HOMOLOGICAL STABILITY FOR COMPLETED HOMOLOGY

FRANK CALEGARI AND MATTHEW EMERTON

Contents

1. Introduction  1
1.1. Acknowledgements  3
2. Arithmetic Manifolds  3
2.1. Cohomology of arithmetic quotients  3
3. Stability for ℓ ≠ p  4
4. Hecke Operators  6
4.1. Proof of Theorem 1.2  7
5. Stability for ℓ = p  8
5.1. Consequences for classical cohomology groups  12
5.2. Relation to K-theory  12
References  12

1. Introduction

Let GL_{N}(Z, M) denote the principal congruence subgroup of GL_{N}(Z) of level M, explicitly, the group of invertible matrices with integer coefficients which are congruent to the identity modulo M. A theorem of Charney [Cha80] and (independently) Maazen [Maa79] shows that, for fixed d, the homology groups

H_{d}(GL_{N}(Z, 1), Z) = H_{d}(GL_{N}(Z), Z)

are stable, that is, they are independent of N for N sufficiently large. Moreover, the natural transition maps induced from the inclusions GL_{N} ⊂ GL_{N+1} in this range are isomorphisms. In [Cha84], Charney extends this result to prove a stability theorem for congruence subgroups. In particular, she proves that the homology groups

H_{d} (GL_{N}(Z, M), Z [\frac{1}{M} ])

are also stable as N → ∞. Note that the analogous result with coefficients in R is (in both cases) a consequence of a theorem of Borel [Bor74]. The result is no longer true if one does not invert primes dividing the level. For example, if p > 2 is prime, it is an elementary consequence of the congruence subgroup property [BLS64, Men65] that the homomorphism

H_{1}(GL_{N}(Z, p), Z) → GL_{N}(Z, p)/GL_{N}(Z, p^{2}) ≃ (Z/pZ)^{N^{2}−1}

is an isomorphism for N ≥ 3, and the latter group is certainly not stable. The starting point for the Church–Farb theory of representation stability [CF11] is the observation that, as a representation of SL_{N}(F_{p}), the right hand side has a description which is independent of N (it is the adjoint representation). Alternatively, one might view this calculation as saying that all the cohomology of GL_{N}(Z, p) in degree 1 (for N ≥ 3) arises via pullback from the homology of congruence subgroups of the local p-adic lie group GL_{N}(Z_{p}). This latter homology is evidently insensitive to the finer arithmetic properties of GL_{N}(Z), and so it is natural to try to excise the cohomology arising for “local” reasons and consider what remains. This is exactly what is achieved by replacing the homology groups by completed homology groups, as first defined and studied in [Eme06, CE09, CE12].

2010 Mathematics Subject Classification. 11F80, 11F75, 19F99.

The first author was supported in part by NSF Career Grant DMS-0846285. The second author was supported in part by NSF Grant DMS-1003339.

Fix a tame level M, and let Γ_{N}(p^{k}) = GL_{N}(Z, Mp^{k}) denote the principal congruence subgroup of level Mp^{k}. Recall that the completed homology and cohomology groups are defined as follows [CE12]:

H̃_{∗}(F_{p}) := \underleftarrow{lim} H_{∗}(Γ_{N}(p^{r}), F_{p})  H̃^{∗}(F_{p}) := \underrightarrow{lim} H^{∗}(Γ_{N}(p^{r}), F_{p}),
H̃_{∗}(Z_{p}) := \underleftarrow{lim} H_{∗}(Γ_{N}(p^{r}), Z_{p})  H̃^{∗}(Q_{p}/Z_{p}) := \underrightarrow{lim} H^{∗}(Γ_{N}(p^{r}), Q_{p}/Z_{p}).

Our main result is as follows:

Theorem 1.1. The modules H̃_{d}(Z_{p}) and H̃_{d}(F_{p}) stabilize as N → ∞ and are finite Z_{p}- modules and F_{p}-vector spaces respectively. Moreover, the action of SL_{N}(Q_{p}) on either module is trivial.

The question of computing these groups and understanding their relationship to arithmetic and K-theory will be taken up in the companion paper [Cal13], although see § 5.1 for some immediate consequences of stability of completed homology for classical homology groups. (We confine ourselves here to the non-obvious remark that all the classical homology groups at level Γ_{N}(p^{r}) can be recovered from the completed homology groups.)

Our second result concerns the action of Hecke operators on cohomology classes in low degree. Let Γ = Γ_{N}(M) for some level M. If p is prime and F is a field of characteristic p, then each of the cohomology groups H^{d}(Γ, F) admits a natural action of a commutative ring T of Hecke operators T_{ℓ,k} for 1 ≤ k ≤ N and for primes ℓ not dividing Mp. Let [c] ∈ H^{d}(Γ, F) denote an eigenclass for the action of T with eigenvalues a(ℓ, k) ∈ F. A theorem of Scholze [Sch13] (Conjecture B of [Ash92]) says that, associated to [c], there exists a continuous semisimple Galois representation ρ : G_{Q} → GL_{N}(F) unramified outside Np such that

(⋆)  ∑(−1)^{k}ℓ^{k(k−1)/2}a(ℓ, k)X^{k} = det(I − ρ(Frob_{ℓ})^{−1}X).

We determine exactly which Galois representations arise in low degree relative to N. Let ω : G_{Q} → F^{×} denote the mod-p cyclotomic character.

Theorem 1.2. Fix an integer d, and suppose that N is sufficiently large compared to d. Then for any eigenclass [c] ∈ H^{d}(Γ, F), there exists a character χ of conductor M and a Galois representation:

ρ = χ ⊗ (1 ⊕ ω ⊕ ω^{2} ⊕ … ⊕ ω^{N−1}) ,

such that ρ(Frob_{ℓ}) satisfies ⋆.

Remark 1.3. It follows from the argument that N ≥ 2d + 6 will suffice.

This theorem should be interpreted as saying that the action of Hecke on stable cohomology is trivial, which is indeed how we shall prove this theorem. Note that if [c] comes from characteristic zero, then the result follows from a theorem of Borel [Bor74], which shows in particular that the only rational cohomology in low degrees arises from the trivial automorphic representation.

Example 1.4. A special case of a construction due to Soulé [Sou79] implies that K_{22}(Z) contains an element of order 691.^{1} Associated to [c] via the Hurewicz map is a stable class [c] in H_{22}(GL_{N}(Z), Z) for all sufficiently large N. Our theorem implies that the class [c] is associated to the representation ρ := 1⊕ω⊕…⊕ω^{N−1} via ⋆. On the other hand, the existence of [c] corresponds — implicitly — to the existence of a non-semisimple Galois representation ϱ with ϱ^{ss} = 1 ⊕ ω^{11} such that the extension class in Ext^{1}(ω^{11}, 1) is unramified everywhere. Is there a generalization of Ash’s conjectures that predicts the existence of a non-semisimple Galois representations associated to Eisenstein Hecke eigenclasses?

Remark 1.5. That Theorem 1.2 (or something similar) might be true was suggested by Akshay Venkatesh in discussions with the first author. (See [Ash12] for a discussion of this conjecture and some partial results.)

1.1. Acknowledgements. The first author would like to thank Akshay Venkatesh, and both authors are grateful to Andy Putman for answering questions about his paper [Put12]. We would also like to thank Konstantin Ardakov, Kevin Buzzard, Thomas Church, Jean-François Dat, Benson Farb, Toby Gee, Florian Herzig, Akshay Venkatesh, and Simon Wadsley for interesting conversations and helpful remarks. An earlier version of this preprint from 2012 established Theorem 1.1 conditionally on Conjecture 5.6; the unconditional proof of Theorem 1.1 was found in October of 2013. The first author would especially like to thank Florian Herzig for some remarks concerning admissible representations that led to the proof of Theorem 1.1.

2. Arithmetic Manifolds

Torsion free arithmetic groups are well known to act freely and properly discontinuously on their associated symmetric spaces. This allows one to translate questions concerning the cohomology of arithmetic groups into questions about cohomology of the associated arithmetic quotients. While one can study such spaces independently of any adelic framework, it is more natural from the perspective of Hecke operators to work in this generality, and this is the approach we adopt in this paper.

2.1. Cohomology of arithmetic quotients. Let K_{∞} denote a fixed maximal compact subgroup of GL_{n}(R), and let K_{∞}^{0} denote the connected component containing the identity. One has isomorphisms K_{∞} ≃ O(n) and K_{∞}^{0} ≃ SO(n). Let 𝔸 denote the adeles of Q. For any finite set of places S, let 𝔸^{S} denote the adeles with the components at the places v|S missing, so (for example) 𝔸^{∞} denotes the finite adeles. Fix a compact open subgroup K^{ℓ} of GL_{n}(𝔸^{ℓ,∞}), let K_{ℓ} denote a compact open subgroup of GL_{n}(Z_{ℓ}), and let K = K^{ℓ}K_{ℓ}. Let

Y (K) = GL_{n}(Q)\GL_{n}(𝔸)/K_{∞}^{0}K^{ℓ}K_{ℓ}

denote the corresponding arithmetic quotient.

Assume that F is a finite field of characteristic p ≠ ℓ. We now let Π_{n,ℓ} denote the direct limit

Π_{n,ℓ} = \underrightarrow{lim}_{K_{ℓ}} H^{d}(Y (K), F).

1Indeed, K_{22}(Z) ≃ Z/691Z.

For K_{ℓ} sufficiently small, the quotient Y (K) is a manifold consisting of a finite number of connected components, all of which are K(π, 1) spaces. In particular, if K_{ℓ} is the level ℓ^{k} congruence subgroup, and K^{ℓ} is the open subgroup of tame level N, then writing the associated space as Y (K, ℓ^{k}) there is an isomorphism

H^{d}(Y (K, ℓ^{k}), F) ≃ ⨁_{A} H^{d}(Γ(ℓ^{k}), F),

where A := Q^{×}\𝔸^{∞,×}/ det(K) is a ray class group of conductor N times a power of ℓ, and Γ is the corresponding classical congruence subgroup of level N.

The module Π_{n,ℓ} is endowed tautologically with an action of GL_{n}(Q_{ℓ}) that is admissible; that is, letting G(ℓ^{k}) denote the full congruence subgroup of GL_{n}(Z_{ℓ}) of level ℓ^{k}, we have that

dim_{F} Π_{n,ℓ}^{G(ℓ^{k})} < ∞

for any k. This property records nothing more than the finite dimensionality of the cohomology groups H^{d}(Y (K, ℓ^{k}), F).

3. Stability for ℓ ≠ p

We use the following two key inputs to prove our result. The first is as follows:

Proposition 3.1. There is an isomorphism Π_{n,ℓ} → Π_{n+1,ℓ} for all n ≥ 2d + 6.

Proof. For any fixed ℓ-power congruence subgroup, this follows immediately from the main result of Charney [Cha84] (in particular, §5.4 Example (iv), p.2118). The theorem then follow by taking direct limits. □

Lemma 3.2. For n ≥ 2d + 6, the action of GL_{n}(Q_{ℓ}) on Π_{n,ℓ} is via the determinant.

Proof. By [Cha84], the action of SL_{n}(Z_{ℓ}) on H^{∗}(Γ(ℓ^{m}), F) is trivial in the stable range (see also Corollary 7 of [Ash12]), and hence the action of SL_{n}(Z_{ℓ}) on Π_{n,ℓ} is also trivial. Let V be an irreducible sub-quotient of Π_{n,ℓ}. The only admissible representations of GL_{n}(Q_{ℓ}) that are trivial on SL_{n}(Z_{ℓ}) are given by characters (the normal closure of SL_{n}(Z_{ℓ}) inside SL_{n}(Q_{ℓ}) is the entire group), and it follows that every irreducible constituent of Π_{n,ℓ} is a character. Since the action of GL_{n}(Q_{ℓ}) on the extension of any two characters still acts through the determinant, the result follows for Π_{n,ℓ}. □

Remark 3.3. One easy way to see that SL_{n}(Z_{ℓ}) must act trivially on H^{∗}(Γ(ℓ^{m}), F) in the stable range is that the latter module has fixed dimension for all n, whereas the smallest non-trivial representation of SL_{n}(Z_{ℓ}) over F has unbounded dimension as n increases.

We now sketch two further proofs of Lemma 3.2 (one incomplete). The first proof “explains” the triviality of the action of SL_{n}(Q_{ℓ}) by showing that Π_{n,ℓ} is very small in a precise way.

Proposition 3.4. For sufficiently large n, there is, for all m, an isomorphism

H^{d}(Γ(ℓ), F) ≃ H^{d}(Γ(ℓ^{m}), F).

Proof. The main result of [Cha84] identifies the stable cohomology groups H^{∗}(Γ(ℓ), F) as well as the groups H^{∗}(Γ(ℓ^{m}), F) with the cohomology of the homotopy fibre of the map BSL(Z)^{+} → BSL(Z/Nℓ)^{+} and BSL(Z)^{+} → BSL(Z/Nℓ^{m})^{+} respectively (under the assumption that F has characteristic prime to Nℓ). The natural map between these fibre sequences induces an isomorphism on F-homology, trivially for BSL(Z)^{+} and for BSL(Z/Nℓ^{m})^{+} → BSL(Z/Nℓ)^{+} by Gabber’s rigidity theorem [Gab92]. By the Zeeman comparison theorem, this implies the result for the homotopy fibre. □

More details of this argument can be found in the companion paper [Cal13] (see Remark 1.13). Alternatively, one can prove the result directly in this case, since the transfer map from Γ(ℓ^{m}) to Γ(ℓ) is an isomorphism over any ring such that ℓ is inverted (the group Γ(ℓ)/Γ(ℓ^{m}) is an ℓ-group), and hence it suffices to show that the SL_{n}(Z_{ℓ}) action on H^{d}(Γ(ℓ^{m}), F) is trivial. Lemma 3.2 is an easy consequence of Proposition 3.4. For our next argument, we begin with the following:

Proposition 3.5. Let V̂ be an irreducible admissible infinite dimensional representation of GL_{n}(Q_{ℓ}) in characteristic zero. Then the Gelfand–Kirillov dimension of V̂ is at least n − 1.

Proof. The Gelfand–Kirillov dimension of V̂ can be interpreted in two ways. On the one hand, it is that value of d ≥ 0 for which dim V^{G(ℓ^{k})} ≍ ℓ^{dk}; on the other hand, if we pull back the Harish–Chandra character to a neighbourhood of 0 in 𝔤, the Lie algebra of GL_{n}(Q_{ℓ}), via the exponential map, then a result of Howe and Harish–Chandra shows that the resulting function χ admits an expansion

χ = ∑_{O} c_{O}μ̂_{O},

where the sum ranges over the nilpotent GL_{n}(Q_{ℓ})-orbits in 𝔤^{∨}, and μ̂_{O} denotes the Fourier transform of a suitably normalized GL_{n}(Q_{ℓ})-invariant measure on O (see e.g. the introduction of [GS05], whose notation we are following here, for a discussion of these ideas); the Gelfand–

Kirillov dimension of V̂ is then also equal \frac{1}{2} max_{O | c_{O}≠0} dim O (as one sees by pairing the

characteristic function of G(ℓ^{k}) — pulled back to 𝔤 — against χ). Since V̂ has Gelfand– Kirillov dimension 0 if and only if it is finite dimensional, and since the minimal non-zero nilpotent orbit in 𝔤 has dimension 2(n − 1) (see §1.3 p. 459 and Table 1 on p. 460 of [Smi83]), the proposition follows. □

Let V be an irreducible sub-quotient of Π_{n,ℓ}. Since the cohomology of pro-ℓ groups vanishes in characteristic p ≠ ℓ, we deduce that, for all k,

dim V^{G(ℓ^{k})} ≤ dim Π_{n,ℓ}^{G(ℓ^{k})} = dim Π_{m,ℓ}^{G(ℓ^{k})},

where m = 2d + 6 is fixed, and where the equality follows from Propostion 3.1. Yet, for a fixed m, there is the trivial inequality relating the growth of cohomology to the growth of the index (up to a constant) and thus

dim Π_{m,ℓ}^{G(ℓ^{k})} ≪ [GL_{m}(Z_{ℓ}) : G(ℓ^{k})] ≪ ℓ^{km^{2}}.

The bound on invariant growth then implies that the Gelfand–Kirillov dimension of V̂ is at most m^{2}. Suppose that V lifted to a (necessarily irreducible) representation V̂ in characteristic 0. We deduce a corresponding bound for the invariants of V. By Proposition 3.5, this is a contradiction for sufficiently large n unless V̂ and thus V is one-dimensional. In general, it follows from the main theorem of Vignéras ([Vig01], p.182) that any irreducible admissible irreducible representation V of GL_{n}(Q_{ℓ}) over F lifts to a virtual representation [V̂] in characteristic 0. We expect that [V̂] may be chosen to contain a unique irreducible representation of largest Gelfand–Kirillov dimension. If this is so, then the bound of Proposition 3.5 applies also to mod-p representations of GL_{n}(Q_{ℓ}), and it would follow that every irreducible constituent of Π_{n,ℓ} is a character. Since the action of GL_{n}(Q_{ℓ}) on the extension of any two characters still acts through the determinant, this would yield a third proof of Proposition 3.2.

Remark 3.6. In a preliminary version of this paper, we made the error of assuming that any irreducible GL_{n}(Q_{ℓ})-representation V in characteristic p lifted to a characteristic zero representation V̂. We thank Jean-François Dat for pointing out to us that this is not always the case, and also for suggesting a possible approach to proving our modified expectation via induction on the ordering of partitions associated to V. Because of the availability of other arguments, however, we have not made a serious attempt to carry out this one. Although this alternate argument is therefore incomplete, it is the one that is most amenable to generalization to the case of ℓ = p (see §5), and so for psychological reasons we have presented it here.

4. Hecke Operators

Let g ∈ GL_{n}(𝔸^{∞}) be invertible. Associated to g one has the Hecke operator T (g), defined by considering the composition:

H^{•}(Y (K), F) → H^{•}(Y (gKg^{−1} ∩ K), F) → H^{•}(Y (K ∩ g^{−1}Kg), F) → H^{•}(Y (K), F),

the first map coming from the obvious inclusion, the final coming from corestriction map. The Hecke operators preserve H^{•}(Y (K), F), but not necessarily the cohomology of the connected components. Indeed, the action on the component group is via the determinant map on 𝔾(𝔸^{∞}) and the natural action of 𝔸^{∞,×} on A. The Hecke operator T_{ℓ,k} is defined by taking g to be the diagonal matrix consisting of k copies of ℓ and n − k copies of 1. The algebra T of endomorphisms generated by T_{ℓ,k} on H^{∗}(Y (K), F) for ℓ prime to the level of K generates a commutative algebra. For any such T = T (g), let ⟨T ⟩ denote the isomorphism of H^{∗}(Y (K), F) that acts by permuting the components according to the image of det(g) in A.

There are many definitions of the notion of “Eisenstein” in many different contexts. For our purposes, the following very restrictive definition is appropriate:

Definition 4.1. A cohomology class [c] ∈ H^{d}(Y (K), F) is Eisenstein if T [c] = ⟨T ⟩ deg(T)[c] for any Hecke operator T. A maximal ideal 𝔪 of T is Eisenstein if and only if 𝔪 contains T − deg(T) for all T with ⟨T ⟩ = 1 in A.

If [c] ∈ H^{d}(Γ, F) is a Hecke eigenclass such that T [c] = deg(T)[c] for all ⟨T ⟩ = 1, then [c] is necessarily Eisenstein, and moreover T [c] = χ(⟨T ⟩) deg(T)[c] for some character χ : A → F^{×} of A. By class field theory, χ corresponds to a finite order character of G_{Q} of conductor dividing M. A easy computation of deg(T_{ℓ,k}) then implies that ⋆ will be satisfied with ρ equals χ ⊗ (1 ⊕ ω ⊕ … ⊕ ω^{n−1}) if and only if [c] is Eisenstein.

Lemma 4.2. If [c] is a Hecke eigenclass such that T [c] = det(T)[c] for almost all T with ⟨T ⟩ = 1 in A, then [c] is Eisenstein.

Proof. It suffices to show that T [c] = det(T)[c] for all T of level prime to Np with ⟨T ⟩ = 1 in A. This is an immediate consequence of the Cebotarev density theorem and the fact that there exists a Galois representation associated to [c] by [Sch13]. □

4.1. Proof of Theorem 1.2. Given any eigenclass [c] ∈ H^{d}(Y (K), F), consider its image [ι(c)] in Π_{n,ℓ}. If this image is non-zero, we can determine the eigenvalues of [c] by determining those of [ι(c)]. Since the GL_{n}(Q_{ℓ})-action on Π_{n,ℓ} factors through det, the action of any g ∈ GL_{n}(Q_{ℓ}) on [ι(c)], as well as the action of the deg(T (g)) representatives in the double coset decomposition of g, is via ⟨T ⟩; hence T [ι(c)] − ⟨T ⟩ deg(T)[ι(c)] is zero in Π_{n,ℓ}. It follows that either [c] is Eisenstein or it lies in the kernel of the map H^{d}(Y (K), F) → Π_{n,ℓ}. Hence Theorem 1.2 follows by induction from the following Lemma.

Lemma 4.3. Suppose that for every non-Eisenstein maximal ideal 𝔪 of T the cohomology groups H^{i}(Y (L), F)_{𝔪} vanish for all i < d, where L = K^{ℓ}L_{ℓ} is any compact normal open subgroup L_{ℓ} ⊂ GL_{n}(Z_{ℓ}). If [c] lies in the kernel of the map H^{d}(Y (K), F) → Π_{n,ℓ}, then [c] is Eisenstein.

Proof. By assumption, the class [c] lies in the kernel of the map

H^{d}(Y (K), F) → H^{d}(Y (L), F)

for some L as above. There is a Hochschild–Serre spectral sequence:

E_{2}^{ij} = H^{i}(K/L, H^{j}(L, F)) → H^{i+j}(K, F).

This spectral sequence is compatible with Hecke operators T away from Np and ℓ. By Lemma 4.2,we may detect whether a class is Eisenstein by considering Hecke operators away from ℓ. Localizing the spectral sequence at a non-Eisenstein maximal ideal 𝔪, we deduce that E_{2}^{ij} vanishes for j < d. Hence the first d rows vanish, and we are left with an isomorphism:

H^{d}(Y (K), F)_{𝔪} = H^{0}(K/L, H^{d}(Y (L), F)_{𝔪}) = H^{0}(Y (L), F)_{𝔪}^{L/K}.

It follows that if [c] is not Eisenstein, then it is trivial. The result follows. □

This completes the proof of Theorem 1.2.

Remark 4.4. By the universal coefficient theorem, one has

Π_{n,ℓ}^{∨} := Hom(Π_{n,ℓ}, F) = \underleftarrow{lim} H_{d}(Y (K), F).

Suppose that one defines

Π_{n,ℓ}^{∨}(Z_{ℓ}) = \underleftarrow{lim} H_{d}(Y (K), Z_{ℓ}).

Then it follows from Lemma 3.2 for d and d + 1 that the action of GL_{n}(Q_{ℓ}) on Π_{n,ℓ}^{∨}(Z_{ℓ}) is via the determinant for sufficiently large n.

Remark 4.5. One can ask whether a Hecke operator T with ⟨T ⟩ trivial in A acts via the degree on the entire cohomology group H^{d}(Γ, Z). Our argument shows that, for such T, the image of T − deg(T) on H^{d}(Γ, Z) is — in the notation of [CV12] — congruence; i.e., it lies in the kernel of the map H^{d}(Γ, Z) → H^{d}(Γ(M), Z) for some M.

Remark 4.6. The main theorem and its proof remain valid, and essentially unchanged, if one replaces GL_{n}(Z) by GL_{n}(O_{F}) for any number field F.

5. Stability for ℓ = p

It is natural to wonder whether the methods of the previous sections can be extended to ℓ = p. One obvious obstruction is that the naïve notion of stability fails, even for d = 1 (as noted in the introduction). We proceed instead by using completed cohomology, whose definition for a finite field F we recall below. It is completed cohomology that is the correct analogue of the modules Π_{n,ℓ} defined above.

We fix, once and for all, a tame level M, and let Γ(p^{k}) = Γ_{n}(p^{k}) denote the principal congruence subgroup of SL_{n}(Z) of level Mp^{k}.

Definition 5.1. The completed cohomology groups H̃_{n}^{d} are defined as follows [CE12]:

H̃_{n}^{d} := \underrightarrow{lim} H^{d}(Γ(p^{k}), F)

Although this definition is formally the same as Π_{n,ℓ}, the theory when ℓ = p is quite different to the ℓ ≠ p case, due to the non-semisimple nature of representations of pro-p groups on mod-p vector spaces. However, we still prove the following result:

Theorem 5.2 (Stability of completed cohomology). For n sufficiently large, the modules H̃_{n}^{d} are finite dimensional over F and are independent of n.

Remark 5.3. We have an isomorphism H̃_{d,n} := Hom(H̃_{n}^{d}, F) ≃ \underleftarrow{lim} H_{d}(Γ(p^{k}), F). If we define

H̃_{d,n}(Z_{p}) := \underleftarrow{lim} H_{d}(Γ(p^{k}), Z_{p}),

then Theorem 5.2 also implies, by Nakayama’s lemma, that H̃_{d,n}(Z_{p}) is a finite Z_{p}-module for sufficiently large n. Hence Theorem 5.2 implies Theorem 1.1.

Remark 5.4. Theorem 5.2 is immediate when d = 0 and d = 1. Indeed, for d = 0 one has H̃_{n}^{0}(F) = F for all n, while for d = 1 one has H̃_{n}^{1}(F) = 0 for all n ≥ 3 by the congruence subgroup property.

The proof of Theorem 5.2 will occupy most of the remainder of the paper. We begin by recalling some facts concerning non-commutative Iwasawa theory. Let K = SL_{n}(Z_{p}) and let K(p^{k}) denote the principal congruence subgroups of K. By construction, the module H̃_{n}^{d} is naturally a module over the completed group ring Λ = Λ_{F_{p}} := F_{p}[[K(p)]]. If M is a Λ-module, then let M^{∨} := Hom(M, F_{p}) be the dual Λ-module. By Nakyama’s Lemma, H̃_{d,n} is finitely generated, which implies (by definition) that H̃_{n}^{d} is co-finitely generated. The ring Λ is Auslander regular [Ven02], which implies that there is a nice notion of dimension and co-dimension of finitely generated Λ-modules. One characterization of dimension for modules is given by the following result of Ardakov and Brown [AB06]:

Proposition 5.5. If M is finitely generated and M^{∨} is the co-finitely generated dual, then M has dimension at most m if and only if, as k increases without bound,

dim(M^{∨})^{K(p^{k})} ≪ p^{mk}.

The following is the natural analogue of Proposition 3.5 in this context.

Conjecture 5.6. Let M be a finitely generated Z_{p}[[K(p)]]-module. If M is infinite dimensional over F, then the dimension of M is at least n − 1.

Remark 5.7. The analogue of this conjecture for Λ_{Q_{p}} is true by Theorem A of [AW12]. In particular, if any finitely generated pure Λ_{F_{p}}-module M is the reduction of of a p-torsion free Λ_{Z_{p}} = Z_{p}[[K(p)]]-module, then the conjecture is true. Such a lifting always exists for commutative regular local rings, but it is unclear whether one should expect it to hold for Λ.

Our arguments proceed in a manner quite similar to the ℓ ≠ p case. One missing ingredient is that Proposition 3.1 is no longer valid. We use the central stability results of Putman [Put12] as a replacement.

Let (V_{n}) be a collection of representations of S_{n}. Recall from [Put12] that a sequence

[figure: …; V_{n−1}; ϕ_{n−1}; V_{n}; ϕ_{n}; V_{n+1}; …]

is centrally stable if:

(1) The ϕ_{∗} are equivariant with respect to the natural inclusions on symmetric groups,

(2) For each n, the morphism Ind_{S_{n}}^{S_{n+1}}V_{n} → V_{n+1} induced by ϕ_{n} identifies V_{n+1} with the (unique) maximal quotient of Ind_{S_{n}}^{S_{n+1}}V_{n} such that the 2-cycle (n, n + 1) acts trivially on ϕ_{n−1}(V_{n−1}).

Lemma 5.8. If Theorem 5.2 holds for j < d, then the modules H̃_{d,n} are centrally stable for sufficiently large n.

Proof. This argument is essentially taken directly from Putman [Put12], and we follow his argument closely. Putman considers a spectral sequence at a fixed level, which we may take to be Γ(p^{k}). Taking the inverse limit, one obtains a corresponding spectral sequence for completed homology. In order for this sequence to degenerate at the relevant terms on page 2, it suffices to show that the appropriate E_{i,j}^{2} are actually zero. By assumption, the modules H̃_{j,n} for j < d are finite dimensional vector spaces that are independent of n and have a trivial S_{n}-action. Thus it suffices (following Putman) to show that the (n+1)st central stability complex for the trivial sequence:

F → F → F → …

of modules for the symmetric group (starting at the trivial group) is exact. This is a very special case of Proposition 6.1 of [Put12], but can be verified directly in this case. Hence one deduces — as in Putman — that the H̃_{d,n} are centrally stable. □

Remark 5.9. It is expected that forthcoming work of Church and Ellenberg will provide a direct proof that the groups H_{d}(Γ(p^{k}), F) are centrally stable. After taking inverse limits in d, this would imply that the H̃_{d} are centrally stable without any induction hypothesis and provide another (although not entirely unrelated) proof of Lemma 5.8.

Corollary 5.10. If Theorem 5.2 holds for j < d, then the dimension of H̃_{d,n} as a Λ_{F_{p}}-module is bounded independently of n.

Proof. This is obvious for any fixed collection of n. Yet, for n sufficiently large, central stability implies that the natural map

Ind_{S_{n}}^{S_{n+1}} H̃_{d,n} → H̃_{d,n+1}

induced by the n+1 embeddings of SL(n) into SL(n+1) is surjective. Passing to cohomology, the natural maps H̃_{n+1}^{d} → H̃_{n}^{d} induce homomorphisms from the Γ_{n+1}(p^{k})-invariants of H̃_{n+1}^{d} to the Γ_{n}(p^{k})-invariants of H̃_{n}^{d}. In particular, the dimension of the Γ_{n+1}(p^{k})-invariants of H̃_{n+1}^{d} is at most n times the dimension of the Γ_{n}(p^{k})-invariants of H̃_{n}^{d}, and so it follows by Proposition 5.5 that H̃_{d,n+1} has Iwasawa dimension bounded by the Iwasawa dimension of H̃_{d,n}. □

Remark 5.11. Using Putman’s spectral sequence, one may prove unconditionally that

dim_{F} H^{d}(Γ(p^{k}), F) ≪ p^{mk}

for some constant m that does not depend on n (although the implied constant does depend on n). This leads to an alternate proof of Corollary 5.10 via the Hochschild–Serre spectral sequence and Proposition 5.5.

We prove Theorem 5.2 by induction. By the proceeding corollary, the dimension of H̃_{d,n} must be bounded for sufficiently large n. If we assume Conjecture 5.6, then we immediately deduce that this dimension is actually zero for sufficiently large n, and hence H̃_{d,n} is finite and the action of SL_{n}(Q_{p}) on H̃_{d,n} is trivial. In particular, the action of S_{n} must also be either trivial, or via the sign character, for sufficiently large n. By Lemma 5.8, we also deduce that the sequence H̃_{d,n} is centrally stable. This rules out the possibility that S_{n} acts via the sign character (for large enough n), since these do not fit into a centrally stable sequence. It follows that we must have a centrally stable sequence of trivial S_{n}-representations. Yet a centrally stable sequence of trivial modules is stable, and hence H̃_{d,n} stabilizes.

How may we upgrade this to an unconditional proof? One thing to observe is that the modules H̃_{d,n} have an action of SL_{n}(Q_{p}) that has not been exploited. For technical reasons, it is actually more convenient to have an action of the group GL_{n}(Q_{p}). In order to upgrade the action of SL_{n}(Q_{p}) to GL_{n}(Q_{p}), one should take the direct limit (in cohomology) as follows:

\underrightarrow{lim}_{K_{p}} H^{d}(Y (K), F).

From now on, we use H̃_{n}^{d} to denote this limit (rather than the limit over the cohomology of congruence subgroups that it denoted up till now).

With this new definition, H̃_{n}^{d} is cofinitely generated over F_{p}[[GL_{n}(Z_{p})]] rather than Λ_{F_{p}} = F_{p}[[SL_{n}(Z_{p})]]. It then suffices to show that these completions are eventually stable and finite over the ring F_{p}[[det(GL_{n}(Z_{p}))]] = F_{p}[[Z_{p}^{×}]]. Instead of appealing to a general conjecture concerning finitely generated Λ_{F_{p}}-modules, we only need consider such modules whose dual admits a smooth admissible action of GL_{n}(Q_{p}). In particular, it suffices to show that the only irreducible GL_{n}(Q_{p})-representations that occur as sub-quotients of these completions are either trivial or have large dimension. So, instead of proving Conjecture 5.6, it suffices to prove the following:

Conjecture 5.12. Let π be an irreducible infinite-dimensional smooth admissible representation of GL_{n}(Q_{p}), and let π^{∨} = Hom(π, F_{p}). Then the dimension of π^{∨} as a Λ_{F_{p}}-module is ≥ n − 1.

Although this conjecture is presumably easier than Conjecture 5.6, we still do not know how to prove it. Instead, we prove the following two lemmas:

Lemma 5.13. Let π be an irreducible infinite-dimensional smooth admissible representation of GL_{n}(Q_{p}), and let π^{∨} = Hom(π, F_{p}). If π is not a supercuspidal representation, then the dimension of π^{∨} as a Λ_{F_{p}}-module is ≥ n − 1.

Lemma 5.14. Any infinite-dimensional irreducible smooth admissible representation π that occurs as a sub-quotient of H̃_{n+1}^{d} is not supercuspidal.

Taken together, these lemmas serve as a replacement for either Conjecture 5.12 or Conjecture 5.6, and imply the main theorem. To recapitulate the argument, from Lemma 5.14, we deduce that every irreducible constitutent of H̃_{n+1}^{d} is not supercuspidal. From Lemma 5.13, we deduce that either H̃_{n+1}^{d} contains a representation of dimension at least n, which contradicts the uniformly bounded dimension of H̃_{n+1}^{d} for all n, or the only irreducible constituents of H̃_{n+1}^{d} are finite, in which case the action of SL_{n}(Q_{p}) is also trivial.

Proof of Lemma 5.13. Let G denote GL_{n}(Q_{p}) and let B denote a Borel of G. The Lemma is easy to verify directly for the Steinberg representation, for which the Iwasawa dimension of the dual is equal to dim(G/B). By the main theorem of [Her11], we may therefore assume that there exists a proper parabolic P with Levi M such that:

π = Ind_{P}^{G}(σ_{1} ⊗ … ⊗ σ_{r}) = Ind_{P}^{G}(σ).

The dual of such an induced representation has dimension equal to the sum of the dimension of the dual of σ and the dimension of G/P, and hence has dimension at least equal to that of G/P. It is easy enough to prove this directly, but we instead give a proof here using the chacacterization of dimension coming from Proposition 5.5 (which applies just as well to GL_{n}(Z_{p}) as it does to SL_{n}(Z_{p})).

Using the Iwasawa decomposition G = KP (where now K denotes GL_{n}(Z_{p})), we see that we may rewrite π (thought of as a K-representation) as π = Ind_{P ∩K}^{K}(σ). A short calculation then shows, since the level p^{k} congruence subgroup K(p^{k}) is normal in K, that

π^{K(p^{k})} = Ind_{P (Z/p^{k})}^{G(Z/p^{k})}(σ^{M∩K(p^{k})}).

Now for k > 0, the representation inside the induction is nonzero. Hence

dim π^{K(p^{k})} ≥ [G(Z/p^{k}) : P (Z/p^{k})] ∼ p^{mk},

where m = dim(G/P). As claimed, this gives a lower bound of dim(G/P) on the Iwasawa dimension of π.

The minimum value of dim(G/P) amongst all proper parabolics is n−1, and so the lemma follows. □

Proof of Lemma 5.14. Let M ⊂ GL_{n+1}(Q_{p}) denote the Levi subgroup Q_{p}^{×} × GL_{n}(Q_{p}) of GL_{n+1}(Q_{p}), and let P be the corresponding parabolic. Associated to P is a natural arithmetic group Γ_{P} ⊂ GL_{n+1}(Z). Let Γ_{P} (p^{k}) = Γ_{P} ∩ Γ_{n+1}(p^{k}). The group Γ_{P} (p^{k}) is a semi-direct product:

Γ_{P} (p^{k}) ≃ (p^{k}Z)^{n} ⋊ Γ_{n}(p^{k}).

Let us compute the completed cohomology groups of Γ_{P}. There is a Hochschild–Serre spectral sequence:

H^{i}(Γ_{n}(p^{k}), H^{j}((p^{k}Z)^{n}, F_{p})) ⇒ H^{i+j}(Γ_{P} (p^{k}), F_{p}).

There is an isomorphism H^{j}(p^{n}Z, F_{p}) = ∧^{j}(F_{p})^{n}, but the composition maps from Γ_{P} (p^{k}) to Γ_{P} (p^{k+1}) induce the zero map on these cohomology groups unless j = 0. Hence, taking direct limits, we get an isomorphism:

H̃_{P}^{d} := \underrightarrow{lim} H^{d}(Γ_{P} (p^{n}), F_{p}) ≃ H̃_{n}^{d}.

Omit the degree d from the notation. We therefore have maps as follows:

H̃_{n+1} → H̃_{P} → H̃_{n}.

The first map is P -equivariant, the second is an isomorphism, and the composition is the natural map. It follows that there exists a G-equivariant map

H̃_{n+1} → Ind_{P}^{G}H̃_{n}.

Note that S_{n+1} ∩ P = S_{n}, and hence the right hand side includes the induction from H̃_{n} from S_{n} to S_{n+1}. Hence, by central stability, we actually have an injection:

H̃_{n+1} → Ind_{P}^{G}H̃_{n}.

To complete the proof of the lemma, it suffices to show that no constituent of the right hand side is supercuspidal. But this also follows immediately from Herzig’s classification of irreducible admissible representations of G [Her11], since H̃_{n} is a smooth admissible Mrepresentation, where M is the Levi of P. □

5.1. Consequences for classical cohomology groups. The homology (or cohomology) of arithmetic groups with F coefficients can be recovered from completed homology via the Hochschild–Serre spectral sequence [CE12]. For example, if we take our arithmetic group to be the principal congruence subgroup Γ(p) of SL_{n}(Z), then, recalling that G(p) is the principal subgroup of SL_{n}(Z_{p}), this spectral sequence has the form

E_{2}^{i,j} := H^{i}(G(p), H̃^{j}) ⟹ H^{i+j}(Γ(p), F).

As noted above, H̃^{0} = F, and Theorem 5.2 implies that each H̃^{j} is finite with trivial G(p)- action. Hence Theorem 5.2 implies that the cohomology group H^{d}(Γ(p), F) has a filtration consisting of three types of classes: those arising via pullback from H^{d}(G(p), F), those arising via (higher) transgressions from H^{j} of G(p) for j < d, and those arising from H̃^{d} (which do not depend on n). In this optic, the notion of representation stability developed by Church and Farb [CF11] is then seen (in this context) to have its origin in the mod-p cohomology of p-adic Lie groups, rather than in the properties of arithmetic groups.

One interpretation of Theorem 5.2 is that the ℓ = p and ℓ ≠ p theories after completion are quite similar. The difference in phenomenology between these two cases is then a consequence of the vanishing of H^{i}(G(ℓ^{k}), F) when k ≥ 1 and the chararacteristic p of F is not ℓ. Another way to state these results is to say that the only irreducible GL_{n}(𝔸^{∞})-representations occurring inside cohomology in sufficiently small degree are one-dimensional.

5.2. Relation to K-theory. Even knowing that H̃_{d} is finite, it is not at all apparent how to compute it. One might wonder whether there is some interpretation of H̃_{d} in terms of K-theory (some related speculation along these lines occurs in §8.3 of [CV12]). This question is taken up in the companion paper [Cal13].

References

[AB06] K. Ardakov and K. A. Brown, Ring-theoretic properties of Iwasawa algebras: a survey, Doc. Math. (2006), no. Extra Vol., 7–33. MR 2290583 (2007k:11185)

[Ash92] Avner Ash, Galois representations attached to mod p cohomology of GL(n, Z), Duke Math. J. 65 (1992), no. 2, 235–255. MR 1150586 (93c:11036)

[Ash12] , Hecke operators and the stable homology of GL(n), Comm. Algebra 40 (2012), no. 4, 1369– 1387. MR 2912991 [AW12] Konstantin Ardakov and S. J. Wadsley, On irreducible representations of compact p-adic analytic

groups, preprint, 2012.

[BLS64] H. Bass, M. Lazard, and J.-P. Serre, Sous-groupes d’indice fini dans SL(n, Z), Bull. Amer. Math. Soc. 70 (1964), 385–392. MR 0161913 (28 #5117)

[Bor74] Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 0387496 (52 #8338)

[Cal13] Frank Calegari, The stable homology of congruence subgroups, preprint, 2013.

[CE09] Frank Calegari and Matthew Emerton, Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms, Ann. of Math. (2) 170 (2009), no. 3, 1437–1446. MR 2600878 (2011c:22032)

[CE12] , Completed cohomology—a survey, Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser., vol. 393, Cambridge Univ. Press, Cambridge, 2012, pp. 239– 257. MR 2905536

[CF11] Thomas Church and Benson Farb, Representation theory and homological stability, preprint, 2011.

[Cha80] Ruth M. Charney, Homology stability for GL_{n} of a Dedekind domain, Invent. Math. 56 (1980), no. 1, 1–17. MR 557579 (81h:18010)

[Cha84] Ruth Charney, On the problem of homology stability for congruence subgroups, Comm. Algebra 12 (1984), no. 17-18, 2081–2123. MR 747219 (86b:20053)

[CV12] Frank Calegari and Akshay Venkatesh, Towards a Torsion Langlands correspondence, draft, 2012.

[Eme06] Matthew Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math. 164 (2006), no. 1, 1–84. MR 2207783 (2007k:22018)

[Gab92] Ofer Gabber, K-theory of Henselian local rings and Henselian pairs, Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502 (93c:19005)

[GS05] Wee Teck Gan and Gordan Savin, On minimal representations definitions and properties, Represent. Theory 9 (2005), 46–93 (electronic). MR 2123125 (2006a:22015)

[Her11] Florian Herzig, The classification of irreducible admissible mod p representations of a p-adic GL_{n}, Invent. Math. 186 (2011), no. 2, 373–434. MR 2845621

[Maa79] Hendrik Maazen, Stabilité de l’homologie de GL_{n}, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 15, 707–708. MR 532394 (80f:20056)

[Men65] Jens L. Mennicke, Finite factor groups of the unimodular group, Ann. of Math. (2) 81 (1965), 31–37. MR 0171856 (30 #2083)

[Put12] Andy Putman, Stability in the homology of congruence subgroups, preprint, 2012.

[Sch13] Peter Scholze, On torsion in the cohomology of locally symmetric varieties, preprint, 2013.

[Smi83] S. P. Smith, Krull dimension of factor rings of the enveloping algebra of a semisimple Lie algebra, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 459–466. MR 698349 (85a:17007)

[Sou79] C. Soulé, K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), no. 3, 251–295. MR 553999 (81i:12016)

[Ven02] Otmar Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 3, 271–311. MR 1924402 (2004h:16029)

[Vig01] Marie-France Vignéras, Correspondance de Langlands semi-simple pour GL(n, F) modulo ℓ ≠ p, Invent. Math. 144 (2001), no. 1, 177–223. MR 1821157 (2003f:11182)