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 GL_{2}(ℚ_{p}), 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 GL_{2} 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], we have decided here to describe some aspects of the p-adic Langlands program that make 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 GL_{2}(ℚ_{p}). 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 completed cohomology (via the Taylor–Wiles–Kisin method) in order to construct a candidate for the p-adic local Langlands correspondence for GL_{n} of an arbitrary p-adic field.

∗The author’s work is partially supported by NSF grant DMS-1303450

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 GL_{2}(ℚ_{p}). 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 ℚ_{p}, contained inside some given algebraic closure \overline{ℚ}_{p}. We let O denote the ring of integers of L, and let ϖ 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, 𝔸_{f} denotes the ring of finite adèles, and 𝔸_{f}^{p} 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(ℚ_{p}). 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, David Geraghty, 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; many of 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 rise to Shimura varieties, we can use algebro-geometric methods to define spaces of p-adic automorphic forms, which however seem less representation-theoretic in nature 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.

Completed cohomology and the p-adic Langlands program 3

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 being of a representation-theoretic nature. For our purposes, it will thus serve as a suitable surrogate for a space of p-adic automorphic forms. For groups that are compact 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 discussion of global p-adic Langlands by recalling the basic definitions and facts related to completed cohomology, referring to [24] and [31] for more details.

2.1. Definitions and basic properties. Let us suppose that 𝔾 is a

reductive 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 choice of maximal compact subgroup of G_{∞}. For any Lie group H, we let H^{∘} denote the subgroup consisting of the connected component of the identity.

The quotient G_{∞}/A_{∞}^{∘}K_{∞}^{∘} is a symmetric space on which G_{∞} acts. We denote its dimension by d. We also write l_{0} := rank of G_{∞} − rank of A_{∞}K_{∞}, and q_{0} := (d − l_{0})/2. If G_{∞} is semisimple (so that in particular A_{∞} is trivial) then these quantities coincide with the quantities denoted by the same symbols in [10] (which is why we notate them as we do). We note that q_{0} 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 K_{f} ⊂ 𝔾(𝔸_{f}) is a compact open subgroup, then we write

Y (K_{f}) := 𝔾(ℚ)\𝔾(𝔸)/A_{∞}^{∘}K_{∞}^{∘}K_{f}.

The double quotient Y (K_{f}) 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 K_{f}^{p} ⊂ 𝔾(𝔸_{f}^{p}). If K_{p} ⊂ G := 𝔾(ℚ_{p}) is a compact open subgroup, then of course K_{p}K_{f}^{p} is a compact open subgroup of 𝔾(𝔸_{f}), and so we may form the space Y (K_{p}K_{f}^{p}). We define the completed (co)homology at the tame level K_{f}^{p} as follows [24]:

H̃^{i} := \underleftarrow{lim}_{s} \underrightarrow{lim}_{K_{p}} H^{i}(Y (K_{p}K_{f}^{p}), O/ϖ^{s}) and H̃_{i} := \underleftarrow{lim}_{K_{p}} H_{i}(Y (K_{p}K_{f}^{p}), O), (1)

where in both limits K_{p} ranges over all compact open subgroups of 𝔾(ℚ_{p}).

We equip each of H̃^{i} and H̃_{i} with its evident inverse limit topology. In the case of H̃^{i}, each of the O/ϖ^{s}-modules \underrightarrow{lim}_{K_{p}} H^{i}(Y (K_{p}K_{f}^{p}), O/ϖ^{s}) appearing in the projective limit is equipped with its discrete topology, and the projective limit topology on H̃^{i} coincides with its ϖ-adic topology; H̃^{i} is complete with respect to this topology. In the case of H̃_{i}, each of the terms H_{i}(Y (K_{p}K_{f}^{p}), O) (which is a finitely generated O-module) appearing in the projective limit is equipped with its ϖ-adic topology. The topology on H̃_{i} is then a pro-finite topology, and so in particular H̃_{i} is compact.

The completed cohomology and homology are related to one another in the usual way by duality over O [24]. In particular, if we ignore O-torsion, then H̃_{i} is the O-dual of H̃^{i} (equipped with its weak topology), and H̃^{i} is the continuous O-dual of H̃_{i}.

We can also form the limit

H^{i} := \underrightarrow{lim}_{K_{p}} \underleftarrow{lim}_{s} H^{i}(Y (K_{p}K_{f}^{p}), O/ϖ^{s}) ≅ \underrightarrow{lim}_{K_{p}} H^{i}(Y (K_{p}K_{f}^{p}), O).  (2)

There is a natural morphism

H^{i} → H̃^{i},  (3)

which induces an embedding

Ĥ^{i} ↪ H̃^{i},  (4)

whose source is the ϖ-adic completion of H^{i}. Note that the morphism (3) need not be injective: although each of the terms H^{i}(Y (K_{p}K_{f}^{p}), 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 then become zero after we pass to the ϖ-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 K′_{p} ⊂ K_{p}, then a trace argument shows that the restriction map

H^{i}(Y (K_{p}K_{f}^{p}), O) → H^{i}(Y (K′_{p}K_{f}^{p}), O)

becomes injective after tensoring with L. Thus the kernel of the natural map H^{i}(Y (K_{p}K_{f}^{p}), O) → H^{i} consists of torsion classes.

On the other hand, the kernel of the natural map

H^{i}(Y (K_{p}K_{f}^{p}), O) → H̃^{i}  (5)

need not consist only of torsion classes; it is possible for non-torsion classes to have infinitely divisible image in H^{i}, and hence have vanishing image in Ĥ^{i} (and so also have vanishing image in H̃^{i}). (Again, we refer to (2.2.3) for an example of this.).

The morphism (4), although injective, need not be surjective. Its cokernel is naturally identified with T_{p}H^{i+1} := \underleftarrow{lim}_{s} H^{i+1}[ϖ^{s}]; in other words we have a short exact sequence

0 → Ĥ^{i} → H̃^{i} → T_{p}H^{i+1} → 0.  (6)

(Here we have written H^{i+1}[ϖ^{s}] to denote the submodule of ϖ^{s}-torsion elements in H^{i+1}, and the projective limit is taken with respect to the multiplication-by-ϖ map from H^{i+1}[ϖ^{s+1}] to H^{i+1}[ϖ^{s}]. The notation “T_{p}” is for Tate module.) In particular, if all the cohomology modules H^{i+1}(Y (K_{p}K_{f}^{p}), O) are torsion free, so that 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 K_{p}K_{f}^{p} via the Hochschild–Serre spectral 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 complicated in general. For instance, infinitely divisible torsion elements in H^{i} give rise to elements of T_{p}H^{i}, which by the discussion of the previous paragraph is naturally a quotient of H̃^{i−1}. However, infinitely divisible non-torsion elements of H^{i} have no obvious incarnation as elements of completed cohomology, and the manner in which 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(ℚ_{p}) on each of H̃^{i} and H̃_{i}, whose key property is encapsulated in the following result.

Theorem 2.1. [24, 31] The G-action on H̃^{i} makes it a ϖ-adically admissible representation of G; i.e. it is ϖ-adically complete as an O-module, and each quotient H̃^{i}/ϖ^{s} (which is a smooth representation of G over O/ϖ^{s}) is admissible in the usual sense (for each compact open subgroup K_{p} ⊂ G, the submodule of K_{p}-invariants is finitely generated over O).

If K_{p} ⊂ G is compact open, then we write O[[K_{p}]] := \underleftarrow{lim}_{K′_{p}} O[K_{p}/K′_{p}], where

the projective limit is taken over all open normal subgroups K′_{p} ⊂ K_{p}. Since G acts continuously on H̃^{i} and H̃_{i}, it follows that for any such K_{p}, the K_{p}-action on each of these modules may be promoted to an action of O[[K_{p}]]. One then has the following reformulation of the admissibility of the G-action on H̃^{i}.

Theorem 2.2. Each of the O[[K_{p}]]-modules H̃_{i} is finitely generated.

(We remark that if this finite generation statement holds for one choice of K_{p} 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 K_{p} ⊂ G, and K_{p} is sufficiently small, so that 𝔾(ℚ) acts with trivial stabilizers on 𝔾(𝔸)/A_{∞}^{∘}K_{∞}^{∘}K_{p}K_{f}^{p}, then, for each open subgroup K′_{p} ⊂ K_{p}, the representation W determines a local system W of O-

modules on Y (K_{p}K_{f}^{p}), defined via W := 𝔾(ℚ)\((𝔾(𝔸_{f})/A_{∞}^{∘}K_{∞}^{∘}K_{f}^{p}) × W )/K′_{p}. Suppose that W is furthermore torsion-free as an O-module, and let W^{∨} denote the O-dual of W, endowed with the contragredient K_{p}-action. There is then [24, 31] a Hochschild–Serre spectral sequence

E_{2}^{i,j} = Ext_{O[[K_{p}]]}^{i}(W^{∨}, H̃^{j}) ⟹ H^{i+j}(Y (K_{p}K_{f}^{p}), W).

This gives a precise sense to the idea that p-adically completed cohomology captures all the cohomology (with arbitrary coefficients) at finite levels. It is the realization, 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 with classical automorphic forms, see (2.1.6) below.)

In particular, if we take W = O (with the trivial K_{p}-action), we obtain a spectral sequence E_{2}^{i,j} = H^{i}(K_{p}, H̃^{j}) ⟹ H^{i+j}(Y (K_{p}K_{f}^{p}), O) (where H^{i} denotes continuous cohomology), which recovers the cohomology at the finite level K_{p}K_{f}^{p} from the completed cohomology. If we take a direct limit over all K_{p}, we obtain a spectral sequence E_{2}^{i,j} = \underrightarrow{lim}_{K_{p}} H^{i}(K_{p}, H̃^{j}) ⟹ H^{i+j}, which recovers the limits

H^{i+j} of the cohomology at finite levels. The edge map H^{i} → \underrightarrow{lim}_{K_{p}} (H̃^{i})^{K_{p}} is induced by the morphism (3). As already noted, the relationship between this spectral sequence, the morphism (3), and the exact sequence (6) is subtle. (See the example of (2.2.3) below.)

More generally, if we write H^{i}(W) := \underrightarrow{lim}_{K′_{p}} H^{i}(Y (K′_{p}K_{f}^{p}), W) (where K′_{p} runs

over all open subgroups of the fixed K_{p}, of which W is a representation), then we have an edge map H^{i}(W) → \underrightarrow{lim}_{K′_{p}} Hom_{K′_{p}}(W^{∨}, H̃^{i}) which relates the cohomology

at finite levels with coefficients in W to the W^{∨}-isotypic parts (for open subgroups K′_{p} of K_{p}) of H̃^{i}.

2.1.4. Hecke actions. There is a finite set of primes Σ_{0} (containing p) such that for ℓ ∉ Σ_{0}, we may factor K_{f}^{p} = K_{f}^{p,ℓ}K_{ℓ} where K_{f}^{p,ℓ} is compact open in 𝔾(𝔸_{f}^{p,ℓ}), and K_{ℓ} is a hyperspecial maximal compact subgroup of 𝔾(ℚ_{ℓ}). (In particular, 𝔾 is unramified at such a prime ℓ, so that it admits a hyperspecial maximal compact subgroup.) We may then consider the spherical Hecke algebra (i.e. the double coset algebra) H_{ℓ} := H(𝔾(ℚ_{ℓ})//K_{ℓ}, O) with coefficients in O. This algebra is commutative [41, 42], and acts naturally (by continuous operators) on any of the cohomology groups H^{i}(Y (K_{p}K_{f}^{p}), W) considered in (2.1.3).

If we let i range over all cohomological degrees, let K_{p} range over all compact open subgroups of G, and let W range over all representations of K_{p} on finitely generated torsion O-modules, then ∏_{i} ∏_{K_{p}} ∏_{W} End H^{i}(Y (K_{p}K_{f}^{p}), W) is a profinite ring. We fix a set of primes Σ containing Σ_{0}, and we define the global Hecke algebra 𝕋_{Σ} to be the closure in this profinite ring of the O-subalgebra generated by the image of H_{ℓ} for all ℓ ∉ Σ. (The reasons for allowing the possibility of Σ being larger than Σ_{0} are essentially technical; it is not misleading to simply imagine that Σ is equal to Σ_{0} 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 K_{f}^{p}.) By construction, 𝕋_{Σ} acts on each of the cohomology groups H^{i}(Y (K_{p}K_{f}^{p}), W) considered in (2.1.3) (this is obvious if W is torsion, and follows in general by writing W as a projective limit W ≅ \underleftarrow{lim}_{s} W/ϖ^{s}), and also on H̃^{i} and H̃_{i}. (In the case of H̃^{i} this follows from its definition in terms of limits of cohomology groups H^{i}(Y (K_{p}K_{f}^{p}), O/ϖ^{s}). It then follows for H̃_{i} by duality; more precisely, we can write H̃_{i} as the projective limit over s and K_{p} of H_{i}(Y (K_{p}K_{f}^{p}), O/ϖ^{s}), and this latter module is the O/ϖ^{s}-dual of H^{i}(Y (K_{p}K_{f}^{p}), O/ϖ^{s}).)

These Hecke actions on H̃^{i} and H̃_{i} commute with the action of G. Thus there is an action of Hecke on the E_{2}-terms, as well the E_{∞}-terms, of the various Hochschild–Serre spectral sequences considered in (2.1.3), and these are in fact spectral sequences of 𝕋_{Σ}-modules.

The O-algebra 𝕋_{Σ} is a complete semi-local ring, i.e. it is a product of finitely many factors 𝕋_{𝔪}, each of which is a complete local ring. (This finiteness statement is proved via the methods of [3].) The topology on 𝕋_{Σ} is the product of the 𝔪-adic topologies on each of the complete local rings 𝕋_{𝔪}.

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 when the quotients Y (K_{f}) are non-compact. (We recall that the quotients Y (K_{f}) are compact 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 define G-representations H̃_{c}^{i}, H_{c}^{i}, Ĥ_{c}^{i}, and T_{p}H_{c}^{i}, and we have Hochschild–Serre spectral sequences for compactly supported cohomology. Similarly, we can define completed Borel–Moore homology H̃_{i}^{BM}, which is related to H̃_{c}^{i} by duality over O.

There is a more subtle duality over O[[K_{p}]] (for a compact open subgroup K_{p} ⊂ G) which relates the usual and compactly supported variants of completed cohomology. It is most easily expressed on the homology side, where it takes the form of the Poincaré duality spectral sequence

E_{2}^{i,j} := Ext_{O[[K_{p}]]}^{i}(H̃_{j}, O[[K_{p}]]) ⟹ H̃_{d−(i+j)}^{BM}.  (7)

Of course, when the quotients Y (K_{f}) are compact, compactly supported and usual cohomology coincide, as do usual and Borel–Moore homology. In general, we can relate them by considering the Borel–Serre compactifications \overline{Y} (K_{f}). If we let ∂(K_{f}) denote the boundary of \overline{Y} (K_{f}), then we may compute the compactly supported cohomology of Y (K_{f}) as the relative cohomology

H_{c}^{i}(Y (K_{f}), O/ϖ^{s}) = H^{i}(\overline{Y} (K_{f}), ∂(K_{f}); O/ϖ^{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 of Y (K_{f}), the compactly supported cohomology of Y (K_{f}), and the cohomology of ∂(K_{f}). Passing to the various limits, we obtain a long exact sequence

⋯ → H̃_{c}^{i} → H̃^{i} → H̃^{i}(∂) := \underleftarrow{lim}_{s} \underrightarrow{lim}_{K_{p}} H^{i}(∂(K_{p}K_{f}^{p}), O/ϖ^{s}) → H̃_{c}^{i+1} → · · · .

The completed cohomology of the boundary H̃^{i}(∂) can be computed in terms of completed cohomology of the various Levi subgroups of 𝔾; see [24] for more details.

We also mention that there is a 𝕋_{Σ}-action on each of the objects introduced here, and that the various spectral sequences and long exact sequences considered here are all compatible with these actions.

2.1.6. The relationship to automorphic forms. We fix an isomorphism ı : \overline{ℚ}_{p} ≅ ℂ. Since L ⊂ \overline{ℚ}_{p}, we obtain induced embeddings O ⊂ L ↪ ℂ.

Suppose that V is an algebraic representation of 𝔾 defined over L, and suppose further that W is a K_{p}-invariant O-lattice in V, for some compact open subgroup K_{p} of G := 𝔾(ℚ_{p}) ⊂ 𝔾(L). Write V_{ℂ} := 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 K′_{p} of K_{p}, we let V_{ℂ} denote the local system of ℂ-vector spaces on Y (K′_{p}K_{f}^{p}) associated to V_{ℂ}, then we obtain a natural isomorphism

ℂ ⊗_{O} H^{i}(W) ≅ \underrightarrow{lim}_{K′_{p}} H^{i}(Y (K′_{p}K_{f}^{p}), V_{ℂ}) =: H^{i}(V_{ℂ}).

We may compute this cohomology using the space of automorphic forms on 𝔾 [39].

Precisely, write A(K_{f}) for the space of automorphic forms on 𝔾(ℚ)\𝔾(𝔸)/K_{f}, for any compact open subgroup K_{f} ⊂ 𝔾(𝔸_{f}), and write A(K_{f}^{p}) = \underrightarrow{lim}_{K_{p}} A(K_{p}K_{f}^{p}), where the direct limit is taken (as usual) over the compact open subgroups K_{p} ⊂ G := 𝔾(ℚ_{p}). For simplicity, suppose that A_{∞}^{∘} acts on V_{ℂ} through some character χ (this will certainly hold if V is absolutely irreducible), and let A(K_{f}^{p})_{χ^{−1}} denote the subspace of A(K_{f}^{p}) on which A_{∞}^{∘} acts through the character χ^{−1}. Let G̃_{∞} denote the group of real points of the intersection of the kernels of all the rational characters of 𝔾, and let 𝔤̃ and 𝔨 denote the Lie algebras of G̃_{∞} and K_{∞} respectively. Then it is proved by J. Franke in [39] that there is a natural isomorphism

H^{i}(V_{ℂ}) ≅ H^{i}(𝔤̃, 𝔨; A(K_{f}^{p})_{χ^{−1}} ⊗ V_{ℂ}).  (8)

(In the case when the the quotients Y (K_{f}) are compact, this result is known as Matsushima’s formula [49]. In the case when 𝔾 = GL_{2} and i = 1, it is known as the Eichler–Shimura isomorphism [54].) The action of 𝕋_{Σ} on H^{i}(W) induces an action of 𝕋_{Σ} on H^{i}(V_{ℂ}), and the resulting systems of Hecke eigenvalues that appear in H^{i}(V_{ℂ}) (which we may think of as being simultaneously ℂ-valued and \overline{ℚ}_{p}-valued, by employing the isomorphism ı) are automorphic, i.e. arise as systems of Hecke eigenvalues on the space of automorphic forms A(K_{f}^{p}).

Thus, to first approximation, we may regard the Hecke algebra 𝕋_{Σ} (or, more precisely, the \overline{ℚ}_{p}-valued points of its Spec) as being obtained by ϖ-adically interpolating the automorphic systems of Hecke eigenvalues that occur in H^{i}. However, as we already noted, the map from H^{i} to Ĥ^{i} can have a non-trivial kernel. More significantly (given that the Hochschild–Serre spectral sequence shows that, despite this possibility, the systems of Hecke eigenvalues appearing in H^{i} can be recovered from the completed cohomology), the inclusion Ĥ^{i} ↪ H̃^{i} can have a non-trivial cokernel T_{p}H^{i+1}, which arises from infinitely divisible torsion in H^{i+1}. Thus Spec 𝕋_{Σ} sees not only the systems of eigenvalues arising from classical automorphic forms (and their ϖ-adic interpolations), but also systems of Hecke eigenvalues arising from torsion cohomology classes (and their ϖ-adic interpolations). We will discuss this point 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(K_{f}^{p})_{χ^{−1}} as the direct sum

A(K_{f}^{p})_{χ^{−1}} = A_{cusp}(K_{f}^{p})_{χ^{−1}} ⊕ A_{Eis}(K_{f}^{p})_{χ^{−1}}  (9)

of the cuspforms and the forms which are orthogonal to the cuspforms (in A(K_{f}^{p})_{χ}) under the Petersson inner product (i.e. the L^{2} inner product of functions on 𝔾(ℚ)\𝔾(𝔸)/A_{∞}^{∘}K_{f}^{p})). We label this complement to the space of cuspforms with the subscript Eis for Eisenstein, although its precise relationship with the space of Eisenstein series can be complicated; see e.g. [47], [39], and [40]. Note that if the quotients Y (K_{f}) are compact, then A(K_{f}^{p})_{χ^{−1}} = A_{cusp}(K_{f}^{p})_{χ^{−1}}.

The decomposition (9) is a direct sum of G_{∞} × G-representations, and so the cohomology H^{i}(𝔤̃, 𝔨; A_{cusp}(K_{f}^{p})_{χ^{−1}} ⊗ V_{ℂ}) is a G-invariant direct summand of the cohomology H^{i}(𝔤̃, 𝔨; A(K_{f}^{p})_{χ^{−1}} ⊗ V_{ℂ}), which via the isomorphism (8) we identify with a G-invariant direct summand H_{cusp}^{i}(V_{ℂ}) of H^{i}(V_{ℂ}). It is also 𝕋_{Σ}-invariant.

We may choose an everywhere positive element of A(K_{f}^{p}) on which 𝔾(𝔸) acts via a character extending χ^{2}, multiplication by which induces an isomorphism between A_{cusp}(K_{f}^{p})_{χ^{−1}} and A_{cusp}(K_{f}^{p})_{χ}. The Petersson (i.e. the L^{2}) inner product between A_{cusp}(K_{f}^{p})_{χ^{−1}} and A_{cusp}(K_{f}^{p})_{χ}, taken together with this isomorphism, thus induces a pre-Hilbert space structure on A_{cusp}(K_{f}^{p})_{χ^{−1}}, with respect to which the G_{∞} × G-action is unitary up to a character. In particular, we find that A_{cusp}(K_{f}^{p})_{χ^{−1}} is semisimple as a G_{∞} × G-representation, and so decomposes as a direct sum A_{cusp}(K_{f}^{p})_{χ^{−1}} = ⨁_{π_{∞}⊗π_{p}} π_{∞} ⊗ π_{p} ⊗ M(π_{∞} ⊗ π_{p}), 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_{∞} ⊗ π_{p} of an irreducible admissible representation π_{∞} of G_{∞} on which A_{∞}^{∘} acts via χ^{−1}, and an irreducible admissible smooth representation π_{p} of G), and M(π_{∞} ⊗ π_{p}) := Hom_{G_{∞}×G}(π_{∞} ⊗ π_{p}, A_{cusp}(K_{f}^{p})_{χ^{−1}}) is the (finite-dimensional) multiplicity space of π_{∞} ⊗ π_{p} in A_{cusp}(K_{f}^{p})_{χ^{−1}}. Consequently, we obtain a direct sum decomposition

H_{cusp}^{i}(V_{ℂ}) ≅ H^{i}(𝔤̃, 𝔨; A_{cusp}(K_{f}^{p})_{χ^{−1}} ⊗ V_{ℂ})

≅ ⨁_{π_{∞}⊗π_{p}} H^{i}(𝔤̃, 𝔨; π_{∞} ⊗ V_{ℂ}) ⊗ π_{p} ⊗ M(π_{∞} ⊗ π_{p}) (10)

(the point here being that the (𝔤̃, 𝔨)-cohomology depends only on the structure of π_{∞} ⊗ π_{p} 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 summands π_{∞} ⊗ π_{p} of A_{cusp}(K_{f}^{p})_{χ^{−1}} to the cohomology H_{cusp}^{i}(V_{ℂ}). In the case when H^{i}(𝔤̃, 𝔨; π_{∞} ⊗V_{ℂ}) is non-zero, so that π_{∞} ⊗π_{p} actually does contribute to cohomology, the multiplicity space M(π_{∞} ⊗ π_{p}) is naturally a 𝕋_{Σ}-module, and the direct sum decomposition (10) is compatible with the actions of G and 𝕋_{Σ} (where G-acts on π_{p}, and 𝕋_{Σ} acts on M(π_{∞} ⊗ π_{p})).

We can now explain the significance of the quantities l_{0} and q_{0} introduced above. Namely, q_{0} is the lowest degree in which tempered π_{∞} can admit nontrivial (𝔤̃, 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 do so precisely in degrees between q_{0} and q_{0} + l_{0} [10, Ch. III, Thm. 5.1]. (This is a range of degrees of length l_{0} + 1 symmetric about d/2 — which is one-half of the dimension of the quotients Y (K_{f}).) A fundamental theorem of Harish-Chandra states that, when 𝔾 is semisimple, the group G_{∞} admits discrete series representations precisely if l_{0} = 0; in this case q_{0} is equal to d/2, and the fundamental tempered representations are precisely the discrete series representations. Key examples for which l_{0} = 0 are given by groups 𝔾 for which G_{∞} is compact, and (the semisimple parts of) groups giving rise to Shimura varieties.

One representation of G_{∞} × G that always appears in A(K_{f}^{p}) is the trivial representation 𝟙. Thus there are induced morphisms

H^{i}(𝔤̃, 𝔨; 𝟙) → H^{i}(𝔤̃, 𝔨; A(K_{f}^{p})_{𝟙}) ≅ ℂ ⊗_{O} H^{i},  (11)

whose sources are naturally identified with the cohomology spaces of the compact dual to the symmetric space G_{∞}^{∘}/A_{∞}^{∘}K_{∞}^{∘}. If the Y (K_{f}) are not compact, then 𝟙 lies in A_{Eis}(K_{f}^{p})_{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 = SL_{N}) in (2.2.3) below.

2.1.7. Dimension theory. For any compact open subgroup K_{p} ⊂ G, the completed group ring O[[K_{p}]] 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[[K_{p}]]-modules: to any such module M we associate the Ext-modules E^{i}(M) := Ext_{O[[K_{p}]]}^{i}(M, O[[K_{p}]]), for i ≥ 0 (which necessarily vanish if i > dim 𝔾 + 1); these are again naturally O[[K_{p}]]-modules. (We use the left O[[K_{p}]]- module structure on O[[K_{p}]] to compute the E^{i}, and then use the right O[[K_{p}]]- module structure on O[[K_{p}]], converted to a left module structure by the usual device of applying the anti-involution k ↦ k^{−1} on K_{p}, to give the E^{i} an O[[K_{p}]]- module structure.)

An important point is that E^{i}(M) is canonically independent of the choice of K_{p} (in the sense that we may regard M as an O[[K′_{p}]]-module, for any open subgroup K′_{p} of K_{p}, but E^{i}(M) is canonically independent of which choice of K′_{p} we make to define it). This has the consequence that if the K_{p}-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 as

E_{2}^{i,j} = E^{i}(H̃_{j}) ⟹ H̃_{d−(i+j)}^{BM}.

All the objects appearing in it are G-representations, and it is G-equivariant.

If K_{p} is p-torsion free (which will be true provided K_{p} is sufficiently small) then the ring O[[K_{p}]] is in fact Auslander regular [55]. If K_{p} is furthermore pro-p, then the ring O[[K_{p}]] is an integral domain (in the sense that it contains no non-trivial left or right zero divisors). This has various implications for the theory of finitely generated O[[K_{p}]]-modules. For example, it is reasonable to define such a module to be torsion if every element has a non-zero annihilator in O[[K_{p}]] (where K_{p} is chosen small enough to be pro-p and p-torsion free). More generally, we may make the following definition.

Definition 2.3. If M is a finitely generated O[[K_{p}]]-module, then the codimension cd(M) of M is defined to be the least value of i for which E^{i}(M) ≠ 0; the dimension dim M of M is defined to be dim 𝔾 + 1 − cd(M).

If 𝔾 is a torus, so that K_{p} is commutative, then dim M is precisely the dimension of the support of the localization of M over Spec O[[K_{p}]]. In general O[[K_{p}]] 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[[K_{p}]]. (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, the support of M is empty) precisely when M = 0.

By Theorem 2.2, completed homology is finitely generated over O[[K_{p}]], and so these dimension-theoretic notions apply to it. Information about the (co)dimension of H̃_{i} can be obtained by analyzing the Poincaré duality spectral sequence, since the functors E^{i} appear explicitly in it, and there is a tension in this spectral sequence between the top degree for the duality of O[[K_{p}]]-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 have some a priori information about the (co)dimensions of the H̃^{i}. This is provided by the following result.

Theorem 2.4. [23] Suppose that 𝔾 is semisimple. Then H̃^{i} is torsion (i.e. its codimension is positive) unless l_{0} = 0 and i = q_{0}, in which case cd H̃_{i} = 0.

We make a much more precise conjecture about the codimensions of the H̃^{i} in Conjecture 3.2 below.

2.2. Examples. We illustrate the preceding discussion with some examples.

2.2.1. GL_{2} of ℚ. If 𝔾 = GL_{2}, then the quotients Y (K_{f}) 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 and 1. Since the cohomology of a curve is torsion free, we have H̃^{0} = Ĥ^{0} ≅ C(Δ×ℤ_{p}^{×}, O), the space of continuous O-valued functions on the product Δ × ℤ_{p}^{×}, where Δ is a finite group that depends on the choice of tame level, and so H̃_{0} (which is simply the O-dual of Ĥ^{0}) is isomorphic to O[[Δ × ℤ_{p}^{×}]], which is of dimension two as a module over O[[K_{p}]] (for any compact open K_{p} ⊂ GL_{2}(ℚ_{p})). We also have H̃^{1} = Ĥ^{1}. Again, H̃_{1} is the O-dual of H̃^{1}, and Theorem 2.4 shows that cd H̃_{1} = 0. (Strictly speaking, we have to apply the theorem to SL_{2} rather than GL_{2}, but it is then easy to deduce the corresponding result for GL_{2}, by considering the cup-product action of H^{0} on H^{1}.)

2.2.2. SL_{2} of an imaginary quadratic field. Suppose that 𝔾 = Res_{F/ℚ} SL_{2}, where F is an imaginary quadratic extension of ℚ. The quotients Y (K_{f}) are then connected, non-compact three-manifolds. The relevant (co)homological degrees are thus i = 0, 1, and 2. Since the Y (K_{f}) are connected we see that H̃^{0} = H̃_{0} = O. Theorem 2.4 implies that H̃_{1} has positive codimension. A consideration of the Poincaré duality spectral sequence then shows that H̃_{2} = 0, and that H̃_{1} is of codimension 1. This computation exploits both the fact that H̃_{0} ≠ 0, and the gap between 3 (the dimension of the Y (K_{f})) and 6 (the dimension of 𝔾).

Since H^{1} with coefficients in O of any space is ϖ-torsion free, we see that H̃^{1} is ϖ torsion-free, and coincides with the O-dual of H̃_{1}, while H̃^{2} is ϖ-torsion, and is naturally identified with the Pontrjagin dual of the O-torsion submodule of H̃_{1} [24, Thm. 1.1]. We conjecture that in fact H̃_{1} is ϖ-torsion free (see Conjecture 3.2), and thus that H̃^{2} = 0.

2.2.3. SL_{N} of ℚ in low degrees. We first discuss H̃^{0} and H̃^{1}, before turning to a discussion of higher degree cohomology from the point of view of homological stability.

The quotients Y (K_{f}) are connected, and so H̃_{0} = O. If N ≥ 3, then SL_{N} satisfies the congruence subgroup property. Furthermore, the groups SL_{N}(ℤ_{ℓ}) are perfect for all values of ℓ. Combining these two facts, we see that if Γ(p^{r}) denotes the usual congruence subgroup of level p^{r} (i.e. the kernel of the surjection SL_{N}(ℤ) → SL_{N}(ℤ/p^{r})), then H_{1}(Γ(p^{r}), O) = O⊗_{ℤ_{p}} Γ(p^{r})^{ab} ≅ O⊗_{ℤ_{p}} Γ(p^{r})/Γ(p^{2r}). Thus, if we take the tame level to be trivial (i.e. “level one”), then we see that the transition maps in the projective system defining H̃_{1} are eventually zero, implying that H̃_{1} = 0. Similarly, we see that H^{1} = H̃^{1} = 0. (If we allowed a more general tame level, then H̃_{1} could be non-zero, but would be finite. Since H^{1} and H̃^{1} are always torsion free, they would continue to vanish.)

We now consider cohomology in higher degree, but in the stable range, in the sense that we now explain. (The tame level can now be taken to be arbitrary.) Borel’s result [9] on homological stability for SL_{N} 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 automorphic representation, in the sense discussed in (2.1.6). If we define H_{stab}^{i} to be the stable value of H^{i}, then Borel shows that ⨁ H_{stab}^{i} (as an algebra under cup product) is isomorphic to an exterior algebra generated by a single element in each degree i = 0, 5, 9, 13, ….

In fact, stability also holds for completed cohomology.

Theorem 2.5. [25] If i is sufficiently small compared to N, then H̃^{i} is independent of N, is finitely generated as an O-module, and affords the trivial representation of G.

We write H̃_{stab}^{i} to denote the stable value of H̃^{i}. F. Calegari has succeeded in computing H̃_{stab}^{i} modulo torsion, contingent on a natural non-vanishing conjecture 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 that appropriate special values of the p-adic zeta function are non-zero. Then there is an isomorphism of graded vector spaces ⨁_{i≥0} L ⊗_{O} H̃_{stab}^{i} ≅ L[x_{2}, x_{6}, x_{10}, …] (where L[x_{2}, x_{6}, x_{10}, …] denotes the graded ring generated by the indicated sequence of polynomial variables x_{i}, i ≡ 2 mod 4, with x_{i} placed in degree i).

Note in particular that ⨁_{i} L ⊗_{O} H̃_{stab}^{i} vanishes in all odd degrees, while ⨁_{i} L ⊗_{O} H_{stab}^{i} is generated by classes in odd degrees. Thus, when 𝔾 = SL_{N}, 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.e. the non-zero elements of L ⊗_{O} H_{stab}^{i}) become infinitely divisible when we pull 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}(SL_{N}(ℤ_{p}), L) coincides with the Lie algebra cohomology of 𝔰𝔩_{N} [48], which is (stably in N) an exterior algebra on generators in degrees 3, 5, 7, …. Since all of the non-zero H̃_{stab}^{i} 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 see (continuing to assume the hypothesis of the theorem) that Ĥ^{i} vanishes (modulo torsion), while H̃^{i} is non-zero. Thus the term T_{p}H^{i+1} must be non-zero; this provides 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 (K_{f}) are simply finite sets. Thus the only interesting degree of cohomology is i = 0. In this case the inverse limit Y (K_{f}^{p}) := \underleftarrow{lim}_{K_{p}} Y (K_{p}K_{f}^{p}) is a pro-finite set, which is in fact a compact p-adic analytic manifold, equipped with an analytic action of G. If K_{p} is taken sufficiently small (small enough that the 𝔾(ℚ)-action on 𝔾(𝔸)/G_{∞}^{∘}K_{p}K_{f}^{p} is fixed-point free), then K_{p} acts with trivial stabilizers on Y (K_{p}K_{f}^{p}), and Y (K_{p}K_{f}^{p}) is the disjoint union of finitely many (say n) principal homogeneous spaces over K_{p}.

One immediately verifies that H̃^{0} ≅ C(Y (K_{f}^{p}), O), the space of continuous O-valued functions on Y (K_{f}^{p}), while H̃_{0} ≅ O[[Y (K_{f}^{p})]], the space of O-valued measures on Y (K_{f}^{p}) (which is the O-dual of C(Y (K_{f}^{p}), O)). In particular, if K_{p} is sufficiently small, then we see that H̃_{0} is free of rank n over O[[K_{p}]].

It is natural (e.g. in light of Gross’s definition of algebraic modular forms in this context [42]) to define C(Y (K_{f}^{p}), O) to be the space of p-adic automorphic forms on 𝔾(𝔸) of tame level K_{f}^{p}, and so in this case we see that completed cohomology does indeed coincide with the natural notion of p-adic automorphic forms.

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 Ga-

lois representations. If π is an automorphic representation of 𝔾(𝔸) for which H^{i}(𝔤̃, 𝔨; π_{∞} ⊗ V_{ℂ}) ≠ 0 for some algebraic representation V of 𝔾 and some degree i, then π is C-algebraic [21, Lemma 7.2.2]. Thus, we expect that associated to π there should be a continuous representation of G_{ℚ} into the \overline{ℚ}_{p}-valued points of the C-group of 𝔾 [21, Conj. 5.3.4]. In light of the isomorphism (8), we thus expect that the systems of Hecke eigenvalues that occur in H^{i}(V) should have associated representations of G_{ℚ} into the C-group of 𝔾. For systems of Hecke eigenvalues occurring on torsion classes in cohomology, conjectures of Ash [2] again suggest that 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 GL_{n} over a totally real or a CM number field. (See [45] in the case of characteristic zero systems of eigenvalues, and [53] in the case of 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 G_{ℚ} 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 𝕋_{Σ}, write 𝔽 := 𝕋_{Σ}/𝔪, and let \overline{𝔽} denote a chosen algebraic closure of 𝔽. Then, by the above discussion, we believe that associated to 𝔪 there should be a continuous representation of G_{ℚ} into the \overline{𝔽}-valued points of the C-group of 𝔾. Assuming that this representation exists, we will denote it by \overline{ρ}_{𝔪}.

To be a little more precise: the representation \overline{ρ}_{𝔪} should have the property that it is unramified at the primes outside Σ, and that for any ℓ ∉ Σ, the semisimple part of \overline{ρ}_{𝔪}(Frob_{ℓ}) should be associated to the local-at-ℓ 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 \overline{ρ}_{𝔪} uniquely; even in the case 𝔾 = GL_{n}, it determines \overline{ρ}_{𝔪} only up to semisimplification; for other choices of 𝔾, in addition to this issue, one can have the additional phenomenon that even a 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 in general here; rather we simply ignore it, and continue our discussion as if \overline{ρ}_{𝔪} were unambiguously defined.

As we observed above, we may regard 𝕋_{𝔪} (or its Spec) as interpolating systems of Hecke eigenvalues associated to classical C-algebraic automorphic forms, and/or to torsion classes in cohomology. Since the deformations of \overline{ρ}_{𝔪} can be interpolated into a formal deformation space, it is then reasonable to imagine that we may deform \overline{ρ}_{𝔪} to a representation of G_{ℚ} into the 𝕋_{𝔪}-valued points of the C-group of 𝔾. Here again there are many caveats: firstly, if \overline{ρ}_{𝔪} is “reducible” (i.e. factors through a proper parabolic of the C-group), then we would have to work with some form of pseudo-character or determinant, as in [28] and [53]; also, there is a question of rationality, or “Schur index” — it may be that if we want our deformation of \overline{ρ}_{𝔪} 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 vanishing that are motivated by the preceding discussion, and then continue by discussing the relationships between completed cohomology and p-adic Hodge theory and a possible p-adic local Langlands correspondence. Finally, we turn to a discussion of some specific examples, where we can make our generalities more precise.

3.1.1. Conjectures on dimension and vanishing. We begin by making a somewhat vague conjecture, which can be thought of as a rough expression of our hopes for reciprocity in the context of global p-adic Langlands: namely (continuing with the notation introduced in the preceding discussion), we conjecture that 𝕋_{𝔪} is universal (in some suitable sense) for parameterizing odd formal deformations of \overline{ρ}_{𝔪} whose ramification away from p is compatible with the tame level structure K_{f}^{p} (via some appropriate form of ϖ-adic local Langlands for the group 𝔾 at primes ℓ ∤ p which, again, we won’t attempt to formulate here; but see [36] in the case when 𝔾 = GL_{n}). The global Euler characteristic formula for Galois cohomology lets us compute the expected dimension of such a universal deformation ring, and this motivates in large part the following concrete conjecture [24, §8].

Conjecture 3.1. Each local factor 𝕋_{𝔪} of 𝕋_{Σ} is Noetherian, reduced, and ϖtorsion free, of Krull dimension equal to dim 𝔹 + 1 − l_{0}, where dim 𝔹 denotes the dimension of a Borel subgroup 𝔹 of 𝔾.

This is known in some cases (which we will recall below). We know of no way to prove the Noetherianness of 𝕋_{𝔪}, or the statement about its Krull dimension, other than to relate the Hecke algebra to a deformation ring of Galois representations, and then use techniques from the theory of Galois representations to compute the dimension.

In some situations we can prove that 𝕋_{𝔪} is torsion free and reduced (see e.g. the discussion 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].

Conjecture 3.2. H̃_{i} = 0 if i > q_{0}, while cd H̃_{q_{0}} = l_{0}, and cd H̃_{i} > l_{0} + q_{0} − i if i < q_{0}. Furthermore, H̃_{q_{0}} is ϖ-torsion free, and 𝕋_{Σ} acts faithfully on H̃_{q_{0}}.

As noted in [24, Thm. 1.6], the truth of this conjecture for 𝔾 and all its Levi subgroups implies the analogous statement for H̃_{i}^{BM}. Also, the vanishing conjecture for 𝔾 × 𝔾 implies (via the Künneth formula) the torsion-freeness of H̃_{q_{0}}.

One of the ideas behind this conjecture is that “all the interesting Hecke eigenvalues should appear in degree q_{0}”. This is inspired by the fact (recalled in (2.1.6)) that tempered automorphic representations don’t contribute to cohomology in degrees below q_{0}, so that the Galois representations associated to the systems of Hecke eigenvalues appearing in degrees below q_{0} 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 form a 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 q_{0} which can’t already be observed in degree q_{0}, and hence we expect that 𝕋_{Σ} will act faithfully on H̃_{q_{0}}.

We also expect that H̃_{i} should be “small” if i < q_{0}, since the possible systems of Hecke eigenvalues which it can carry are (or should be) constrained. However, if the H̃_{i} are small enough for i < q_{0}, then the Poincaré duality spectral sequence (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) implies the vanishing of H̃_{i} in degrees > q_{0}.

If we believe that 𝕋_{Σ} acts faithfully on H̃_{q_{0}}, and that 𝕋_{Σ} has dimension dim 𝔹 + 1 − l_{0} (as predicted by Conjecture 3.1), then conjecturing that H̃_{q_{0}} has codimension l_{0} is morally equivalent to conjecturing that the fibres of H̃_{q_{0}} over the points of Spec 𝕋_{Σ} are of dimension dim 𝔾/𝔹. This latter statement fits nicely with the fact that generic irreducible representations of G := 𝔾(ℚ_{p}) have Gelfand– Kirillov dimension equal to dim 𝔾/𝔹, and the analogy between the dimension of O[[K_{p}]]-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 𝕋_{Σ}, the dimension of H̃_{q_{0}}, and the dimension of the fibres of the latter over points of Spec 𝕋_{Σ}, even assuming that 𝕋_{Σ} acts faithfully on H̃_{q_{0}}. Nevertheless, the idea that these dimensions should be related is an important motivation for the conjecture.

We note that if 𝕋_{Σ} acts faithfully on H̃_{q_{0}}, and H̃_{q_{0}} is ϖ-torsion free, then 𝕋_{Σ} is ϖ-torsion free. This motivates the conjecture of ϖ-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 < q_{0}.)

We mention one more example here, namely the case when 𝔾 is the restriction of scalars of SL_{2} from an imaginary quadratic field. In this case we have l_{0} = 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 𝕋_{Σ}, and that the associated representation \overline{ρ}_{𝔪} (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 non-Eisenstein. For any 𝕋_{Σ}-module M, we write M_{𝔪} := 𝕋_{𝔪} ⊗_{𝕋_{Σ}} M to denote the localization 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 proper parabolic subgroup, and so we conjecture that, when 𝔪 is a non-Eisenstein maximal ideal, H^{i}(Y (K_{p}K_{f}^{p}), W)_{𝔪} = 0 for all i < q_{0} and all local systems W associated to K_{p}-representations W on finitely generated O-modules (where K_{p} 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 of compactly supported cohomology in the usual cohomology. However, the interior cohomology is contained in the L^{2}-cohomology, and now Arthur’s conjectures imply that any automorphic representation π contributing to L^{2}-cohomology that is nontempered at ∞ — as must be the case for an automorphic representation that contributes to cohomology in degree < q_{0} — is non-tempered at every prime, and hence should give rise to a reducible Galois representation.) We believe that this statement should be true for torsion cohomology as well.

Let us suppose, then, that H^{i}(Y (K_{p}K_{f}^{p}), W)_{𝔪} = 0 for all i < q_{0}. Just considering the case when W = O (the trivial representation), we find that H̃_{𝔪}^{i} = 0 for i < q_{0}. Certainly it should be the case that H^{i}(∂(K_{p}K_{f}^{p}), W)_{𝔪} = 0 when 𝔪 is non-Eisenstein. We then find that H_{c}^{i}(Y (K_{p}K_{f}^{p}), W)_{𝔪} = 0 for all i < q_{0} as well. Suppose now that l_{0} = 0, so that q_{0} = d/2. Classical Poincaré duality then gives that H^{i}(Y (K_{p}K_{f}^{p}), W)_{𝔪} = 0 for all i > q_{0}. Presuming that Conjecture 3.2 is true, so that H̃^{i} = 0 for i > q_{0} and H̃^{q_{0}} is ϖ-torsion free, a consideration of the Hochschild–Serre spectral sequence shows that Ext_{O[[K_{p}]]}^{i}(W^{∨}, H̃_{𝔪}^{q_{0}}) = 0 for all i > 0 and all representations of K_{p} on finitely generated torsion free O-modules. From this one easily deduces that (H̃_{q_{0}})_{𝔪} is projective as an O[[K_{p}]]-module.

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 𝕋_{Σ}, and if l_{0} = 0, then (H̃_{i})_{𝔪} = 0 for i ≠ q_{0}, and (H̃_{q_{0}})_{𝔪} is a projective O[[K_{p}]]-module, for any sufficiently small subgroup K_{p} of G. (Here sufficiently small means that 𝔾(ℚ) acts with trivial stabilizers on 𝔾(𝔸)/A_{∞}^{∘}K_{∞}^{∘}K_{p}K_{f}^{p}.)

If l_{0} = 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 K_{p} (as in the statement of the conjecture) and any representation of K_{p} on a finitely generated torsion free O-module W, we have H^{i}(Y (K_{p}K_{f}^{p}), W)_{𝔪} = 0 if i ≠ q_{0}, while H^{q_{0}}(Y (K_{p}K_{f}^{p}), W)_{𝔪} ≅ Hom_{O[[K_{p}]]}(W^{∨}, H̃_{𝔪}^{q_{0}}). In particular, we see that H^{q_{0}}(Y (K_{p}K_{f}^{p}), W)_{𝔪} is ϖ-torsion free. We also see that Ĥ_{𝔪}^{q_{0}} = H̃_{𝔪}^{q_{0}} (since H_{𝔪}^{q_{0}+1} vanishes, and hence so does T_{p}H_{𝔪}^{q_{0}+1}), and that 𝕋_{𝔪} acts faithfully on H̃_{𝔪}^{q_{0}}.

Presuming that H^{i}(∂(K_{p}K_{f}^{p}), W)_{𝔪} = 0 (which should certainly be true when 𝔪 is non-Eisenstein), we conclude that the natural map

H_{c}^{q_{0}}(Y (K_{p}K_{f}^{p}), W)_{𝔪} → H^{q_{0}}(Y (K_{p}K_{f}^{p}), W)_{𝔪}

is an isomorphism, and thus that H^{q_{0}}(Y (K_{p}K_{f}^{p}), W)_{𝔪} consists entirely of interior cohomology. In particular (being torsion free) it embeds into the L^{2}-cohomology. From this we deduce that the image of 𝕋_{Σ} acting on H^{q_{0}}(Y (K_{p}K_{f}^{p}), W)_{𝔪} is reduced, and hence (considering all possible K_{p} and W) that 𝕋_{𝔪} itself is reduced, as well as being ϖ-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 the case when 𝔾 = GL_{2}, 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 = GL_{N}, this notion is again well-defined, and the vanishing of (H̃_{i})_{𝔪} is proved for i in the stable range in [25]. Again, if 𝔾 is a form of U(n − 1, 1) over ℚ, then the notion of a non-Eisenstein maximal ideal is well-defined, and in [34] we prove vanishing of (H̃_{i})_{𝔪} for i in a range of low degrees, for certain maximal ideals 𝔪. In particular, in the 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 \overline{ρ}_{𝔪} 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 l_{0} = 0), then Conjecture 3.3 holds. Indeed, in this case we saw in (2.2.4), for sufficiently small K_{p}, that H̃_{0} is free over O[[K_{p}]], even without localizing at a non-Eisenstein maximal ideal.

If l_{0} ≠ 0, then we don’t expect that (H̃_{q_{0}})_{𝔪} should be projective over O[[K_{p}]]; indeed, this would be incompatible with Conjecture 3.2. Rather, we expect that it should be pure of codimension l_{0}, in the sense of [55, Def. 3.1].

3.1.3. The relationship with p-adic Hodge theory. Let us continue to suppose that l_{0} = 0 and that 𝔪 is a non-Eisenstein maximal ideal in 𝕋_{Σ}, and let us suppose that Conjecture 3.3 holds. As we have observed, this implies that for any (sufficiently small) compact open subgroup K_{p} of G, and any representation W of K_{p} on a finitely generated torsion free O-module, we have

H^{i}(Y (K_{p}K_{f}^{p}), W)_{𝔪} ≅ Hom_{O[[K_{p}]]}(W^{∨}, H̃_{𝔪}^{q_{0}}) ≅ Hom_{O[[K_{p}]]}((H̃_{q_{0}})_{𝔪}, W )

(the first isomorphism being given by the Hochschild–Serre spectral sequence, and the 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 (K_{p}K_{f}^{p}), W)_{𝔪} is ϖ-torsion free, it is a lattice in H^{i}(Y (K_{p}K_{f}^{p}), V)_{𝔪}. Considering the description of this cohomology in terms of 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 H^{i}(Y (K_{p}K_{f}^{p}), V)_{𝔪} should be potentially semistable locally at p, with Hodge–Tate weights 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 R_{\overline{ρ}_{𝔪}} of \overline{ρ}_{𝔪}, and we conjecture that there is a natural isomorphism 𝕋_{𝔪} ≅ R_{\overline{ρ}_{𝔪}}. We see that the 𝕋_{𝔪}-modules H^{i}(Y (K_{p}K_{f}^{p}), 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 R_{\overline{ρ}_{𝔪}} corresponding to deformations of \overline{ρ}_{𝔪} that 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 (K_{p}K_{f}^{p}), W)_{𝔪}, in purely Galois-theoretic terms. Namely, any potentially semistable deformation of \overline{ρ}_{𝔪} 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 K_{p}K_{f}^{p}, 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 (K_{p}K_{f}^{p}), W)_{𝔪} is as predicted by the Fontaine–Mazur–Langlands conjecture, then we can deduce this conjecture for deformations of \overline{ρ}_{𝔪} that are classified by Spec 𝕋_{𝔪}. We recall in (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 this strategy was employed in [32] and [46] in the case when 𝔾 = GL_{2}.

Let us fix a sufficiently small open subgroup K_{p}. Since (H̃_{q_{0}})_{𝔪} is projective over O[[K_{p}]] by assumption, it is a direct summand of a finitely generated free O[[K_{p}]]- module, and hence H̃_{𝔪}^{q_{0}} is a direct summand of C(K_{p}, O)^{⨁ n}, for some n > 0. Thus L ⊗_{O} H̃_{𝔪}^{q_{0}} is a direct summand of C(K_{p}, L)^{⊕n}, the space of continuous L-valued functions on K_{p}. Now the theory of Mahler expansions shows that the affine ring L[𝔾] embeds with dense image in C(K_{p}, L) [51, Lemma A.1]. Recall that, as a 𝔾-representation, we have L[𝔾] ≅ ⨁_{V} V ⊗_{L} Hom_{𝔾}(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 \overline{ℚ}_{p} are in fact defined over L; thus these irreducible V are in fact absolutely irreducible.) Now the inclusion of L[𝔾] into C(K_{p}, L) induces an isomorphism Hom_{𝔾}(V, L[𝔾]) ≅ Hom_{K_{p}}(V, C(K_{p}, L)) (both are naturally identified with V^{∨}, the contragredient of V). We conclude that the natural morphism ⨁_{V} V ⊗_{L} Hom_{K_{p}}(V, C(K_{p}, L)) → C(K_{p}, L) is injective with dense image. This property is clearly preserved under passing to direct sums and direct summands, and hence the natural morphism ⨁_{V} V ⊗_{L}Hom_{K_{p}}(V, H̃_{𝔪}^{q_{0}}⊗_{O}L) → H̃_{𝔪}^{q_{0}}⊗_{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 recalling that (by Hochschild–Serre) Hom_{K_{p}}(V^{∨}, H̃_{𝔪}^{q_{0}} ⊗_{O} L) ≅ H^{q_{0}}(Y (K_{p}K_{f}^{p}), V)_{𝔪}, we find that 𝕋_{𝔪} acts faithfully on ⨁_{V} H^{q_{0}}(Y (K_{p}K_{f}^{p}), V)_{𝔪} (since, as we noted above, it follows from Conjecture 3.3 that 𝕋_{𝔪} acts faithfully on H̃_{𝔪}^{q_{0}}).

Assuming that we can identify 𝕋_{𝔪} with a deformation space of Galois representations, and that we know that H^{q_{0}}(Y (K_{p}K_{f}^{p}), 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 𝕋_{𝔪}.

In forthcoming work [37], the author and V. Paškūnas will apply a more sophisticated version of this argument to deduce additional Zariski density statements for various collections of potentially semistable loci in global Galois deformation spaces.

Note that these density results rely crucially on the projectivity statement of Conjecture 3.3 for their proof. As we already noted, we can’t expect such a statement to be true when l_{0} > 0, and (at least if 𝔾 is semisimple, or, more generally, if the semisimple part of 𝔾 satisfies l_{0} > 0) we don’t expect the potentially semistable points to be Zariski dense in global deformation spaces in this case. (See [26] for an elaboration on this point.) In particular, in such contexts, we don’t expect that Ĥ^{q_{0}} will equal H̃^{q_{0}}, and so H̃^{q_{0}}, and Spec 𝕋_{Σ}, should receive a non-trivial contribution from torsion classes (via the term T_{p}H^{q_{0}+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 that if π_{∞} ⊗ π_{p} is a direct summand of A_{cusp}(K_{f}^{p})_{χ^{−1}} which contributes to cohomology, lying in a Hecke eigenspace that corresponds to some p-adic Galois representation ρ, then the local factor π_{p} 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 π_{p} 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 same should 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 \overline{ρ}_{𝔪}, and that π_{∞} and π_{p} are tempered, so that we have an embedding π_{p} ↪ H^{q_{0}}(V)_{𝔪} ≅ \underrightarrow{lim}_{K_{p}} Hom_{K_{p}}(V^{∨}, L⊗_{O} H̃_{𝔪}^{q_{0}}). We

may rewrite this as an embedding V^{∨} ⊗_{L} π_{p} ↪ L⊗_{O} H̃_{𝔪}^{q_{0}} and we can then take the closure of V^{∨} ⊗ π_{p} in H^{q_{0}}(V)_{𝔪}, to obtain a unitary Banach space representation { \atop V^{∨} ⊗_{L} π}_{p} of G over L. A slightly more refined procedure is to form the intersection (V^{∨} ⊗_{L} π_{p})^{∘} := (V^{∨} ⊗_{L} π_{p})∩H̃_{𝔪}^{q_{0}} (the intersection being taken in L⊗_{O} H̃_{𝔪}^{q_{0}}). This is a G-invariant O-lattice contained in V^{∨} ⊗_{L} π_{p}, and the Banach space { \atop V^{∨} ⊗_{L} π}_{p} is then obtained by completing V^{∨} ⊗_{L} π_{p} with respect to this lattice. A natural question to ask, then, in the spirit of a local Langlands correspondence and localglobal compatibility, is whether { \atop V^{∨} ⊗_{L} π}_{p} depends only on the restriction to p of 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} π_{p})^{∘} 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 = GL_{2} (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 might entertain regarding a p-adic local Langlands correspondence is that for any formal deformation ring R\overline{r} parameterizing the deformations of a representation \overline{r} of the local Galois group G_{ℚ_{p}} into the \overline{𝔽}-valued points of the C-group of G, there is a profinite R\overline{r}-module M, finitely generated over R\overline{r}[[K_{p}]] for some (and hence every) compact open subgroup K_{p} ⊂ G, equipped with a continuous G-action extending its K_{p}-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 continuing with the notation of that discussion), the fibre over the restriction to p of ρ^{∘} (which is a point in Spec R\overline{r}, if \overline{r} is the restriction of \overline{ρ}_{𝔪} to G_{ℚ_{p}}) is isomorphic to the O-dual of (V^{∨} ⊗_{L} π_{p})^{∘}. (Here we suppress the issue of whether R\overline{r} should be understood to be a framed deformation ring, or a pseudo-deformation ring, or ….) In short, M should be a local analogue of the completed homology space (H̃_{q_{0}})_{𝔪}.

Since M is finitely generated over R\overline{r}[[K_{p}]], for any representation W of K_{p} on a finitely generated torsion free O-module, the R\overline{r}-module Hom_{K_{p}}^{cont}(M, W)^{d} (where d denotes the continuous O-dual) is a finitely generated R\overline{r}-module. Another property one might require of M is that when W is a K_{p}-invariant O-lattice in an irreducible algebraic 𝔾-representation V, then Hom_{K_{p}}^{cont}(M, W)^{d} is supported on an appropriate locus of potentially semistable representations, corresponding to the fact that Hom_{O[[K_{p}]]}((H̃_{q_{0}})_{𝔪}, W ) (which is isomorphic to H^{q_{0}}(Y (K_{p}K_{f}^{p}), W)) is supported on a locus of potentially semistable representations in Spec 𝕋_{𝔪}.

Ideally, one might ask for the support of Hom_{K_{p}}^{cont}(M, W)^{d} to be the full potentially semistable locus of appropriate Hodge–Tate weights (corresponding to the highest weight of V) and Weil–Deligne representations (corresponding to the particular choice of K_{p} 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 compatibility between M and (H̃_{q_{0}})_{𝔪}, would prove that any global Galois representation corresponding to a point of Spec 𝕋_{𝔪} which is potentially semistable of appropriate Hodge–Tate weights and with an appropriate Weil–Deligne representation, in fact arises from a system of Hecke eigenvalues occurring in H̃^{q_{0}}(Y (K_{p}K_{f}^{p}), V)_{𝔪}, thus verifying the Fontaine–Mazur–Langlands conjecture for such points.

Note that a local Galois representation lies in the support of Hom_{K_{p}}^{cont}(M, W)^{d} precisely if the dual to the fibre of M at this point receives a non-zero K_{p}- equivariant homomorphism from W. In particular, the dual to the fibre at such a point contains locally algebraic vectors. In the case of GL_{2}, the idea of describing p-adic local Langlands for potentially semistable representations in terms of locally algebraic vectors goes back to C. Breuil’s first work on the subject [11, 12].

The optimistic scenario described here has been realized for the group G = GL_{2}(ℚ_{p}); this is the theory of p-adic local Langlands for GL_{2}(ℚ_{p}) [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. GL_{2} of ℚ. As already mentioned, the theory of p-adic local Langlands for GL_{2}(ℚ_{p}) provides a structure satisfying all the desiderata of (3.2.2). The localglobal compatibility between this structure and H̃_{𝔪}^{1} (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 behaviour of \overline{ρ}_{𝔪}). In particular, the strategy of (3.1.3) then applies to prove the Fontaine– Mazur–Langlands conjecture for points of Spec 𝕋_{𝔪}.

Under the slightly stronger hypotheses that p is odd and \overline{ρ}_{𝔪} remains irreducible on restriction to G_{ℚ(ζ_{p})} (the Taylor–Wiles condition) one can prove that 𝕋_{𝔪} ≅ R_{\overline{ρ}_{𝔪}} [8], [32, Thm. 1.2.3]. Since Conjecture 3.3 holds in this context [32, Cor. 5.3.19], the method of (3.1.3) (applied with K_{p} = GL_{2}(ℤ_{p})) allows us to deduce the density of the crystalline loci in Spec R_{\overline{ρ}_{𝔪}}; the extension of this method to be described in [37] will allow us to deduce other density results about various potentially semistable loci. Here is one such simple variant: instead of considering the restriction to GL_{2}(ℤ_{p}) of the family V of algebraic representations of GL_{2}, 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 𝕋_{𝔪} (and hence in R_{\overline{ρ}_{𝔪}}, 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 𝕋_{𝔪} is a quotient of R_{\overline{ρ}_{𝔪}}.

Assuming that p is odd and \overline{ρ}_{𝔪} satisfies the Taylor–Wiles condition, we conclude that in fact 𝕋_{𝔪} ≅ R_{\overline{ρ}_{𝔪}}. Indeed, we saw in the preceding example that the potentially crystalline points of supercuspidal type are Zariski dense in R_{\overline{ρ}_{𝔪}}. 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 𝕋_{𝔪} contains a Zariski dense set of points in Spec R_{\overline{ρ}_{𝔪}}, and so the two Specs coincide. As one interesting consequence of this, we note that (assuming that the Taylor–Wiles condition holds) the Hecke algebras at 𝔪 for GL_{2} and D^{×} are naturally isomorphic (both being isomorphic to R_{\overline{ρ}_{𝔪}}). One can think of this as a p-adic interpolation of the classical Jacquet–Langlands correspondence. It is interesting to note that even though under the classical Jacquet–Langlands correspondence automorphic forms on GL_{2} that are principal series at p don’t match with a corresponding Hecke eigenform on D^{×}, they are not excluded from this p-adic Jacquet–Langlands correspondence.

As another consequence, note that if we choose K_{p} to be the units in the maximal order of (D ⊗ℚ_{p})^{×}, and apply the density argument of (3.1.3), we deduce that the representations which are genuinely semistable at p (i.e. semistable, and not crystalline) are Zariski dense in Spec 𝕋_{𝔪}, and hence in Spec R_{\overline{ρ}_{𝔪}}, provided the Taylor–Wiles condition holds.

3.3.3. Definite quaternion algebras over totally real fields. Consider the case 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 the situation of (3.2.1), with the additional assumption that V is the trivial representation, and π_{p} is a tamely ramified principal series. In this case π_{p} contains, as a GL_{2}(ℤ_{p} ⊗_{ℤ} O_{F})-subrepresentation, a principal series type σ, which is a representation of GL_{2}(O_{F} /p) induced from a character of the Borel subgroup. In the paper [16], Breuil gave a conjectural description of the isomorphism class of the lattice σ^{∘} := σ ∩ H̃_{𝔪}^{0}, purely in terms of the restriction to p of ρ^{∘}. Under mild assumptions on \overline{ρ}_{𝔪}, this was proved in [35]. This gives some evidence towards the possibility that (V^{∨} ⊗_{L} π_{p})^{∘} may be of a purely local nature.

3.3.4. Compact unitary groups. In [27], we apply Taylor–Wiles patching to pass 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 adjoining a certain number of formal variables to a local deformation ring. More precisely: for any finite extension K of ℚ_{p}, any n such that p ∤ 2n, and any representation \overline{r} : G_{K} → GL_{n}(\overline{𝔽}_{p}) that admits a potentially crystalline lift which is potentially diagonalizable (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(ℚ_{p}) is isomorphic to a product of copies of GL_{n}(K), and such that the restriction of \overline{ρ}_{𝔪} to G_{K} is equal to \overline{r}. The Grepresentation M_{∞}, which is a module over R\overline{r}[[x_{1}, …, x_{n}]] 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 generated over R_{∞}[[K_{p}]], and the modules Hom_{K_{p}}^{cont}(M, W)^{d} (for K_{p}-invariant lattices W in algebraic representations) are supported on a union of components of the appropriate potentially semistable loci. Many questions about M_{∞} remain open, however: whether the support of M_{∞} equals the entirety of Spec R_{∞}; whether all potentially semistable points are contained in the support of Hom_{K_{p}}^{cont}(M, W)^{d} (for an appropriate choice of W and K_{p}); 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[[K_{p}]]-modules. In [37], we hope to show that M_{∞} is in fact of full support on Spec R\overline{r}[[x_{1}, …, x_{n}]] 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.

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