Bounds for Multiplicities of Unitary Representations of
Cohomological Type in Spaces of Cusp Forms
Frank Calegari

Matthew Emerton

February 29, 2008

1

Introduction

b ∞ . The goal of this note is to produce
Let G∞ be a semisimple real Lie group with unitary dual Gb ∞ of cohomological typenew upper bounds for the multiplicities with which representations π ∈ Gappear 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 theintersection of G∞ with the congruence subgroup of GLN (OF ) of full level q. We also fix anarithmetic lattice Γ in
T G∞ (i.e. a subgroupbcommensurable with the congruence subgroups G(q))
and write Γ(q) := Γ G(q). For any π ∈ G∞ , let m(π, Γ(q)) denote the multiplicity with which
π occurs in the decomposition of the regular representation of G∞ on L2cusp (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.
b ∞ be of cohomological type. Suppose
Theorem 1.1. Let p be a prime ideal in OF . Let π ∈ Geither that G∞ does not admit discrete series, or, if G∞ admits discrete series, that π contributes
1
to cohomology in degrees other than dim(G∞ /K∞ ). Then2m(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(pk )\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 suppose1
furthermore that n 6= dim(G∞ /K∞ ). Then2

dim H n 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
∞ , VW,k may be computed in terms of automorphic forms.

H n Γ(pk )\G1

In the remainder of this introduction, we discuss the relation of Theorem 1.1 to prior results inthis 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 iscocompact. In particular (ibid, Corollary 3.2), they showed thatZ−12m(π, Γ) ≤|φ(g)| dg
vol(Γ\G∞ ),(1.1)
Bwhere φ(g) = hπ(g)v, vi is a matrix coefficient, and B is the preimage in Γ\G∞ of a ball inΓ\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 thatlim

V (q)−1 · m(π, Γ(q)) = 0.

(1.2)

NF/Q (q)→∞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 formm(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] haveconjectured an inequality of the following form (in the case of cocompact Γ):b ∞ fixed,
Conjecture 1.3 (Sarnak–Xue). For π ∈ Gm(π, Γ(q))  V (q)(2/p(π))+ ,for all  > 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 fixedprime 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 theform (1.3) for any µ > 0.
As we already noted, our two main theorems are closely related. Indeed, Theorem 1.1 is aneasy corollary of Theorem 1.2 (see the end of Section 3 below), and most of our efforts will beconcentrated on establishing the latter result.
When studying the Betti numbers of arithmetic quotients of symmetric spaces, it is natural totry to use tools such as Euler characteristics and the Lefschetz trace formula. When applied to2

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, thesemethods 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 symmetricspace is a real manifold of odd dimension n, Poincare duality leads to cancellations in the naturalnXsum(−1)k dim(H k ). One is thus forced to find different techniques. The proof of Theorem 1.2
k=0

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 + pk MN (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 · pdk , 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 noncommutative 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 ) isa finite-dimensional left L -vector space; we define the rank of M to be the L -dimension of thisvector 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 0 (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 categoryof finitely generated ΛE -modules.) We define the corank of an admissible G-representation to be
the rank of the finitely generated ΛE -module V 0 .
A coadmissible G-representation V is not determined by the collection of subspaces of invariantsGkV(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 the3continuous Gk -cohomology on the category of admissible continuous G-representations may also becomputed 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, equippedwith the right regular G-action. The module C has corank one (indeed, it is cofree – its topologicaldual 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, thenthere 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 longexact sequence of cohomology we obtain the following:- V Gk0- (C Gk )n- W Gk- H 1 (Gk , V )- 0,

H i (Gk , V ) ' H i−1 (Gk , W ), i ≥ 2.

(2.1)
(2.2)

The lemma for i = 1 follows from a consideration of (2.1), taking into account Theorem 2.1 and thefact 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 fromthe 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 overF , 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 )←k(where Fp denotes the completion of F at p); alternatively, we may define G to be the closure of Γ inG(Fp ). If we replace Γ by Γ(pk ) 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 bean 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 todenote the closure of Γ(pek ) in G. Alternatively, if we consider the embedding
G(Fp ) ,→ GLN (Fp ) ,→ GLefN (Qp ),then Gk = G ∩ (1 + pk MefN (Zp )); thus, our notation is compatible with that of the preceding
section. We let d denote the dimension of G; note that d = [Fef:Q] · dim(G∞ ).
4

For each k ≥ 0, we write
Yk := Γ(pek )\G∞ /K∞ .
There is a natural action of G on Yk through its quotient G/Gk (∼= Γ/Γ(pek )), which is compatible
with the projections Yk → Yk0 for 0 ≤ k 0 ≤ 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 .
If 0 ≤ k 0 ≤ k, then the sheaf Vk on Yk is naturally isomorphic to the pull-back of the sheaf Vk0 on
Yk0 under the projection Yk → Yk0 . In particular the sheaf Vk is G/Gk -equivariant.
Recall the following definitions from from [4], p. 21:
e n (V) := lim lim H n (Yk , Vk /pl ),
H
←
− −
→
l

e n (V)E := E ⊗O H
e n (V).
HEke n (V) is a p-adically complete OE -module, equipped with a left G-action in a natural way, andEach He n (V)E has a natural structure of E-Banach space and is equipped with a continuous
hence, each H
left G-action. In fact, they are admissible continuous representations of G ([4], Thm. 2.1.5 (i)), andin particular, Theorem 2.1 and Lemma 2.2 apply to them. (Note that the results of [4] are statedin the adèlic language. We leave it to the reader to make the easy translation to the more classicallanguage we are using in this paper.)The following result, which is Theorem 2.1.5 (ii) of [4], p. 22, is a “control theorem” relatinge n (V)Gk to the classical cohomology classes H n (Yk , Vk ) ⊗ E.
Gk invariants in H
E
Theorem 3.1. Fix an integer k. There is a spectral sequencee j (V)E ) =⇒ H i+j (Yk , Vk )E .
E2i,j (Yk ) = H i (Gk , H
One should view this spectral sequence as a version of the Hochschild-Serre spectral sequence“compatible in the G-tower.”e n (V), thenTheorem 3.2. For any n ≥ 0, if rn denotes the corank of H
dimE H n (Yk , Vk )E = rn · c · pdk + 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 Eli,j (Yk ) denote the terms in the spectral sequence ofe j (V) is admissible, Lemma 2.2 implies that dimE H i (Gk , H
e j (V)E )  p(d−1)kTheorem 3.1. Since Hi,ji,jif i > 0, and thus that dimE El (Yk )  p(d−1)k as k → ∞, if i > 0 (since El (Yk ) is a subquotiente j (V)E ) for l ≥ 2). Theorem 2.1 shows thatof E2i,j (Yk ) := H i (Gk , H
e n (V)Gk = rn · c · pdk + O(p(d−1)k ).
dimE E20,n (Yk ) = dimE H
E
On the other hand, since the spectral sequence of Theorem 3.1 is an upper right quadrant exact
0,nsequence, E∞
is obtained by taking finitely many successive kernels of differentials dl to Eli+l,j−l+1 ,which all have order  p(d−1)k by the first part of our argument. Thus,
0,ndimE E∞= rn · c · pdk + O(p(d−1)k ).

Since H n (Yk , Vk )E admits a finite length filtration whose associated graded pieces are isomorphic
i,j
to E∞
for i + j = n, we conclude that dimE H n (Yk , Vk )E = rn · c · pdk + O(p(d−1)k ), as claimed.
5

The following lemma quantifies the precise relationship between multiplicities and the dimensions 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] ofb ∞ that contribute to cohomology with coefficients in V in degree n. Thenπ∈GXnm(π, Gk )  dimE Hcusp(Yk , Vk )E .
[π]∈S

Proof. Since the set S is finite, there is an integer d ≥ 1 so that1 ≤ dim H n (g, k; π ⊗ W ) ≤ dfor each isomorphism class [π] ∈ S. This implies thatXX
nm(π, Gk )m(π, Gk ) ≤ dimE Hcusp(Yk , Vk )E ≤ d
[π]∈S

[π]∈S

for each k ≥ 0.
We can now prove our main result.
Theorem 3.4. Let n ≥ 0, and suppose either that G∞ does not admit discrete series, or that1n 6= dim(G∞ /K∞ ). Then2
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
1
only in the dimension dim(G∞ /K∞ ) ([1], Thm. 5.1, p. 101). Thus, under the assumptions
2n (Y , V ). The inequalof the theorem, there is no contribution from the discrete series to Hcuspkkity (1.2) of [3] and [10], together
with
Lemma
3.3
and
the
main
resultof[8](whichstates thatn (Y , V ) = o(pdk )), thus shows that dim H n (Y , V ) = o(pdk ) as k → ∞,dimE H n (Yk , Vk )/Hcusp
E
k
k
k
k E
e n (V) has corank 0. Another application
for all n ≥ 0. From Theorem 3.2, we then infer that each Hof the same theorem now gives our result.
Note that V (pek ) ∼ [G : Gk ] ∼ c · pdk ; thus Theorem 3.4 implies Theorem 1.2 sincedim(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 obtaina corresponding improvement in the bounds of Theorems 1.1 and 1.2, namely (in the notation of
their statements), thatm(π, Γ(pk ))anddimE H i (Y (pk ), VW,k )E  V (pn )1−1/ dim(G)(where, as we noted above, dim(G) = [Fef:Q] · dim(G∞ ), with e and f being the ramification andinertial index of p respectively).
6

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 acongruence subgroup Γ. Assume that p = pp splits in OF , and apply Theorem 3.4 to the p-power
tower. We obtain the inequality
1
Hcusp(Yk , Vk )  p2kas 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
1
Hcusp(Yk , C) = 0

for all k. On the other hand, if there exists at least one newform on Γ(pk ) for some k, then aconsideration of the associated oldforms shows that
1
Hcusp
(Yk , Vk )  pkas 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 theb ∞ lies in the discrete series, thencocompact case): if π ∈ Gm(π, Γ(pk )) = d(π)V (pk ) + o(V (pk ))(3.2)

b ∞ (W )d denote the subset
as k → ∞. Fix a finite-dimensional representation W of G∞ , and let Gb ∞ consisting of discrete series representations that contribute to cohomology with coefficientsof G
b ∞ (W )d , we obtain the formulain W . Summing over all π ∈ G
1Xb ∞ (W )d ||Gm(π, Γ(pk )) = d(π)V (pk ) + o(V (pk )).

(3.3)

b ∞ (W )dπ∈G(A result first proved in [8].) The following result provides an improvement in the error termof (3.3).
Theorem 3.8. There exists µ > 0 such that1b ∞ (W )d ||GXm(π, Γ(pk )) = d(π)V (pk ) + O(V (pk )1−µ ).

b ∞ (W )dπ∈G

1dim(G∞ /K∞ ). As already noted, it follows from [1], Thm. 5.1, p. 101, that all2n (Y , V ) are non-tempered. The same result shows that
non-discrete series contributions to Hcuspkkeach discrete series has one-dimensional (g, k)-cohomology in dimension n. As we recalled in the
Proof. Let n =7introduction, the multiplicity of any non-tempered representations is bounded by V (pk )1−µ forsome µ > 0 [9], and thus, Theorem 3.2 and (the proof of) Lemma 3.3 show that
1X

b ∞ (W )d ||Gm(π, Γ(pk )) = C · V (pk ) + O(V (pk )1−µ ).

b ∞ (W )dπ∈G

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(π, Γ(pk )) = 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 ofTokyo 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.

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