Bounds for Multiplicities of Unitary Representations of
Cohomological Type in Spaces of Cusp Forms
Frank Calegari Matthew Emerton
November 4, 2018
1 Introduction
Let G∞ be a semisimple real Lie group with unitary dual Ĝ∞. The goal of this note is to produce
new upper bounds for the multiplicities with which representations π ∈ Ĝ∞ of cohomological type
appear in certain spaces of cusp forms on G∞.
More precisely, we suppose that G∞ := G(R ⊗Q F ) for some connected semisimple linear
algebraic group G over a number field F . Let K∞ be a maximal compact subgroup of G∞. We
fix an embedding G →֒ GLN for some N , and for any ideal q of OF , we let G(q) denote the
intersection of G∞ with the congruence subgroup of GLN (OF ) of full level q. We also fix an
arithmetic lattice Γ in G∞ (i.e. a subgroup commensurable with the congruence subgroups G(q))
and write Γ(q) := Γ
G(q). For any π ∈ Ĝ∞, let m(π,Γ(q)) denote the multiplicity with which
π occurs in the decomposition of the regular representation of G∞ on L
cusp(Γ(q)\G∞). Let V (q)
denote the volume of the arithmetic quotient Γ(q)\G∞.
In terms of this notation, we may state our main results.
Theorem 1.1. Let p be a prime ideal in OF . Let π ∈ Ĝ∞ be of cohomological type. Suppose
either that G∞ does not admit discrete series, or, if G∞ admits discrete series, that π contributes
to cohomology in degrees other than
dim(G∞/K∞). Then
m(π,Γ(pk)) ≪ V (pk)1−1/ dim(G∞)
as k → ∞, with the implied constant depending on Γ and p.
Theorem 1.2. Let p be a prime ideal in OF . Let W be a finite-dimensional representation of G∞,
and let VW,k denote the local system on Γ(p
k)\G∞ induced by W (assuming that k is taken large
enough for Γ(pk) to be torsion free). Let n ≥ 0, and if G∞ admits discrete series, then suppose
furthermore that n 6=
dim(G∞/K∞). Then
dimHn
Γ(pk)\G∞,VW,k
≪ V (pk)1−1/ dim(G∞)
as k → ∞, with the implied constant depending on Γ and p.
These two theorems are evidently closely related, in light of the results of [5], which show that
Γ(pk)\G∞,VW,k
may be computed in terms of automorphic forms.
http://arxiv.org/abs/0704.0662v1
In the remainder of this introduction, we discuss the relation of Theorem 1.1 to prior results in
this direction before briefly describing the main ingredients in the proof of the two theorems.
DeGeorge and Wallach [3] established general upper bounds for m(π,Γ) in the case where Γ is
cocompact. In particular (ibid, Corollary 3.2), they showed that
m(π,Γ) ≤
|φ(g)|2dg
vol(Γ\G∞), (1.1)
where φ(g) = 〈π(g)v, v〉 is a matrix coefficient, and B is the preimage in Γ\G∞ of a ball in
Γ\G∞/K∞ of radius equal to the injectivity radius of Γ\G∞/K∞. Suppose, however, that π is
not a discrete series. In particular, the corresponding matrix coefficients of π are then not square
integrable. If Γ(q) denotes the mod q congruence subgroup of Γ, then inj.rad(Γ(q)\G∞) → ∞ as
NF/Q(q) → ∞, and thus, the formula of DeGeorge–Wallach implies that
NF/Q(q)→∞
V (q)−1 ·m(π,Γ(q)) = 0. (1.2)
For non-cocompact Γ, an analogous result was established by Savin [10].
It is natural to try to improve this result so as to obtain an estimate on the rate of decay in (1.2)
as NF/Q(q) → ∞. If π is non-tempered, then (1.1) itself implies an estimate of the form
m(π,Γ(q)) ≪ V (q)1−µ (1.3)
for some µ > 0. (See [9], Lemma 1 and displayed equation (6).) In fact Sarnak and Xue in [9] have
conjectured an inequality of the following form (in the case of cocompact Γ):
Conjecture 1.3 (Sarnak–Xue). For π ∈ Ĝ∞ fixed,
m(π,Γ(q)) ≪ V (q)(2/p(π))+ǫ, for all ǫ > 0,
where p(π) is the infimum over p ≥ 2 such that the K-finite matrix coefficients of π are in Lp(G∞).
Sarnak and Xue proved their conjecture for arithmetic lattices in SL2(R) and SL2(C), obtaining
partial results in the direction of this conjecture for SU(2, 1). Note, however, that their conjecture
is non trivial only for non-tempered representations, since for tempered representations, p(π) = 2.
In particular, in the tempered but non-discrete series case, Conjecture 1.3 is weaker than the known
result (1.2).
In Theorem 1.1, we restrict our attention to congruence covers of the form Γ(pk) for the fixed
prime p. For such covers we obtain a quantitative improvement of (1.2) even in the case of tempered
representations (at least for those of cohomological type; note that non-discrete series tempered
representations of cohomological type exist precisely when G∞ admits no discrete series – see [1],
Thm. 5.1, p. 101). For such representations, our result provides the first general bound of the
form (1.3) for any µ > 0.
As we already noted, our two main theorems are closely related. Indeed, Theorem 1.1 is an
easy corollary of Theorem 1.2 (see the end of Section 3 below), and most of our efforts will be
concentrated on establishing the latter result.
When studying the Betti numbers of arithmetic quotients of symmetric spaces, it is natural to
try to use tools such as Euler characteristics and the Lefschetz trace formula. When applied to
analyzing contributions from the discrete series, such methods tend to be very powerful; for example,
the (g, k)-cohomology of a discrete series representation is concentrated in a single dimension [1],
and so no cancellations occur when taking alternating sums. However, in other situations, these
methods can be useless. For example, if π is tempered but not discrete series, then the Euler
characteristic of its (g, k)-cohomology vanishes [1]. Similarly, in situations where the symmetric
space is a real manifold of odd dimension n, Poincare duality leads to cancellations in the natural
(−1)k dim(Hk). One is thus forced to find different techniques. The proof of Theorem 1.2
takes as input the inequality (1.2) of [3] and [10] and a spectral sequence from [4], proceeding via
a bootstrapping argument relying on non-commutative Iwasawa theory.
Acknowledgments. The second author would like to thank Peter Sarnak for a very stimulating
conversation on the subject of this note.
2 Iwasawa Theory
Let G ⊆ GLN (Zp) be an analytic pro-p group. Let Gk = G ∩ (1 + p
kMN (Zp)) ⊆ GLN (Zp). The
subgroups Gk form a fundamental set of open neighbourhoods of the identity in G, and moreover,
for large k, there exists a constant c such that [G : Gk] = c · p
dk, where d = dim(G).
Fix a finite extension E of Qp with ring of integers OE . Write Λ = OE [[G]] and ΛE = E⊗OE Λ.
The module theory of Λ falls under the rubric of Iwasawa theory. A fundamental result of Lazard [7]
states that Λ is Noetherian; the same is thus true of the ring ΛE . The rings Λ and ΛE are non-
commutative domains admitting a common field of fractions which we will denote by L . Thus,
L is a division ring which contains Λ and ΛE and is flat over each of them (on both sides). If
M is a finitely generated left Λ-module (resp. ΛE-module), then L ⊗Λ M (resp. L ⊗ΛE M) is
a finite-dimensional left L -vector space; we define the rank of M to be the L -dimension of this
vector space. Note that rank is additive in short exact sequences of finitely generated Λ-modules
(resp. ΛE-modules), by virtue of the flatness of L over Λ and ΛE .
Recall that a continuous representation of G on an E-Banach space V is called admissible if its
topological E-dual V ′ (which is naturally a ΛE-module) is finitely generated over ΛE . (See [11]; a
key point is that since ΛE is Noetherian, the category of admissible continuous G-representations
is abelian. Indeed, passing to topological duals yields an anti-equivalence with the abelian category
of finitely generated ΛE-modules.) We define the corank of an admissible G-representation to be
the rank of the finitely generated ΛE-module V
A coadmissible G-representation V is not determined by the collection of subspaces of invariants
V Gk (r ≥ 1). However, the following result (Theorem 1.10 of Harris [6]) shows that its corank is
so determined.
Theorem 2.1 (Harris). Let V be an E-Banach space equipped with an admissible continuous G-
representation and let d = dim(G). Then as k → ∞,
dimE V
Gk = r · [G : Gk] +O(p
(d−1)k) = r · c · pdk +O(p(d−1)k),
where r is the corank of V and c depends only on G.
Using this result, we may obtain bounds on the dimensions of the continuous cohomology groups
H i(Gk, V ) in terms of k for admissible continuous G-representations V . (Let us remark that the
continuous Gk-cohomology on the category of admissible continuous G-representations may also be
computed as the right derived functors of the functor of Gk-invariants; see Prop. 1.1.3 of [4].)
Lemma 2.2. Let V be an admissible continuous G-representation. For each i ≥ 1,
dimE H
i(Gk, V ) ≪ p
(d−1)k,
as k → ∞.
Proof. Let C := C(G,E) denote the Banach space of continuous E-valued functions on G, equipped
with the right regular G-action. The module C has corank one (indeed, it is cofree – its topological
dual is free of rank one over ΛE). Moreover, C is injective in the abelian category of admissible
G-representations and is therefore acyclic. If V is an admissible continuous G-representation, then
there exists an exact sequence
0 −→ V −→ Cn −→ W −→ 0
of admissible continuous G-representations for some integer n ≥ 0. Since C is acyclic, from the long
exact sequence of cohomology we obtain the following:
0 −→ V Gk −→ (CGk)n −→ WGk −→ H1(Gk, V ) −→ 0, (2.1)
H i(Gk, V ) ≃ H
i−1(Gk,W ), i ≥ 2. (2.2)
The lemma for i = 1 follows from a consideration of (2.1), taking into account Theorem 2.1 and the
fact that corank of W is equal to n minus the corank of V (since corank is additive in short exact
sequences). We now proceed by induction on i. Assume the result for i ≤ m and all admissible
continuous representations, in particular for W . The result for i = m+1 then follows directly from
the isomorphism (2.2). This completes the proof.
3 Cohomology of Arithmetic Quotients of Symmetric Spaces
We now return to the situation considered in the introduction and use the notation introduced there.
In particular, we fix a connected semisimple linear group G over F , an embedding G →֒ GLN over
F , an arithmetic lattice Γ of the associated real group G∞, and a prime p of F .
If we write G := lim
Γ/Γ(pk), then G is a compact open subgroup of the p-adic Lie group G(Fp)
(where Fp denotes the completion of F at p); alternatively, we may define G to be the closure of Γ in
G(Fp). If we replace Γ by Γ(p
k) for some sufficiently large value of k (i.e. discarding finitely many
initial terms in the descending sequence of lattices Γ(pk)), then G will be pro-p and hence, will be
an analytic pro-p-group. Note that Γ is a dense subgroup of G. Let e and f denote respectively
the ramification and inertial indices of p in F (so that [Fp : Qp] = ef). For each k ≥ 0, write Gk to
denote the closure of Γ(pek) in G. Alternatively, if we consider the embedding
G(Fp) →֒ GLN (Fp) →֒ GLefN (Qp),
then Gk = G ∩ (1 + p
kMefN (Zp)); thus, our notation is compatible with that of the preceding
section. We let d denote the dimension of G; note that d =
[F :Q]
· dim(G∞).
For each k ≥ 0, we write
Yk := Γ(p
ek)\G∞/K∞.
There is a natural action of G on Yk through its quotient G/Gk (∼= Γ/Γ(p
ek)), which is compatible
with the projections Yk → Yk′ for 0 ≤ k
′ ≤ k.
Fix a finite-dimensional representation W of G over E, and let W0 denote a G-invariant OE-
lattice in W . Let Vk denote the local system of free finite rank OE-modules on Yk associated to
W0, and denote by Vk the pull-back of V0 to Yk for any k ≥ 0. If 0 ≤ k
′ ≤ k, then the sheaf Vk on
Yk is naturally isomorphic to the pull-back of the sheaf Vk′ on Yk′ under the projection Yk → Yk′ .
In particular the sheaf Vk is G/Gk-equivariant.
Recall the following definitions from from [4], p. 21:
H̃n(V) := lim
Hj(Yk,Vk/p
s), H̃n(V)E := E ⊗OE H̃
n(V).
Each H̃n(V) is a p-adically completeOE-module, equipped with a leftG-action in a natural way, and
hence, each H̃n(V)E has a natural structure of E-Banach space and is equipped with a continuous
left G-action. In fact, they are admissible continuous representations of G ([4], Thm. 2.1.5 (i)), and
in particular, Theorem 2.1 and Lemma 2.2 apply to them. (Note that the results of [4] are stated
in the adèlic language. We leave it to the reader to make the easy translation to the more classical
language we are using in this paper.)
The following result, which is Theorem 2.1.5 (ii) of [4], p. 22, is a “control theorem” relating
Gk invariants in H̃
E to the classical cohomology classes H
j(Yk,Vk)⊗ E.
Theorem 3.1. Fix an integer k. There is a spectral sequence
2 (Yk) = H
i(Gk, H̃
j(V)E) =⇒ H
i+j(Yk,Vk)E .
One should view this spectral sequence as a version of the Hochschild-Serre spectral sequence
“compatible in the G-tower.”
Theorem 3.2. For any n ≥ 0, if rn denotes the corank of H̃
n(V), then
dimE H
n(Yk,Vk)E = rn · c · p
dk +O(p(d−1)k)
as k → ∞. (Here c denotes the constant appearing in the statement of Theorem 2.1; it depends
only on G.)
Proof. For each i, j ≥ 0 and l ≥ 2, let E
l (Yk) denote the terms in the spectral sequence of
Theorem 3.1. Since H̃j(V) is admissible, Lemma 2.2 implies that dimE H
i(Gk, H̃
j(V)E) ≪ p
(d−1)k,
and thus, dimE E
l (Yk) ≪ p
(d−1)k, as k → ∞ (since E
l (Yk) is a subquotient of E
2 (Yk) :=
H i(Gk, H̃
j(V)E) for l ≥ 2). Theorem 2.1 shows that
dimE E
2 (Yk) = dimE H̃
E = rn · c · p
dk +O(p(d−1)k).
On the other hand, since the spectral sequence of Theorem 3.1 is an upper right quadrant exact
sequence, E
∞ is obtained by taking finitely many successive kernels of differentials dl to E
i+l,j−l+1
which all have order ≪ p(d−1)k by the first part of our argument. Thus,
dimE E
= rn · c · p
dk +O(p(d−1)k).
Since Hn(Yk,Vk)E admits a finite length filtration whose associated graded pieces are isomorphic
∞ for i+ j = n, we conclude that dimE H
n(Yk,Vk)E = rn · c · p
dk +O(p(d−1)k), as claimed.
The following lemma quantifies the precise relationship between multiplicities and the dimen-
sions of cohomology groups that we will require to deduce Theorem 1.1 from Theorem 1.2.
Lemma 3.3. Fix a cohomological degree n, and let S denote the set of isomorphism classes [π] of
π ∈ Ĝ∞ that contribute to cohomology with coefficients in V in degree n. Then
[π]∈S
m(π,Gk) ≍ dimE H
cusp(Yk,Vk)E .
Proof. Since the set S is finite, there is an integer d ≥ 1 so that
1 ≤ dimHn(g, k;π ⊗W ) ≤ d
for each isomorphism class [π] ∈ S. This implies that
[π]∈S
m(π,Gk) ≤ dimE H
cusp(Yk,Vk)E ≤ d
[π]∈S
m(π,Gk)
for each k ≥ 0.
We can now prove our main result.
Theorem 3.4. Let n ≥ 0, and suppose that either G∞ does not admit discrete series or else that
dim(G∞/K∞). Then
dimE H
n(Yk,Vk)E ≪ p
(d−1)k
as k → ∞, for all n ≥ 0.
Proof. In the case when G∞ admits discrete series, recall that these contribute to cohomology
only in the dimension
dim(G∞/K∞) ([1], Thm. 5.1, p. 101). Thus, under the assumptions
of the theorem, there is no contribution from the discrete series to Hncusp(Yk,Vk). The inequal-
ity (1.2) of [3] and [10], together with Lemma 3.3 and the main result of [8] (which states that
Hn(Yk,Vk)/H
cusp(Yk,Vk)
= o(pdk)), thus shows that dimE H
n(Yk,Vk)E = o(p
dk) as k → ∞,
for all n ≥ 0. From Theorem 3.2, we then infer that each H̃n(V) has corank 0. Another application
of the same theorem now gives our result.
Note that V (pek) ∼ [G : Gk] ∼ c · p
dk; thus Theorem 3.4 implies Theorem 1.2 since
dim(G) ≤ dim(G∞). (3.1)
Theorem 1.2 and Lemma 3.3 together imply Theorem 1.1.
Remark 3.5. We have equality in (3.1) precisely when p is the unique prime lying over p in OF . If
there is more than one prime lying over p, then dim(G) is strictly less than dim(G∞), and we obtain
a corresponding improvement in the bounds of Theorems 1.1 and 1.2, namely (in the notation of
their statements), that
m(π,Γ(pk)) and dimE H
i(Y (pk),VW,k)E ≪ V (p
n)1−1/ dim(G)
(where, as we noted above, dim(G) =
[F :Q]
· dim(G∞), with e and f being the ramification and
inertial index of p respectively).
Example/Question 3.6. Let F/Q be an imaginary quadratic field, and let G = SL2/F . The
corresponding symmetric space G∞/K∞ = SL2(C)/SU(2) is a real hyperbolic three space H,
and the quotients Y are commensurable with the Bianchi manifolds H/PGL2(OK). Choose a
local system V0 associated to some finite-dimensional representation W of G∞ = GL2(C) and a
congruence subgroup Γ. Assume that p = pp splits in OF , and apply Theorem 3.4 to the p-power
tower. We obtain the inequality
H1cusp(Yk,Vk) ≪ p
as k → ∞. It is natural to ask how tight this inequality is.
The main result of Calegari–Dunfield [2] shows that there exists at least one (F,Γ, p) for which
H1cusp(Yk,C) = 0
for all k. On the other hand, if there exists at least one newform on Γ(pk) for some k, then a
consideration of the associated oldforms shows that
H1cusp(Yk,Vk) ≫ p
as k → ∞. Are there situations in which this lower bound gives the true rate of growth?
Remark 3.7. Our results are most interesting in the case when G∞ does not admit any discrete
series, since, as we noted in the introduction, in this case (and only in this case), G∞ admits
(non-discrete series) tempered representations of cohomological type.
On the other hand, Theorem 3.2 does have a consequence in the case when G∞ admits discrete
series which may be of some interest. Recall the following result from [10] (established in [3] in the
cocompact case): if π ∈ Ĝ∞ lies in the discrete series, then
m(Γ(πk), π) = d(π)V (pk) + o(V (pk)) (3.2)
as k → ∞. Fix a finite-dimensional representation W of G∞, and let Ĝ∞(W )d denote the subset
of Ĝ∞ consisting of discrete series representations that contribute to cohomology with coefficients
in W . Summing over all π ∈ Ĝ∞(W )d, we obtain the formula
|Ĝ∞(W )d|
π∈ bG∞(W )d
m(Γ(πk), π) = d(π)V (pk) + o(V (pk)). (3.3)
(A result first proved in [8].) The following result provides an improvement in the error term
of (3.3).
Theorem 3.8. There exists µ > 0 such that
|Ĝ∞(W )d|
π∈ bG∞(W )d
m(Γ(πk), π) = d(π)V (pk) +O(V (pk)1−µ).
Proof. Let n =
dim(G∞/K∞). As already noted, it follows from [1], (Thm. 5.1, p. 101), that
all non-discrete series contributions to Hncusp(Yk,Vk) are non-tempered. The same result shows
that each discrete series has one-dimensional (g, k)-cohomology in dimension n. As we recalled in
the introduction, the multiplicity of any non-tempered representations is bounded by V (pk)1−µ for
some µ > 0 [9], and thus, Theorem 3.2 and (the proof of) Lemma 3.3 show that
|Ĝ∞(W )d|
π∈ bG∞(W )d
m(Γ(πk), π) = C · V (pk) +O(V (pk)1−µ).
Comparing this formula with (3.3) yields the theorem.
Question 3.9. Does the result of Theorem 3.8 hold term-by-term? That is, does (3.2) admit an
improvement of the form
m(Γ(πk), π)
= d(π)V (pk) +O(V (pk)1−µ)
for some µ > 0?
References
[1] Borel, A.; Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive
groups. Annals of Mathematics Studies, 94. Princeton University Press, Princeton, N.J.; University of
Tokyo Press, Tokyo, 1980. xvii+388 pp.
[2] Calegari, F.; Dunfield, N. Automorphic forms and rational homology 3-spheres. Geom. Topol. 10 (2006),
295–329.
[3] DeGeorge, D.; Wallach, N. Limit formulas for multiplicities in L2(Γ\G). Ann. Math. (2) 107 (1978),
no. 1, 133–150.
[4] Emerton, M. On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms.
Invent. Math. 164 (2006), no. 1, 1–84.
[5] Franke, J. Harmonic Analysis in Weighted L2-spaces. Ann. Sci. cole Norm. Sup. (4) 31 (1998), no. 2,
181–279.
[6] Harris, M. Correction to: “p-adic representations arising from descent on abelian varieties” [Compositio
Math. 39 (1979), no. 2, 177–245]. Compositio Math. 121 (2000), no. 1, 105–108.
[7] Lazard, M. Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965).
[8] Rohlfs, J.; Speh, B. On limit multiplicities of representations with cohomology in the cuspidal spectrum.
Duke Math. J. 55 (1987), no. 1, 199–211.
[9] Sarnak, P.; Xue, X. Bounds for multiplicities of automorphic representations. Duke Math. J. 64 (1991),
no. 1, 207–227.
[10] Savin, G. Limit multiplicities of cusp forms. Invent. Math. 95 (1989), no. 1, 149–159.
[11] Schneider, P.; Teitelbaum, J. Banach space representations and Iwasawa theory, Israel. J. Math. 127
(2002), 359–380.
	Introduction
	Iwasawa Theory
	Cohomology of Arithmetic Quotients of Symmetric Spaces

  Let $\Goo$ be a semisimple real Lie group with unitary dual $\Ghat$. The goal
of this note is to produce new upper bounds for the multiplicities with which
representations $\pi \in \Ghat$ of cohomological type appear in certain spaces
of cusp forms on $\Goo$.

Introduction
Let G∞ be a semisimple real Lie group with unitary dual Ĝ∞. The goal of this note is to produce
new upper bounds for the multiplicities with which representations π ∈ Ĝ∞ of cohomological type
appear in certain spaces of cusp forms on G∞.
More precisely, we suppose that G∞ := G(R ⊗Q F ) for some connected semisimple linear
algebraic group G over a number field F . Let K∞ be a maximal compact subgroup of G∞. We
fix an embedding G →֒ GLN for some N , and for any ideal q of OF , we let G(q) denote the
intersection of G∞ with the congruence subgroup of GLN (OF ) of full level q. We also fix an
arithmetic lattice Γ in G∞ (i.e. a subgroup commensurable with the congruence subgroups G(q))
and write Γ(q) := Γ
G(q). For any π ∈ Ĝ∞, let m(π,Γ(q)) denote the multiplicity with which
π occurs in the decomposition of the regular representation of G∞ on L
cusp(Γ(q)\G∞). Let V (q)
denote the volume of the arithmetic quotient Γ(q)\G∞.
In terms of this notation, we may state our main results.
Theorem 1.1. Let p be a prime ideal in OF . Let π ∈ Ĝ∞ be of cohomological type. Suppose
either that G∞ does not admit discrete series, or, if G∞ admits discrete series, that π contributes
to cohomology in degrees other than
dim(G∞/K∞). Then
m(π,Γ(pk)) ≪ V (pk)1−1/ dim(G∞)
as k → ∞, with the implied constant depending on Γ and p.
Theorem 1.2. Let p be a prime ideal in OF . Let W be a finite-dimensional representation of G∞,
and let VW,k denote the local system on Γ(p
k)\G∞ induced by W (assuming that k is taken large
enough for Γ(pk) to be torsion free). Let n ≥ 0, and if G∞ admits discrete series, then suppose
furthermore that n 6=
dim(G∞/K∞). Then
dimHn
Γ(pk)\G∞,VW,k
≪ V (pk)1−1/ dim(G∞)
as k → ∞, with the implied constant depending on Γ and p.
These two theorems are evidently closely related, in light of the results of [5], which show that
Γ(pk)\G∞,VW,k
may be computed in terms of automorphic forms.
http://arxiv.org/abs/0704.0662v1
In the remainder of this introduction, we discuss the relation of Theorem 1.1 to prior results in
this direction before briefly describing the main ingredients in the proof of the two theorems.
DeGeorge and Wallach [3] established general upper bounds for m(π,Γ) in the case where Γ is
cocompact. In particular (ibid, Corollary 3.2), they showed that
m(π,Γ) ≤
|φ(g)|2dg
vol(Γ\G∞), (1.1)
where φ(g) = 〈π(g)v, v〉 is a matrix coefficient, and B is the preimage in Γ\G∞ of a ball in
Γ\G∞/K∞ of radius equal to the injectivity radius of Γ\G∞/K∞. Suppose, however, that π is
not a discrete series. In particular, the corresponding matrix coefficients of π are then not square
integrable. If Γ(q) denotes the mod q congruence subgroup of Γ, then inj.rad(Γ(q)\G∞) → ∞ as
NF/Q(q) → ∞, and thus, the formula of DeGeorge–Wallach implies that
NF/Q(q)→∞
V (q)−1 ·m(π,Γ(q)) = 0. (1.2)
For non-cocompact Γ, an analogous result was established by Savin [10].
It is natural to try to improve this result so as to obtain an estimate on the rate of decay in (1.2)
as NF/Q(q) → ∞. If π is non-tempered, then (1.1) itself implies an estimate of the form
m(π,Γ(q)) ≪ V (q)1−µ (1.3)
for some µ > 0. (See [9], Lemma 1 and displayed equation (6).) In fact Sarnak and Xue in [9] have
conjectured an inequality of the following form (in the case of cocompact Γ):
Conjecture 1.3 (Sarnak–Xue). For π ∈ Ĝ∞ fixed,
m(π,Γ(q)) ≪ V (q)(2/p(π))+ǫ, for all ǫ > 0,
where p(π) is the infimum over p ≥ 2 such that the K-finite matrix coefficients of π are in Lp(G∞).
Sarnak and Xue proved their conjecture for arithmetic lattices in SL2(R) and SL2(C), obtaining
partial results in the direction of this conjecture for SU(2, 1). Note, however, that their conjecture
is non trivial only for non-tempered representations, since for tempered representations, p(π) = 2.
In particular, in the tempered but non-discrete series case, Conjecture 1.3 is weaker than the known
result (1.2).
In Theorem 1.1, we restrict our attention to congruence covers of the form Γ(pk) for the fixed
prime p. For such covers we obtain a quantitative improvement of (1.2) even in the case of tempered
representations (at least for those of cohomological type; note that non-discrete series tempered
representations of cohomological type exist precisely when G∞ admits no discrete series – see [1],
Thm. 5.1, p. 101). For such representations, our result provides the first general bound of the
form (1.3) for any µ > 0.
As we already noted, our two main theorems are closely related. Indeed, Theorem 1.1 is an
easy corollary of Theorem 1.2 (see the end of Section 3 below), and most of our efforts will be
concentrated on establishing the latter result.
When studying the Betti numbers of arithmetic quotients of symmetric spaces, it is natural to
try to use tools such as Euler characteristics and the Lefschetz trace formula. When applied to
analyzing contributions from the discrete series, such methods tend to be very powerful; for example,
the (g, k)-cohomology of a discrete series representation is concentrated in a single dimension [1],
and so no cancellations occur when taking alternating sums. However, in other situations, these
methods can be useless. For example, if π is tempered but not discrete series, then the Euler
characteristic of its (g, k)-cohomology vanishes [1]. Similarly, in situations where the symmetric
space is a real manifold of odd dimension n, Poincare duality leads to cancellations in the natural
(−1)k dim(Hk). One is thus forced to find different techniques. The proof of Theorem 1.2
takes as input the inequality (1.2) of [3] and [10] and a spectral sequence from [4], proceeding via
a bootstrapping argument relying on non-commutative Iwasawa theory.
Acknowledgments. The second author would like to thank Peter Sarnak for a very stimulating
conversation on the subject of this note.
2 Iwasawa Theory
Let G ⊆ GLN (Zp) be an analytic pro-p group. Let Gk = G ∩ (1 + p
kMN (Zp)) ⊆ GLN (Zp). The
subgroups Gk form a fundamental set of open neighbourhoods of the identity in G, and moreover,
for large k, there exists a constant c such that [G : Gk] = c · p
dk, where d = dim(G).
Fix a finite extension E of Qp with ring of integers OE . Write Λ = OE [[G]] and ΛE = E⊗OE Λ.
The module theory of Λ falls under the rubric of Iwasawa theory. A fundamental result of Lazard [7]
states that Λ is Noetherian; the same is thus true of the ring ΛE . The rings Λ and ΛE are non-
commutative domains admitting a common field of fractions which we will denote by L . Thus,
L is a division ring which contains Λ and ΛE and is flat over each of them (on both sides). If
M is a finitely generated left Λ-module (resp. ΛE-module), then L ⊗Λ M (resp. L ⊗ΛE M) is
a finite-dimensional left L -vector space; we define the rank of M to be the L -dimension of this
vector space. Note that rank is additive in short exact sequences of finitely generated Λ-modules
(resp. ΛE-modules), by virtue of the flatness of L over Λ and ΛE .
Recall that a continuous representation of G on an E-Banach space V is called admissible if its
topological E-dual V ′ (which is naturally a ΛE-module) is finitely generated over ΛE . (See [11]; a
key point is that since ΛE is Noetherian, the category of admissible continuous G-representations
is abelian. Indeed, passing to topological duals yields an anti-equivalence with the abelian category
of finitely generated ΛE-modules.) We define the corank of an admissible G-representation to be
the rank of the finitely generated ΛE-module V
A coadmissible G-representation V is not determined by the collection of subspaces of invariants
V Gk (r ≥ 1). However, the following result (Theorem 1.10 of Harris [6]) shows that its corank is
so determined.
Theorem 2.1 (Harris). Let V be an E-Banach space equipped with an admissible continuous G-
representation and let d = dim(G). Then as k → ∞,
dimE V
Gk = r · [G : Gk] +O(p
(d−1)k) = r · c · pdk +O(p(d−1)k),
where r is the corank of V and c depends only on G.
Using this result, we may obtain bounds on the dimensions of the continuous cohomology groups
H i(Gk, V ) in terms of k for admissible continuous G-representations V . (Let us remark that the
continuous Gk-cohomology on the category of admissible continuous G-representations may also be
computed as the right derived functors of the functor of Gk-invariants; see Prop. 1.1.3 of [4].)
Lemma 2.2. Let V be an admissible continuous G-representation. For each i ≥ 1,
dimE H
i(Gk, V ) ≪ p
(d−1)k,
as k → ∞.
Proof. Let C := C(G,E) denote the Banach space of continuous E-valued functions on G, equipped
with the right regular G-action. The module C has corank one (indeed, it is cofree – its topological
dual is free of rank one over ΛE). Moreover, C is injective in the abelian category of admissible
G-representations and is therefore acyclic. If V is an admissible continuous G-representation, then
there exists an exact sequence
0 −→ V −→ Cn −→ W −→ 0
of admissible continuous G-representations for some integer n ≥ 0. Since C is acyclic, from the long
exact sequence of cohomology we obtain the following:
0 −→ V Gk −→ (CGk)n −→ WGk −→ H1(Gk, V ) −→ 0, (2.1)
H i(Gk, V ) ≃ H
i−1(Gk,W ), i ≥ 2. (2.2)
The lemma for i = 1 follows from a consideration of (2.1), taking into account Theorem 2.1 and the
fact that corank of W is equal to n minus the corank of V (since corank is additive in short exact
sequences). We now proceed by induction on i. Assume the result for i ≤ m and all admissible
continuous representations, in particular for W . The result for i = m+1 then follows directly from
the isomorphism (2.2). This completes the proof.
3 Cohomology of Arithmetic Quotients of Symmetric Spaces
We now return to the situation considered in the introduction and use the notation introduced there.
In particular, we fix a connected semisimple linear group G over F , an embedding G →֒ GLN over
F , an arithmetic lattice Γ of the associated real group G∞, and a prime p of F .
If we write G := lim
Γ/Γ(pk), then G is a compact open subgroup of the p-adic Lie group G(Fp)
(where Fp denotes the completion of F at p); alternatively, we may define G to be the closure of Γ in
G(Fp). If we replace Γ by Γ(p
k) for some sufficiently large value of k (i.e. discarding finitely many
initial terms in the descending sequence of lattices Γ(pk)), then G will be pro-p and hence, will be
an analytic pro-p-group. Note that Γ is a dense subgroup of G. Let e and f denote respectively
the ramification and inertial indices of p in F (so that [Fp : Qp] = ef). For each k ≥ 0, write Gk to
denote the closure of Γ(pek) in G. Alternatively, if we consider the embedding
G(Fp) →֒ GLN (Fp) →֒ GLefN (Qp),
then Gk = G ∩ (1 + p
kMefN (Zp)); thus, our notation is compatible with that of the preceding
section. We let d denote the dimension of G; note that d =
[F :Q]
· dim(G∞).
For each k ≥ 0, we write
Yk := Γ(p
ek)\G∞/K∞.
There is a natural action of G on Yk through its quotient G/Gk (∼= Γ/Γ(p
ek)), which is compatible
with the projections Yk → Yk′ for 0 ≤ k
′ ≤ k.
Fix a finite-dimensional representation W of G over E, and let W0 denote a G-invariant OE-
lattice in W . Let Vk denote the local system of free finite rank OE-modules on Yk associated to
W0, and denote by Vk the pull-back of V0 to Yk for any k ≥ 0. If 0 ≤ k
′ ≤ k, then the sheaf Vk on
Yk is naturally isomorphic to the pull-back of the sheaf Vk′ on Yk′ under the projection Yk → Yk′ .
In particular the sheaf Vk is G/Gk-equivariant.
Recall the following definitions from from [4], p. 21:
H̃n(V) := lim
Hj(Yk,Vk/p
s), H̃n(V)E := E ⊗OE H̃
n(V).
Each H̃n(V) is a p-adically completeOE-module, equipped with a leftG-action in a natural way, and
hence, each H̃n(V)E has a natural structure of E-Banach space and is equipped with a continuous
left G-action. In fact, they are admissible continuous representations of G ([4], Thm. 2.1.5 (i)), and
in particular, Theorem 2.1 and Lemma 2.2 apply to them. (Note that the results of [4] are stated
in the adèlic language. We leave it to the reader to make the easy translation to the more classical
language we are using in this paper.)
The following result, which is Theorem 2.1.5 (ii) of [4], p. 22, is a “control theorem” relating
Gk invariants in H̃
E to the classical cohomology classes H
j(Yk,Vk)⊗ E.
Theorem 3.1. Fix an integer k. There is a spectral sequence
2 (Yk) = H
i(Gk, H̃
j(V)E) =⇒ H
i+j(Yk,Vk)E .
One should view this spectral sequence as a version of the Hochschild-Serre spectral sequence
“compatible in the G-tower.”
Theorem 3.2. For any n ≥ 0, if rn denotes the corank of H̃
n(V), then
dimE H
n(Yk,Vk)E = rn · c · p
dk +O(p(d−1)k)
as k → ∞. (Here c denotes the constant appearing in the statement of Theorem 2.1; it depends
only on G.)
Proof. For each i, j ≥ 0 and l ≥ 2, let E
l (Yk) denote the terms in the spectral sequence of
Theorem 3.1. Since H̃j(V) is admissible, Lemma 2.2 implies that dimE H
i(Gk, H̃
j(V)E) ≪ p
(d−1)k,
and thus, dimE E
l (Yk) ≪ p
(d−1)k, as k → ∞ (since E
l (Yk) is a subquotient of E
2 (Yk) :=
H i(Gk, H̃
j(V)E) for l ≥ 2). Theorem 2.1 shows that
dimE E
2 (Yk) = dimE H̃
E = rn · c · p
dk +O(p(d−1)k).
On the other hand, since the spectral sequence of Theorem 3.1 is an upper right quadrant exact
sequence, E
∞ is obtained by taking finitely many successive kernels of differentials dl to E
i+l,j−l+1
which all have order ≪ p(d−1)k by the first part of our argument. Thus,
dimE E
= rn · c · p
dk +O(p(d−1)k).
Since Hn(Yk,Vk)E admits a finite length filtration whose associated graded pieces are isomorphic
∞ for i+ j = n, we conclude that dimE H
n(Yk,Vk)E = rn · c · p
dk +O(p(d−1)k), as claimed.
The following lemma quantifies the precise relationship between multiplicities and the dimen-
sions of cohomology groups that we will require to deduce Theorem 1.1 from Theorem 1.2.
Lemma 3.3. Fix a cohomological degree n, and let S denote the set of isomorphism classes [π] of
π ∈ Ĝ∞ that contribute to cohomology with coefficients in V in degree n. Then
[π]∈S
m(π,Gk) ≍ dimE H
cusp(Yk,Vk)E .
Proof. Since the set S is finite, there is an integer d ≥ 1 so that
1 ≤ dimHn(g, k;π ⊗W ) ≤ d
for each isomorphism class [π] ∈ S. This implies that
[π]∈S
m(π,Gk) ≤ dimE H
cusp(Yk,Vk)E ≤ d
[π]∈S
m(π,Gk)
for each k ≥ 0.
We can now prove our main result.
Theorem 3.4. Let n ≥ 0, and suppose that either G∞ does not admit discrete series or else that
dim(G∞/K∞). Then
dimE H
n(Yk,Vk)E ≪ p
(d−1)k
as k → ∞, for all n ≥ 0.
Proof. In the case when G∞ admits discrete series, recall that these contribute to cohomology
only in the dimension
dim(G∞/K∞) ([1], Thm. 5.1, p. 101). Thus, under the assumptions
of the theorem, there is no contribution from the discrete series to Hncusp(Yk,Vk). The inequal-
ity (1.2) of [3] and [10], together with Lemma 3.3 and the main result of [8] (which states that
Hn(Yk,Vk)/H
cusp(Yk,Vk)
= o(pdk)), thus shows that dimE H
n(Yk,Vk)E = o(p
dk) as k → ∞,
for all n ≥ 0. From Theorem 3.2, we then infer that each H̃n(V) has corank 0. Another application
of the same theorem now gives our result.
Note that V (pek) ∼ [G : Gk] ∼ c · p
dk; thus Theorem 3.4 implies Theorem 1.2 since
dim(G) ≤ dim(G∞). (3.1)
Theorem 1.2 and Lemma 3.3 together imply Theorem 1.1.
Remark 3.5. We have equality in (3.1) precisely when p is the unique prime lying over p in OF . If
there is more than one prime lying over p, then dim(G) is strictly less than dim(G∞), and we obtain
a corresponding improvement in the bounds of Theorems 1.1 and 1.2, namely (in the notation of
their statements), that
m(π,Γ(pk)) and dimE H
i(Y (pk),VW,k)E ≪ V (p
n)1−1/ dim(G)
(where, as we noted above, dim(G) =
[F :Q]
· dim(G∞), with e and f being the ramification and
inertial index of p respectively).
Example/Question 3.6. Let F/Q be an imaginary quadratic field, and let G = SL2/F . The
corresponding symmetric space G∞/K∞ = SL2(C)/SU(2) is a real hyperbolic three space H,
and the quotients Y are commensurable with the Bianchi manifolds H/PGL2(OK). Choose a
local system V0 associated to some finite-dimensional representation W of G∞ = GL2(C) and a
congruence subgroup Γ. Assume that p = pp splits in OF , and apply Theorem 3.4 to the p-power
tower. We obtain the inequality
H1cusp(Yk,Vk) ≪ p
as k → ∞. It is natural to ask how tight this inequality is.
The main result of Calegari–Dunfield [2] shows that there exists at least one (F,Γ, p) for which
H1cusp(Yk,C) = 0
for all k. On the other hand, if there exists at least one newform on Γ(pk) for some k, then a
consideration of the associated oldforms shows that
H1cusp(Yk,Vk) ≫ p
as k → ∞. Are there situations in which this lower bound gives the true rate of growth?
Remark 3.7. Our results are most interesting in the case when G∞ does not admit any discrete
series, since, as we noted in the introduction, in this case (and only in this case), G∞ admits
(non-discrete series) tempered representations of cohomological type.
On the other hand, Theorem 3.2 does have a consequence in the case when G∞ admits discrete
series which may be of some interest. Recall the following result from [10] (established in [3] in the
cocompact case): if π ∈ Ĝ∞ lies in the discrete series, then
m(Γ(πk), π) = d(π)V (pk) + o(V (pk)) (3.2)
as k → ∞. Fix a finite-dimensional representation W of G∞, and let Ĝ∞(W )d denote the subset
of Ĝ∞ consisting of discrete series representations that contribute to cohomology with coefficients
in W . Summing over all π ∈ Ĝ∞(W )d, we obtain the formula
|Ĝ∞(W )d|
π∈ bG∞(W )d
m(Γ(πk), π) = d(π)V (pk) + o(V (pk)). (3.3)
(A result first proved in [8].) The following result provides an improvement in the error term
of (3.3).
Theorem 3.8. There exists µ > 0 such that
|Ĝ∞(W )d|
π∈ bG∞(W )d
m(Γ(πk), π) = d(π)V (pk) +O(V (pk)1−µ).
Proof. Let n =
dim(G∞/K∞). As already noted, it follows from [1], (Thm. 5.1, p. 101), that
all non-discrete series contributions to Hncusp(Yk,Vk) are non-tempered. The same result shows
that each discrete series has one-dimensional (g, k)-cohomology in dimension n. As we recalled in
the introduction, the multiplicity of any non-tempered representations is bounded by V (pk)1−µ for
some µ > 0 [9], and thus, Theorem 3.2 and (the proof of) Lemma 3.3 show that
|Ĝ∞(W )d|
π∈ bG∞(W )d
m(Γ(πk), π) = C · V (pk) +O(V (pk)1−µ).
Comparing this formula with (3.3) yields the theorem.
Question 3.9. Does the result of Theorem 3.8 hold term-by-term? That is, does (3.2) admit an
improvement of the form
m(Γ(πk), π)
= d(π)V (pk) +O(V (pk)1−µ)
for some µ > 0?
References
[1] Borel, A.; Wallach, N. Continuous cohomology, discrete subgroups, and representations of reductive
groups. Annals of Mathematics Studies, 94. Princeton University Press, Princeton, N.J.; University of
Tokyo Press, Tokyo, 1980. xvii+388 pp.
[2] Calegari, F.; Dunfield, N. Automorphic forms and rational homology 3-spheres. Geom. Topol. 10 (2006),
295–329.
[3] DeGeorge, D.; Wallach, N. Limit formulas for multiplicities in L2(Γ\G). Ann. Math. (2) 107 (1978),
no. 1, 133–150.
[4] Emerton, M. On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms.
Invent. Math. 164 (2006), no. 1, 1–84.
[5] Franke, J. Harmonic Analysis in Weighted L2-spaces. Ann. Sci. cole Norm. Sup. (4) 31 (1998), no. 2,
181–279.
[6] Harris, M. Correction to: “p-adic representations arising from descent on abelian varieties” [Compositio
Math. 39 (1979), no. 2, 177–245]. Compositio Math. 121 (2000), no. 1, 105–108.
[7] Lazard, M. Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965).
[8] Rohlfs, J.; Speh, B. On limit multiplicities of representations with cohomology in the cuspidal spectrum.
Duke Math. J. 55 (1987), no. 1, 199–211.
[9] Sarnak, P.; Xue, X. Bounds for multiplicities of automorphic representations. Duke Math. J. 64 (1991),
no. 1, 207–227.
[10] Savin, G. Limit multiplicities of cusp forms. Invent. Math. 95 (1989), no. 1, 149–159.
[11] Schneider, P.; Teitelbaum, J. Banach space representations and Iwasawa theory, Israel. J. Math. 127
(2002), 359–380.
	Introduction
	Iwasawa Theory
	Cohomology of Arithmetic Quotients of Symmetric Spaces
