doesn’t seem possible to rule out the existence of an injection π → B(V )lalg whose image is not contained in B(V )exp.lalg , since the possible
admissible unitary completions of π have not been clasified. (If V is the restriction
to GQp of the global Galois representation attached to a finite slope overconvergent
eigenform, then we can rule out this possibility, and thus verify condition (7) forsuch V – see Remark 7.10.6 below.)
Our knowledge regarding condition (8) is similarly partial. Namely, we have thefollowing result.
6.6.5. Proposition. Let V be a trianguline irreducible two dimensional continuousb
GQp -representation, and η ⊗ ψ ∈ T(E)+.
(1) If η ⊗ ψ is not ultracritical, and if ηψ −1 is not of the form εz n for somen > 0, thendim Ref η⊗ψ (V ) = dim Expη|(2) If ηψ

−1

|⊗ψε|

|−1

|⊗ψε|

|−1

(B(V )an ).

n= εz for some n > 0, thendim Ref η⊗ψ (V ) ≤ dim Expη|(B(V )an ),with equality if B(W )exp.lalg = B(W )lalg for all twists W of V .
(3) If η ⊗ ψ is ultracritical, thendim Ref η⊗ψ (V ) ≥ dim Expη|

|⊗ψε|

|−1

(B(V )an ).

Proof. If η⊗ψ is unitary or critical then Ref η⊗ψ (V ) = ∅ by Lemmas 4.4.1 and 4.5.3,−1while Expη| |⊗ψ| |ε (V ) = ∅ by Lemma 6.6.4. Thus we may suppose that η ⊗ ψ
is neither unitary nor critical.
We begin by establishing the inequality
(41)

−1

dim Ref η⊗ψ (V ) ≤ dim Expη| |⊗ψε| | (B(V )an )for all non-ultracritical characters η⊗ψ (assumed to be neither unitary nor critical).
First note that if Ref η⊗ψ (V ) is empty, then this inequality is trivially satisfied. Thus
we may as well suppose that there exists a a non-ultracritical refinement R of V
such that σ(R) = η ⊗ ψ. Lemma 6.6.3, applied to the identity map from B(V ) to
itself, then shows that the right-hand side of (41) is non-negative. Thus (41) holds
in all cases.
Now suppose that η ⊗ ψ satisfies the conditions of (1). It then follows from
Remark 5.3.12 that B(η⊗ψε) is topologically irreducible and admissible unitary. By
Theorem 4.5.4 there exits a unique (up to isomorphism) irreducible trianguline twodimensional representation W of GQp such that Ref η⊗ψ (W ) 6= ∅; then by definition
B(W ) = B(η ⊗ ψε). If V and W are not isomorphic then by construction of W wehave Ref η⊗ψ (V ) = ∅. Also Theorem 6.6.1 (2) shows that B(V ) and B(W ) are not−1
isomorphic, and so by Corollary 5.3.14 we see that Expη| |⊗ψε| | (B(V )an ) = ∅.
Thus the equality of (1) holds in this case. Suppose on the other hand that V and−1
W are isomorphic. Then we must show that dim Expη| |⊗ψε| | (B(V )an ) = 0.

56

MATTHEW EMERTON

This follows from Theorem 5.2.5 and the fact that B(V ) satisfies Schur’s lemma.
This completes the proof of (1).
Suppose next that ηψ −1 = εz n for some n > 0. The inequality of (2) is then aspecial case of (41). Write π := η ⊗ St ⊗ (Symn E 2 )∨ . Twisting both V and (η, ψ)appropriately, we may assume that η is of integral Hodge-Tate weight, and thusthat π is locally algebraic. We then have−1

dim Expη| |⊗ψε| | (B(V )an ) = dim HomGL2 (Qp ) (B(η ⊗ ψε), B(V )) − 1= dim HomGL2 (Qp ) (π, B(V )) − 1 = dim HomGL2 (Qp ) (π, B(V )lalg ) − 1,
the first and second equality following from Theorem 5.2.5 and the fact that B(η ⊗ψε) is the universal unitary completion of π, and the third from the fact that π
is a locally algebraic representation. If we assume that B(V )lalg has the structure
predicted by Conjecture 3.3.1 (7), then one easily checks that dim Ref η⊗ψ (V ) =dim HomGL2 (Qp ) (π, B(V )lalg ) − 1. (Both sides of the equation equal 0.) This gives
equality in (2).
Finally, let us assume that η ⊗ ψ is ultracritical, and let w > 0 denote the
Hodge-Tate weight of ηψ −1 . Proposition 5.2.6 yields an inequality
(42)

−1

dim Expη| |⊗ψε| | (B(V )an ) ≤ dim Expη| |z−w⊗ψε| |−1 z w(B(V )an ).

On the other hand, we have an equality
dim Ref η⊗ψ (V ) = dim Ref ηz

−w

⊗ψz w

(V )

(as follows from Proposition 4.2.4 and the fact that both dimensions are necessarily−1 if V is not Hodge-Tate up to twist, or is potentially semi-stable up to twist).
Thus (3) follows from (1), applied to ηz −w ⊗ ψz w .
If V is the restriction to GQp of the GQ -representation attached to a finite slope
overconvergent eigenform, then we will show in Remark 7.10.6 below that in fact
B(V ) does satisfy condition (8).
Note that our inability to prove an equality rather than an inequality in generalin part (2) of this result is again due to our lack of knowledge of the possible
admissible unitary completions of representations of the form Special ⊗ algebraic.
One way to bypass this difficulty, and to establish conditions (7) and (8) completely
for the representations B(V ) under consideration, would be to have an explicit
description of B(V )an (which would then allow us to calculate both JP(Qp ) (B(V )an )and B(V )lalg ). In the following subsection we will present a conjectural descriptionof B(V )an of the desired type.
6.7. Conjectures on locally analytic vectors. In this subsection we describe
the expected structure of the locally analytic GL2 (Qp )-representation B(V )an associated to any trianguline continuous two dimensional representation V of GQp over
E. We consider various cases in turn.
6.7.1. V is irreducible, and does not admit an L-invariant. According to Definition 6.1.2, in this case B(V ) is defined to be B(R) for an appropriately chosenrefinement R of V . (Any refinement that is not ultracritical will do.) Thus, if wewrite σ(R) = (η, ψ), then B(V ) is equal to the universal unitary completion of thelocally analytic induction
GL (Q )
(43)
IndP(Q2 ) p η ⊗ ψε an ,pLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

57

and so there is a natural morphismGL (Q )
IndP(Q2 ) p η ⊗ ψε an → B(V )an ,

(44)

p

whose image is dense in B(V ) by construction.
6.7.2. Lemma. The map (44) is injective (provided that B(V ) is non-zero).
Proof. If ηψ −1 is not of positive integral Hodge-Tate weight, then the locally analytic induction (43) is topologically irreducible, and the lemma immediately follows.
If instead ηψ −1 is of Hodge-Tate weight w, for some integer w > 0, then (43) contains the locally algebraic induction
GL (Q )
(45)
IndP(Q2 ) p η ⊗ ψε lalgp

as its unique non-zero proper submodule. Since V is irreducible, R is not an
ordinary refinement. Also, R is not ultracritical by assumption. It follows from
Lemma 4.5.3 that 1 >| η(p) |>| p |w , and thus by Proposition 5.3.8 we see that the
locally algebraic induction (45) is itself dense in B(V ). Thus the lemma follows inthis case also.

As we have noted in the preceding subsection, it is known in almost all casesthat B(V ) is indeed non-zero.
GL (Q )
6.7.3. Conjecture. The cokernel of (44) is isomorphic to IndP(Q2 ) p ψ ⊗ ηε an ,pand thus the space of locally analytic vectors B(V )an sits in a short exact sequenceof locally analytic GL2 (Qp )-representations
GL (Q )GL (Q )
(46)
0 → IndP(Q2 ) p η ⊗ ψε an → B(V )an → IndP(Q2 ) p ψ ⊗ ηε an → 0.
pp

6.7.4. Lemma. Any extension of the form (46) is necessarily non-split.
Proof. If the extension (46) were to split, then the locally analytic induction
GL (Q )
(47)
IndP(Q2 ) p ψ ⊗ ηε anpwould admit an embedding into the unitary representation B(V ). On the other
hand, since the pair of characters (η, ψ) arises from a non-ordinary refinement(non-ordinary because V is assumed irreducible), we see that | η(p) |< 1, and thusthat | ψ(p) |> 1. A consideration of Jacquet modules (more precisely, Lemma 5.2.4)
shows that (47) does not admits a GL2 (Qp )-invariant norm. Thus the extension
under consideration must be non-split, as claimed.

Since we are in the case when V does not admit an L-invariant, the trianguline
representation V is completely determined by R, and so we expect that the extension appearing in Conjecture 6.7.3 should not depend on any additional invariant.
This suggests that, up to isomorphism, there is a unique non-trivial extension ofGL (Q )GL (Q )
IndP(Q2 ) p ψ⊗ηε an by IndP(Q2 ) p η⊗ψε an in the category of locally analytic repppresentations of GL2 (Qp ). (Reasoning analogous to that presented in Remark 6.3.2
shows that any such extension would necessarily admit a central character.)

58

MATTHEW EMERTON

6.7.5. Justification for and elaboration on Conjecture 6.7.3. Some justification forthis conjecture arises by comparison with the reducible case; see [6, Cor 7.2.6]
and 6.7.10 below. In Proposition 7.6.5 below, we will show in the case when V
arises from a twist of a finite slope overconvergent eigenform that is non-classical,
but of integral weight (so that V is Hodge-Tate, but not potentially semi-stable, up
to twist, by [41, Thm. 6.6]), that the space B(V )an does contain as a subrepresenGL (Q )
tation an extension of the locally algebraic representation IndP(Q2 ) p ψ ⊗ ηε lalgpGL (Q )by IndP(Q2 ) p η ⊗ ψε an . This provides some evidence for the conjecture.
p

Consider now the case when V is potentially crystalline and generic (so that it
satisfies our assumption of not admitting an L-invariant) and also Frobenius semiGL (Q )simple. In this case it is known that B(V )an contains an extension of IndP(Q2 ) p ψ⊗p
GL (Q )
ηε an by IndP(Q2 ) p η ⊗ ψε an , as we now recall.
p

It will be harmless to replace V by a twist (since this just replaces B(V ) by thesame twist), and so we may and do assume that V is in fact potentially crystalline
with Hodge-Tate weights 0 and w, for some integer w < 0. Since V is Frobenius
semi-simple, it admits a unique (up to equivalence) pair of inequivalent refinements
R1 and R2 . If we write σ(R1 ) = (η, ψ), then σ(R2 ) = (ψz −w , ηz w ). Considering (44)with R taken to be R1 and R2 in turn, we obtain maps
GL (Q )
IndP(Q2 ) p η ⊗ ψε an → B(V )an

(48)

p

and

GL (Q )
IndP(Q2 ) p ψz −w ⊗ ηz w ε an → B(V )an ,

(49)

p

which are injective by Lemma 6.7.2 (since B(V ) is non-zero by Theorem 6.6.1 (1)),
and which coincide (up to multiplication by a non-zero scalar) on the common
locally algebraic subrepresentation of their sources∼
GL (Q )GL (Q )
IndP(Q2 ) p η ⊗ ψε lalg −→ IndP(Q2 ) p ψz −w ⊗ ηz w ε lalgpp(the isomorphism being provided by the theory of intertwining operators for smooth
principal series). Thus we obtain an injection
(50)

 M

g
GL (Q )GL (Q )
IndP(Q2 ) p ψz −w ⊗ ηz w ε an → B(V )an ,IndP(Q2 ) p η ⊗ ψε anppLwhere f indicates that we form the amalgamated sum of the two summands overtheir common locally algebraic subrepresentation. Since the quotient
GL (Q )GL (Q )
IndP(Q2 ) p ψz −w ⊗ ηz w ε an / IndP(Q2 ) p ψz −w ⊗ ηz w ε lalgppGL (Q )
is isomorphic to IndP(Q2 ) p ψ ⊗ηε an , we see that the source of (50) is an extensionpGL (Q )GL (Q )of IndP(Q2 ) p ψ⊗ηε an by IndP(Q2 ) p η⊗ψε an . Thus in this case Conjecture 6.7.3
pp

is equivalent to conjecturing that (50) is an isomorphism. The author learnt thiscase of Conjecture 6.7.3 from Breuil. (See [8, Conj. 5.4.4].)

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

59

6.7.6. V is irreducible, and admits an L-invariant. As in 5.1.7, we replace V by atwist, extending E if necessary, so that V has Hodge-Tate weights 0 and w, and sothat B(V ) is equal to the universal unitary completion of Σ(1 − w, L).
It follows directly from Lemma 5.1.9 that the tautological map
Σ(1 − w, L) → B(V )an(51)

is an injection. We then make the following conjecture, which is the natural analogue of Conjecture 6.7.3 in the present context.
GL (Q )

6.7.7. Conjecture. The cokernel of (51) is isomorphic to IndP(Q2 ) p | |(−1+w)/2p
z w ⊗ | |(3+w)/2 z an , and thus the space of locally analytic vectors B(V )an sits in ashort exact sequence of locally analytic GL2 (Qp )-representations(52)
GL (Q )
0 → Σ(1 − w, L) → B(V )an → IndP(Q2 ) p | |(−1+w)/2 z w ⊗ | |(3+w)/2 z an → 0.
p

6.7.8. Lemma. Any extension of the form (46) is necessarily non-split.
Proof. This follows by the same argument as used in the proof of Lemma 6.7.4.



6.7.9. V is reducible. The description of B(V )an in this case is straightforward, if
we assume that B(V ) has the structure conjectured in Subsections 6.2 through 6.5,
using the following two facts: the space of locally analytic vectors in a continuous
GL (Q )
induction IndP(Q2 ) p χ1 ⊗χ2 cont coincides with the corresponding locally analyticpGL (Q )
induction IndP(Q2 ) p χ1 ⊗ χ2 an ; passing to locally analytic vectors is an exactpfunctor [51,L
Thm. 7.1].
If V = η ψ and ηψ −1 6= ε±1 , so that B(V ) is as defined in Definition 6.2.3 (1),then we will have MGL (Q )GL (Q )
B(V )an = IndP(Q2 ) p η ⊗ ψε an
IndP(Q2 ) p ψ ⊗ ηε an .
pp
L
If V = η ηε−1 , so that B(V ) is as defined in Definition 6.2.3 (2), then we willhaveMGL (Q )
IndP(Q2 ) p ηε−1 ⊗ ηε an .
B(V )an = η ◦ det ⊗B(2, ∞)anp

As is discussed below, for any value of L ∈ P1 (E), the space η ◦ det ⊗B(2, L)an isb an .
an extension of η ◦ det by η ◦ det ⊗St
If V is a non-split extension 0 → η → V → ψ → 0, and ηψ −1 6= ε, so that
B(V ) is a non-split extension of the form (32), then B(V )an is an extension of theform (46) which is itself necessarily non-split (since B(V ) may be recovered from
B(V )an by passing to universal unitary completions).
Finally, if V is an extension of the form 0 → η → V → ηε−1 → 0, so that B(V )is a non-split extension
GL (Q )

0 → η ◦ det ⊗B(2, L) → B(V ) → η ◦ det ⊗(IndP(Q2 ) p ε−1 ⊗ ε)cont → 0,pwhere η ◦ det ⊗B(2, L) sits in the non-split short exact sequenceb = IndGL2 (Qp ) η ⊗ η /η ◦ det → η ◦ det ⊗B(2, L) → η ◦ det → 0
0 → η ◦ det ⊗St
an
P(Q )
p

whose isomorphism class is classified by the L-invariant of V (in P1 (E)), thenB(V )an and η ◦ det ⊗B(2, L)an sit in extensions
GL (Q )

0 → η ◦ det B(2, L)an → B(V )an → (IndP(Q2 ) p ηε−1 ⊗ ηε)an → 0,p

60

MATTHEW EMERTON

and
b an = IndGL2 (Qp ) η ⊗ η /η ◦ det
0 → η ◦ det ⊗St
an
P(Q )
p

→ η ◦ det ⊗B(2, L)an → η ◦ det → 0,
both of which are non-split (for the same reason as in the previous case, namely that
B(V ) (resp. η◦det ⊗B(2, L)) can be recovered from B(V )an (resp. η◦det ⊗B(2, L)an )by passing to universal unitary completions).
6.7.10. Additional remarks. In the case when V is indecomposable (but possibly
reducible) and does not admit an L-invariant, we have seen that B(V )an should
always be a non-split extension of the form (46). When V is not of Hodge-Tate
type, or is of Hodge-Tate type but the two Hodge-Tate weights of V coincide, sothat neither η ⊗ ψε nor ψ ⊗ ηε is of non-negative integral Hodge-Tate weight, boththe locally analytic inductions appearing in (46) are topologically irreducible [49,Thm. 6.1], and so there is (conjecturally) no further “internal structure” in B(V )an .
However, if V is Hodge-Tate up to twist, with distinct weights, then either
GL (Q )
(53)
IndP(Q2 ) p η ⊗ ψε lalg
p

or
(54)


GL (Q )
IndP(Q2 ) p ψ ⊗ ηε anpcontains a locally algebraic subrepresentation, and so B(V )an has (conjecturally)
topological length three. Set w 6= 0 to be the Hodge-Tate weight of ηψ −1 .
In the case when w < 0, so that V is furthermore potentially crystalline up to
twist, it is (53) that contains a locally algebraic representation, and in 6.7.5 wesaw that (when V is irreducible) B(V )an should be the amalgamated sum of two
locally analytic inductions over their common locally algebraic subrepresentation.
In the case when V is reducible, so that B(V ) is defined by Definition 6.3.3, one
can compute directly that B(V )an has this same structure [6, Cor. 7.2.6]. The
lattice of closed subrepresentations of B(V )an thus (conjecturally) has the following
structure:
B(V )an6
 666

66 IndGL2 (Qp ) ηzw ⊗ψz−w εGL (Q )
Ind 2 p ψ⊗ηε66 P(Qp )P(Qp )
an 
an
66


66

66



•7
•
77

77


77

7



77
GL2 (Qp )GL (Q )
 IndP(Qψ⊗ηεInd 2 p ηz w ⊗ψz −w ε77
P(Qp ))ananp77 
•

GL2 (Qp )η⊗ψεP(Qp )
lalg

Ind

0

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

61(where the labels indicate the corresponding topological Jordan-Hölder factors).
In the case when w > 0, so that V is not potentially crystalline up to twist, itis (54) that contains a locally algebraic subrepresentation. Thus the extension (46)contains a submodule W which is an extension
GL (Q )GL (Q )
0 → IndP(Q2 ) p η ⊗ ψε an → W → IndP(Q2 ) p ψ ⊗ ηε lalg → 0,ppwhich must also be non-split. (When V is irreducible, this follows by the same
argument that proves lemma 6.7.4. When V is reducible, so that ψ ⊗ ηε is unitary,
we see that if this extension were split, then the same would be true of (46) itself,by Proposition 5.3.8.) The lattice of closed subrepresentations of B(V )an thus
(conjecturally) has the following structure:
B(V )anGL2 (Qp )ψz −w ⊗ηz w εP(Qp )
an

Ind

•


GL2 (Qp )ψ⊗ηεP(Qp )
lalg

Ind

•


GL2 (Qp )η⊗ψεP(Qp )anInd

0(where again we have labelled the various topological Jordan-Hölder factors). We
will see in Subsection 7.6 below (see Proposition 7.6.5 and its proof) that in thecase when V arises from an overconvergent eigenform of finite slope, the appearance of the locally algebraic representation in the middle of the composition series,
rather than at the bottom, corresponds to the existence of the operator θw on
overconvergent forms of weight 1 − w.
7. Local-global compatibility
The goal of this section is to discuss in some detail Conjecture 1.1.1 of theintroduction.
7.1. p-adic GL2 (Af )-representations associated to p-adic representationsof GQ . Suppose that V is a two dimensional continuous representation of GQ defined over E, which is unramified away from a finite number of primes. As in Subsection 2.2, we may attach the admissible smooth GL2 (Apf )-representation π m, p (V )to V , which by Remark 2.2.1 is defined over E.
Assuming that the local p-adic Langlands correspondence predicted by Conjecture 3.3.1 exists, we may also associate to V |Dp the admissible unitary Banach space
representation B(V |Dp ) of GL2 (Qp ) over E.

62

MATTHEW EMERTON

7.1.1. Definition. Write Π(V ) := B(V |Dp ) ⊗E π m, p (V ), equipped with the inductive tensor product topology (in the sense of [48, §17]).
The locally convex E-space Π(V ) is equipped with an action of GL2 (Af ) =GL2 (Qp ) × GL2 (Apf ) which makes it an admissible continuous representation of thisgroup (in the sense of [31, Def. 7.2.1]).
7.1.2. Remark. We remark for future reference that if V = Vf for some classicalcuspidal newform f , then π p (V ) = π m, p (V ); equivalently, the local representationsπ` (V ) ∼= π` (f ) are all generic [40, p. 354]. Thus for such V we may also write
Π(V ) = B(V |Dp )⊗E π p (V ). Since the formation of Π(V ) is compatible with twistingin an evident sense, the same remark hold for those V that are twists of Vf .
Among all the admissible continuous representations of GL2 (Af ) one would liketo single out those representations of the form Π(V ) as being “automorphic”. Ideally, one would have an a priori definition of what it means for an admissible continuous GL2 (Af )-representation to be automorphic (perhaps in terms of a spaceof “p-adic automorphic forms” equipped with an admissible continuous GL2 (Af )action), and then one could phrase a global p-adic Langlands conjecture to theeffect that these are precisely the representations of the form Π(V ). Unfortunately,
no such a priori definition is known as of yet; more precisely, one does not actually
have a candidate for the space of p-adic automorphic forms for GL2 over Q. However, in the discussion that follows we will present a workable substitute for thisspace, which one hopes should detect those Π(V ) for which V is an odd irreducible
representation.
7.2. Completed cohomology of modular curves. We recall some construcb p ); wetions from [33]. To begin with, fix a compact open subgroup K p of GL2 (Zprefer to K as the “tame level”. Fix a finite extension E of Qp with ring of integers OE . For A = OE , OE /$s (for some s > 0), or E, write H∗1 (K p )A :=lim
H∗1 (Y (K p Kp )/Q , A), where the inductive limit is taken over all open subgroups−→
Kpof GL2 (Zp ), the cohomology is étale cohomology, and ∗ = ∅ or c (i.e. we are considering either cohomology with unrestricted supports or cohomology with compact
supports.)
7.2.1. Lemma. The OE -module H∗1 (K p )OE is torsion free and p-adically separated.
Proof. That H∗1 (K p )OE is torsion free is clear, since this is true of the étale cohomology with coefficients in OE of any curve over an algebraically closed field.
The claim of separatedness follows from the fact that the map [33, (4.3.4)] is an
isomorphism.
There is a natural inclusion H∗1 (K p )OE ⊂ H∗1 (K p )E . Since H∗1 (K p )OE spans
1H∗ (K p )E as an E-vector space, Lemma 7.2.1 shows that we may put a normon H∗1 (K p )E – the so-called gauge of H∗1 (K p )OE – whose unit ball is equal toH∗1 (K p )OE .
b ∗1 (K p )E to be the p-adic Banach space obtained by
7.2.2. Definition. Define H1
pb ∗1 (K p )O
completing H∗ (K )E with respect to the gauge of H∗1 (K p )OE . We let H
E1pb ∗ (K )E ; it is naturally identified with the p-adic completion
denote the unit ball of H
of H∗1 (K p )OE .

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

63

7.2.3. Lemma. For any s > 0, there is a natural isomorphism∼b ∗1 (K p )O /$s −→H
lim H∗1 (Y (Kp K p ), OE /$s )
E

−→
Kp

(where Kp runs over the directed set of all compact open subgroups of GL2 (Qp )).
∼ e1
b ∗1 (K p )O −→Proof. This follows from the isomorphism H
H∗ (K p )OE discussed inE[33, §4.1].
The following corollary results immediately from the preceding lemma, and theb ∗1 (K p )O .
definition of H
E
7.2.4. Corollary. There is a natural isomorphism∼b ∗1 (K p )O −→b ∗1 (Y (Kp K p ), OE /$s ).
H
lim lim H
E

←− −→
s
Kp

There is a natural action of GL2 (Qp ) on H∗1 (K p )E , induced by the right actionof GL2 (Qp ) on the tower of modular curves Y (Kp K p ) (for varying Kp and fixedK p ). This action preserves H∗1 (K p )OE , and thus extends to an action of GL2 (Qp )b 1 (K p )E which preserves H
b 1 (K p )O .
on H
∗
∗
E
b 1 (K p )E equips this Banach space with
7.2.5. Lemma. The GL2 (Qp )-action on H∗
an admissible unitary representation of GL2 (Qp ).
Proof. This is a special case of [33, Thm. 2.2.11].



7.2.6. Definition. If Σ is a fixed finite set of primes, we write T (or T(Σ) if wewish to emphasize Σ) to denote the polynomial algebra OE [{T` , S` }`6∈Σ ].
Let Σ(K p ) denote the set of primes at which K p is ramified, together with p.
For any finite set of primes Σ containing Σ(K p ), the usual action of T := T(Σ) on
H∗1 (K p )E via Hecke operators commutes with the GL2 (Qp )-action and preservesb ∗1 (K p )E
the lattice H∗1 (K p )OE , and so extends to a continuous action of T on H
b 1 (K p )O .
that commutes with the GL2 (Qp )-action and preserves H∗Epp
If K1 ⊂ K2 is an inclusion of tame levels, then the injection H∗1 (K2p )E →H∗1 (K1p )E induced by pulling back cohomology classes is GL2 (Qp )-equivariant, andinduces a closed GL2 (Qp )-equivariant embedding(55)b ∗1 (K p )E → H
b ∗1 (K p )E .
H
2
1

(See [33, Prop. 2.2.13].) If Σ is any finite set of primes containing Σ(K2p ), then thisembedding is also T(Σ)-equivariant.
7.2.7. Definition. Define1b ∗,Eb ∗1 (K p )EH:= limH−→
Kp(the inductive limit being taken over all tame levels K p ), endowed with the inductive
limit GL2 (Qp )-action, and with the locally convex inductive limit topology.
b 1 commuting with the GL2 (Qp )There is a smooth action of GL2 (Apf ) on H∗,Epaction, induced by the action of GL2 (Af ) on the tower of modular curves Y (Kf ).
(See [33, Thm. 2.2.16] for more details.) This action is compatible with the actionb ∗1 (K p )E , in a manner thatof the Hecke algebras T Σ(K p ) on the various spaces H
the following lemma makes precise.

64

MATTHEW EMERTONb ∗1 (K p )E
7.2.8. Lemma. If K p is any tame level, then the natural injection of H1binto H
induces an isomorphism of continuous GL2 (Qp )-representations
∗,E

p
∼
1
b 1 (K p )E −→
b ∗,E
H
(H
)K .
∗

Furthermore, this isomorphism intertwines
 theaction of T` (resp.
 S` ) on the source
p ` 0
p
p ` 0
with the action of the double coset K
K (resp. KK p ) on the
0 1
0 `target, for each prime ` 6= p that is unramified in K p .
Proof. The first claim follows from [33, Thm. 2.2.16 (ii)]. For the second claim, see
the formulas of [33, Prop. 4.4.2].

b 1 . Each of the spaces H∗1 (K p )E is also equipped with
7.2.9. The GQ -action on H∗,E
a continuous action of GQ that preserves H∗1 (K p )OE , and so extends to a continuousb ∗1 (K p )E that preserves Hb ∗1 (K p )O . This action commutes withaction of GQ on HE
the actions of GL2 (Qp ) and T, and is compatible with the embeddings (55). It isunramified outside of Σ(K p ). Passing to the inductive limit, we obtain a continuousb 1 , commuting with the GL2 (Qp ) and GL2 (Ap )-actions.
action of GQ on H∗,Ef
7.2.10. Forgetting supports. Forgetting supports induces GQ ×GL2 (Af )-equivariant
morphisms
(56)

b1
b1
H
c,OE → H OE

and
(57)

1
b c,E
b 1E
H
→H

that are furthermore surjective. Indeed, for each tame level K p , the maps
(58)

b c1 (K p )O → H
b 1 (K p )O
H
E
E

and
(59)

b c1 (K p )E → H
b 1 (K p )EH

are surjective. (See [33, Prop. 4.3.9] and its proof).
7.2.11. Completions of H 0 . We close this subsection with a brief discussion of theb 0 (K p )E and H
b 0 , which are defined in a completely analogous fashion to
spaces HE
b 1 (K p )E and H
b 1 (replacing the superscript 1 by 0 at each point). The spaceHE
b 0 (K p )E is equipped with commuting actions of GQ × GL2 (Qp ) and T(Σ(K p )),H
b 0 is equipped with an action of GQ × GL2 (Af ). The cup product defines awhile HE
b0 ⊗ Hb1 → Hb 1 for ∗ = ∅ or c [33, Prop. 2.2.21].
pairing H
E∗,E∗,E
Unlike the completions of H 1 , these completions of H 0 can be described very
explicitly. Namely, there is an isomorphism
(60)

∼
×
b 0E −→H
C (Q>0 \Af , E),

where the target of the isomorphism is the space of functions that are continuous
in the p-adic variable, and smooth in the prime-to-p-adic variables. We may deb 0 concretely in terms of this isomorphism:
scribe the GQ × GL2 (Af )-action on H
E∼
global class field theory yields an isomorphism GQab −→ Q>0 \A×f , and GQ acts on
××
C (Q>0 \Af , E) via the composite of this isomorphism with the action of Q>0 \AfLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

65

by translation; the group GL2 (Af ) act on C (Q>0 \A×
f , E) via the composite of thedeterminant map det : GL2 (Af ) → A×with
theactionof A×ff by translation.
7.3. Systems of Hecke eigenvalues. We will recall some basic terminology andfacts related to systems of Hecke eigenvalues. Fix a finite set of primes Σ containingp, and write T := T(Σ) as above.
7.3.1. Definition. If A is an OE -algebra, then a system of Hecke eigenvalues definedover A (and outside of Σ) is a homomorphism of OE -algebras λ : T → A.
If λ is a system of Hecke eigenvalues defined over A, and if M is a T-module,
λthen we write A ⊗OE M to denote the subspace of A ⊗OE M on which T acts
through the character λ.
7.3.2. Definition. Let A be an OE -algebra, and let V be a free A-module of rank
two equipped with a GQ -representation that is unramified outside Σ. We then definea system of Hecke eigenvalues over A via the formula: λ(T` ) = trace(Frob`−1 | V ),λ(`S` ) = det(Frob−1| V ), for all ` 6∈ Σ. We refer to λ as the system of Hecke`
eigenvalues associated to V .
Recall that F denotes the residue field OE /$.
7.3.3. Remark. As is well-known, if A is a finite extension of either E or F, and if
the representation V is continuous, then V is determined up to semi-simplificationby λ (by the Čebotarev density and Brauer-Nesbitt theorems).
7.3.4. Definition. If λ is a system of Hecke eigenvalues (outside of Σ) defined overb × → A× is a character, unramified outside of Σ,an OE -algebra A, and if ψ : Z
then we define λ ⊗ ψ (the twist of λ by ψ) to be the system of Hecke eigenvaluesT` 7→ ψ(`)λ(T` ), S` 7→ ψ(`)2 λ(S` ) (for ` 6∈ Σ).
Note that if λ is the system of Hecke eigenvalues attached to a two dimensional
GQ -representation V , then λ ⊗ ψ is the system of eigenvalues attached to the twistV ⊗ ψ (where we identify ψ with a character of GQ via global class field theory).
7.3.5. Definition. Let A be one of the fields E or F, and let A denote its algebraic
closure. We say that a system of Hecke eigenvalues λ defined over a finite extension
b × → A× ,of A is Eisenstein if there is a pair of continuous characters ψ1 , ψ2 : Z
unramified
outside of Σ, such that λ is the system of Hecke eigenvalues attached toLψ1 ψ2 , regarded as a GQ -representation via global class field theory. (Concretely,
this amounts to requiring that λ(T` ) = ψ1 (`) + ψ2 (`) and λ(`S` ) = ψ1 (`)ψ2 (`) forall primes ` 6∈ Σ.)
7.3.6. Remark. Remark 7.3.3 shows that the system of Hecke eigenvalues associated to a continuous two dimensional GQp -representation V defined over a finite
extension of E (resp. F) is Eisenstein if and only if V becomes reducible over afinite extension of E (resp. F).
7.3.7. Definition. We say that a p-torsion free T-module M is Eisenstein if any
system of Hecke eigenvalues λ defined over a finite extension E 0 of E in Qp forwhich (E 0 ⊗OE M )λ 6= 0 is Eisenstein.
There is a simple criterion for a T-module to be Eisenstein.

66

MATTHEW EMERTON

7.3.8. Lemma. Suppose that M is p-adically separated p-torsion free T-module withb× →the following property: there is a pair of commuting representations Ψ1 , Ψ2 : Z
Aut(M ), each unramified outside Σ and continuous when Aut(M ) is endowed withthe weak topology (i.e. the topology of pointwise convergence), such that for all ` 6∈ Σ
we have T` acts on M via Ψ1 (`) + Ψ2 (`) and `S` acts on M via Ψ1 (`)Ψ2 (`). Then
M is Eisenstein.
Proof. This is essentially what is shown in the proof of [14, Cor. 3.1.3]. We recall
the argument. Suppose that λ is a system of Hecke eigenvalues defined over afinite extension E 0 of E for which (E 0 ⊗OE M )λ 6= 0. We again write Ψi (`) todenote the extension of scalars of Ψi (`) to each of OE 0 ⊗OE M and E 0 ⊗OE M(i = 1, 2). The formulas for the action of T` and S` in terms of the automorphisms
Ψi (`) show that these automorphisms commute with the action of T, and thusthat (E 0 ⊗OE M )λ is stable under the action of the Ψi (`). Furthermore, on thisspace we have λ(T` ) = Ψ1 (`) + Ψ2 (`) and λ(`S` ) = Ψ1 (`)Ψ2 (`) for all ` 6∈ Σ.
In particular, multiplication by λ(T` ) and λ(S` ) preserves the non-zero p-adicallyseparated module (OE 0 ⊗OE M )λ , and so λ must take values in OE 0 .
Replacing E by E 0 and M by (OE 0 ⊗OE M )λ , we may assume that E 0 = E, that
M is a non-zero T-module, and that λ is an OE -valued system of Hecke eigenvalues
such that T acts on M through λ. We then find that for each i = 1, 2, we haveΨi (`)2 − λ(T` )Ψi (`) + λ(`S` ) = 0. Replacing E by the compositum of its (finitely
many) quadratic extensions, we may assume that each of these quadratics splitsb that topologically generate Z
b × /(Zb Σ )× . Since M
over E. Choose `1 , . . . , `r ∈ Z
is non-zero, and since the automorphisms Ψi (`j ) commute among themselves, we
may find a non-zero element m of M such that each Ψi (`j ) acts via a scalar on m,
and so deduce (since Ψ1 and Ψ2 are weakly continuous) that there exist continuousb × /(Zb Σ )× → O× such that Ψi (`)m = ψi (`)m (i = 1, 2). We
characters ψ1 , ψ2 : Z
E
conclude that λ(T` ) = ψ1 (`) + ψ2 (`) and λ(`S` ) = ψ1 (`)ψ2 (`), and thus that λ is
Eisenstein.
For example, one can use this criterion to show that for any tame level K p
unramified outside Σ, the kernel of the map (57) is an Eisenstein T-module. (SeeLem. 3.1.2 and Cor. 3.1.3 of [14].)7.3.9. Definition. If K p is a tame level unramified outside of Σ, then we writee ∗ (K p ) to denote the weak closure of the image of T in End Hb ∗1 (K p )E . (OfT
course here End denotes continuous endomorphisms.)e ∗ (K p ). For any compact open
We can give a more concrete description of Te ∗ (Kp K p )s to denote thesubgroup Kp of GL2 (Qp ), and any integer s > 0, write T11ps
image of T in End H∗ (Y (Kp K ), OE /$ ) . Since H∗ (Y (Kp K p ), OE /$s ) is a finitee ∗ (Kp K p )s is a finite OE -algebra. Corollary 7.2.4 gives a
OE -module, we see that T
topological isomorphism(61)∼e ∗ (Kp ) −→e ∗ (Kp K p )s .
T
lim
T
←−
s,Kp

The surjection (59) induces a continuous homomorphism(62)

e c (K p ) → T(Ke p ),
T

which has dense image by construction.

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

67

7.3.10. Lemma. The map (62) is an isomorphism of reduced, compact topologicalOE -algebras.
Proof. The isomorphism (61) shows that both the source and target of (62) are
profinite, and so compact. Since this map is continuous with dense image, it must bee ∗ (Kp K p ) := lim Te ∗ (K p Kp )s , thensurjective. If we fix the level Kp K p , and write T←−se ∗ (K p Kp ) is naturally identified with the image of T in End H 1 (Y (Kp K p ), OE ) .
T∗
This image is well-known to be reduced. Thus, rewriting (61) in the form∼e ∗ (Kp ) −→e ∗ (Kp K p ),
T
lim
T
←−
Kp

e ∗ (K p ) as the projective limit of a projective
we obtain a description of the algebra Te ∗ (K p ) is reduced.
system of reduced rings. This shows that T
cE ⊂ Hb c1 (K p )E denote the kernel of (59), and write
Let M
\
\
e c (K p ) Hom(He c (K p ) Hom(Hb c1 (K p )E , M
cE ), J = T
b 1 (K p )E , M
cE )I=T(where Hom denotes continuous homomorphisms.) Clearly I is the kernel of (62),e c (K p ) that annihilates
while J 2 = 0. We claim that I = J, i.e. that any element of Tb 1 (K p ) also annihilates M
cE . It will follow that I 2 = 0, and thus that I = 0, sinceHe(as we have just shown) Tc (K p ) is reduced.
b × → Q× , unramified
Let Eis denote the set of pairs of characters ψ1 , ψ2 : Zpoutside of Σ, associated to the various classical Eisenstein series of weight 2 andtame level K p . For each (ψ1 , ψ2 ) ∈ Eis, there is a continuous homomorphism λψ1 ,ψ2 :e c (K p ) → Qp uniquely determined by the fact that its composite with the naturalTe c (K p ) is equal to the system of Hecke eigenvalues associated to ψ1 L ψ2 .
map T → T
(This is just the homomorphism describing the Hecke action on the Eisenstein series
attached to (ψ1 , ψ2 ); cf. Remark 7.3.14 below.) If Iψ1 ,ψ2 denotes the kernel of λψ1 ,ψ2 ,e c (K p ) is equal to T
cE in T
then the annihilator of M
(ψ1 ,ψ2 )∈Eis Iψ1 ,ψ2 . (This followscfor example from the explicit description of ME given in [14].) Thus we must showe c (K p ) annihilates H
b 1 (K p )E , then t ∈ Iψ ,ψ for all (ψ1 , ψ2 ). This isthat if t ∈ T
1
2
clear, however, since the classical cohomology space H 1 (K p )E contains a class withannihilator equal to Iψ1 ,ψ2 (since the system of eigenvalues λψ1 ,ψ2 appears in thecokernel of the map Hc1 (K p )E → H 1 (K p )E ).

7.3.11. Definition. Let A be a finite field extension of E (resp. of F). We say thata system of Hecke eigenvalues λ defined over A is promodular (resp. modular) if
e p ) with a continit can be written as the composite of the natural map T → T(Kpe : T(Keuous homomorphism λ) → A, for some tame level K p that is unramified
outside Σ.
e p ) with T
e c (K p ) in the
7.3.12. Remark. Lemma 7.3.10 shows that replacing T(Kpreceding definition would yield an equivalent notion.
7.3.13. Lemma. Let λ be a system of eigenvalues defined over a finite extensionb 1 (K p )E 0 )λ 6= 0, then λ is promodular.
E 0 of E. If (He p ), we see that any element of T(Ke p ) acts
Proof. Since T has dense image in T(K
1
λ
p
1
e
b
b
on the vectors of (H (Kp )E 0 ) through a scalar. Since T(K ) acts on H (Kp )E 0 via

68

MATTHEW EMERTON

e : T(Ke p ) → E 0 is continuous.
continuous operators, the resulting homomorphism λe we seee p ) with λ,Since λ is obtained as the composite of the natural map T → T(Kthat λ is promodular.

7.3.14. Remark. If f is a Hecke eigenform of weight k ≥ 2 and tame level K p ,either cuspidal or an Eisenstein series, then the system of Hecke eigenvalues λattached to f is promodular. If f is of weight 2, then this follows directly from thepreceding lemma and Eichler-Shimura theory, which shows that (H 1 (K p )E 0 )λ 6= 0.
If k > 2, then (H 1 (K p )E 0 )λ = 0. Nevertheless, Theorem 7.4.2 below shows thatb 1 0 )λ 6= 0, and so the preceding lemma still applies. Any weight 1 eigenform(HE
also gives rise to a promodular system of Hecke eigenvalues: taking its productwith the members of a sequence of positive weight Eisenstein series that converges
p-adically to the constant q-expansion 1, we obtain a sequence of modular forms ofweights ≥ 2 whose members are eigenforms modulo increasingly large powers of pthat converges (on the level of q-expansions) to f .
7.3.15. Definition. Let A be a finite extension of E (resp. of F), and let V bea continuous two dimensional GQp -representation defined over A and unramified
outside of Σ. We say that V is promodular (resp. modular) if its associated system
of Hecke eigenvalues is promodular (resp. modular).
If V is given, but the set of primes Σ is not specified, we will say that V is
promodular (or modular) if it is so with respect to some set of primes Σ containing
p together with all the ramified primes of V .
7.3.16. Remark. Let V be an absolutely irreducible continuous two dimensional
GQ -representation over F. Then V is modular in the above sense if and only if it ismodular in the usual sense of being obtained as the reduction mod $ of the Galois
representation attached to some newform. Furthermore, let X be the deformation
space associated to V (parameterizing deformations of V to GQ -representations
over Artinian local OE -algebras with residue field equal to F), which is a formal
scheme over OE , and let X be its rigid analytic generic fibre. Let X mod denote therigid analytic Zariski closure in X of the set of points arising from deformations of
V attached to classical modular forms of weight ≥ 2 and level unramified outsideof Σ. One can check that a lift of V to a representation V over a finite extension Aof E is promodular in the above sense if and only if it corresponds to an A-valued
point of X mod .
7.3.17. Proposition. Let A be a finite field extension of E (resp. of F). If λ :T → A is a promodular (resp. modular) system of Hecke eigenvalues, then there isa continuous two dimensional Galois representation V defined over A, unramified
outside of Σ, and such that λ is the system of Hecke eigenvalues associated to V .
Any such V is necessarily odd.
e the homoProof. This result is well-known, but we recall the proof. Denote by λpemorphism T(K ) → A associated to λ (for an appropriate choice of tame level K p ).
e factors
In the case when A is an extension of F, the isomorphism (61) shows that λpthrough T(Kp K )1 → A for some sufficiently small Kp . Thus (by the Deligne-Serrelemma [30, Lem. 6.11]) λ can be regarded as the reduction mod $ of the system of
Hecke eigenvalues attached to some weight 2 newform f , and we take V to be thereduction mod $ of Vf . (Note that V is odd, since Vf is.)

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

69

Now suppose that A is an extension of E. For each newform (either cuspidal
or Eisenstein) f of weight 2 and tame level K p , Remark 7.3.14 shows that theree p ) → Qp describing the action of the Hecke operators on
is a homomorphism T(Kb 1 (K p )E , Eichler-Shimura
f . Let If denote its kernel. Since H 1 (K p )E is dense in H
theory show that\(63)
If = 0f(the intersection running over all f ). Attached to each f is an odd continuous two dimensional representation Vf of GQ , unramified outside of Σ, and defined over (T/If )[1/p]. (When f isLthe Eisenstein series associated to a pairof characters (ψ1 , ψ2 ), set Vf = ψ1 ψ2 .) Taking the product of all these, we
Q e pobtain an odd continuous representation V of GQ into GL2 ( f (T(K)/If )[1/p])that is unramified outside Σ. From (63) we see that the natural (diagonal) mape p ) → Q (T(Ke p )/If )[1/p] realizes T(Ke p ) as a subalgebra of the product. SinceT(Kfe p ), one sees that the continuous pseudothe traces on V of elements of GQ lie in T(Ke p ) (or in T(Ke p ) ⊗O E, if
representation attached to V actually takes values in T(KE
p = 2); cf. the formulas of [24, p. 71]. Specializing this pseudo-representation withe yields a continuous pseudo-representation of GQ
respect to the homomorphism λover A that is unramified outside Σ. This continuous pseudo-representation thenunderlies a continuous two dimensional odd GQ -representation, unramified outside
Σ, which is our desired representation V .
Note that (in either case) if W is any other representation associated to the same
system of Hecke eigenvalues λ then V and W have isomorphic semi-simplifications(by Remark 7.3.3), and thus W is also odd.

7.3.18. Remark. In the context of the preceding proposition, it follows from Remarks 7.3.3 and 7.3.6 that if λ is not Eisenstein then V is uniquely determined up
to isomorphism, and is absolutely irreducible.
7.3.19. Lemma. If V is a promodular continuous two dimensional representationof GQ over E, and if ψ : GQ → E × is a continuous character, then V ⊗ ψ is again
promodular.
Proof. Choose the tame level K p so that the system of Hecke eigenvalues λ attachede : T(Ke p ) → E; further shrink K p
to V arises from a continuous homomorphism λif necessary so that ψ, regarded as a character of Q>0 \A×f by global class fieldptheory, is trivial on det(K ). Thinking of ψ as an element of C (Q>0 \A×
f , E), let
b 0 that corresponds to ψ under the isomorcψ denote the cohomology class in HEb ∗1 (K p )E , which interphism (60). Cupping with cψ induces an automorphism of H
twines the Hecke operator T` (resp. S` ) with the operator ψ(`)T` (resp. ψ(`)2 S` ) forany ` ∈ Σ(K p ). Thus the automorphsm T` 7→ ψ(`)T` , S` 7→ ψ(`)2 S` of T(Σ(K p ))e withe p ). Composing λinduces a corresponding continuous automorphism of T(K0e , whose restriction tothis automorphism yields a continuous homomorphism λT(Σ(K p )) is immediately seen to coincide with λ ⊗ ψ. Thus λ ⊗ ψ, and so alsoV ⊗ ψ, is promodular.

b 1 . We will give an explicit description of
7.4. Locally algebraic vectors in H∗,Eb1
the space H∗,E lalg .

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MATTHEW EMERTON

7.4.1. Definition. Let E(1) denote a one dimensional vector space over E on
which GQ × GL2 (Af ) acts through the character ε ⊗ (ε ◦ det). For any n ∈ Z let
E(n) = E(1)⊗n .
Recall the definition of H∗1 (Vk ) (for k ≥ 2) from Subsection 2.5.
7.4.2. Theorem. For ∗ = ∅ or c there is a natural GQ × GL2 (Af )-equivariant
isomorphism
M
∨

∼
1
b ∗,E
.
H∗1 (Vk )E ⊗E Symk−2 E 2 ⊗E E(n) −→ Hlalg
k≥2,n∈Z∨(Here GQ × GL2 (Af ) acts on Symk−2 E 2 through its quotient GL2 (Qp ), and thesubscript “ lalg” indicates the subspace of locally GL2 (Qp )-algebraic vectors.)
Proof. This follows from the isomorphism of [33, (4.3.4)], once one notes that∨
{ Symk−2 E 2 ⊗ detn }k≥2,n∈Z
is a complete set of isomorphism class representatives of the irreducible algebraic
representations of GL2 (Qp ).

If f is a cuspidal newform of some weight k defined over E then the two dimensional GQ -representation Vf may also be defined over E, as may the associatedGL2 (Af )-representation π(Vf ) (see Remark 2.2.1). From Theorem 2.5.1 we see that
there is a GQ × GL2 (Af )-equivariant embedding
Vf ⊗E π(Vf ) → H∗1 (Vk )E ,for ∗ = ∅ or c. (We are justified in taking either value of ∗, since the ManinDrinfeld theorem provides a GQ × GL2 (Af )-equivariant splitting of the surjection1
Hc1 (Vk )E → Hpar
(Vk )E .)7.4.3. Corollary. If f is a cuspidal newform of weight k defined over E, then for
either value of ∗ (i.e. ∅ or c) there is a GQ × GL2 (Af )-equivariant embedding
∨
1
b ∗,E
Vf ⊗E π(Vf ) ⊗E Symk−2 E 2 → H.
∨(Here GQ × GL2 (Af ) acts on Symk−2 E 2 through the projection onto GL2 (Qp ).)Proof. This follows from the preceding remarks together with Theorem 7.4.2.



b 1 (K p )E . Fix a tame level K p . As we recalled
7.5. The Jacquet module of Hb 1 (K p )E is admissible unitary, its Jacquetin the proof of Proposition 5.3.1, since H1pb (K )E,an is an essentially admissible T(Qp )-representation, andmodule JP(Qp ) Hb (Recallso corresponds to a coherent sheaf E(K p ) on the rigid analyic space T.
pthat the global sections of E(K ) are naturally identified with the topological dualb 1 (K p )E,an .) Write Σ := Σ(K p ). The action of T(Σ) on H
b 1 (K p )E
to JP(Qp ) Hb 1 (K p )E,an , and hence on E(K p ). We let
induces an action of T(Σ) on JP(Qp ) Hb generated by theA(K p ) denote the coherent sheaf of commutative algebras over Tp
image of T(Σ) in the sheaf of endomorphisms of E(K ), and let Spec A(K p ) denote
its relative spectrum over T. Thus Spec A(K p ) is a rigid analytic space over E,
equipped with a closed immersion
b × Spec E ⊗O T(Σ) ,Spec A(K p ) → T
E

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

71

b is a finite morphism. We
whose composite with the projection onto the factor Tpb p ),
denote a typical point of Spec A(K ) (Qp ) by (χ1 ⊗χ2 , λ), where χ1 ⊗χ2 ∈ T(Qand λ : T(Σ) → Qp .
7.5.1. Definition. Let W denote the rigid analytic space parameterizing the continuous characters of Z×p (i.e. “weight space”, in the usual terminology).
b × Spec E ⊗O T(Σ) via twisting, given on
There is a natural action of W on TE
the level of Qp -valued points by the formulaψ × (χ1 ⊗ χ2 , λ) 7→ (χ1 ψ ⊗ χ2 ψ, λ ⊗ ψ)
(where we extend ψ to a character of Q×
by setting ψ(p) = 1). The closed subspaces
pbSpec A(K p ) of T×Spec
E ⊗OE T(Σ) are invariant under twisting [33, Prop. 4.4.6].
The following results show that there are GQ -representations attached to theb 1 (K p )E .
points of Spec A(K p ), and that these representations appear in H
7.5.2. Proposition. If x = (χ1 ⊗ χ2 , λ) is a Qp -valued point of Spec A(K p ) (recall
that x is then in fact an E 0 -valued point for some finite extension E 0 of E), thenthe system of Hecke eigenvalues λ is promodular.
χ1 ⊗χ2b 1 (K p )E 0 )λ 6= 0. In particular,
Proof. The existence of x shows that JP(Q(Hanp)
b 1 (K p )E 0 )λ 6= 0, and so Lemma 7.3.13 shows that λ is promodular.
(H

an

7.5.3. Proposition. Let (χ1 ⊗ χ2 , λ) be a point of Spec A(K p ) (E) for whichχ1 | |−1 ⊗χ2 | | ε−1 is neither critical nor ultracritical and λ is non-Eisenstein.
Let V be the GQ -representation over E associated to λ (by the previous Proposition
together with Proposition 7.3.17).
(1) If χ1 | |−1 ⊗χ2 | | is unitary then there is a non-zero GQ × GL2 (Qp )equivariant continuous map
GL (Q )b 1 (K p )λE .
V ⊗ IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | cont → H
p−1(2) If χ1 | | ⊗χ2 | | is not unitary then there is a non-zero GQ × GL2 (Qp )equivariant continuous map
b 1 (K p )λE .
V ⊗ B(χ1 | |−1 ⊗χ2 | |) → H(Recall from Definition 5.3.10 that B(χ1 | |−1 ⊗χ2 | |) denotes the uniGL (Q )
versal unitary completion of IndP(Q2 ) p χ1 | |−1 ⊗χ2 | | an .)pProof. The assumption that (χ1 ⊗ χ2 , λ) lies in Spec A(K p ) (E) implies that theχ1 ⊗χ2b 1 (K p )E )an )λ is non-zero,(a priori finite dimensional) GQ -representation JP(Q(Hp)
and so by the Eichler-Shimura relations (and the irreducibility of V ) contains a copyof V . Theorem 5.2.5 thus shows that there is a non-zero GQ × GL2 (Qp )-equivariantmap
GL (Q )b 1 (K p )λE .
V ⊗ Ind 2 p χ1 | |−1 ⊗χ2 | |→H
P(Qp )an

The proposition follows (taking into account Proposition 5.3.4 (1) in case (1)). 
b 1 (K p )lalg → H
b 1 (K p )an induces a corresponding continuous injecThe injection H
tion on Jacquet modules. If we take into account the isomorphism of Theorem 7.4.2

72

MATTHEW EMERTON

and the compatibility of the formation of Jacquet modules with countable directsums [32, Lem. 3.4.7 (iv)], we may write this injection in the form
M

p
(64)
JP(Qp ) H 1 (Vk )K ⊗E (Symk−2 E 2 )∨ ⊗E E(n)k≥2,n∈Z
b 1 (K p )E,an .
→ JP(Qp ) H
pIf we fix a value of k and n, then the injection H 1 (Vk )K ⊗E (Symk−2 E 2 )∨ ⊗Eb 1 (K p )E
E(n) → H
is a closed embedding [31, Prop. 6.3.6] (the source beinganequipped with its finest convex topology), and so induces a closed embedding onb we obtain aJacquet modules. Passing to the associated coherent sheaves on T,
surjection of coherent sheaves of T(Σ)-modules E(K p ) → Ek,n (where we let Ek,npdenote the coherent sheaf associated to JP(Qp ) H 1 (Vk )K ⊗E (Symk−2 E 2 )∨ ⊗E
E(n) ). Thus if we let Ak,n denote the coherent algebra of endomorphisms of Ek,ninduced by T(Σ), we obtain a closed embedding Spec Ak,n → Spec A(K p ). Taking
the union over all k ≥ 2, n ∈ Z, we find that (64) induces an injective morphism ofbrigid analytic spaces over TaSpec Ak,n → Spec A(K p ),(65)
k≥2,n∈Z

which induces a closed embedding when restricted to each term of the disjoint unionin the source.
7.5.4. Lemma. For any fixed value of k and n, the sheaf Ak,n is reduced, and hasbdiscrete support on T.
Proof. Recall that the locally analytic Jacquet module functor is compatible withthe classical Jacquet module functor [32, Prop. 4.3.6], and that the classical Jacquet
module of an admissible smooth representation of GL2 (Qp ) is an admissible smooth
representation of T(Qp ), which corresponds to a sheaf with discrete support onb Since H 1 (Vk )K p is an admissible smooth GL2 (Qp )-representation, we conclude
T.
E
that Ek,n , and so also Ak,n , has discrete support. Since the action of T(Σ) onp
H 1 (Vk )K
E is semi-simple (as one sees from Eichler-Shimura theory), the sheaf Ak,nis furthermore reduced.

7.5.5. Definition. We say that a point of Spec A(K p ) (Qp ) is classical if it is in
the image of the map on Qp -points induced by (65).
e p ) the rigid analytic Zariski closure of the set
7.5.6. Definition. We denote by D(Kb × Spec E ⊗O T(Σ) ).
of classical points in Spec A(K p ) red (or equivalently, in TEe p ) as the “eigensurface” of tame level K p .
We refer to D(K7.5.7. Remark. Since the source of (65) is reduced (Lemma 7.5.4), we may alsoe p ) as the “scheme-theoretic image” (so to speak) of that map.
define D(KFor a fixed pair (k, n) the space Spec Ak,n is invariant under twisting by locally constant characters in W, while twisting by εm induces an isomorphism∼e p ) is invariant under twistSpec Ak,n −→ Spec Ak,m+n . Thus we see that D(King by characters of integral Hodge-Tate weight in W. Since these characters aree p ) is invariant under the twisting action of W.
Zariski dense in W, we see that D(KLOCAL-GLOBAL COMPATIBILITY IN p-ADIC LANGLANDS FOR GL2/Q

73e p ) factors as a productThe discussion of [33, §4.4] shows that in fact D(Ke p) ∼D(K= D(K p ) × W(66)

where D(K p ) denotes the reduced eigencurve parameterizing finite slope overcone p)
vergent p-adic eigenforms of tame level K p . (This justifies our designation of D(Kas an eigensurface.) This factorization is W-equivariant, with respect to the twisting action by W on the source, and the action of W by multiplication on itself on
the target.
Let us describe the isomorphism (66) more precisely. Suppose that f is a finite
slope overconvergent T(Σ)[Up ]-eigenform of tame level K p and integral weight k.
Let α ∈ Qp denote Up -eigenvalue of f , let χ denote the nebentypus of f (so thatb × → Q× is a locally constant character unramified outside of Σ(K p )), andχ : Zplet λ : T(Σ) → Qp denote the system of Hecke eigenvalues attached to f . Let
cf ∈ D(K p )(Qp ) be the point attached to f (so that cf depends only on α and λ –but note that λ encodes χ), and fix some ψ ∈ W(Qp ). Under the isomorphism (66),the point (cf , ψ) ∈ D(K p ) × W (Qp ) corresponds to the pointe p )(Qp ) ⊂ Spec Ac (K p ) (Qp ).
(ur(α) | | ψ ⊗ur(α)−1 χp−1 | |−1 ε2−k ψ, λ⊗ψ) ∈ D(KHere χp denotes the local component of χ at p, thinking of χ as a character of A×f
via the isomorphism (1) of Subsection 1.3, and as above we extend ψ to a character×of Qp by setting ψ(p) = 1; thus the first coordinate
ur(α) | | ψ ⊗ ur(α)−1 χp−1 | |−1 ε2−k ψb as it should. (See the discussion ofof this point makes sense as an element of T,[33, §4.4], and in particular the formulas in the statement of [33, Prop. 4.4.2].)e p )(Qp ) lying in the
We also note that under (66) the classical points in D(Kimage of (Spec Ak,n (Qp ) under the map (65) correspond to the points (cf , ψ) ∈(D(K p )×W)(Qp ) for which f is a finite slope p-stabilized classical Hecke eigenformof weight k, and ψ is a character of integral Hodge-Tate weight n. (Thus thediscreteness assertion of Lemma 7.5.4 is simply a rephrasing of the fact that thecollection of classical eigenforms of finite slope and fixed weight k gives rise to adiscrete set of points on the eigencurve.)e p ) andThe following
result clarifies the relationship between the spaces D(KpSpec A(K ) red . In its statement we employ the following notational convention:
if ψ1 and ψ2 are a pair of continuous characters GQ → E, we write ψ1 ⊕ψL
2 to denotethe system of Hecke eigenvalues attached to the GQ -representation ψ1 ψ2 .
7.5.8. Theorem. The image of the closed embeddinge p ) → Spec A(K p )
(67)
D(K
red
is a union of connected components of the target, and its complement decomposes
as the disjoint union of a collection of one dimensional irreducible connected components. Furthermore, the system of Hecke eigenvalues attached to any point inthis complement is Eisenstein.
Proof. We begin by recalling some
 of the known result about the structure ofp thee p ) and Spec A(K p )spaces D(K. It is known that the eigencurve D(K ) is
red

74

MATTHEW EMERTONe p ) is equidimensional of diequidimensional of dimension one [16], and thus D(Kmension two. On the other hand, it follows from [33, Prop. 2.3] that the spaces
Spec A(K p ) red are at most two dimensional. Since they are invariant under twisting, each irreducible component is of thus of dimension either one or two. Furthermore, the one dimensional irreducible components (if any exist) are also connectedcomponents, since they consist of single W-orbits under twisting. The non-criticalslope criterion for classicality[33, Prop. 2.3.6] shows that any two dimensional component of Spec A(K p ) red contains a Zariski dense set of classical points, and thuse p ). If we write C to denote the complement of theis in fact a component of D(Kimage of the closed embedding (67), then the preceding discussion
shows that Cconsists of a union of connected components of Spec A(K p ) red , each consisting ofa single W-orbit, and containing no classical points.
We now recall some notation and constructions from the proof of [33, Thm. 2.1.5].
That argument, specialized to our particular case, yields the existence of a length
two complex Se0 → Se1 of admissible Banach Kp -representations (for some sufficiently small compact open subgroup Kp of GL2 (Zp )) whose cohomology groupsb 0 (K p )E and H
b 1 (K p )E respectively, and such thatare naturally isomorphic to H∼ri
Se −→ C (Kp , E) i as a Kp -representation for some ri ≥ 0 (i = 0, 1). (Actually,e i rather than the completions H
b i unthat proposition discuss certain spaces Hiibeder discussion here. However, in our situation the spaces H and H coincide –see [33, §4.1].) Passing to locally analytic vectors (which we remind the reader•
is an exact functor [51, Thm. 7.1]) yields a complex Seanof locally analytic Kp b • (K p )E , and such
representations, with cohomology groups isomorphic to H
an∼ithat Sean−→ C an (Kp , E)ri . Taking topological duals we obtain a complex of fi• 0nite rank free Dan (Kp , E)-modules (Sean) , with cohomology groups isomorphic to
0
•
p
an
b
H (K )E,an . (Here D (Kp , E) denotes the nuclear Fréchet algebra of locallyanalytic E-valued distributions on Kp .)b 0 (K p )E,an vanished. We would then have a short
Imagine for a moment that H
exact sequence

• 0
• 0
b 1 (K p )E,an 0 → (Sean) → (Sean) → 0,
0→ H
of Dan (K p , E)-modules in which the second and third terms are free. It wouldb 1 (K p )E,an 0 is a direct summand of a free Dan (K p , E)-module, andfollow that Hb anda variant of [32, Prop. 4.2.36] would show that the support of E(K p ) in T,phence also Spec A(K ), is equidimensional of dimension two. The discussion of thefirst paragraph would then show that (67) is an isomorphism.
b 0 (K p )E,an 6= 0. However, it is Eisenstein. So, heuristically, if
Now of course H
we localize away from the Eisenstein systems of Hecke eigenvalues, we may proceedb 0 (K p )E,an were zero, and so conclude that any component containing a
as if H
point Spec A(K p ) red associated to a non-Eisenstein system of Hecke eigenvalues
is equidimensional of dimension two, yielding the theorem. The details of theargument will appear in a future publication of the author.

7.5.9. Remark. It was proved in [14, Prop. 5.3.2] that the embedding (67) inducesa bijection on ordinary points.
7.6. Refinements and the eigensurface. The following result is essentially areformulation of [41, Thm., p. 375].

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

75

7.6.1. Theorem. Let V be an odd irreducible continuous two dimensional representation of GQ over E, unramified outside of a finite set of primes. Let K p be atame level such that Σ(K p ) contains all the primes of ramification of V , and let
λ : T(Σ(K p )) → E denote the system of Hecke eigenvalues attached to V .
(1) The following are equivalent:(a) There is a Galois character ψ : GQ → E × , unramified outside ofp, such that the twist V ⊗ ψ is the Galois representation attached to an
overconvergent modular form of finite slope and of tame level K p definedover E.
bsuch that (χ1 ⊗ χ2 , λ) lies in(b) There is a character χ1 ⊗ χ2 ∈ T(E)pe
D(K )(E).
(2) Furthermore, for any character χ1 ⊗ χ2 satisfying condition (1)(b), there
is a refinement R of V |Dp such that σ(R) = (χ1 | |−1 , χ2 | | ε−1 ). (Inparticular, V |Dp is trianguline.)(3) If V |Dp is indecomposable, then the converse to (2) holds: If V satisfies theequivalent conditions of (1), then for any refinement R of V |Dp , writinge p )(Qp ).
σ(R) = (η, ψ), the point (η | | ⊗ψε | |−1 , λ) lies in D(KProof. The equivalence of the two conditions of (1) follows directly from the dee p ). Clearly the assertion of (2) is invariant under twisting.
scription (66) of D(KThus we may assume that χ1 ⊗ χ2 = ur(α) | | ⊗ ur(α)−1 ε | |−1 det V |Dp , andthat V is the Galois representation attached to an overconvergent eigenform f with
Up -eigenvalue α. It then follows from [41, Thm., p. 375] that V admits a refinement
R such that σ(R) = (ur(α), ur(α)−1 det V |Dp ). This proves (2).
We turn to proving (3). Thus V |Dp is assumed to be indecomposable, and so
Propositions 4.2.2 and 4.4.4 (4) show that V admits a unique equivalence classof refinements unless V |Dp is a twist of a Hodge-Tate representation with distinct
Hodge-Tate weights. Thus, taking into account (1)(b), and applying a twist if
necessary, we see that we may assume that V is attached to an overconvergent
eigenform f of integral weight k 6= 1. Since V |Dp is assumed to be indecomposable,
it follows from [41, Thm. 6.6] that f is classical if and only if V |Dp is potentiallysemi-stable. Let us consider this case first. Since f is classical, we may and do
assume that it is the p-stabilization of a newform.
If V |Dp is not potentially crystalline, or is potentially crystalline but not Frobenius semi-simple, then it admits a unique equivalence class of refinements, and so
there is nothing to prove. If V |Dp is potentially crystalline and Frobenius semie p)
simple, then it has two equivalence classes of refinements. The point on D(Kgiven by f corresponds to one of these equivalence classes, and we must find asecond point lying over V that corresponds to the other. If V |Dp is in fact crystalline, then f is the p-stabilization of a newform of prime-to-p level. If fe denotes
the p-stabilized twin of f , then fe gives the sought-after second point. If V |Dp is
not crystalline, then f is itself a newform, whose nebentypus ω is non-trivial at p.
Let ωp be the p-part of ω. If we take fe to be the newform associated to the twistf ⊗ ωp−1 , then fe is p-stabilized and has finite slope, and gives rise to a point one p ) with system of Hecke eigenvalues λ⊗ω. Twisting that point by ω −1 ∈ W(E)
D(Kgives the second point lying over V . (See [14, §4.1] for a representation theoretic
point of view on this discussion.)

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MATTHEW EMERTON

Now suppose that V |Dp is Hodge-Tate but not potentially semi-stable. If k > 1,
then let g be an overconvergent eigenform of weight 2 − k such that θk−1 g = f.
(Such a g exists by [21, 22].) If k < 1, then let g = θ1−k f. Let x denote the point ine p )(E) corresponding to f , and let y denote the point in D(Ke p )(E) obtained byD(K1−ktwisting the point corresponding to g by the character ε. Then V is the Galois
representation attached to both x and y, and we have found two distinct pointse p )(E) giving rise to V , which by (2) must account for the two equivalenceof D(K
classes of refinements of V . This completes the proof of (3).

7.6.2. Conjecture. Part (3) of the preceding theorem continues to hold in the casewhen V |Dp is the direct sum of two characters.
7.6.3. Remark. The preceding conjecture can be proved in many cases. Supposethat V satisfies the Lequivalent conditions of part (1) of the preceding theorem, andalso that V |Dp = η ψ for two characters η and ψ. If η = ψ then σ(R) = (η, η)for every refinement of V and there is nothing to prove. Thus we may also assume
that η and ψ are distinct. There are then two cases to consider, namely the twocases of Proposition 4.4.5.
In the first case, V |Dp admits just its two equivalence classes of ordinary refinements. Part (2) of the preceding theorem then shows that the overconvergent
eigenform f to whose twist V is attached must be ordinary. Twisting and relabelling
if necessary, we may assume that V is attached to f , that η is unramified, and thate p ) given by f corresponds
η(p) is the Up -eigenvalue of f . Thus the point of D(Kto the ordinary refinement R1 of V |Dp for which σ(R1 ) = (η, ψ). Let ψ0 denote, regarded as a character of GQ unramified outside of p. To
the restriction ψ|Z×p
prove (3), we must show that there is a (necessarily ordinary) overconvergent form
g with Up -eigenvalue equal to ψ(p), and with associated Galois representation equale p ) obtained by twisting the point corresponding toto V ⊗ ψ0−1 . (The point of D(Kg by ψ0 will then have V as associated Galois representation, and will correspondto the refinement R2 for which σ(R2 ) = (ψ, η).) If the mod $ representation V
attached to V satisfies the necessary hypothesis, then results identifying ordinary
universal deformation rings with ordinary Hecke algebras (such as the one stated in[17, §1]) will provide the required form g, and thus verify part (3) of the preceding
theorem in this case.
Now suppose that we are in case (2) of Proposition 4.4.5, so that V is potentiallycrystalline, with distinct Hodge-Tate weights, up to a twist. Since V has threee p ) thatequivalence classes of refinements, we must construct three points on D(Kgive rise to V . Twisting V if necessary, we may thus assume that V is potentiallycrystalline, and is attached to an overconvergent eigenform f of integral weightk 6= 1. If k < 1, then V ⊗ εk−1 is attached to θ1−k f , which is of weight 2 − k. Thus
we may assume that in fact f is of weight k > 1. The Fontaine-Mazur conjecture [37]implies that in fact f should be classical. If this is so, then the argument that proves[14, Thm. 1.1.2] in the split case provides the necessary three points. Essentially,
we have two classical points, namely an ordinary form f and an associated slopek − 1 form fe (just as in the proof of classical case of the preceding theorem) as well
as a weight 2 − k ordinary form g such that θk−1 g = fe. (The existence of g is aconsequence of the fact that V |Dp is split [14, Thm. 1.1.3].) The results of Skinnerand Wiles [55, 56] show that in fact the Fontaine-Mazur conjecture for V is true
provided that η and ψ have distinct reductions modulo $. (The reader should bear

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

77

in mind that V is assumed to arise from an overconvergent eigenform, so that theresidual modularity hypothesis of [56] holds for V .) Thus under this hypothesis (3)is also true. (Note that (3) is actually equivalent to the Fontaine-Mazur conjecture
for V : if (3) holds, then we see that some twist of V must arise from an ordinary
form of weight k > 1, which is necessarily classical.)
7.6.4. Remark. Consider the projectione p )(Qp ) → Spec E ⊗O T(Σ) (Qp )pr : D(KE
which maps a point to its associated system of Hecke eigenvalues. If λ is a nonEisenstein system of Hecke eigenvalues lying in the image, corresponding to theirreducible GQ -representation V , then we see from the preceding discussion thatthe fibre pr−1 (λ) consists of one, two, or (admitting Conjecture 7.6.2) three points,
depending on whether V |Dp satisfies condition (1), (2), or (3) of the following list:(1) Either(a) V |Dp is indecomposable, and is not Hodge-Tate up to a twist.
(b) V |Dp is indecomposable, and is potentially semi-stable, but notpotentially crystalline and Frobenius semi-simple, up to twist.
(c) V |Dp is the direct sum of two copies of the same character.
(2) Either(a) V |Dp is indecomposable, and is Hodge-Tate, but not potentiallysemi-stable, up to a twist.
(b) V |Dp is indecomposable, and is potentially crystalline and Frobenius semi-simple up to a twist.
(c) V |Dp is a direct sum of two characters, but is not Hodge-Tate up
to a twist.
(3) V |Dp is a direct sum of two distinct characters, and is Hodge-Tate up to a
twist.
Recall that if one restricts the projection pr to the eigencurve D(K p ) then the
behaviour is simpler: the preimage of a non-Eisenstein system of eigenvalues will
have either one or two points. Furthermore the second case occurs precisely when
the corresponding Galois representation V is Frobenius semi-simple and crystalline,
and thus attached to a newform of prime-to-p conductor (with the proviso that we
admit Conjecture 7.6.2, or equivalently, the Fontaine-Mazur conjecture, in the casewhen V |Dp is a direct sum of distinct characters whose reductions mod $ coincide;
this is the exceptional case of [41, Thm. 6.6]).
We now state a result that gives some evidence for Conjecture 6.7.3 (cf. thediscussions of 6.7.5 and 6.7.10). Suppose that V is an irreducible continuous twodimensional GQp -representation that is trianguline and Hodge-Tate, but not potentially semi-stable, up to a twist. Suppose furthermore that V is (the restriction toe p )(E). Let R be theGQp of) the Galois representation attached to a point of D(Knon-ultracritical refinement of V , and write σ(R) = (η, ψ).
7.6.5. Proposition. With the preceding hypotheses and notation, the locally analytic GL2 (Qp )-representation B(V )an contains a subrepresentation W that sits ina short exact sequenceGL (Q )GL (Q )
0 → IndP(Q2 ) p η ⊗ ψε an → W → IndP(Q2 ) p ψ ⊗ ηε lalg → 0.
p

p

78

MATTHEW EMERTON

Proof. Let R0 denote the ultracritical refinement of V , so that σ(R0 ) = (ηz −w , ψz w ),where w < 0 is the Hodge-Tate weight of ηψ −1 . If λ denotes the system of Hecke
eigenvalues attached to the overconvergent eigenform giving rise to V , then Theorem 7.6.1 (3) (applied to R0 ) shows that the point (ηz −w | | ⊗ψz w ε | |−1 , λ) lies in
−w
w
−1


e p )(E) ⊂ Spec Ac (K p ) (E). In particular J ηz | |⊗ψz ε| | H
b c1 (K p )E,an 6=
D(KP(Qp )

b c1 (K p )E,an denote the canonical lift (in the sense of [32, 0.9]) of a non0. Let u ∈ Hzero element of this space, and let U denote the closed GL2 (Qp )-subrepresentationb c1 (K p )E generated by u. As was observed in the proof of Proposition 5.2.6,of H
b 1 (K p )E,an is the canonical lift of a non-zero element of
the element (X− )−w u ∈ H c
η| |⊗ψε| |−1 b 1
p
JP(Qp )Hc (K )E . Since (X− )−w u ∈ U , it is in fact the canonical lift of aη| |⊗ψε| |−1
non-zero element of JP(Qp )U . By Theorem 5.2.5 there is thus a non-zeroGL (Q )
map IndP(Q2 ) p η ⊗ ψε an → U. Since U is unitary (being a closed subspace ofpb c1 (K p )E ), this extends to a non-zero map
HB(η ⊗ ψε) → U.

(68)

GL (Q )

(Recall that B(η ⊗ ψε) denotes the universal unitary completion of IndP(Q2 ) p η ⊗pψε an .)
Since B(η ⊗ ψε) is topologically irreducible and admissible (by Theorem 5.3.7)
the map (68) must be a closed embedding. Let u denote the image of u in thecokernel of this embedding. Since (X− )−w u = 0, we see that if u is non-zero, thenit generates a copy of


GL2
GL2
IndP(Qηz −w ⊗ ψz w ε lalg ∼ψ ⊗ ηε lalg
= IndP(Q))
pp(the isomorphism being supplied by an intertwiner on the smooth factors of these
irreducible locally algebraic representations). However Lemma 5.2.4 shows that
this locally algebraic representation cannot be contained in a unitary GL2 (Qp )representation. Thus necessarily u = 0, and so (68) is an isomorphism. If W denotesb c1 (K p )E , thenthe closed GL2 (Qp )-subrepresentation generated by u in Uan ⊂ Han
W has the extension structure stipulated in the statement of the proposition (since
the closed subrepresentation of Uan generated by (X− )−w u is precisely a copy of
GL (Q )
IndP(Q2 ) p η ⊗ ψε an ). The proposition follows, once we recall that B(V ) = B(η ⊗pψε) by definition.

b 1 . Throughout this subsection
7.7. Mapping Galois representations into H∗,E
V will denote an irreducible continuous two dimensional representation of GQ over
E. We let V denote the semi-simplification of the reduction modulo $ of (some
GQ -invariant lattice in) V . (The resulting semi-simple GQ -representation over F isindependent of the choice of lattice.)b 1 ).
7.7.1. Definition. If ∗ ∈ c, ∅, then we write M∗ (V ) := HomGQ (V, H∗,E
Note that M∗ (V ) is a GL2 (Af )-invariant closed subspace of the admissible conb 1 , and so is again an admissible contintinuous GL2 (Af )-representation V ∨ ⊗E H∗,Euous GL2 (Af )-representation [31, Prop. 7.2.2]. The following lemma states somebasic properties of M∗ (V ).

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

79

b 1 is a
7.7.2. Lemma.
(1) The natural evaluation map V ⊗E M∗ (V ) → H∗,E
GQ × GL2 (Af )-equivariant embedding.
(2) If ψ is any continuous character ψ : GQ → E × , then there is a GL2 (Qp )∼equivariant topological isomorphism M(V ⊗ ψ)∗ −→ M∗ (V ) ⊗ (ψ ◦ det)
(where on the right hand side we regard ψ as a character of A×f via globalclass field theory).
(3) The formation of M∗ (V ) is compatible with extension of scalars to finite
extensions of E.
Proof. The evaluation map is tautologically GQ × GL2 (Af )-equivariant, and is aninjection since V is irreducible (and thus absolutely irreducible, being odd). The
proof of (2) is similar to the proof of Lemma 7.3.19. As in that proof, let cψ denoteb 0 that corresponds to ψ under the isomorphism (60).
the cohomology class in H
E
Then GQ acts on cψ through the character ψ, while GL2 (Af ) acts on cψ through ψ ◦det. Cupping with cψ thus induces the claimed isomorphism of (2) (with the inverse
isomorphism being provided by cupping with cψ−1 ). Claim (3) is immediate.

7.7.3. Lemma. If M∗ (V )(K p ) 6= 0 (for ∗ = ∅ or c) and V is absolutely irreducible10
then V is modular.
b 1 (K p )E for some tame level K p , then Lemma 7.2.3 shows
Proof. If V appears in H∗
1
that V appears in H∗ (Y (Kp K p ), F) for some sufficiently small compact open subgroup Kp of GL2 (Qp ). The space M := HomGQ V , H∗1 (Y (Kp K p ), F) is naturally
a T := T(Σ(K p ))-module. Furthermore, the T-action on this space factors throughe p K p )1 (as defined in Subsection 7.3). Since M is finite dimensionalits quotient T(Kover F, we must have that (F0 ⊗F M )λ 6= 0 for some finite extension F0 of F and somesystem of Hecke eigenvalues λ defined over F0 . The system λ is necessarily modue p K p )1 . Since V is absolutelylar, since it is in fact a system of eigenvalues of T(K0irreducible, the natural evaluation map V ⊗F (F ⊗F M )λ → H∗1 (Y (Kp K p ), F0 ) is
injective, and the Eichler-Shimura relations then imply that λ coincides with thesystem of eigenvalues associated to V . Thus V is modular, as claimed.
7.7.4. Corollary. If M∗ (V ) 6= 0, if V is absolutely irreducible, and if p 6= 2, thenV is odd.
Proof. The preceding proposition shows that V is modular, and thus odd. Since
p 6= 2, we may conclude that V is odd.

The following result gives some control over the action of GL2 (Apf ) on M∗ (V ).
p

7.7.5. Lemma. If K p is a tame level for which M∗ (V )K 6= 0, then V is unramified
outside of Σ(K p ).
p

Proof. To say that M∗ (V )K 6= 0 is to say that there is a GQ -equivariant embeddingb ∗1 (K p )E . The lemma thus follows from the fact that the GQ -action on
V → H
1
p
b ∗ (K )E is unramified away from Σ(K p ).
H
7.7.6. Corollary. If M∗ (V ) 6= 0 then V is unramified at all but finitely many
primes.
10We will see in Proposition 7.7.13 below that under this assumption the spaces M (V ) for∗
either choice of ∗ coincide; in particular, if one is non-zero, so is the other.

80

MATTHEW EMERTONp

Proof. If M∗ (V ) 6= 0, then M∗ (V )K 6= 0 for some tame level K p . The claim thus
follows from the preceding lemma.

If K p is a tame level, and Σ := Σ(K p ), then the Hecke algebra T(Σ) acts naturallypon M∗ (V )K .
7.7.7. Proposition. Suppose that we are in the situation of Lemma 7.7.5, andthat furthermore V is not a twist of a representation with finite image. Let λ :T(Σ(K p )) → E denote the system of Hecke eigenvalues attached to V (which is dep
fined, since V is unramified outside of Σ(K p )). Then T(Σ(K p )) acts on M∗ (V )Kthrough the character λ.
Proof. The following lemma shows that our hypothesis on V is equivalent to requiring that the projective representation associated to V has infinite image. The
proposition thus follows from the (proof of) [14, Prop. 3.2.3].

7.7.8. Lemma. The following conditions on a continuous two dimensional Galois
representation V defined over Qp are equivalent:(1) The projective representation attached to V has finite image;(2) V is a twist of a representation with finite image.
Proof. The claimed equivalence follows from the discussion of [53, §6]. Indeed, if
the projective image of V is finite, then [53, Cor., p. 227] shows that we may finda continuous two dimensional representation W of GQp over Qp which has finite
image, and whose associated projective representation coincides with that of V .
The discussion of [53, p. 226] then shows that V is a twist of W .
Proposition 7.7.7 has the following corollary.
7.7.9. Corollary. If M∗ (V ) 6= 0 for ∗ = ∅ or c and if V is not a twist of arepresentation with finite image,11 then V is promodular (and in particular, is odd).
p

Proof. If M∗ (V ) 6= 0 then M∗ (V )K 6= 0 for some tame level K p . If λ denotes thesystem of Hecke eigenvalues attached to V , then the preceding proposition showsb ∗1 (K p )E )λ 6= 0. It follows from Lemma 7.3.13 that λ is promodular.
that (H
7.7.10. Remark. Presumably any irreducible two dimensional Galois representation V over E for which M∗ (V ) 6= 0 (for ∗ = ∅ or c) is promodular, and so also
odd. In fact these two conditions on V are conjecturally equivalent (assuming always that M∗ (V ) 6= 0). Indeed, suppose that V is odd and that M∗ (V ) 6= 0. We
know that V is promodular unless it is a twist of a representation with finite image.
By Lemmas 7.7.2 (2) and 7.3.19, we may assume that V does in fact have finite image. The Artin conjecture, together with standard converse theorems, then implies
that V ∼= Vf for some modular form f of weight 1. (This is the so-called Strong
Artin conjecture, which is actually a conjecture of Langlands.) Thus Remark 7.3.14
shows that V is promodular.
The surjection (57) induces a GL2 (Af )-equivariant continuous map(69)

Mc (V ) → M(V ).

11We will see in Proposition 7.7.13 below that under this assumption the spaces M (V ) for∗
either choice of ∗ coincide; in particular, if one is non-zero, so is the other.

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

81

7.7.11. Conjecture. The map (69) is a GL2 (Af )-equivariant topological isomorphism.
This conjecture was originally formulated by Breuil in the case when V = Vf fora modular form f of weight ≥ 2. It has been proved in that case by Breuil and theauthor (see Proposition 7.7.13 (2) below).
7.7.12. Remark. Since the Galois action on the kernel of (57) is abelian, it follows
directly that (69) is injective; the difficulty is to show that it is surjective. Noteppthat (69) is an isomorphism if and only if the induced map Mc (V )K → M(V )Kis an isomorphism for each tame level K p .
The following proposition establishes Conjecture 7.7.11 in many cases. Part (1)is due to Breuil; part (2) to Breuil and the author [14, Prop. 3.2.4].
7.7.13. Proposition. Conjecture 7.7.11 is true provided V satisfies either one of
the following two conditions:(1) V is absolutely irreducible.
(2) V is not a twist of a representation with finite image.
Proof. In case (1), it follows from Lemma 7.7.3 that V is modular. Let λ : T(Σ) → F
denote the corresponding system of eigenvalues, let m denote the kernel of λ, andlet subscript m denote localization at m. It is clear that the injectionb ∗1 (K p )E =: M∗ (V )b ∗1 (K p )E,m → HomG V, H
HomG V, H
Q

Q

is in fact an isomorphism (cf. the proof of Lemma 7.7.3). Also, since V is absolutely
c of (59) is Eisenstein [14, Cor. 3.1.3], we see that
irreducible, and since the kernel M1pbb 1 (K p )E,m is an isomorphism. This proves (1).
the natural map Hc (K )E,m → H
Case (2) is proved in [14]. We present a variant on that proof here. Let λ :e p ) → OE denote the (promodular) system of Hecke eigenvalues attached to V .
T(KIt follows from Proposition 7.7.7 thatb ∗1 (K p )E )λ .
M∗ (V ) = HomG V, (HQ

We claim that the natural map
(70)



b c1 (K p )E λ → H
b 1 (K p )E λH

is an isomorphism.
cO denotes the kernel of (58), then (70) sits in the short exact sequence of
If M
E
p
e
T(K )-modules
b 1 (K p )E λcλ → Hb c1 (K p )E λ → H
0→Me p )/I, M
cO /π s ⊗O E,→ lim
Ext1 T(KE
E←−s

where I denotes the kernel of λ, and the Ext1 s are computed in the category ofe p )-modules. On the one hand, the final term in this sequence is annihilated byT(KcO satisfies the conditions of Lemma 7.3.8, by [14,I. On the other hand, since M
E
Lem. 3.1.2], the same is also true of this term, and so that lemma implies that thiscλterm is Eisenstein. Since λ is not Eisenstein, this term must vanish. The term M
cmust also vanish, since M is Eisenstein. This shows that (70) is an isomorphism,
as required.


82

MATTHEW EMERTON

The following result describes the structure of the space M∗ (V )lalg of locally
GL2 (Qp )-algebraic vectors in M∗ (V ).
7.7.14. Proposition. The space M∗ (V )lalg is non-zero (for either choice of ∗ ∈{∅, c}) if and only if V is the twist by a power of ε of the representation associatedto a classical newform f over E of weight k ≥ 2.
If these equivalent conditions hold (so that in particular V |Dp is potentially semistable with distinct Hodge-Tate weights, and the locally algebraic representationπep (V |Dp ) is defined – see Conjecture 3.3.1 (7)) then the following additional results
also hold:(1) There is a canonical GL2 (Apf )-equivariant isomorphism∼(71)
π p (V ) −→ HomGL2 (Qp ) π
ep (V |Dp ), M(V ) ,which induces a GL2 (Af )-equivariant isomorphism
(72)

∼

π
ep (V |Dp ) × π p (V ) −→ M(V )lalg .

(2) For any 0 6= ξ ∈ π p (V ), regarded as an element of
HomGL2 (Qp ) π
ep (V |Dp ), M(V )
via the isomorphism of (1), the closure of ξ π
ep (V |Dp ) in M(V ) is anadmissible unitary GL2 (Qp )-representation which is independent of ξ, up
to a canonical isomorphism. Denote this completion of π
ep (V |Dp ) by πbp (V ).
(3) There is a canonical closed GL2 (Af )-equivariant embedding(73)πbp (V ) ⊗E π p (V ) → M(V ),whose image coincides with the closure in M(V ) of M(V )lalg .
(4) The isomorphism (71) induces a canonical GL2 (Af )-equivariant isomor∼phism π p (V ) −→ HomGL2 (Qp ) π
bp (V ), M(V ) (where Hom denotes continuous homomorphisms).
∼b1
Proof. There is a natural isomorphism M∗ (V )lalg −→ HomGQ V, H∗,E,lalg . The
first claim of the theorem thus follows from Theorems 7.4.2 and 2.5.1, once we1recall that the kernel of the natural map Hc1 (Vk )E → Hpar
(Vk )E (resp. the cokernel
11of the natural map Hpar (Vk )E → H (Vk )E ) contains no absolutely irreducible twodimensional Galois representations, for any k ≥ 2.
Suppose for the remainder of the proof that in fact V ∼= Vf ⊗ εn for some
newform f of weight k ≥ 2 and some integer n. The isomorphism (72) then follows
directly from the same theorems as were cited above. Since π
ep (V |Dp ) is a locallyalgebraic representation of GL2 (Qp ), the inclusion M(V )lalg ⊂ M(V ) induces anisomorphism ∼
HomGL2 (Qp ) π
ep (V |Dp ), M(V )lalg −→ HomGL2 (Qp ) π
ep (V |Dp ), M(V ) .
Combined with (72), this yields the isomorphism (71).
The isomorphism (71) in turn induces a GL2 (Qp )-equivariant embeddingπep (V |Dp ) → HomGL2 (Apf ) (π p (V ), M(V )).
Since π p (V ) is irreducible, and so in particular finitely generated, we see that thetarget of this embedding is an admissible unitary GL2 (Qp )-representation. Thus if

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

83π
bp (V ) denotes the closure of the image of this embedding, we see that πbp (V ) is anadmissible unitary completion of π
ep (V |Dp ). The natural evaluation map
HomGL2 (Apf ) (π p (V ), M(V )) ⊗E π p (V ) → M(V )is tautologically GL2 (Af )-equivariant, and is injective, since π p (V ) is absolutely
irreducible. Thus it restricts to a GL2 (Af )-equivariant injection (73), which mustin fact be a closed embedding, by [31, Prop. 7.2.2 (iii)]. Its image is the closureof M(V )lalg , as follows from the explicit description of M(V )lalg provided by (72).
This proves part (3).

If 0 6= ξ ∈ π p (V ), then the closure of ξ π
ep (V |Dp ) coincides with the image ofπbp (V ) ⊗ ξ under (73). Since that map is an injection, we see that this closure iscanonically identified with πbp (V ). This gives (2).
Finally, we have the series of maps
π p (V ) → HomGL2 (Qp ) π
bp (V ), M(V ) ∼→ HomGL2 (Qp ) π
ep (V |Dp ), M(V ) −→ π p (V ),where the first map is the injection induced by the embedding (73), the second map is induced by restriction, which is also an injection (since the inclusionπep (V |Dp ) → π
bp (V ) has dense image), and the third map is the isomorphism (71).
By construction, the composite of these maps is the identity, and so each must be
an isomorphism. This gives (4).

7.7.15. Remark. If f is any classical cuspidal newform f over E of weight k ≥ 2,then πbp (Vf ) is conjectured to be topologically irreducible. Indeed, when Vf |Dp
is irreducible, it is conjectured to coincide with B(Vf |Dp ), the admissible unitary
GL2 (Qp )-Banach space representation attached to Vf |Dp via Conjecture 3.3.1 (seeConjecture 7.8.1 below). This is known when Vf |Dp is trianguline and Frobenius
semi-simple (using the definition of B(Vf |Dp ) given in Subsection 6.1); see Theorem 7.10.1 below. On the other hand, if V |Dp is reducible, then π
bp (V ) coincides withthe universal unitary completion of π
ep (V Dp ) (since this universal unitary completionis admissible unitary and topologically irreducible, by Lemma 5.3.3 and Proposition 5.3.4). Thus in this case we find that πbp (V ) is a topologically irreducible closed
subrepresentation of the reducible representation B(Vf |Dp ) (at least in those cases
when a definition of B(Vf |Dp ) has been given in Section 6.)
7.7.16. Remark. Let π be a locally algebraic representation of GL2 (Qp ) of the form
π := U ⊗ (Symk−2 E 2 )∨ , where U is an absolutely irreducible admissible smooth
representation of GL2 (Qp ). The preceding proposition shows that if π can berealized as π
ep (Vf |Dp ) for some cuspidal newform f of weight k, then π has a nonzero universal unitary completion, and in fact admits a non-zero admissible unitarycompletion, namely πbp (Vf ). If we suppose that U is cuspidal and that π has unitarycentral character, then we can always realize π in the form π
ep (Vf |Dp ) (at least after
making a twist and replacing E with a finite extension); thus we have verified theremarks made following the proof of Proposition 5.1.18.
Our final result of this subsection provides a characterization of the represene p ) in terms of the structure
tations V attached to points of the eigensurface D(KKpof M(V ) .

84

MATTHEW EMERTON

7.7.17. Proposition. For any tame level K p , the set of exponentsp
Exp M(V )Kan
is non-empty if and only if V is a twist of a representation associated to a finite
slope overconvergent eigenform of tame level K p defined over E.
pProof. If M(V )K vanishes, there is nothing to prove. Thus we may as well assume
that this space is non-zero, and thus (by Lemma 7.7.5) that V is unramified outsideΣ(K p ). Write T := T(Σ(K p )), and λ denote the system of Hecke eigenvalues
associated to V .
p
χ1 ⊗χ2b
Suppose first that JP(Q
M(V )K
6= 0 for some χ1 ⊗ χ2 ∈ T(E).
This isanp)then a non-trivial finite dimensional E-vector space equipped with an action of T.
Lemma 7.7.2 (1) yields a GQ -equivariant T-linear injectionp
χ1 ⊗χ2 b 1
χ1 ⊗χ2(74)
V ⊗ JP(Q
M(V )K→ JP(Q
H (K p )E .
anp)p)
Since the source of (74) is non-trivial and finite dimensional, the Eichler-Shimura
relations show that it contains a T-eigenvector
with system of eigenvalues λ. Con
χ1 ⊗χ2 b 1
p λsequently, we find that JP(Qp ) H (K )E 6= 0, and thus that the point (χ1 ⊗ χ2 , λ)e p )(E) (by Theorem 7.5.8, since λ islies in Spec A(K p ) (E), and hence in D(Knot Eisenstein). Thus V is a twist of a GQ -representation attached to a finite
slope overconvergent eigenform of tame level K p defined over E. Conversely,
if V is such a twist, then Proposition 7.5.3 (or better, its proof) shows thatpExp(M(V )Kan ) 6= ∅.
7.8. The local-global compatibility conjecture. If V is an irreducible continuous representation of GQ over E, unramified outside of a finite number of primes,
then as in the preceding subsection we will let V denote the semi-simplification of
the reduction modulo $ of (some GQ -invariant lattice in) V .
We now restate Conjecture 1.1.1.
7.8.1. Conjecture. If V is an odd irreducible continuous representation of GQ
over E, unramified outside of a finite number of primes, then there is a GL2 (Af )equivariant topological isomorphism∼ Π(V )
M(V ) =
(where Π(V ) is the admissible continuous GL2 (Af )-representation associated to V
via Definition 7.1.1).
7.8.2. Remark. In light of Remark 7.7.10 and Conjecture 7.7.11, we have restricted
our attention to odd Galois representations and to the multiplicity space M(V ).
We also state the following conjecture, which is well-posed independent of any
assumption about the existence of a local p-adic correspondence defined on all GQp representations.
7.8.3. Conjecture. If V is an odd irreducible continuous representation of GQ
over E, then there is an admissible unitary GL2 (Qp )-representation Πp (V ) such∼that M(V ) −→ Πp (V ) ⊗E π m, p (V ). Furthermore, the association of Πp (V ) to V
satisfies the following properties:(1) If V and V 0 are two odd irreducible continuous two dimensional representations of GQ over E, then V |Dp ∼= V 0|Dp as GQp -representations if and only
if there is a GL2 (Qp )-equivariant topological isomorphism Πp (V ) ∼= Πp (V 0 ).

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

85(2) The representation Πp (V ) has central character corresponding to the local
Galois character (det V )|GQp ε via local class field theory.
(3) The representation Πp (V ) satisfies the hypothesis of Definition 3.2.2, andΠp (V ) is associated to the semi-simplification of V |Dp with respect to the
local mod $ Langlands correspondence of [10].
(4) If V |Dp is irreducible then Πp (V ) is topologically irreducible.
(5) The representation V |Dp is potentially semi-stable, with distinct Hodge-Tateweights, if and only if the subspace Πp (V )lalg of locally algebraic vectors ofΠp (V ) is non-zero. Furthermore, if these conditions hold, then the subspace Πp (V )lalg coincides with the locally algebraic representation π
ep (V |Dp )attached to V |Dp as defined in condition (7) of Conjecture 3.3.1.
b(6) For any character η ⊗ ψ ∈ T(E)there is an equality of dimensions
dim Ref η⊗ψ (V ) = dim Expη|

|⊗ψε|

|−1

(Πp (V )an ).

(7) If V |Dp is trianguline, then there is a GL2 (Qp )-equivariant topological isomorphism Πp (V ) ∼= B(V |Dp ), where B(V |Dp ) is the admissible unitary
GL2 (Qp )-representation attached to V |Dp by the discussion of Section 6.12
7.8.4. Remark. Lemma 7.7.2 shows that the formation of Πp (V ) would necessarily
be compatible with twisting, and with change of scalars.
7.8.5. Remark. As follows from the discussion of the preceding sections, condition (7) is by no means independent of the preceding conditions. For example,
if V |Dp is irreducible and trianguline, and is either not potentially semi-stable up
to a twist, or else is potentially crystalline and Frobenius semi-simple up to a
twist, then conditions (4) and (6) imply that Πp (V ) ∼= B(V |Dp ). Similarly, if V |Dp
is the direct sum of two characters, then conditions (3), (5), and (6) imply thatΠp (V ) ∼= B(V |Dp ).
7.8.6. Remark. If V = Vf for some classical newform f of weight k ≥ 2, then
Proposition 7.7.14 shows that Πp (V ), if it exists, must contain π
bp (V ). If furthermore V |Dp is irreducible (equivalently, if f is not a twist of a form that is ordinaryat p), then condition (4) of the conjecture implies that Πp (V ) should equal πbp (V ),
or equivalently, that M(V )lalg should be dense in M(V ).
The following result provides some evidence for Conjecture 7.8.3 at the primes
away from p.
7.8.7. Proposition. If V is not a twist of a representation of finite image, thenpfor any tame level K p , the inclusion M(V )K ⊂ M(V ), together with the actionp
Σ(K )Σ(K p )of GL2 (Af) on M(V ), induces a natural GL2 (Af)-equivariant map
O

p
0
(75)
π`m (V ) ⊗E M(V )K → M(V )`6∈Σ(K p )Σ(K p )

(where GL2 (Af) acts through its action on the first factor). If V is furthermore
absolutely irreducible, then (75) is an embedding.
12For those reducible indecomposable V|Dp for which B(V |Dp ) has not yet been defined, we
understand this statement to mean just that Πp (V ) is an extension of the form conjectured in
Subsections 6.4 and 6.5 above.

86

MATTHEW EMERTONp

Proof. Write Σ := Σ(K p ). If M(V )K vanishes then there is nothing to prove, sopwe may assume that M(V )K 6= 0. Lemma 7.7.5 then shows that V is unramified
outside Σ. Thus, letting λ denote the system of Hecke eigenvalues attached to V ,we have thatO
GL2 (AΣf)0 m(76)π` (V ) ∼= c − IndbΣ 1 sm,λ .
GL2 (Z )`6∈Σ(Here c − Ind denotes the compactly supported induction, and the subscript λ indicates that we take the maximal quotient on which T(Σ) acts through λ.) The
induced representation appearing in (76) is universal for maps to unramified representations of GL2 (AΣf ) on which T(Σ) acts via λ; more precisely, for any smoothΣ
GL2 (Af )-representation W over E, there is a natural isomorphism ∼GL2 (AΣf)bΣ ) 1)sm,λ , W −→GL2 (ZHomGL2 (AΣf ) (c − Indb Σ λW GL2 (Z ).

b Σ ), andWe write K p as the product of its ramified and unramified parts KΣp ×GL2 (ZpKΣthen apply this isomorphism with W taken to be M(V ) . Since Proposition 7.7.7
pshows that the Hecke algebra T(Σ) acts on the non-zero space M(V )K through λ,we obtain the map (75).
Suppose now that V is absolutely irreducible. For any fixed value of `, eitherπ`m (V ) = π` (V ) is irreducible, or else π`m (V ) is the non-split extension of a character of GL2 (Q` ) by an irreducible (special) representation. Thus if (75) is notp
an injection, we find that M(V )K contains a one dimensional GL2 (Q` )-invariant
subspace for some ` 6∈ Σ. If we denote this one dimensional subspace by L, thenb 1 (K p )E .
Lemma 7.7.2 (1) gives a GQ ×GL2 (Q` )-equivariant embedding V ⊗E L → H
If we let L denote the reduction mod $ of L, then reducing this embedding mod $(and taking into account Lemma 7.2.3) yields a GQ × GL2 (Q` )-invariant embedding
V ⊗F L → lim
H 1 (Y (Kp K p ), F).
−→

(77)

Kp

Thus for some sufficiently small compact open subgroup Kp of GL2 (Qp ) we seethat H 1 (Y (Kp K p ), F) contains a copy of V which is invariant under GL2 (Q` ).
This contradicts Ihara’s lemma [39] (see also the proof of Theorem 4.1 in [46]).
(We require the hypothesis that V is absolutely irreducible because Ihara’s lemma
is a statement about the cohomology of closed modular curves.)
7.9. Consequences of the local-global compatibility conjecture. The following result makes explicit some of the consequences of the preceding conjecture.
(The ? are to indicate that each assertion is dependent on Conjecture 7.8.3.)
7.9.1. Proposition. Conjecture 7.8.3 has the following consequences. (As usual welet V denote an odd irreducible continuous two dimensional representation of GQ
over E.)(1)? If N is the tame conductor of V , then V appears as a GQ-subrepresentationb p ) | g ≡ ∗0 ∗1 mod N }.)b 1 (K p (N ))E . (Here K 1 (N ) := {g ∈ GL2 (Z
of H
1

p

(2)? V is pro-modular.
(3)? The representation V |Dp of GQp is potentially semi-stable with distinct
Hodge-Tate weights if and only if V is a twist by a power of ε of the Galois

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

87

representation attached to a classical cuspidal Hecke eigenform of weightk ≥ 2.
(4)? The representation V |Dp of GQp is trianguline if and only if V is a twist ofa Galois representation attached to a p-adic overconvergent cuspidal Hecke
eigenform of finite slope.
p

Proof. If V has tame conductor N , then π p (V )K1 (N ) 6= 0. Thus the conjecturepimplies that M(V )K1 (N ) 6= 0, giving (1)? . If Σ denotes the set of primes dividing
N p, and if λ denotes the system of eigenvalues attached to V , then T(Σ) actsλpp
on π m, p (V )K1 (N ) through λ. The conjecture thus implies that M(V )K1 (N ) 6=0, and so (2)? follows from Lemma 7.3.13. Part (3)? follows immediately fromcondition (5) of the conjecture together with Proposition 7.7.14, while part (4)?
follows from condition (6) of the conjecture together with Proposition 7.7.17. 
7.9.2. Remark. Note that part (1)? of the preceding proposition provides a strong
converse to Lemma 7.7.5.
7.9.3. Remark. Part (2)? is in many cases a theorem of Böckle. Indeed, he has
shown in a large number of cases for which V is absolutely irreducible that themodular points are Zariski dense in the deformation space of V , and thus that V
is necessarily promodular [11].
7.9.4. Remark. As we remarked in the introduction, part (3)? is a consequenceof the Fontaine-Mazur conjecture [37, Conj. 3c], while part (4)? is related to a
conjecture of Kisin [41, Conj. 11.8].
7.9.5. Remark. The converse to (2)? above seems quite plausible: namely, if one
assumes that V is promodular, it seems reasonable to expect that M(V ) 6= 0.
When combined with the results of Böckle mentioned in Remark 7.9.3, this provides
a strong source of motivation for Conjecture 7.8.3.
7.9.6. Remark. If we combine (1)? with the proof of (4)? , we obtain the following
result: if V is an irreducible GQ -representation attached to a finite slope overconvergent eigenform of some tame level, and if N is the tame conductor of V , then V
is in fact attached to a finite slope overconvergent eigenform of tame level N (givinga kind of theory of newforms for finite slope overconvergent eigenforms). Can one
prove this unconditionally? Note that in many cases when V is absolutely irreducible, the work of Böckle mentioned in Remark 7.9.3 identifies the localization ofT(K1p (N )) at the maximal ideal corresponding to V with the universal deformation
ring of V . Thus one can conclude in these cases that V is attached to a p-adic
modular form of tame level N , and the problem is to show that this p-adic modular
form can be taken to be overconvergent. If V |Dp is reducible then one can do this,since the ordinary part of T(K1p (N )) becomes identified with the ordinary universal
deformation ring (under appropriate assumptions on V – see [17, §1] for example).
7.10. Some partial results. Recall from Theorem 7.6.1 that if V is a twist of the
Galois representation attached to an overconvergent cuspidal Hecke eigenform offinite slope, then V |Dp is trianguline, so that we may define B(V |Dp ) (at least formost such V |Dp ) according to the discussion of Subsection 6.
The next two results are in the direction of part (7) of Conjecture 7.8.3. Notethough that they only apply to Galois representations V that are assumed to be

88

MATTHEW EMERTON

attached to finite slope overconvergent eigenforms (up to a twist), and so do notshed any light on part (4)? of Proposition 7.9.1.
The first theorem deals with the case when V actually arises (up to a twist)from a classical newform (which by [41, Thm. 6.6] is essentially equivalent to assuming that V |Dp is potentially semi-stable up to a twist). In the case of V satisfying condition (a) of the theorem, it strengthens a part of [13, Thm. 1.1.2] (thisstrengthening being made possible by virtue of the results of [8, 26]). For V satisfying condition (b), it is essentially a restatement of [14, Thm. 1.1.2 (i)]. (Tofacilitate the comparison with Conjecture 7.8.3, we recall from Remark 7.1.2 that
π p (V ) = π m, p (V ) in the context of the theorem.)
7.10.1. Theorem. Let V be a twist of the Galois representation attached to a classical cuspidal Hecke eigenform of finite slope and weight k ≥ 2. Assume furthermorethat the following conditions hold:(a) If V |Dp is irreducible, then one (equivalently every) potentially semi-stable
twist of V |Dp is Frobenius semi-simple.
(b) If V |Dp is reducible, then it is indecomposable, and is potentially crystalline
up to a twist.
Then there is a GL2 (Apf )-equivariant isomorphism
π p (V ) → HomGL2 (Qp ) B(V |Dp ), M(V ) .
Furthermore, all the non-zero morphisms in the target of this isomorphism are
injective.
Proof. Replacing V by an appropriate twist, and appealing to Lemma 7.7.2 (2),we see that it is no loss of generality to assume that V is attached to a classicalnewform f of weight k ≥ 2, and that either V |Dp is irreducible and Frobenius semisimple, or else is reducible, indecomposable, and potentially crystalline. Let πbp (V )
denote the completion of π
ep (V |Dp ) attached to V by Proposition 7.7.14.
Suppose first that V |Dp is irreducible. We will show that B(V |Dp ) ∼bp (V ). The=πclaimed isomorphism will then follow from Proposition 7.7.14 (4).
If V |Dp is potentially crystalline, then B(V |Dp ) is equal to the universal unitarycompletion of π
ep (V |Dp ) (cf. Remark 6.1.4), and is topologically irreducible (Theorem 5.1.6), and so we are done. If V |Dp is not potentially crystalline, then themain result of [13] gives a non-zero homomorphism B(1 − w, L) ⊗ η → π
bp (V ), where
1 − w is the weight of f , η is a character, and L is an (a priori unknown) elementof E. By construction the image of this map contains π
ep (V |Dp ), and so is densein π
bp (V ). Since B(1 − w, L) is admissible unitary and topologically irreducible(Theorem 5.1.13) this map must be an isomorphism. In particular, πbp (V ) is also
topologically irreducible. As Colmez explained to me, it then follows from [26] that
L must in fact equal the L-invariant of V |Dp , and thus that B(V |Dp ) ∼bp (V ).
=π±Colmez’s argument is as follows: Let µf denote the Mazur-Tate-Teitelbaum
distributions defining the p-adic L-function attached to f . Essentially by definition
these distributions may be regarded as elements of the topological dual to theuniversal unitary completion B of π
ep (V ). As is explained in [13, §5] (see also[34, §5]) it follows from the construction of µ±
f via modular symbols that they infact lie in the topological dual of the quotient πbp (V ) of B. On the other hand,
Kato’s approach to defining p-adic L-functions, via the method of “Coleman power

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

89

series” and explicit reciprocity laws, when reinterpreted in the language of (φ, Γ)modules (as in [25]) and combined with the main result of [26], shows that thesesame distributions µ±f also lie in the topological dual of the quotient B(V |Dp ) of B.
Since both B(V |Dp ) and πbp (V ) are topologically irreducible quotients of B, we seethat they must be quotients by the same closed subrepresentation, and thus thatπbp (V ) ∼= B(V |Dp ).
Consider now the case when V |Dp is reducible, but indecomposable. Recall from
Remark 7.7.15 that πbp (V ) is a closed subrepresentation of B(V |Dp ). It follows from
[14, Thm. 1.1.2 (ii)] that any non-zero element of HomGL2 (Qp ) B(V |Dp ), M(V )must be injective, and thus that restricting homomorphisms yields an injection ∼(78) HomGL2 (Qp ) B(V |Dp ), M(V ) → HomGL2 (Qp ) π
bp (V ), M(V ) −→ π p (V )(the isomorphism being provided by Proposition 7.7.14 (4)). Since π p (V ) is anirreducible GL2 (Apf )-representation, and since [14, Thm. 1.1.2 (i)] shows that thesource is non-zero, we see that (78) is in fact an isomorphism. Thus in this case weagain obtain the desired isomorphism.

7.10.2. Remark. Suppose in the context of the preceding
theorem that V |Dp isL
the direct sum of two characters, say V |Dp = η ψ. Since V is associated toa classical cuspform, all of the local factors of the representation of GL2 (Af ) associated to V are generic [40, p. 354], and so in particular ηψ −1 6= ε±1 . Thus
L
GL (Q )GL (Q )
IndP(Q2 ) p ψ ⊗ ηε cont . If we label the sumB(V |Dp ) := IndP(Q2 ) p η ⊗ ψε contppmands η and ψ in such a way that ηψ −1 has Hodge-Tate weight k − 1, then thefirst of these summands is the universal unitary completion of π
ep (V |Dp ). One thusp
has isomorphisms of smooth GL2 (Af )-representations(79)
GL (Q )
HomGL2 (Qp ) (IndP(Q2 ) p η ⊗ ψε)cont , M(V )p ∼∼−→ HomGL2 (Qp ) π
ep (V |Dp ), M(V ) −→ πp(the first isomorphism following from the universal property of the universal unitarycompletion, and the second from Theorem 7.4.2). Part (7) of Conjecture 7.8.3
implies that there should be a similar isomorphism of GL2 (Apf )-representations?
 ∼GL (Q )
HomGL2 (Qp ) (IndP(Q2 ) p ψ ⊗ ηε)cont , M(V ) −→ πp .
p

It follows from [14, Thm. 1.1.2 (i)] that the source of this conjectural isomorphism
is non-zero (and in fact that it has a non-zero space of K p -invariants, if K p denotes
the tame level of the newform f giving rise to V ). However neither that result
nor its proof seems to extend in any immediate way to determine the structure of
GL (Q )
HomGL2 (Qp ) (IndP(Q2 ) p ψ ⊗ ηε)cont , M(V ) as a GL2 (Apf )-representation.
p

The next result deals with the case when V is attached to a non-classical finite
slope overconvergent eigenform.
7.10.3. Theorem. If V is a twist of a Galois representation attached to a nonclassical overconvergent eigenform of finite slope and tame level K p , and if V |Dp ispirreducible, then there is a GL2 (Qp )-equivariant embedding B(V |Dp ) → M(V )K .

90

MATTHEW EMERTON

Proof. If R is a non-ultracritical refinement of V |Dp , then Theorem 7.6.1 and Propop
sition 7.5.3 together yield a map B(R) → M(V )K . Since V |Dp is irreducible, andsince the eigenform to which it is associated (up to a twist) is not classical, we inferfrom [41, Thm. 6.6] that V |Dp is not potentially semi-stable up to a twist. Thus
B(V |Dp ) := B(R), and the theorem follows.

7.10.4. Remark. The various restrictions on the structure of V |Dp in the statements of the preceding two theorems are not actually so serious. For example, if
Conjecture 5.1.5 were to be proved in full generality, then we could eliminate condition (a) from the statement of Theorem 7.10.1. In any case, it is conjectured that
V |Dp is always Frobenius semi-simple when V is the Galois representation attached
to a classical newform.
Let us now consider the case when V |Dp is reducible, but does not satisfy condition (b) of Theorem 7.10.1. If V |Dp is indecomposable, and if the conjectured
extensions of Subsections 6.4 and 6.5 were constructed, so that B(V |Dp ) could be
defined, then an argument analogous to that of [14, §5.6] would provide an embedding B(V |Dp ) ⊂ M(V ) (at least if V |Dp is an extension of distinct characters).
(Note that in the case when V admits an L-invariant, one would build on the main
result of [13], which provides an embedding of the appropriate twist of B(2, L) into
M(V ).)
Suppose on the other hand that V |Dp is split. If V is attached to a classicalform of weight k ≥ 2, then (as was implicitly recalled in Remark 7.10.2 above)it follows from [14, Thm. 1.1.2 (i)] that the analogue of Theorem 7.10.3 holds
for V . If instead V is attached either to a form of weight 1 or to a non-classicalform, then the method of proof of [14, Thm. 1.1.2 (i)] will extend to establish an
analogue of Theorem 7.10.3 for V provided that one can prove the existence of
p-adic companion forms in these contexts. (Essentially, one is given an ordinary
eigenform corresponding to one equivalence class of ordinary refinements, and one
has to construct an ordinary eigenform that corresponds to the other equivalenceclass of ordinary refinements.) The isomorphism between the ordinary deformation
ring and the ordinary Hecke ring attached to V provided by [17, §1] (for example)
will establish the existence of such a companion form, provided that V satisfies thenecessary hypotheses for that result to apply (and so Theorem 7.10.3 does extend
to cover these cases).
7.10.5. Remark. In contrast to the cases considered in Theorem 7.10.1, in Theorem 7.10.3 we do not have any control over the multiplicity with which B(V |Dp )appears in M(V ). (As was noted in Remark 7.9.6, it seems that one doesn’t evenpknow for which tame levels K p one has M(V )K 6= 0.)
7.10.6. Remark. The preceding results, combined with Proposition 7.7.14, showthat if V is the GQ -representation attached to a finite slope overconvergent eigenform for which V |Dp is irreducible, and is not a twist of a potentially semi-stable
Frobenius non-semi-simple representation, then the representation B(V |Dp ) satisfies
condition (7) of Conjecture 3.3.1. It follows that B(V |Dp ) also satisfies condition (8)of that conjecture (cf. Proposition 6.6.5, and note that Proposition 7.6.5 allows us
to replace the inequality in part (3) of that result by equality for the representations
under consideration).

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

91

The final result of our paper gives a kind of converse to the preceding theorems.
It includes as a special case a part of [13, Thm. 1.1.2].
7.10.7. Theorem. Assume that V is not a twist of a representation having finite
image, and let W be a trianguline continuous two dimensional representation of
GQp for which B(W ) has been defined (i.e. W satisfies one of the three conditions
stated at the beginning of Subsection 6.6).
(1) If there exists a non-zero continuous GL2 (Qp )-equivariant map B(W ) →M(V ), then V is a twist of a representation attached to an overconvergent
eigenform of finite slope, and there is an isomorphism (V |Dp )ss ∼= W ss .
(2) If V |Dp is indecomposable, if W is not the direct sum of two copies of thesame character, and if the map of (1) is furthermore an embedding, thenV |Dp ∼
= W.
Proof. Suppose that we are given a non-zero map B(W ) → M(V ). Our assumption
on V , together with Proposition 7.7.7, shows that T(Σ(K p )) acts on M(V ) throughthe system of eigenvalues λ associated to V . Thus there is induced a non-zero map
b 1 (K p )λE .
(80)
B(W ) → H
Suppose first that W is irreducible, and let R be a non-ultracritical refinementof W . If we write σ(R) = (η, ψ), then Lemma 6.6.3 shows that
η| |⊗ψε| |−1 b 1
H (K p )λE,an 6= 0.
(81)
JP(Qp )
e p )(E) (byThus (η | | ⊗ψε | |−1 , λ) lies in Spec A(K p ) (E), and hence in D(KTheorem 7.5.8, since λ is non-Eisenstein). In particular we see that V is attached
to a twist of a finite slope overconvergent eigenform. Theorem 7.6.1 (2) then showsthat there is a refinement R0 of V |Dp such that σ(R0 ) = (η, ψ). Thus Theorem 4.5.4
implies that V |Dp and W are isomorphic, unless ηψ −1 ε−1 = z w for some w > 0,in which case V |Dp and W each admit an L-invariant. In this case we can apply
the argument of Colmez given in the proof of Theorem 7.10.1. Namely, after twisting each of V and W , we may assume that V |Dp is semi-stable, and thus (by [41,Thm. 6.6]) that V is attached to a classical newform f . The Mazur-Tate-Teitelbaum
distributions µ±
f , which are naturally distributions on B(R), are then seen to induce distributions on both B(V |Dp ) and B(W ). Since these are both topologicallyirreducible quotients of B(R), they must coincide, and so again V |Dp ∼= W . (Ifthe L-invariants of V |Dp and W are finite, then we could instead appeal to [13,Thm. 1.1.2].)
We summarize the case when W is reducible, leaving the reader to fill in thedetails. As we observed in Subsection 6.6, the representation B(W ) satisfies condition (8) of Conjecture 3.3.1. Thus if the map (80) is injective, then condition (81)holds for every refinement R of W , and hence (again applying Theorem 7.6.1 (2))
we find that Ref σ (W ) 6= ∅ for some σ ∈ Homcont (Wp , T(E)) implies that also
Ref σ (V |Dp ) 6= ∅. If we assume furthermore that V |Dp is indecomposable, and that
W is not the direct sum of two copies of the same character, then, given our otherrestrictions on W , and recalling Propositions 4.5.5 and 4.5.6 and Lemmas 4.4.1
and 4.4.3, we conclude that indeed V |Dp ∼= W.
Suppose now that (80) is not necessarily injective. Writing W as an extensionof ψ by η (and interchanging ψ and η if necessary in the case when W is split),
and taking into account the definition of B(W ), we may then assume that either

92

MATTHEW EMERTONGL (Q )(a) the map (80) induces an injection IndP(Q2p ) p η ⊗ ψε cont → M(V ), (b) the
GL (Q )map (80) induces an injection IndP(Q2p ) p ψ ⊗ ηε cont → M(V ), (c) ηψ −1 = ε,and (80) induces an injection η ◦ det ⊗B(2, ∞) → M(V ), or (d) W = ηψ −1 = ε,b → M(V ).
and (80) induces an injection η ◦ det = η ◦ det ⊗B(2, ∞) / η ◦ det ⊗St
In cases (a) and (c) (resp. (b)), a consideration of Jacquet modules shows that V |Dp
admits a refinement R such that σ(R) = (η, ψ) (resp. σ(R) = (ψ, η)). This implies
that V is reducible, and that (V |Dp )ss ∼= W ss . In case (d), M(V ) would containthe one dimensional representation η ◦ det of GL2 (Qp ), which is impossible. (Sucha representation is locally algebraic up to a twist, but Theorem 7.4.2, together withthe fact that the local factors of the GL2 (Af )-representation attached to a cuspidalb 1 (K p )E,lalg cannot contain a one
newform are generic [40, p. 354], shows that H
dimensional GL2 (Qp )-representation.) Thus case (d) cannot occur.


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Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston,
IL 60208
E-mail address: emerton@math.northwestern.edu