Completed cohomology and the p-adic Langlands program
Matthew Emerton ∗Abstract. We discuss some known and conjectural properties of the completed cohomology of congruence quotients associated to reductive groups over Q. We also discuss
the conjectural relationships to local and global Galois representations, and a possible
p-adic local Langlands correspondence.
Mathematics Subject Classification (2010). Primary 11F70; Secondary 22D12.
Keywords. p-adic Langlands program, p-adic Hodge theory, completed cohomology,
Galois representations.

1. Introduction
The idea that there could be a p-adic version of the local Langlands correspondence
was originally proposed by C. Breuil, in the case of the group GL2 (Qp ), in his
papers [11, 12, 13]. In [13], he also proposed a local-global compatibility between
this (at the time conjectural) correspondence and p-adically completed cohomology
of modular curves. Since then, the theory of p-adic Langlands has been extensively
developed in the case of the group GL2 over Q; see e.g. [6, 29, 50, 30, 32, 46, 52].
Since excellent expositions of much of this material already exist [5, 14, 15], wehave decided here to describe some aspects of the p-adic Langlands program thatmake sense for arbitrary groups. This has led us to focus on completed cohomology,
since this is a construction that makes sense for arbitrary groups, and about which
it is possible to establish some general results, and make some general conjectures.
Many of these conjectures are very much motivated by the conjectural relationship with Galois representations, and we have also tried to outline our expectations
regarding this relationship, while trying not get bogged down in the myriad technical details that would necessarily attend a more careful discussion of this topic.
Finally, we have tried to indicate how completed cohomology may be related, by
the principle of local-global compatibility, to a still largely conjectural p-adic local Langlands correspondence for groups other than GL2 (Qp ). Our focus is on
drawing inferences about the possible structure of the local correspondence by interpolating from our expectations regarding completed cohomology. In this regard
we mention also the paper [27], which somewhat literally interpolates p-adically∗ The author’s work is partially supported by NSF grant DMS-1303450

2

Matthew Emerton

completed cohomology (via the Taylor–Wiles–Kisin method) in order to constructa candidate for the p-adic local Langlands correspondence for GLn of an arbitrary
p-adic field.
We close this introduction by simply listing a number of additional important
papers [17, 18, 19, 20] which also have the goal of extending the p-adic local Langlands correspondence, and local-global compatibility, beyond the case of GL2 (Qp ).
Unfortunately, lack of space prevents us from saying more about them here.

1.1. Notation. Throughout, we fix a prime p. We also fix a finite extension
L of Qp , contained inside some given algebraic closure Qp . We let O denote thering of integers of L, and let $ denote a uniformizer of O. The ring O, and field L,
will serve as our coefficients.
As usual, A denotes the ring of adèles over Q, Af denotes the ring of finiteadèles, and Apf denotes the ring of prime-to-p finite adéles. For a given algebraic
group G over Q, we will consistently write G∞ := G(R), and G := G(Qp ). Additional notation will be introduced as necessary.

1.2. Acknowledgements. The ideas expressed here owe much to the many
discussions with my colleagues and collaborators that I’ve enjoyed over the years,
and I would like to thank them, especially Ana Caraiani, Pierre Colmez, DavidGeraghty, Michael Harris, David Helm, Florian Herzig, Mark Kisin, David Savitt,and Sug Woo Shin. I am particularly indebted to Christophe Breuil, whose ideas
regarding the p-adic Langlands program have been so influential and inspiring.
Special thanks are owed to Frank Calegari, Toby Gee, and Vytas Paškūnas; manyof the ideas described here were worked out in collaboration with them.
I am also grateful to Frank Calegari, Tianqi Fan, Toby Gee, Michael Harris,
Daniel Le, David Savitt, and the Imperial College Study Group (Rebecca Bellovin,
Kevin Buzzard, Toby Gee, David Helm, Judith Ludwig, James Newton, and Jack
Shotton) for their careful reading of various preliminary versions of this note.

2. Completed cohomology
Ideally, in the p-adic Langlands program, we would like to define spaces of p-adic
automorphic forms, to serve as p-adic analogues of spaces of classical automorphic
forms in the usual Langlands program. Unfortunately, for general reductive groups,
no such definition is currently available. For groups that are compact at infinity,
we can make such a definition (see (2.2.4) below), while for groups giving riseto Shimura varieties, we can use algebro-geometric methods to define spaces of
p-adic automorphic forms, which however seem less representation-theoretic innature than classical automorphic forms. Since one of our main goals is to employ
representation-theoretic methods, we are thus led to find an alternative approach.
The approach we take here is to work with p-adically completed cohomology.
This has the advantages of being definable for arbitrary groups, and of beingof a representation-theoretic nature. For our purposes, it will thus serve as aCompleted cohomology and the p-adic Langlands program

3

suitable surrogate for a space of p-adic automorphic forms. For groups that arecompact at infinity, it does recover the usual notion of p-adic automorphic forms
that was already mentioned. (Its relationship to algebro-geometric notions of padic automorphic forms in the Shimura variety context is less clear; P. Scholze’s
paper [53] makes fundamental progress in this — and many other! — directions,but we won’t attempt to discuss this here.) We therefore begin our discussionof global p-adic Langlands by recalling the basic definitions and facts related tocompleted cohomology, referring to [24] and [31] for more details.

2.1. Definitions and basic properties. Let us suppose that G is areductive linear algebraic group over Q. We write G∞ := G(R) for the group of
real-valued points of R; this is a reductive Lie group. We let A∞ denote the Rpoints of the maximal Q-split torus in the centre of G, and let K∞ denote a choiceof maximal compact subgroup of G∞ . For any Lie group H, we let H ◦ denote thesubgroup consisting of the connected component of the identity.
◦
The quotient G∞ /A◦∞ K∞
is a symmetric space on which G∞ acts. We denoteits dimension by d. We also write l0 := rank of G∞ − rank of A∞ K∞ , andq0 := (d − l0 )/2. If G∞ is semisimple (so that in particular A∞ is trivial) thenthese quantities coincide with the quantities denoted by the same symbols in [10](which is why we notate them as we do). We note that q0 is in fact an integer.
(For the role played by these two quantities in our discussion, see (2.1.6), as well
as Conjectures 3.1 and 3.2, below.)
If Kf ⊂ G(Af ) is a compact open subgroup, then we write◦
Y (Kf ) := G(Q)\G(A)/A◦∞ K∞
Kf .

The double quotient Y (Kf ) is a finite union of quotients of the symmetric space◦G◦∞ /A◦∞ K∞
by cofinite volume discrete subgroups of G∞ .
2.1.1. Definitions. We fix a tame level, i.e. a compact open subgroup Kfp ⊂G(Apf ). If Kp ⊂ G := G(Qp ) is a compact open subgroup, then of course Kp Kfp is
a compact open subgroup of G(Af ), and so we may form the space Y (Kp Kfp ). We
define the completed (co)homology at the tame level Kfp as follows [24]:e i := lim lim H i Y (Kp K p ), O/$s
H
f
←− −→
s

Kp




e i := lim Hi Y (Kp K p ), O ,
and H
f
←−

(1)

Kp

where in both limits Kp ranges over all compact open subgroups of G(Qp ).
e i and H
e i with its evident inverse limit topology. In theWe equip each of Hie , each of the O/$s -modules lim H i Y (Kp K p ), O/$s appearing incase of HfK−→ p
the projective limit is equipped with its discrete topology, and the projective limite i coincides with its $-adic topology; He i is complete with respect
topology on Heto this topology. In the case of Hi , each of the terms Hi Y (Kp Kfp ), O (which isa finitely generated O-module) appearing in the projective limit is equipped withe i is then a pro-finite topology, and so in
its $-adic topology. The topology on He i is compact.
particular H

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Matthew Emerton

The completed cohomology and homology are related to one another in theeiusual way by duality over O [24]. In particular, if we ignore O-torsion, then H
e i (equipped with its weak topology), and He i is the continuousis the O-dual of H
ei.
O-dual of H
We can also form the limit(2)
H i Y (Kp Kfp ), O .
H i := lim lim H i Y (Kp Kfp ), O/$s ∼
= lim
−→
−→ ←−
Kp

s

Kp

There is a natural morphisme i,
Hi → H(3)b i ,→ H
e i,
H

(4)

which induces an embedding
whose source is the $-adic completion of H i . Note that the morphism (3) neednot be injective: although each of the terms H i Y (Kp Kfp ), O in the direct limit
defining H i is a finitely generated O-module, the limit H i is merely countably
generated as an O-module, and hence may contain divisible elements. These thenbecome zero after we pass to the $-adic completion to obtain the embedding (4).
(We give an example of this in (2.2.3) below.)
We furthermore remark that the transition maps in the direct limits (1) and (2)need not be injective. (This is also illustrated by the example of (2.2.3).) However,
if Kp0 ⊂ Kp , then a trace argument shows that the restriction map
H i Y (Kp Kfp ), O) → H i Y (Kp0 Kfp ), O)
becomes injective after tensoring with L. Thus the kernel of the natural map
H i Y (Kp Kfp ), O) → H i consists of torsion classes.
On the other hand, the kernel of the natural mapei
H i Y (Kp Kfp ), O) → H(5)need not consist only of torsion classes; it is possible for non-torsion classes to haveb i (and so also
infinitely divisible image in H i , and hence have vanishing image in H
e i ). (Again, we refer to (2.2.3) for an example of this.).
have vanishing image in H
The morphism (4), although injective, need not be surjective. Its cokernel isnaturally identified with Tp H i+1 := lims H i+1 [$s ]; in other words we have a short←−
exact sequenceib →He i → Tp H i+1 → 0.
0→H
(6)
(Here we have written H i+1 [$s ] to denote the submodule of $s -torsion elements
in H i+1 , and the projective limit is taken with respect to the multiplication-by-$
map from H i+1 [$s+1 ] to H i+1 [$s ]. The notation “Tp ” is for Tate module.) In
particular, if all the cohomology modules H i+1 Y (Kp Kfp ), O are torsion free, sothat H i+1 is torsion free, then the morphism (4) is an isomorphism.
Although the restriction maps (5) can have non-trivial kernels, one can recover
the cohomology at the various finite levels Kp Kfp via the Hochschild–Serre spectral

Completed cohomology and the p-adic Langlands program

5

sequence discussed in (2.1.3) below. The precise manner in which the cohomology
at finite levels gets encoded in the completed cohomology is somewhat complicatedin general. For instance, infinitely divisible torsion elements in H i give rise toelements of Tp H i , which by the discussion of the previous paragraph is naturally
e i−1 . However, infinitely divisible non-torsion elements of H i havea quotient of H
no obvious incarnation as elements of completed cohomology, and the manner inwhich they are recovered by the Hochschild–Serre spectral sequence can be subtle.
2.1.2. Group actions. There is a natural continuous action of G := G(Qp ) one i and H
e i , whose key property is encapsulated in the following result.
each of He i makes it a $-adically admissible
Theorem 2.1. [24, 31] The G-action on H
representation of G; i.e. it is $-adically complete as an O-module, and each quoe i /$s (which is a smooth representation of G over O/$s ) is admissible
tient Hin the usual sense (for each compact open subgroup Kp ⊂ G, the submodule ofKp -invariants is finitely generated over O).
If Kp ⊂ G is compact open, then we write O[[Kp ]] := limK 0 O[Kp /Kp0 ], where←− p
the projective limit is taken over all open normal subgroups Kp0 ⊂ Kp . Since G
e i and H
e i , it follows that for any such Kp , the Kp -action on
acts continuously on H
each of these modules may be promoted to an action of O[[Kp ]]. One then has thee i.
following reformulation of the admissibility of the G-action on He i is finitely generated.
Theorem 2.2. Each of the O[[Kp ]]-modules H(We remark that if this finite generation statement holds for one choice of Kp
then it holds for any such choice.)
2.1.3. Cohomology of local systems and the Hochschild–Serre spectral
sequence. If W is a finitely generated O-module equipped with a continuous
representation of an open subgroup Kp ⊂ G, and Kp is sufficiently small, so◦that G(Q) acts with trivial stabilizers on G(A)/A◦∞ K∞
Kp Kfp , then, for each open0
subgroup Kp ⊂ Kp , the representation W determines a local system W of O◦
modules on Y (Kp Kfp ), defined via W := G(Q)\ G(Af )/A◦∞ K∞
Kfp × W /Kp0 .
Suppose that W is furthermore torsion-free as an O-module, and let W ∨ denote theO-dual of W , endowed with the contragredient Kp -action. There is then [24, 31]a Hochschild–Serre spectral sequencee j ) =⇒ H i+j Y (Kp K p ), W .
E2i,j = ExtiO[[Kp ]] (W ∨ , H
fThis gives a precise sense to the idea that p-adically completed cohomology captures all the cohomology (with arbitrary coefficients) at finite levels. It is therealization, in the context of completed cohomology, of the general philosophy(brought out especially in Hida’s work, e.g. [44]) that when working with p-adic
automorphic forms, by passing to infinite p-power level one automatically encompasses automorphic forms of all possible weights. (For the precise relationship withclassical automorphic forms, see (2.1.6) below.)6

Matthew Emerton

In particular, if we take W = O (with the trivial Kp -action), we obtain ae j =⇒ H i+j Y (Kp K p ), O (where H i denotes
spectral sequence E2i,j = H i Kp , H
fcontinuous cohomology), which recovers the cohomology at the finite level Kp Kfp
from the completed cohomology. If we take a direct limit over all Kp , we obtain ae j =⇒ H i+j , which recovers the limits
spectral sequence E2i,j = limK H i Kp , H−→ pe i )Kp is
H i+j of the cohomology at finite levels. The edge map H i → limK (H−→ pinduced by the morphism (3). As already noted, the relationship between thisspectral sequence, the morphism (3), and the exact sequence (6) is subtle. (Seethe example of (2.2.3) below.)More generally, if we write H i (W) := limK 0 H i Y (Kp0 Kfp ), W (where Kp0 runs−→ pover all open subgroups of the fixed Kp , of which W is a representation), then wee i which relates the cohomology
have an edge map H i (W) → limK 0 HomKp0 (W ∨ , H−→ p
at finite levels with coefficients in W to the W ∨ -isotypic parts (for open subgroupse i.
Kp0 of Kp ) of H
2.1.4. Hecke actions. There is a finite set of primes Σ0 (containing p) such thatfor ` 6∈ Σ0 , we may factor Kfp = Kfp,` K` where Kfp,` is compact open in G(Ap,`f ),and K` is a hyperspecial maximal compact subgroup of G(Q` ). (In particular, G isunramified at such a prime `, so that it admits a hyperspecial maximal compact
subgroup.) We may then consider the spherical Hecke algebra (i.e. the double
coset algebra) H` := H G(Q` )//K` , O with coefficients in O. This algebra iscommutative [41, 42], and acts naturally (by continuous operators) on any of thecohomology groups H i Y (Kp Kfp ), W) considered in (2.1.3).
If we let i range over all cohomological degrees, let Kp range over all compact
open subgroups of G, and let W range
Q QoverQall representations of Kp on finitely
generated torsion O-modules, then i Kp W End H i Y (Kp Kfp ), W) is a profinite ring. We fix a set of primes Σ containing Σ0 , and we define the global Hecke
algebra TΣ to be the closure in this profinite ring of the O-subalgebra generated by
the image of H` for all ` 6∈ Σ. (The reasons for allowing the possibility of Σ being
larger than Σ0 are essentially technical; it is not misleading to simply imagine that
Σ is equal to Σ0 and is fixed once and for all, e.g. by being taken to be as small
as possible, given our fixed choice of tame level Kfp .) By construction, TΣ acts
on each of the cohomology groups H i Y (Kp Kfp ), W) considered in (2.1.3) (this is
obvious if W is torsion, and follows in general by writing W as a projective limit
e i and H
e i . (In the case of He i this follows from its
W ∼W/$s ), and also on H
= lim
←−s

definition in terms of limits of cohomology groups H i Y (Kp Kfp ), O/$s . It thene i by duality; more precisely, we can write He i as the projective limit
follows for Hpsover s and Kp of Hi Y (Kp Kf ), O/$ , and this latter module is the O/$s -dual
of H i Y (Kp Kfp ), O/$s .)e i and He i commute with the action of G. Thus
These Hecke actions on H
there is an action of Hecke on the E2 -terms, as well the E∞ -terms, of the various
Hochschild–Serre spectral sequences considered in (2.1.3), and these are in fact

Completed cohomology and the p-adic Langlands program7

spectral sequences of TΣ -modules.
The O-algebra TΣ is a complete semi-local ring, i.e. it is a product of finitely
many factors Tm , each of which is a complete local ring. (This finiteness statement
is proved via the methods of [3].) The topology on TΣ is the product of the m-adic
topologies on each of the complete local rings Tm .
2.1.5. Variants in the non-compact case, and Poincaré duality. There
are some variants of completed (co)homology which are useful in the case whenthe quotients Y (Kf ) are non-compact. (We recall that the quotients Y (Kf ) arecompact precisely when the semisimple part of the group G is anisotropic, i.e.
contains no torus that is split over Q.)
Namely, replacing cohomology by cohomology with compact supports, we can
e i, H i, Hb i , and Tp H i , and we have Hochschild–Serredefine G-representations Hccccspectral sequences for compactly supported cohomology. Similarly, we can define
e BM , which is related to He ci by duality over O.
completed Borel–Moore homology H
iThere is a more subtle duality over O[[Kp ]] (for a compact open subgroup
Kp ⊂ G) which relates the usual and compactly supported variants of completed
cohomology. It is most easily expressed on the homology side, where it takes theform of the Poincaré duality spectral sequencee j , O[[Kp ]]) =⇒ H
e BM
E2i,j := ExtiO[[Kp ]] (Hd−(i+j) .

(7)

Of course, when the quotients Y (Kf ) are compact, compactly supported andusual cohomology coincide, as do usual and Borel–Moore homology. In general,
we can relate them by considering the Borel–Serre compactifications Y (Kf ). If we
let ∂(Kf ) denote the boundary of Y (Kf ), then we may compute the compactly
supported cohomology of Y (Kf ) as the relative cohomology
Hci Y (Kf ), O/$s = H i Y (Kf ), ∂(Kf ); O/$s(and similarly we may compute Borel–Moore homology as relative homology),
and so we obtain the long exact sequence of the pair relating the cohomology ofY (Kf ), the compactly supported cohomology of Y (Kf ), and the cohomology of∂(Kf ). Passing to the various limits, we obtain a long exact sequence
e ci → H
ei → He i (∂) := lim lim H i ∂(Kp K p ), O/$s → H
e ci+1 → · · · .
··· → H
f
←− −→
s

Kp

e i (∂) can be computed in terms of
The completed cohomology of the boundary H
completed cohomology of the various Levi subgroups of G; see [24] for more details.
We also mention that there is a TΣ -action on each of the objects introduced
here, and that the various spectral sequences and long exact sequences consideredhere are all compatible with these actions.
2.1.6. The relationship to automorphic forms. We fix an isomorphism ı :Qp ∼= C. Since L ⊂ Qp , we obtain induced embeddings O ⊂ L ,→ C.

8Matthew Emerton

Suppose that V is an algebraic representation of G defined over L, and suppose
further that W is a Kp -invariant O-lattice in V, for some compact open subgroup
Kp of G := G(Qp ) ⊂ G(L). Write VC := C ⊗L V = C ⊗O W (the tensor products
being taken with respect to the embeddings induced by ı). If, for any compact
open subgroup Kp0 of Kp , we let VC denote the local system of C-vector spaces on
Y (Kp0 Kfp ) associated to VC , then we obtain a natural isomorphism
H i Y (Kp0 Kfp ), VC =: H i (VC ).
C ⊗O H i (W) ∼
= lim
−→
0
Kp

We may compute this cohomology using the space of automorphic forms on G [39].
Precisely, write A(Kf ) for the space of automorphic forms on G(Q)\G(A)/Kf ,for any compact open subgroup Kf ⊂ G(Af ), and write A(Kfp ) = limK A(Kp Kfp ),−→ pwhere the direct limit is taken (as usual) over the compact open subgroups Kp ⊂G := G(Qp ). For simplicity, suppose that A◦∞ acts on VC through some characterχ (this will certainly hold if V is absolutely irreducible), and let A(Kfp )χ−1 denotee∞the subspace of A(Kfp ) on which A◦∞ acts through the character χ−1 . Let G
denote the group of real points of the intersection of the kernels of all the rationale ∞ and K∞ respectively.
characters of G, and let eg and k denote the Lie algebras of G
Then it is proved by J. Franke in [39] that there is a natural isomorphism
H i (VC ) ∼g, k; A(Kfp )χ−1 ⊗ VC .
(8)
= Hi e(In the case when the the quotients Y (Kf ) are compact, this result is known as
Matsushima’s formula [49]. In the case when G = GL2 and i = 1, it is known
as the Eichler–Shimura isomorphism [54].) The action of TΣ on H i (W) induces
an action of TΣ on H i (VC ), and the resulting systems of Hecke eigenvalues thatappear in H i (VC ) (which we may think of as being simultaneously C-valued andQp -valued, by employing the isomorphism ı) are automorphic, i.e. arise as systemsof Hecke eigenvalues on the space of automorphic forms A(Kfp ).
Thus, to first approximation, we may regard the Hecke algebra TΣ (or, more
precisely, the Qp -valued points of its Spec) as being obtained by $-adically interpolating the automorphic systems of Hecke eigenvalues that occur in H i . However, asb i can have a non-trivial kernel. More signifwe already noted, the map from H i to H
icantly (given that the Hochschild–Serre spectral sequence shows that, despite thispossibility, the systems of Hecke eigenvalues appearing in H i can be recovered fromb i ,→ He i can have a non-trivial cokernel
the completed cohomology), the inclusion H
i+1
Tp H , which arises from infinitely divisible torsion in H i+1 . Thus Spec TΣ sees
not only the systems of eigenvalues arising from classical automorphic forms (andtheir $-adic interpolations), but also systems of Hecke eigenvalues arising fromtorsion cohomology classes (and their $-adic interpolations). We will discuss thispoint further in (3.1.3) below.
It will be helpful to say a little more about how one can use (8) to analyze
cohomology. To this end, we decompose the space of automorphic forms A(Kfp )χ−1
as the direct sumA(Kfp )χ−1 = Acusp (Kfp )χ−1 ⊕ AEis (Kfp )χ−1

(9)

Completed cohomology and the p-adic Langlands program

9of the cuspforms and the forms which are orthogonal to the cuspforms (in A(Kfp )χ )under the Petersson inner product (i.e. the L2 inner product of functions on
G(Q)\G(A)/A◦∞ Kfp )). We label this complement to the space of cuspforms withthe subscript Eis for Eisenstein, although its precise relationship with the space ofEisenstein series can be complicated; see e.g. [47], [39], and [40]. Note that if thequotients Y (Kf ) are compact, then A(Kfp )χ−1 = Acusp (Kfp )χ−1 .
The decomposition (9) is a direct sum of G∞ × G-representations, and so thecohomology H i eg, k; Acusp (Kfp )χ−1 ⊗ VC is a G-invariant direct summand of the
cohomology H i eg, k; A(Kfp )χ−1 ⊗ VC , which via the isomorphism (8) we identify
iwith a G-invariant direct summand Hcusp(VC ) of H i (VC ). It is also TΣ -invariant.
We may choose an everywhere positive element of A(Kfp ) on which G(A) acts
via a character extending χ2 , multiplication by which induces an isomorphism between Acusp (Kfp )χ−1 and Acusp (Kfp )χ . The Petersson (i.e. the L2 ) inner productbetween Acusp (Kfp )χ−1 and Acusp (Kfp )χ , taken together with this isomorphism,
thus induces a pre-Hilbert space structure on Acusp (Kfp )χ−1 , with respect to which
the G∞ × G-action is unitary up to a character. In particular, we find that
Acusp (Kfp )χ−1 is semisimple as a G∞ × G-representation, and so decomposes asLa direct sum Acusp (Kfp )χ−1 = π∞ ⊗πp π∞ ⊗ πp ⊗ M (π∞ ⊗ πp ), where the direct
sum is taken over (a set of isomorphism class representatives of) all the irreducible
admissible representations of G∞ ⊗ G over C on which A◦∞ acts via χ−1 (which
factor as the tensor product π∞ ⊗ πp of an irreducible admissible representation
π∞ of G∞ on which A◦∞ acts via χ−1 , and an irreducible admissible smooth representation πp of G), and M (π∞ ⊗ πp ) := HomG∞ ×G π∞ ⊗ πp , Acusp (Kfp )χ−1 is the(finite-dimensional) multiplicity space of π∞ ⊗ πp in Acusp (Kfp )χ−1 . Consequently,
we obtain a direct sum decomposition

i
Hcusp
(VC ) ∼
g, k; Acusp (Kfp )χ−1 ⊗ VC
= Hi eM∼
H i (eg, k; π∞ ⊗ VC ) ⊗ πp ⊗ M (π∞ ⊗ πp )=(10)π∞ ⊗πp(the point here being that the (eg, k)-cohomology depends only on the structure ofπ∞ ⊗ πp as a G∞ -representation, which is to say, only on π∞ ).
In this way, we can speak of the contribution of each of the irreducible sumi
mands π∞ ⊗ πp of Acusp (Kfp )χ−1 to the cohomology Hcusp(VC ). In the case wheniH (eg, k; π∞ ⊗ VC ) is non-zero, so that π∞ ⊗ πp actually does contribute to cohomology, the multiplicity space M (π∞ ⊗ πp ) is naturally a TΣ -module, and the direct
sum decomposition (10) is compatible with the actions of G and TΣ (where G-acts
on πp , and TΣ acts on M (π∞ ⊗ πp )).
We can now explain the significance of the quantities l0 and q0 introduced
above. Namely, q0 is the lowest degree in which tempered π∞ can admit nontrivial (eg, k)-cohomology. Furthermore, among the tempered representations of
G∞ , it is precisely the fundamental tempered representations (i.e. those which are
induced from discrete series representations of the Levi subgroup of a fundamental
parabolic subgroup of G∞ ) which can admit non-zero cohomology at all, and they

10

Matthew Emerton

do so precisely in degrees between q0 and q0 + l0 [10, Ch. III, Thm. 5.1]. (This is arange of degrees of length l0 + 1 symmetric about d/2 — which is one-half of thedimension of the quotients Y (Kf ).) A fundamental theorem of Harish-Chandra
states that, when G is semisimple, the group G∞ admits discrete series representations precisely if l0 = 0; in this case q0 is equal to d/2, and the fundamental
tempered representations are precisely the discrete series representations. Key examples for which l0 = 0 are given by groups G for which G∞ is compact, and (thesemisimple parts of) groups giving rise to Shimura varieties.
One representation of G∞ × G that always appears in A(Kfp ) is the trivial
representation 1. Thus there are induced morphisms(11)g, k; A(Kfp )1 ∼H i (eg, k; 1) → H i e= C ⊗O H i ,whose sources are naturally identified with the cohomology spaces of the compact◦
dual to the symmetric space G◦∞ /A◦∞ K∞. If the Y (Kf ) are not compact, then 1 liespin AEis (Kf )1 , but it is typically not a direct summand, and so the morphisms (11)are typically not injective. Nevertheless, when i is small they will be injective,
and indeed (if e.g. G is split, semisimple, and simply connected) an isomorphism,
and these isomorphisms play a key role in the theory of homological stability [9].
We discuss the interaction of these isomorphisms with the theory of completed
cohomology (in the case when G = SLN ) in (2.2.3) below.
2.1.7. Dimension theory. For any compact open subgroup Kp ⊂ G, the completed group ring O[[Kp ]] is (left and right) Noetherian, and of finite injective
dimension as a module over itself; more precisely, its injective dimension is equal
to dim G + 1. This allows us to consider a (derived) duality theory for finitely generated O[[Kp ]]-modules: to any such module M we associate the Ext-modules
E i (M ) := ExtiO[[Kp ]] (M, O[[Kp ]]), for i ≥ 0 (which necessarily vanish if i >dim G + 1); these are again naturally O[[Kp ]]-modules. (We use the left O[[Kp ]]module structure on O[[Kp ]] to compute the E i , and then use the right O[[Kp ]]module structure on O[[Kp ]], converted to a left module structure by the usual
device of applying the anti-involution k 7→ k −1 on Kp , to give the E i an O[[Kp ]]module structure.)
An important point is that E i (M ) is canonically independent of the choice of Kp(in the sense that we may regard M as an O[[Kp0 ]]-module, for any open subgroup
Kp0 of Kp , but E i (M ) is canonically independent of which choice of Kp0 we make todefine it). This has the consequence that if the Kp -representation on M extends
to a G-representation, then the E i (M ) are also naturally G-representations.
The Poincaré duality spectral sequence (7) may thus be rewritten ase j ) =⇒ H
e BM
E2i,j = E i (Hd−(i+j) .
All the objects appearing in it are G-representations, and it is G-equivariant.
If Kp is p-torsion free (which will be true provided Kp is sufficiently small) thenthe ring O[[Kp ]] is in fact Auslander regular [55]. If Kp is furthermore pro-p, thenthe ring O[[Kp ]] is an integral domain (in the sense that it contains no non-trivial

Completed cohomology and the p-adic Langlands program

11

left or right zero divisors). This has various implications for the theory of finitely
generated O[[Kp ]]-modules. For example, it is reasonable to define such a module
to be torsion if every element has a non-zero annihilator in O[[Kp ]] (where Kp ischosen small enough to be pro-p and p-torsion free). More generally, we may makethe following definition.
Definition 2.3. If M is a finitely generated O[[Kp ]]-module, then the codimension
cd(M ) of M is defined to be the least value of i for which E i (M ) 6= 0; the dimension
dim M of M is defined to be dim G + 1 − cd(M ).
If G is a torus, so that Kp is commutative, then dim M is precisely the dimension of the support of the localization of M over Spec O[[Kp ]]. In general O[[Kp ]]is non-commutative, and hence doesn’t have a Spec over which we can localize
a finitely generated module M . Nevertheless, the quantity dim M behaves as if
it were the dimension of the support of M in the (non-existent) Spec of O[[Kp ]].
(See e.g. [55, Prop. 3.5].) As one example, we note that M is torsion (in the sense
defined above) if and only if cd(M ) ≥ 1. As another, we note that cd E i (M ) ≥ i,with equality when i = cd M . Also, we note that cd M = ∞ (so, morally, thesupport of M is empty) precisely when M = 0.
By Theorem 2.2, completed homology is finitely generated over O[[Kp ]], and sothese dimension-theoretic notions apply to it. Information about the (co)dimensione i can be obtained by analyzing the Poincaré duality spectral sequence, since theof Hfunctors E i appear explicitly in it, and there is a tension in this spectral sequence
between the top degree for the duality of O[[Kp ]]-modules (which is dim G + 1)
and the top degree for usual Poincaré duality (which is d), which can sometimes be
exploited. (See e.g. (2.2.2) below.) In making any such analysis, it helps to havee i . This is provided
some a priori information about the (co)dimensions of the Hby the following result.
e i is torsion (i.e. its
Theorem 2.4. [23] Suppose that G is semisimple. Then H
e i = 0.
codimension is positive) unless l0 = 0 and i = q0 , in which case cd H
e i inWe make a much more precise conjecture about the codimensions of the H
Conjecture 3.2 below.

2.2. Examples. We illustrate the preceding discussion with some examples.
2.2.1. GL2 of Q. If G = GL2 , then the quotients Y (Kf ) are classical modular
curves. In particular, they have non-vanishing cohomology only in degrees 0 and 1,
and so we consider completed cohomology in degrees 0 and1. Since the cohomology
e0 = Hb0 ∼of a curve is torsion free, we have H= C ∆×Z×
p , O , the space of continuous
×
O-valued functions on the product ∆ × Zp , where ∆ is a finite group that depends
e 0 (which is simply the O-dual of Hb 0 ) ison the choice of tame level, and so H
×isomorphic to O[[∆ × Zp ]], which is of dimension two as a module over O[[Kp ]]e1 = Hb 1 . Again, He 1 is the(for any compact open Kp ⊂ GL2 (Qp )). We also have H
1e , and Theorem 2.4 shows that cd H
e 1 = 0. (Strictly speaking, we haveO-dual of H

12Matthew Emerton

to apply the theorem to SL2 rather than GL2 , but it is then easy to deduce thecorresponding result for GL2 , by considering the cup-product action of H 0 on H 1 .)2.2.2. SL2 of an imaginary quadratic field. Suppose that G = ResF/Q SL2 ,where F is an imaginary quadratic extension of Q. The quotients Y (Kf ) are thenconnected, non-compact three-manifolds. The relevant (co)homological degrees
e0 = He 0 = O.
are thus i = 0, 1, and 2. Since the Y (Kf ) are connected we see that He
Theorem 2.4 implies that H1 has positive codimension. A consideration of thee 2 = 0, and that H
e 1 is ofPoincaré duality spectral sequence then shows that Hecodimension 1. This computation exploits both the fact that H0 6= 0, and the gap
between 3 (the dimension of the Y (Kf )) and 6 (the dimension of G).
e 1 isSince H 1 with coefficients in O of any space is $-torsion free, we see that H2e 1 , while H
e is $-torsion, and is$ torsion-free, and coincides with the O-dual of He 1 [24,naturally identified with the Pontrjagin dual of the O-torsion submodule of HeThm. 1.1]. We conjecture that in fact H1 is $-torsion free (see Conjecture 3.2),e 2 = 0.
and thus that H
e 0 and H
e 1 , before turning
2.2.3. SLN of Q in low degrees. We first discuss H
to a discussion of higher degree cohomology from the point of view of homologicalstability.
e 0 = O. If N ≥ 3, then SLN
The quotients Y (Kf ) are connected, and so H
satisfies the congruence subgroup property. Furthermore, the groups SLN (Z` )are perfect for all values of `. Combining these two facts, we see that if Γ(pr )denotes the usual congruence subgroup of level pr (i.e. the kernel of the surjection
SLN (Z) → SLN (Z/pr )), then H1 Γ(pr ), O) = O⊗Zp Γ(pr )ab ∼= O⊗Zp Γ(pr )/Γ(p2r ).
Thus, if we take the tame level to be trivial (i.e. “level one”), then we see that thee 1 are eventually zero, implying
transition maps in the projective system defining H
e 1 = 0. Similarly, we see that H 1 = H
e 1 = 0. (If we allowed a more generalthat He 1 could be non-zero, but would be finite. Since H 1 and H
e 1 aretame level, then H
always torsion free, they would continue to vanish.)We now consider cohomology in higher degree, but in the stable range, in thesense that we now explain. (The tame level can now be taken to be arbitrary.)
Borel’s result [9] on homological stability for SLN shows that when i is sufficiently
small (compared to N ), the cohomology L ⊗O H i is independent of N ; indeed,
it consists precisely of the contributions to cohomology arising from the trivial
iautomorphic representation, in the sense discussed L
in (2.1.6). If we define Hstabii
to be the stable value of H , then Borel shows that
Hstab (as an algebra undercup product) is isomorphic to an exterior algebra generated by a single element ineach degree i = 0, 5, 9, 13, . . . .
In fact, stability also holds for completed cohomology.
e i is independent
Theorem 2.5. [25] If i is sufficiently small compared to N , then H
of N , is finitely generated as an O-module, and affords the trivial representation
of G.
ei
ei
We write H
stab to denote the stable value of H . F. Calegari has succeededCompleted cohomology and the p-adic Langlands program13

e i modulo torsion, contingent on a natural non-vanishing conjecin computing H
stabture for certain special values of the p-adic ζ-function (a conjecture which holds
automatically if p is a regular prime).
Theorem 2.6. [22, Thm. 2.3] Suppose either that p is a regular prime, or thatappropriate special values of the p-adic zeta
non-zero. Then there
L function are∼ei
is an isomorphism of graded vector spacesi≥0 L ⊗O Hstab = L[x2 , x6 , x10 , . . .](where L[x2 , x6 , x10 , . . .] denotes the graded ring generated by the indicated sequenceof polynomial variables xi , i ≡ 2 mod 4, with xi placed in degree i).
L
eii L ⊗O Hstab vanishes in all odd degrees, while
L Note in iparticular thati L ⊗O Hstab is generated by classes in odd degrees. Thus, when G = SLN ,the map (3) is identically zero (modulo torsion) when i lies in the stable range!
Assuming the hypothesis of the theorem, we conclude that the Borel classes
i(i.e. the non-zero elements of L ⊗O Hstab) become infinitely divisible when wepull them back up the p-power level tower. It is interesting to consider how they
reappear in the Hochschild–Serre
spectral sequence. If we again ignore torsion,then H i SLN (Zp ), L coincides with the Lie algebra cohomology of slN [48], whichis (stably in N ) an exterior algebra on generators in degrees 3, 5, 7, . . . . Sinceei
all of the non-zero H
stab have trivial G-action, we see that we obtain non-trivial
Ext terms in odd degrees in the Hochschild–Serre spectral sequence, and the Borel
classes are recovered from these.
If we consider the short exact sequence (6) for non-zero even degrees, we seeb i vanishes (modulo(continuing to assume the hypothesis of the theorem) that Hii+1
e is non-zero. Thus the term Tp H
torsion), while H
must be non-zero; thisprovides a rather compelling illustration of the manner in which torsion classes
can accumulate as we pass to the limit of the p-power level tower.
2.2.4. Groups that are compact at infinity. If G∞ (or, more generally, the◦quotient G∞ /A∞ ) is compact, then A◦∞ K∞
equals G◦∞ , and so the quotients
Y (Kf ) are simply finite sets. Thus the only interesting degree of cohomology
is i = 0. In this case the inverse limit Y (Kfp ) := limK Y (Kp Kfp ) is a pro-finite←− pset, which is in fact a compact p-adic analytic manifold, equipped with an analytic
action of G. If Kp is taken sufficiently small (small enough that the G(Q)-action
on G(A)/G◦∞ Kp Kfp is fixed-point free), then Kp acts with trivial stabilizers on
Y (Kp Kfp ), and Y (Kp Kfp ) is the disjoint union of finitely many (say n) principal
homogeneous spaces over Kp .
pe0 ∼One immediately verifies that H= C Y (Kf ), O , the space of continuous∼ O[[Y (K p )]], the space of O-valued
e0 =O-valued functions on Y (Kfp ), while Hf pmeasures on Y (Kf ) (which is the O-dual of C Y (Kfp ), O ). In particular, if Kp is
e 0 is free of rank n over O[[Kp ]].
sufficiently small, then we see that H
It is natural (e.g. in light of Gross’s definition of algebraic modular forms in thiscontext [42]) to define C Y (Kfp ), O to be the space of p-adic automorphic formson G(A) of tame level Kfp , and so in this case we see that completed cohomology
does indeed coincide with the natural notion of p-adic automorphic forms.

14

Matthew Emerton

3. p-adic Langlands
We describe the manner in which we expect completed cohomology, and the Hecke
algebra acting on it, to be related to deformation rings of global Galois representations. This conjectural relationship suggests various further conjectures, as we
explain, as well as a relationship to a hypothetical p-adic local Langlands correspondence.

3.1. The connection between completed cohomology and Galois representations. If π is an automorphic representation of G(A) for
which H i (eg, k; π∞ ⊗ VC ) 6= 0 for some algebraic representation V of G and somedegree i, then π is C-algebraic [21, Lemma 7.2.2]. Thus, we expect that associatedto π there should be a continuous representation of GQ into the Qp -valued points of
the C-group of G [21, Conj. 5.3.4]. In light of the isomorphism (8), we thus expectthat the systems of Hecke eigenvalues that occur in H i (V) should have associatedrepresentations of GQ into the C-group of G. For systems of Hecke eigenvalues
occurring on torsion classes in cohomology, conjectures of Ash [2] again suggestthat there should be associated Galois representations.
These expectations have been proved correct in many cases; for example if G is
the restriction of scalars to Q of GLn over a totally real or a CM number field. (See[45] in the case of characteristic zero systems of eigenvalues, and [53] in the caseof torsion systems of eigenvalues. See also [43] for an overview of these results.)
Returning for a moment to the general case, one further expects that the Galois
representations obtained should be odd, in the sense that complex conjugation in
GQ maps to a certain prescribed conjugacy class in the C-group of G. (See [7,Prop. 6.1] for a description of the analogous conjugacy class in the L-group of G.)
Let m denote a maximal ideal in TΣ , write F := TΣ /m, and let F denote a chosen
algebraic closure of F. Then, by the above discussion, we believe that associatedto m there should be a continuous representation of GQ into the F-valued pointsof the C-group of G. Assuming that this representation exists, we will denote itby ρm .
To be a little more precise: the representation ρm should have the property that
it is unramified at the primes outside Σ, and that for any ` 6∈ Σ, the semisimple
part of ρm (Frob` ) should be associated to the local-at-` part of the system of
Hecke eigenvalues by a suitable form of the Satake isomorphism (see [21] and [42]).
In general, this condition may not serve to determine ρm uniquely; even in thecase G = GLn , it determines ρm only up to semisimplification; for other choicesof G, in addition to this issue, one can have the additional phenomenon that evena representation which doesn’t factor through any parabolic subgroup of the Cgroup is not determined by its pointwise conjugacy classes (“local conjugacy does
not determine global conjugacy”). We do not attempt to deal with this issue ingeneral here; rather we simply ignore it, and continue our discussion as if ρm wereunambiguously defined.
As we observed above, we may regard Tm (or its Spec) as interpolating systemsof Hecke eigenvalues associated to classical C-algebraic automorphic forms, and/or
to torsion classes in cohomology. Since the deformations of ρm can be interpolatedCompleted cohomology and the p-adic Langlands program

15

into a formal deformation space, it is then reasonable to imagine that we may
deform ρm to a representation of GQ into the Tm -valued points of the C-groupof G. Here again there are many caveats: firstly, if ρm is “reducible” (i.e. factors
through a proper parabolic of the C-group), then we would have to work with someform of pseudo-character or determinant, as in [28] and [53]; also, there is a questionof rationality, or “Schur index” — it may be that if we want our deformation ofρm to be defined over F, then we may have to extend our scalars, or else replace
the C-group by an inner twist. Again, we don’t attempt to address these issues
here.
Rather, we begin with some general conjectures about dimension and vanishingthat are motivated by the preceding discussion, and then continue by discussing
the relationships between completed cohomology and p-adic Hodge theory and apossible p-adic local Langlands correspondence. Finally, we turn to a discussion ofsome specific examples, where we can make our generalities more precise.
3.1.1. Conjectures on dimension and vanishing. We begin by making asomewhat vague conjecture, which can be thought of as a rough expression of ourhopes for reciprocity in the context of global p-adic Langlands: namely (continuingwith the notation introduced in the preceding discussion), we conjecture that Tm
is universal (in some suitable sense) for parameterizing odd formal deformations ofρm whose ramification away from p is compatible with the tame level structure Kfp(via some appropriate form of $-adic local Langlands for the group G at primes` - p which, again, we won’t attempt to formulate here; but see [36] in the casewhen G = GLn ). The global Euler characteristic formula for Galois cohomology
lets us compute the expected dimension of such a universal deformation ring, andthis motivates in large part the following concrete conjecture [24, §8].
Conjecture 3.1. Each local factor Tm of TΣ is Noetherian, reduced, and $torsion free, of Krull dimension equal to dim B + 1 − l0 , where dim B denotes thedimension of a Borel subgroup B of G.
This is known in some cases (which we will recall below). We know of no way toprove the Noetherianness of Tm , or the statement about its Krull dimension, otherthan to relate the Hecke algebra to a deformation ring of Galois representations,and then use techniques from the theory of Galois representations to compute thedimension.
In some situations we can prove that Tm is torsion free and reduced (see e.g. thediscussion of (3.1.2) below), and it seems reasonable to conjecture these properties
in general. Indeed, these properties are closely related to the following conjecture[24, Conj. 1.5].
e i = 0 if i > q0 , while cd H
e q = l0 , and cd H
e i > l0 + q0 − i if
Conjecture 3.2. H
0eeq .
i < q0 . Furthermore, Hq0 is $-torsion free, and TΣ acts faithfully on H
0
As noted in [24, Thm. 1.6], the truth of this conjecture for G and all its Levie BM . Also, the vanishing conjecture
subgroups implies the analogous statement for H
ieq .
for G × G implies (via the Künneth formula) the torsion-freeness of H0

16Matthew Emerton

One of the ideas behind this conjecture is that “all the interesting Hecke eigenvalues should appear in degree q0 ”. This is inspired by the fact (recalled in (2.1.6))
that tempered automorphic representations don’t contribute to cohomology in degrees below q0 , so that the Galois representations associated to the systems of
Hecke eigenvalues appearing in degrees below q0 should be “reducible” (i.e. factor through a proper parabolic subgroup of the C-group). In a Galois deformation
space with unrestricted ramification at p, the reducible representations should forma proper closed subset, and so should be approximable by “irreducible” Galois representations (i.e. representations which don’t factor through a proper parabolic).
Thus we don’t expect to see any systems of Hecke eigenvalues in degrees below q0
which can’t already be observed in degree q0 , and hence we expect that TΣ willeq .
act faithfully on H0e i should be “small” if i < q0 , since the possible systems
We also expect that H
of Hecke eigenvalues which it can carry are (or should be) constrained. However,e i are small enough for i < q0 , then the Poincaré duality spectral sequence
if the H(combined with an analysis of the completed cohomology of the boundary, which
can be treated inductively, by reducing to the case of lower rank groups) impliese i in degrees > q0 .
the vanishing of He q , and that TΣ has dimension
If we believe that TΣ acts faithfully on H0e q hasdim B + 1 − l0 (as predicted by Conjecture 3.1), then conjecturing that H0e
codimension l0 is morally equivalent to conjecturing that the fibres of Hq0 over the
points of Spec TΣ are of dimension dim G/B. This latter statement fits nicely withthe fact that generic irreducible representations of G := G(Qp ) have Gelfand–
Kirillov dimension equal to dim G/B, and the analogy between the dimensionof O[[Kp ]]-modules and Gelfand–Kirillov dimension [24, Remark 1.19]. Unfortunately, we don’t know how to make this idea precise, since we don’t know how
to prove (in any generality) this relationship between the Krull dimension of TΣ ,e q , and the dimension of the fibres of the latter over points ofthe dimension of H0e q . Nevertheless, the idea thatSpec TΣ , even assuming that TΣ acts faithfully on H
0these dimensions should be related is an important motivation for the conjecture.
e q , and He q is $-torsion free, then TΣ
We note that if TΣ acts faithfully on H00
is $-torsion free. This motivates the conjecture of $-torsion freeness in Conjecture 3.1.
Conjecture 3.2 has essentially been proved by P. Scholze in many cases for which
the group G gives rise to Shimura varieties [53, Cor. IV.2.3]. (He has non-strict
inequalities rather than strict inequalities for the codimension of the cohomology
in degrees < q0 .)We mention one more example here, namely the case when G is the restrictionof scalars of SL2 from an imaginary quadratic field. In this case we have l0 = 1,
and the codimension statement of Conjecture 3.2 follows from the computation
sketched in (2.2.2).
3.1.2. Localization at a non-Eisenstein system of Hecke eigenvalues.
Suppose that m is a maximal ideal in TΣ , and that the associated representa-

Completed cohomology and the p-adic Langlands program

17

tion ρm (which we assume exists) is “irreducible”, i.e. does not factor through any
proper parabolic subgroup of the C-group of G. We refer to such a m as nonEisenstein. For any TΣ -module M , we write Mm := Tm ⊗TΣ M to denote thelocalization of M at m.
As already noted, we expect that all the systems of Hecke eigenvalues appearing in cohomological degrees < i are “reducible”, i.e. do factor through a properparabolic subgroup, and so we conjecture that, when m is a non-Eisenstein maximal ideal, H i Y (Kp Kfp ), W)m = 0 for all i < q0 and all local systems W associatedto Kp -representations W on finitely generated O-modules (where Kp is any sufficiently small compact open subgroup of G).
We remark that if we ignore torsion, then this follows (at least morally) from
Arthur’s conjectures [1]. (Namely, the result is true for the boundary cohomology,
and hence it suffices to check it for the interior cohomology, i.e. the image ofcompactly supported cohomology in the usual cohomology. However, the interior
cohomology is contained in the L2 -cohomology, and now Arthur’s conjectures imply
that any automorphic representation π contributing to L2 -cohomology that is nontempered at ∞ — as must be the case for an automorphic representation that
contributes to cohomology in degree < q0 — is non-tempered at every prime, andhence should give rise to a reducible Galois representation.) We believe that thisstatement should be true for torsion cohomology as well.
Let us suppose, then, that H i Y (Kp Kfp ), W)m = 0 for all i < q0 . Just coniemsidering the case when W = O (the trivial representation), we find that H=0pi
for i < q0 . Certainly it should be the case that H ∂(Kp Kf ), W)m = 0 when m
is non-Eisenstein. We then find that Hci Y (Kp Kfp ), W)m = 0 for all i < q0 as
well. Suppose now that l0 = 0, so that q0 = d/2. Classical Poincaré duality thengives that H i Y (Kp Kfp ), W)m = 0 for all i > q0 . Presuming that Conjecture 3.2
e i = 0 for i > q0 and H
e q0 is $-torsion free, a consideration ofis true, so that H
q0em
the Hochschild–Serre spectral sequence shows that ExtiO[[Kp ]] (W ∨ , H) = 0 for alli > 0 and all representations of Kp on finitely generated torsion free O-modules.
e q )m is projective as an O[[Kp ]]-module.
From this one easily deduces that (H0
In short, we have given a plausibility argument for the following conjecture,
which refines Conjecture 3.2 in the context of localizing at a non-Eisenstein maximal ideal.
Conjecture 3.3. If m is a non-Eisenstein maximal ideal in TΣ , and if l0 = 0,e i )m = 0 for i 6= q0 , and (He q )m is a projective O[[Kp ]]-module, for any
then (H0sufficiently small subgroup Kp of G. (Here sufficiently small means that G(Q) acts◦with trivial stabilizers on G(A)/A◦∞ K∞
Kp Kfp .)If l0 = 0, and if Conjecture 3.3 holds for some non-Eisenstein maximal ideal m,then we see (from the Hochschild–Serre spectral sequence) that for any sufficiently
small Kp (as in the statement of the conjecture) and any representation of Kp on
a finitely generated torsion free O-module W , we have H i Y (Kp Kfp ), W)m = 0 if
q0em
i 6= q0 , while H q0 Y (Kp K p ), W)m ∼). In particular, we see= HomO[[K ]] (W ∨ , H
f

p

q0
q0
bm
em
that H q0 Y (Kp Kfp ), W)m is $-torsion free. We also see that H= H(since18

Matthew Emerton

q0 +1
q0 +1
q0
em
Hm
vanishes, and hence so does Tp Hm), and that Tm acts faithfully on H.
pi
Presuming that H ∂(Kp Kf ), W m = 0 (which should certainly be true whenm is non-Eisenstein), we conclude that the natural map

Hcq0 Y (Kp Kfp ), W)m → H q0 Y (Kp Kfp ), W)mis an isomorphism, and thus that H q0 Y (Kp Kfp ), W)m consists entirely of interior
cohomology. In particular (being torsion free) it embeds into the L2 -cohomology.
From this we deduce that the image of TΣ acting on H q0 Y (Kp Kfp ), W)m is reduced, and hence (considering all possible Kp and W ) that Tm itself is reduced,
as well as being $-torsion free. This provides some evidence for the reducedness
and torsion-freeness statements in Conjecture 3.1.
We now recall some known results in the direction of Conjecture 3.3. In thecase when G = GL2 , the concept of a non-Eisenstein maximal ideal is well-defined,
and the above conjecture holds [32, Cor. 5.3.19]. In the case when G = GLN ,e i )m is proved for i in thethis notion is again well-defined, and the vanishing of (Hstable range in [25]. Again, if G is a form of U (n − 1, 1) over Q, then the notion ofa non-Eisenstein maximal ideal is well-defined, and in [34] we prove vanishing ofe i )m for i in a range of low degrees, for certain maximal ideals m. In particular, in(Hthe case of U (2, 1), we are able to deduce Conjecture 3.3, provided not only that m
is non-Eisenstein, but that the associated Galois representation ρm has sufficiently
large image, and is irreducible locally at p, satisfying a certain regularity condition.
As one more example, note that if G∞ /A∞ is compact (in which case certainly
l0 = 0), then Conjecture 3.3 holds. Indeed, in this case we saw in (2.2.4), fore 0 is free over O[[Kp ]], even without localizing at a
sufficiently small Kp , that H
non-Eisenstein maximal ideal.
e q )m should be projective over O[[Kp ]];If l0 6= 0, then we don’t expect that (H0indeed, this would be incompatible with Conjecture 3.2. Rather, we expect thatit should be pure of codimension l0 , in the sense of [55, Def. 3.1].
3.1.3. The relationship with p-adic Hodge theory. Let us continue to suppose that l0 = 0 and that m is a non-Eisenstein maximal ideal in TΣ , and let us
suppose that Conjecture 3.3 holds. As we have observed, this implies that for any(sufficiently small) compact open subgroup Kp of G, and any representation W of
Kp on a finitely generated torsion free O-module, we have

q0 ∼
em
e q )m , W
H i Y (Kp Kfp ), W)m ∼) = HomO[[Kp ]] (H= HomO[[Kp ]] (W ∨ , H
0(the first isomorphism being given by the Hochschild–Serre spectral sequence, andthe second by duality).
We now suppose that W is an O-lattice in an irreducible algebraic representation V of G over L. Since H i Y (Kp Kfp ), W)m is $-torsion free, it is a latticein H i Y (Kp Kfp ), V)m . Considering the description of this cohomology in termsof automorphic forms (as in (2.1.6)) and the conjectures regarding Galois representations associated to automorphic forms [21, Conj. 5.3.4], we infer that the
Galois representations associated to the systems of Hecke eigenvalues appearing inCompleted cohomology and the p-adic Langlands program

19

H i Y (Kp Kfp ), V)m should be potentially semistable locally at p, with Hodge–Tateweights related to the highest weight of the algebraic representation V .
Since m is non-Eisenstein, it should make sense to speak of the formal deformation ring Rρm of ρm , and we conjecture that there is a natural isomorphismpTm ∼= Rρm . We see that the Tm -modules H i Y (Kp Kf ), W)m (where W ranges
over the various lattices in the various algebraic representations V ) should then be
supported on the set of points of Spec Rρm corresponding to deformations of ρmthat are potentially semistable at p.
The Fontaine–Mazur conjecture [38], when combined with Langlands reciprocity, should give a precise description of the support of H i Y (Kp Kfp ), W)m ,in purely Galois-theoretic terms. Namely, any potentially semistable deformationof ρm of the appropriate Hodge–Tate weights should be motivic, and hence should
be associated to an automorphic form on the quasi-split inner form of G. Whether
this automorphic form can then be transferred to G, and can contribute to cohomology at level Kp Kfp , should be answered by an analysis of the local Langlands
correspondence for G.
Conversely, if we can prove directly that the support of H i Y (Kp Kfp ), W)m is
as predicted by the Fontaine–Mazur–Langlands conjecture, then we can deduce
this conjecture for deformations of ρm that are classified by Spec Tm . We recallin (3.2.2) below how this strategy is compatible with an optimistic view-point on
how p-adic local Langlands might behave. We briefly recall in (3.3.1) how thisstrategy was employed in [32] and [46] in the case when G = GL2 .
e q )m is projective overLet us fix a sufficiently small open subgroup Kp . Since (H0O[[Kp ]] by assumption, it is a direct summand of a finitely
generated
free O[[Kp ]]L
q0ne
module, and hence Hm is a direct summand of C(Kp , O)
, for some n > 0. Thusq0em
L ⊗O His a direct summand of C(Kp , L)⊕n , the space of continuous L-valued
functions on Kp . Now the theory of Mahler expansions shows that the affine ring
L[G] embeds with dense image in L
C(Kp , L) [51, Lemma A.1]. Recall that, as a
G-representation, we have L[G] ∼=V V ⊗L HomG (V, L[G]), where V runs over(a set of isomorphism class representatives of) all irreducible representations of G.
(At this point, we assume for simplicity that L is chosen so that all the irreducible
representations of G over Qp are in fact defined over L; thus these irreducible V
are in fact absolutely irreducible.) Now the inclusion ofL[G] into C(Kp , L) induces
an isomorphism HomG (V, L[G]) ∼= HomKp V, C(Kp , L) (both are naturally iden∨tifiedwithV,thecontragredient of V ). We conclude that the natural morphism
L
V,
C(K
,
L)
→ C(Kp , L) is injective with dense image. This
V
⊗
Hom
L
Kp
p
V
property is clearly preserved underand direct summands,
L passing to direct esumsq0
e q0
and hence the natural morphism V V ⊗L HomKp (V, H⊗m
O L) → Hm ⊗O L is also∨
injective with dense image. Replacing V by V (which clearly changes nothing,since we are summing over all irreducible algebraic representations of G), and re
q0pem
calling that (by Hochschild–Serre) HomKp (V ∨ , H⊗O L) ∼
= H q0 Y (Kp Kf ), V m ,
Lq0
Y (Kp Kfp ), V m (since, as we noted
we find that Tm acts faithfully on
V H
q0
em
above, it follows from Conjecture 3.3 that Tm acts faithfully on H).
Assuming that we can identify Tm with a deformation space of Galois repre-

20

Matthew Emertonsentations, and that we know that H q0 Y (Kp Kfp ), V m is indeed supported on an
appropriate locus of potentially semistable deformations, the preceding analysis
shows that these loci (as V varies) are Zariski dense in Spec Tm .
In forthcoming work [37], the author and V. Paškūnas will apply a more sophisticated version of this argument to deduce additional Zariski density statementsfor various collections of potentially semistable loci in global Galois deformation
spaces.
Note that these density results rely crucially on the projectivity statement ofConjecture 3.3 for their proof. As we already noted, we can’t expect such a statement to be true when l0 > 0, and (at least if G is semisimple, or, more generally, if
the semisimple part of G satisfies l0 > 0) we don’t expect the potentially semistable
points to be Zariski dense in global deformation spaces in this case. (See [26] foran elaboration on this point.) In particular, in such contexts, we don’t expect thatb q0 will equal He q0 , and so H
e q0 , and Spec TΣ , should receive a non-trivial contriH
bution from torsion classes (via the term Tp H q0 +1 in the exact sequence (6)).

3.2. p-adic local Langlands. Until now, our discussion has been entirely
global in nature. We now turn to describing how these global considerations might
be related to a possible p-adic local Langlands correspondence.
3.2.1. The basic idea. Let us return for a moment to the direct sum decomposition (10). Local-global compatibility for classical Langlands reciprocity says thatif π∞ ⊗ πp is a direct summand of Acusp (Kfp )χ−1 which contributes to cohomology,
lying in a Hecke eigenspace that corresponds to some p-adic Galois representation ρ, then the local factor πp and the Weil–Deligne representation attached to ρ
(which should be defined, since ρ should be potentially semistable at p) should
correspond via the local Langlands correspondence (or perhaps more generally via
the local form of Arthur’s conjectures [1], if π∞ and πp are not tempered). In
particular, the local factor at p and the local Galois representation at p should be
related in a purely local manner.
The basic idea of a p-adic local Langlands correspondence is that the sameshould be true when we take into account the structure of completed cohomology.
To explain this, we continue the discussion of the preceding paragraph, and, in
addition, we place ourselves in the context introduced in (3.1.3) (in particular, we
continue to assume that the hypotheses and conclusions of Conjecture 3.3 hold).
Thus, we assume that ρ is a deformation of ρm , and that π∞ and πp are tempered,q0
em
so that we have an embedding πp ,→ H q0 (V)m ∼HomKp (V ∨ , L⊗O H
). We
= lim
−→Kp
q
e m0 and we can then take the
may rewrite this as an embedding V ∨ ⊗L πp ,→ L ⊗O H∨
q0
closure of V ⊗ πp in H (V)m , to obtain a unitary Banach space representation∨ ⊗ π of G over L. A slightly more refined procedure is to form the intersection
V\
L p
q0
q0
em
em
(V ∨ ⊗L πp )◦ := (V ∨ ⊗L πp ) ∩ H(the intersection being taken in L ⊗O H). This∨⊗ π
is a G-invariant O-lattice contained in V ∨ ⊗L πp , and the Banach space V \L p
is then obtained by completing V ∨ ⊗L πp with respect to this lattice. A natural
question to ask, then, in the spirit of a local Langlands correspondence and local-

Completed cohomology and the p-adic Langlands program21∨ ⊗ π depends only on the restriction to p ofglobal compatibility, is whether V \L p
the associated Galois representation ρ. Since by assumption m is non-Eisenstein,
the Galois representation ρ should admit an essentially unique integral model ρ◦ ,and we could further ask whether (V ∨ ⊗L πp )◦ depends only on the restriction to
p of ρ◦ . This question goes back to C. Breuil’s first work on the p-adic Langlands
correspondence in the case when G = GL2 (see especially the introduction of [13]).
It has been largely resolved in that case, but remains open in general.

3.2.2. An optimistic scenario. The most optimistic conjecture that one mightentertain regarding a p-adic local Langlands correspondence is that for any formal
deformation ring Rr parameterizing the deformations of a representation r of thelocal Galois group GQp into the F-valued points of the C-group of G, there is a
profinite Rr -module M , finitely generated over Rr [[Kp ]] for some (and hence every)
compact open subgroup Kp ⊂ G, equipped with a continuous G-action extending
its Kp -module structure, which realizes the p-adic local Langlands correspondence
for the group G in the following sense: in the context of (3.2.1) (and continuingwith the notation of that discussion), the fibre over the restriction to p of ρ◦ (which
is a point in Spec Rr , if r is the restriction of ρm to GQp ) is isomorphic to the O-dualof (V ∨ ⊗L πp )◦ . (Here we suppress the issue of whether Rr should be understood
to be a framed deformation ring, or a pseudo-deformation ring, or . . . .) In short,e q )m .
M should be a local analogue of the completed homology space (H0Since M is finitely generated over Rr [[Kp ]], for any representation W of Kp on adfinitely generated torsion free O-module, the Rr -module Homcont
Kp (M, W ) (whered denotes the continuous O-dual) is a finitely generated Rr -module. Anotherproperty one might require of M is that when W is a Kp -invariant O-lattice indan irreducible algebraic G-representation V , then Homcont
Kp (M, W ) is supported
on an appropriate locus of potentially semistable representations, corresponding toe q )m , W (which is isomorphic to H q0 Y (Kp K p ), W )the fact that HomO[[Kp ]] (H0f
is supported on a locus of potentially semistable representations in Spec Tm .
d
Ideally, one might ask for the support of Homcont
Kp (M, W ) to be the full potentially semistable locus of appropriate Hodge–Tate weights (corresponding to thehighest weight of V ) and Weil–Deligne representations (corresponding to the particular choice of Kp and the nature of the local Langlands correspondence for G).
As we indicated in (3.1.3), such a result, combined with the local-global compatie q )m , would prove that any global Galois representation
bility between M and (H0corresponding to a point of Spec Tm which is potentially semistable of appropriate
Hodge–Tate weights and with an appropriate Weil–Deligne representation,
 in fact
e q0 Y (Kp K p ), V , thus
arises from a system of Hecke eigenvalues occurring in Hfmverifying the Fontaine–Mazur–Langlands conjecture for such points.
d
Note that a local Galois representation lies in the support of Homcont
Kp (M, W )precisely if the dual to the fibre of M at this point receives a non-zero Kp equivariant homomorphism from W . In particular, the dual to the fibre at such a
point contains locally algebraic vectors. In the case of GL2 , the idea of describing
p-adic local Langlands for potentially semistable representations in terms of locallyalgebraic vectors goes back to C. Breuil’s first work on the subject [11, 12].

22

Matthew Emerton

The optimistic scenario described here has been realized for the group G =GL2 (Qp ); this is the theory of p-adic local Langlands for GL2 (Qp ) [6, 29, 50, 30](see also [5, 14]). Whether it can be realized for other groups, or is overly optimistic,
remains to be seen.

3.3. Examples. We again illustrate our discussion with some examples.
3.3.1. GL2 of Q. As already mentioned, the theory of p-adic local Langlands forGL2 (Qp ) provides a structure satisfying all the desiderata of (3.2.2). The local1em
global compatibility between this structure and H(the localization of completed
cohomology at a non-Eisenstein maximal ideal of the Hecke algebra) has been
proved in [32] (see also [14, 15]) (under a mild hypothesis on the local behaviourof ρm ). In particular, the strategy of (3.1.3) then applies to prove the Fontaine–
Mazur–Langlands conjecture for points of Spec Tm .
Under the slightly stronger hypotheses that p is odd and ρm remains irreducible
on restriction to GQ(ζp ) (the Taylor–Wiles condition) one can prove that Tm ∼= Rρm[8], [32, Thm. 1.2.3]. Since Conjecture 3.3 holds in this context [32, Cor. 5.3.19], themethod of (3.1.3) (applied with Kp = GL2 (Zp )) allows us to deduce the density of
the crystalline loci in Spec Rρm ; the extension of this method to be described in [37]will allow us to deduce other density results about various potentially semistableloci. Here is one such simple variant: instead of considering the restriction toGL2 (Zp ) of the family V of algebraic representations of GL2 , we instead consider
the family of representations σ ⊗ V , where σ is some fixed supercuspidal type
and V is algebraic. This allows us to show that potentially crystalline points of(any fixed) supercuspidal type are Zariski dense in Tm (and hence in Rρm , if the
Taylor–Wiles condition is satisfied).
3.3.2. A definite quaternion algebra ramified at p. Let D be the quaternion algebra over Q that is ramified at ∞ and p, and let G = D× . Then G∞ /A∞is compact, and so, as noted above, Conjecture 3.3 holds. The classical Jacquet–
Langlands correspondence allows us to attach odd two-dimensional Galois representations to systems of Hecke eigenvalues. In particular, if m is non-Eisenstein,then Tm is a quotient of Rρm .
Assuming that p is odd and ρm satisfies the Taylor–Wiles condition, we conclude that in fact Tm ∼= Rρm . Indeed, we saw in the preceding example that thepotentially crystalline points of supercuspidal type are Zariski dense in Rρm . Since
Fontaine–Mazur–Langlands holds in this context, they all arise from classical modular forms, and hence (by Jacquet–Langlands) from classical automorphic forms on
D× . Thus Spec Tm contains a Zariski dense set of points in Spec Rρm , and so thetwo Specs coincide. As one interesting consequence of this, we note that (assumingthat the Taylor–Wiles condition holds) the Hecke algebras at m for GL2 and D×
are naturally isomorphic (both being isomorphic to Rρm ). One can think of thisas a p-adic interpolation of the classical Jacquet–Langlands correspondence. It isinteresting to note that even though under the classical Jacquet–Langlands correspondence automorphic forms on GL2 that are principal series at p don’t match

Completed cohomology and the p-adic Langlands program

23with a corresponding Hecke eigenform on D× , they are not excluded from thisp-adic Jacquet–Langlands correspondence.
As another consequence, note that if we choose Kp to be the units in the
maximal order of (D ⊗ Qp )× , and apply the density argument of (3.1.3), we deducethat the representations which are genuinely semistable at p (i.e. semistable, andnot crystalline) are Zariski dense in Spec Tm , and hence in Spec Rρm , provided the
Taylor–Wiles condition holds.
3.3.3. Definite quaternion algebras over totally real fields. Consider thecase where G is the restriction of scalars to Q of the units D× in a totally definite
quaternion algebra D over a totally real field F ; and suppose that F is unramified at p, and D is split at every prime in F above p. Put ourselves in thesituation of (3.2.1), with the additional assumption that V is the trivial representation, and πp is a tamely ramified principal series. In this case πp contains, asa GL2 (Zp ⊗Z OF )-subrepresentation, a principal series type σ, which is a representation of GL2 (OF /p) induced from a character of the Borel subgroup. In thepaper [16], Breuil gave a conjectural description of the isomorphism class of the0emlattice σ ◦ := σ ∩ H, purely in terms of the restriction to p of ρ◦ . Under mild
assumptions on ρm , this was proved in [35]. This gives some evidence towards thepossibility that (V ∨ ⊗L πp )◦ may be of a purely local nature.
3.3.4. Compact unitary groups. In [27], we apply Taylor–Wiles patching topass from completed cohomology over a unitary group that is compact at infinity
to a G-representation on a module M∞ over the ring R∞ obtained by adjoininga certain number of formal variables to a local deformation ring. More precisely:
for any finite extension K of Qp , any n such that p - 2n, and any representationr : GK → GLn (Fp ) that admits a potentially crystalline lift which is potentiallydiagonalizable (in the sense of [4]), we can (by [33, Cor. A.7]) choose a unitary group
G and Hecke maximal ideal m so that G := G(Qp ) is isomorphic to a product ofcopies of GLn (K), and such that the restriction of ρm to GK is equal to r. The Grepresentation M∞ , which is a module over Rr [[x1 , . . . , xn ]] for some some n ≥ 0,is then obtained by patching the completed cohomology for G.
The module M∞ satisfies several of the desiderata of (3.2.2): it is finitely genderated over R∞ [[Kp ]], and the modules Homcont
Kp (M, W ) (for Kp -invariant lattices
W in algebraic representations) are supported on a union of components of theappropriate potentially semistable loci. Many questions about M∞ remain open,
however: whether the support of M∞ equals the entirety of Spec R∞ ; whether alldpotentially semistable points are contained in the support of Homcont
Kp (M, W ) (foran appropriate choice of W and Kp ); and whether M∞ is in fact of a purely local
nature.
One further property of M∞ is that it is projective in the category of profinite topological O[[Kp ]]-modules. In [37], we hope to show that M∞ is in fact offull support on Spec Rr [[x1 , . . . , xn ]] in many cases, and thus extend the method
described in (3.1.3) to deduce density results for potentially semistable representations in local deformation spaces.

24

Matthew Emerton

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Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637
E-mail: emerton@math.uchicago.edu