Torsion in symmetric powers

# The torsion in symmetric powers on congruence subgroups of Bianchi groups

Jonathan Pfaff and Jean Raimbault Universität Bonn
Mathematisches Institut
Endenicher Alle 60
D – 53115 Bonn, Germany
Institut de Mathématiques de Toulouse ; UMR5219
Université de Toulouse ; CNRS
UPS IMT, F-31062 Toulouse Cedex 9, France
###### Abstract.

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the -th symmetric power of the standard representation of grows exponentially in . We give upper and lower bounds for the growth rate. Our result extends a a result of W. Müller and S. Marshall, who proved the corresponding statement for closed arithmetic 3-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for twisted combinatorial Reidemeister torsion on higher dimensional hyperbolic manifolds.

## 1. Introduction

The torsion in the cohomology of arithmetic groups has recently attracted new interest from number theorists. Without aiming at completeness, we refer for example to [BV13], [CV12], [Eme14] and [Sch13]. In this paper, we study the twisted cohomological torsion quantitatively for a fixed principal congruence subgroup of a Bianchi group under a variation of the local system. Bianchi groups represent all classes of non-uniform lattices in ; thus our result complements the study of this question for arithmetic lattices in defined over imaginary quadratic fields done by Simon Marshall and Werner Müller in [MM13] (where the authors give an equality for the asymptotic torsion size, while we only get upper and lower bounds for the growth rate).

To state our main result more precisely, we need to introduce some notation. Let be square-free and let be the associated imaginary quadratic number field with ring of integers . Let . Then is an arithmetic subgroup of which acts on and the quotient is a hyperbolic orbifold of finite volume. If is a non-zero ideal of , we let denote the principal congruence subgroup of level . This is a finite-index subroup of which is neat (i.e. none of its non-unipotent elements have a root of unity as an eigenvalue, in particular it is torsion-free) as soon as the norm is sufficiently large ( suffices). We shall assume this from now on. Thus, is an arithmetic hyperbolic manifold of finite volume. It is never compact and has finitely many cusps, whose number we shall denote by . For let be the natural representation of on the th symmetric power of it’s standard representation on . Then there exists a -lattice in which is preserved by (one can simply take ).

Now one can form the integral cohomology groups of the -modules . These are finitely generated abelian groups and thus they split as

 H∗(Γ(a);Λ(m))=H∗(Γ(a);Λ(m))free⊕H∗(Γ(a);Λ(m))tors,

where the first group in this decomposition is a free finite-rank -module and the second group is a finite abelian group. Moreover, is a lattice in the real cohomology . In this paper we are interested in the behaviour of the size of the cohomology group as goes to infinity. First we note that

 (1.1) dimH1(Γ(a),V(m))=dimH2(Γ(a),V(m))=κ(Γ(a)

for each . This is easy to verify, see for example section 4. On the other hand, we will show that in degree 2 the size of the torsion part grows exponentially in as and we will specify the growth rate. More precisely, the main result of this paper is the following theorem (we stated it for the lattices but we prove that the growth rates are the same for any sequence of -invariant lattices in —see Proposition 7.3).

###### Theorem A.

There exist constants , , which depend only on such that for each non-zero ideal of with one has

 (1.2) liminfm→∞log|H2(Γ(a);Λ(m))tors|m2≥12⋅vol(Xa)π(1−C1(ΓD)N(a))>0

and

 (1.3) limsupm→∞log|H2(Γ(a);Λ(m))tors|m2≤vol(Xa)π(1+C2(ΓD)N(a)).

Finally, we also have

 (1.4) |H1(Γ(a);Λ(m))tors|=O(mlogm),

as .

If the class number of is one and , we can take (this is valid only for and by the Stark–Heegner Theorem). We shall now briefly sketch our method to prove our main result. The main point in our case is to establish the lower bound on the torsion given in (1.2), i.e. to establish its exponential growth in . Let us point out that there are two severe difficulties in the present non-compact case which are not present in the case of compact arithmetic 3-manifolds mentioned above. Firstly, the use of analytic torsion as a main tool is more complicated. Secondly, the real cohomology does not vanish in our situation. We shall now describe these issues in more detail.

As already observed by Nicolas Bergeron and Akshay Venkatesh in [BV13], the size of cohomological torsion is closely related to the Reidemeister torsion of the underlying manifold with coefficients in the underlying local system. In the present finite volume case, one has to work with the twisted Reidemeister torsion of the Borel–Serre compactification of . For technical reasons, in most of the paper we also symmetrize the lattice to a lattice in which it self-dual over (it is then not hard to deduce the estimates in Theorem A from their analogues for -coefficients). The Reidemeister torsion is then defined with respect to a canonical basis in the cohomology using Eisenstein cohomology classes following Günter Harder [Har75]. By a gluing formula for the Reidemeister torsion, which was obtained by the first author in [Pfa13] building on work of Matthias Lesch [Les13], the Reidemeister torsion can be compared to the regularized analytic torsion of the manifold . The asymptotic behaviour of the regularized analytic torsion in the finite volume case for a variation of the local system has already been studied by Müller and the first author in [MP12] using the Selberg trace formula. Thus we have to study the error term which occured in [Pfa13] in the comparison formula between analytic and Reidemeister torsion. It turns out that our study of the error term can be performed without any changes also in the higher dimensional situations. While we do not compute the error term explicitly, we bring it in a form which is sufficient for the application to cohomological torsion.The main point is that the error term depends only on the geometry of the cusps which is very restricted on such manifolds. Also, along the line we can establish limit multiplicity formulae for twisted Reidemeister torsion in the spirit of [BV13] on arithmetic hyperbolic manifolds of finite volume of arbitrary dimension, see Corollary 2.4.

Now we turn to the second aforementioned difficulty. Since the real cohomology of with coefficients in does not vanish in the finite-volume case, we also have to study certain volume factors occuring in the comparison formula between Reidemeister torsion and the size of cohomological torsion. Since our basis in the real cohomology is given by Eisenstein series, this leads to the question about the integrality of certain quotients of -functions evaluated at positive integers. In the 3-dimensional case, these are Hecke L-functions and we can use the work of R. Damerell [Dam70],[Dam71] to control these quotients. At the moment, we do not know how to do this in higher dimensions.

Concerning the other statements of our main theorem, at least with a worse constant, the upper bound (1.3) can be established in an elementary and completely combinatorial way, without referring to analytic or Reidemeister torsion, see Proposition 8.3. In fact, our approach for the upper bound on the combinatorial torsion generalizes easily to arbitrary dimensions and to arbitrary rays in the weight lattice; it is similar to that used by V. Emery [Eme14] or R. Sauer [Sau14]. We note that if we were able to obtain a proper limit for the Redemeister torsion instead of the quotient of two such, we would obtain an optimal upper bound for the exponential growth rate of the torsion in the second cohomology, by an argument similar to that used in the proof of the easy part of [Rai13, Lemma 6.14]. The last estimate (1.4) in our theorem is the easiest one to prove, and does not require analytic torsion or any sophisticated tool. Since we work with a -split group the representations are easier to analyze that in the nonsplit case which was dealt with in [MM13, section 4]—we can work globally from the beginning.

We finally remark that Theorem A also holds with the same proof for slightly more general rays of local systems. Namely, the finite dimensional irreducible representations of are parametrized as , where and where is the complex conjugate of . Each such representation space carries a canonical -lattice preserved by the action of . If we fix and with and let be the representation , then the analog of Theorem A holds if we replace the factor by which grows as if both and are not zero. However, we can by no means remove the assumption . In other words, the ray , which is the ray carrying cuspidal cohomology, cannot be studied by our methods. For a fixed , we can only show that the size of cohomological torsion with coefficients in the canonical lattice associated to grows at most as , but we can say nothing about the existence of torsion along this ray, i.e. we cannot establish any bound from below. The reason is that here belongs to the essential spectrum of the twisted Laplacian in the middle dimension. Therefore, essentially none of the results on analytic torsion we use in our proof is currently available for and also the regulator would be more complicated. We remark that, as far as we know, even in the compact case no result for the growth of torsion along this particular ray has been obtained. For an investigation of the dimension of the (cuspidal) cohomology along this ray, we refer to [FGT10].

This paper is organized as follows : in section 2 we introduce the analytic torsion for cusped manifolds, then in section 3 we study it further for congruence subgroups of Bianchi groups. We introduce the combinatorial (Reidemeister) torsion in section 4 and recall the Cheeger–Müller equality proven in [Pfa13] there. Then we study the intertwining operators in section 5 and 6, first computing them using adeles and then bounding their denominators. The last sections contain the proof of the main theorem: in section 7 we combine the results of the previous sections to prove (1.2), and we prove (1.3) in section 8.

Acknowledgement. The first author was financially supported by the DFG-grant PF 826/1-1. He gratefully acknowledges the hospitality of Stanford University in 2014 and 2015.

## 2. The regularized analytic torsion for coverings

In this section we shall review the definition of regularized traces and the regularized analytic torsion of hyperbolic manifolds of finite volume. These objects are defined in terms of a fixed choice of truncation parameters on and there are two different ways to perform such a trunctation which are relevant in the present paper. Firstly, one can define a truncation of via a fixed choice of -cuspidal parabolic subgroups of . Secondly, if is a finite covering of a hyperbolic orbifold , then a choice of truncation parameters on gives a truncation on in terms of which one can define another regularized analytic torsion. We shall compute the difference between the associate regularized analytic torsions explicitly. For more details we refer to [MP12], [MP14a].

We denote by the identity-component of the isometry group of the standard quadratic form of signature on . Let denote the universal covering of . We let either or . We assume that is odd and write . Let , if or , if . We let be the Lie algebra of . Let denote the standard Cartan involution of and let denote the associated Cartan decomposition of , where is the Lie algebra of . Let be the Killing form. Then

 (2.1) ⟨X,Y⟩:=−12(d−1)B(X,θY)

is an inner product on . Moreover, the globally symmetric space , equipped with the -invariant metric induced by the restriction of (2.1) to is isometric to the -dimensional real hyperbolic space . Let be a discrete, torsion-free subgroup. Then , equipped with the push-down of the metric on , is a -dimensional hyperbolic manifold. We let be a fixed parabolic subgroup of with Langlands decomposition as in [MP12]. Let denote the Lie algebra of and the exponential map. Then we fix a restricted root of in , let be such that and define by . If is another parabolic subgroup of , we fix with and define , . Moreover, for we define . For we let .

A parabolic subgroup of is called -cuspidal if is a lattice in . From now on, we assume that is finite and that is neat in the sense of Borel, i.e. that for each -cuspdail . If is -cuspidal, then for we put

 FP1;Γ(Y):=(Γ∩NP1)∖NP1×AP1[Y]≅(Γ∩NP1)∖NP1×[Y,∞).

We equip with the metric where is the push-down of the invariant metric on induced by the innerer product (2.1) restricted to .

Let denote the set of finite-dimensional irreducible representations of . For the associated vector space posesses a distinugished inner product which is called admissible and which is unique up to scaling. We shall fix an admissible inner product on each . If , then the restriction of to induces a flat vector bundle . This bundle is canonically isomorphic to the locally homogeneous bundle induced by the restriction of to . In particular, since is a unitary representation on , the inner product induces a smooth bundle metric on and therefore on . For let denote the flat Hodge Laplacian acting on the smooth -valued -forms of . Since is complete, with domain the smooth, compactly supported -valued -forms is essentially selfadjoint and its -closure shall be denoted by the same symbol. Let denote the heat semigroup of and let

 Kρ,pX(t,x,y)∈C∞(X×X;Eρ⊠E∗ρ)

be the integral kernel of .

We let be a fixed set of -cuspidal parabolic subgroups of . Then is non-empty if and only if is non-compact. Moreover, equals the number of cusps of which from now on we assume to be nonzero. The choice of and of a fixed base-point in determine an exhaustion of by smooth compact manifolds with boundary, . This exhaustion depends on the choice of . Then one can show that the integral of over has an asymptotic expansion

 (2.2) ∫XPΓ(Y)TrKρ,pX(t,x,x)dx=α−1(t)logY+α0(t)+o(1),

as , [MP12][section 5]. Now one can define the regularized trace of with respect to the choice of by , where is the constant term in the asymptotic expansion in (2.2).

From now on, we also assume that there is a hyperbolic orbifold such that is a finite covering of . Let denote the covering map. Then if a set of truncation parameters on , or in other words a set of representatives of -cuspidal parabolic subgroups are fixed, one obtains truncation parameters on by pulling back the truncation on via . One can again show that there is an asymptotic expansion

 (2.3) ∫π−1XPΓ0(Y)TrKρ,pX(t,x,x)dx=~α−1(t)logY+~α0(t)+o(1),

as and one can define the regulraized trace with respect to the truncation parameters on as . This regulararized trace depends only on the choice of a set of representatives of -cuspidal parabolic subgroups of . Put

 (2.4) KX;PΓ(t,ρ):=∑p(−1)ppTrreg;PΓe−tΔp(ρ);KX;X0(t,ρ):=∑p(−1)ppTrreg;X;X0e−tΔp(ρ).

Now assume that satisfies . Then one defines the analytic torsion with respect to the two truncations of by

 (2.5) (2.6) logTX;X0(ρ):=12dds∣∣∣s=0(1Γ(s)∫∞0ts−1KX;X0(t,ρ)).

Here the integrals converge absolutely and locally uniformly for and are defined near by analytic continuation, [MP12, section 7], [MP14a, section 9].

To compare the two analytic torsions in (2.5) and (2.6), we need to introduce some more notation. We fix and . Then for each there exists a unique and a such that . Write

 (2.7) γj=n0,l(j)ιP0,l(j)(tPj)k0,l(j),

, , as above, and . Since equals its normalizer, the projection of to is unique. Moreover, since is -cuspidal, one has . Thus depends only on and .

Now the analytic torsions and are compared in the following proposition.

###### Proposition 2.1.

One has

 logTPΓ(X;Eρ)=logTX0(X;Eρ)+∑Pj∈PΓlog(tPj)(n∑k=0(−1)kdim(σρ,k)λρ,k2).
###### Proof.

This follows by an application of a theorem of Kostant [Kos61] on nilpotent Lie algebra cohomology. For define a representation of on by . Let , which is a homogeneous vector bundle over . Let be the Casimir element of . Then induces canonically a Laplace-type operator which acts on the smooth sections of . The heat semigroup of is canonically represented by a smooth function , [MP12, section 4, section 7]. Let and put

 kp,ρt:=∑p(−1)pphp,ρt.

Then by the definition of the regularized traces, one has

 (2.8) ∫XPΓ(Y)⎛⎝∑γ∈Γkp,ρt(x−1γx)⎞⎠dx=α−1(t;ρ)logY+KX;PΓ(t,ρ)+o(1),

as , resp.

 (2.9) ∫π−1(XPΓ0(Y))⎛⎝∑γ∈Γkp,ρt(x−1γx)⎞⎠dx=~α−1(t;ρ)logY+KX;X0(t,ρ)+o(1),

as , where we use the notation (2.4). On the other hand, for let be defined as in [MP12, (8.7)]. If we apply the same considerations as in [MP14a, section 6] to the functions , then combining [MP12, Proposition 8.2], (2.8) and (2.9) we obtain

 KX;X0(t,ρ)=KX;PΓ(t,ρ)−∑Pj∈PΓlogtPj⎛⎜⎝n∑k=0(−1)kdim(σρ,k)e−tλ2ρ,k√4πt⎞⎟⎠.

Taking the Mellin transform, the Proposition follows. ∎

Next, as in [Pfa13], for each and for one can define the regularized analytic torsion of and the bundle , where one takes relative boundary conditions. For different , these torsions are compared by the following gluing formula.

###### Lemma 2.2.

Let be as in [Pfa13, equation 15.10]. Then for and one has

 logT(FPj;Γ(Y1),∂FPj;Γ(Y1);Eρ)−c(n)vol(∂FPj;Γ(Y1))rk(Eρ) = logT(FPj;Γ(Y2),∂FPj;Γ(Y2);Eρ)−c(n)vol(∂FPj;Γ(Y2))rk(Eρ) + n∑k=0(−1)k+1λρ,kdim(σρ,k)(log(Y2)−log(Y1)).
###### Proof.

This follows immediately from [Pfa13, Corollary 15.4, equation (15.11) and Corollary 16.2]. ∎

We let denote the Borel-Serre compactification of and we let be the Reidemeister torsion of with coefficients in , defined as in [Pfa13, section 9]. For simplicity, we assume that is normal in . Then for the torsion , the main result of [Pfa13] can be restated as follows.

###### Proposition 2.3.

For the analytic torsion one has

 logτEis(X;Eρ)=logTX0(X;Eρ)−14d−1∑k=0(−1)klog|λρ,k|dimHk(∂¯¯¯¯¯X;Eρ) −κ(Γ0)∑l=1#{Pj∈PΓ:γjPjγ−1j=P0,l}(logT(FP0,l;Γ(1),∂FP0,l;Γ(1);Eρ) −c(n)vol(∂FP0,l;Γ(1))rk(Eρ)).
###### Proof.

By [Pfa13, Theorem 1.1] and by Proposition 2.1 we have

 logτEis(X;Eρ)=logTX0(X;Eρ)+∑Pj∈PΓlog(tPj)(n∑k=0(−1)kdim(σρ,k)λρ,k2) − ∑Pj∈PΓ(logT(FPj;Γ(1),∂FPj;Γ(1);Eρ)−c(n)vol(∂FPj;Γ(1))rk(Eρ)) −14d−1∑k=0(−1)klog|λρ,k|dimHk(∂¯¯¯¯¯X;Eρ)

where we recall that the regularized analytic torsion used in [Pfa13, Theorem 1.1] is the torsion denoted here. Using that is normal in , it easily follows from the definition of that for each one has a canonical isometry . It is easy to see that also is isometric to . Thus we have

 logT(FPj;Γ(1),∂FPj;Γ(1);Eρ)−c(n)vol(∂FPj;Γ(1))rk(Eρ) = logT(FP0,l(j);Γ(tPj),∂FP0,l(j);Γ(tPj);Eρ)−c(n)vol(∂FP0,l(j);Γ(tPj))rk(Eρ).

Applying Lemma 2.2, the Proposition follows.

Although the main topic of this paper is the behaviour of cohomological torsion of congruence subgroups of Bianchi groups under a variation of the local system, we now state the following limit multiplicity formula for twisted Reidemeister torsion in arithmetic hyperbolic congruence towers of arbitrary odd dimension, since the latter is an easy corollary.

###### Corollary 2.4.

Let , odd, and for let denote the principal congruence subgroup of level . Let . Then for any with one has

 limq→∞logτEis(Xq;Eρ)vol(Xq)=t(2)Hd(ρ),

where is the -invariant associated to and which is defined as in [BV13], [MP14a] and which is never zero. The same holds for ever sequence of arithmetic hyperbolic 3-manifolds associated to princiapl congruence subgroups of Bianchi groups if .

###### Proof.

First we assue that . Let and . By [MP14a, Corollary 1.3], for the analytic torsion one has

 (2.10) limq→∞logTX0(Xq;Eρ)vol(Xq)=t(2)Hd(ρ),

as . Next it is well-known that for the number of cusps of one has

 (2.11) limq→∞κ(Γ(q))vol(Xq)=0,

see for example [MP14a, Proposition 8.6]. For we let , which is a lattice in . By a result of Deitmar and Hoffmann [DH99, Lemma 4], for each , there exists a finite set of lattices in such that for each the lattice arises by scaling one of the lattices , , see [MP14a, Lemma 10.1]. For we let , equipped with the flat metric (2.1) restricted to which we shall denote by . Then we let , equipped with the metric , . If with and , then by Lemma 2.2 one has

 logT(FP0,l;Γ(q)(1),∂FP0,l;Γ(q)(1);Eρ)−c(n)vol(∂FP0,l;Γ(q)(1))rk(Eρ) = logT(FΛj(P0,l)(1),∂FΛj(P0,l)(1);Eρ)−c(n)vol(Λj(P0,l))rk(Eρ) +logμq;P0,ln∑k=0(−1)kλρ,kdim(σρ,k).

We can obviously estimate , where is a constant which is independent of . Thus there exists a constant such that for all one can estimate

 ∣∣logT(FP0,l;Γ(q)(1),∂FP0,l;Γ(q)(1);Eρ)−c(n)vol(∂FP0,l;Γ(q)(1))rk(Eρ)∣∣ ≤C2log[Γ0∩NP0,l:Γ(q)∩NP0,l].

For each one has

 #{Pj∈PΓ(q):γjPjγ−1j=P0,l}[Γ0:Γ(q)]=#(Γ(q)∖Γ0/(Γ0∩P0,l))[Γ0:Γ(q)]≤1[Γ0∩NP0,l:Γ(q)∩NP0,l].

Thus for all one can estimate

 κ(Γ0)∑l=1#{Pj∈PΓ(q):γjPjγ−1j=P0,l}[Γ0:Γ(q)] ×|logT(FP0,l;Γ(q)(1),∂FP0,l;Γ(1);Eρ)−c(n)vol(∂FP0,l;Γ(q)(1))rk(Eρ)| (2.12) ≤C2κ(Γ0)∑l=1log[Γ0∩NP0,l:Γ(q)∩NP0,l][Γ0∩NP0,l:Γ(q)∩NP0,l].

Since , as and thus the last term in (2) goes to zero as tends to infinity. Since , the corollary follows by applying Proposition 2.3, equation (2.10) and equation (2.11). For a sequence associated to principal congruence subgroups of Bianchi groups, one argues in the same way. ∎

## 3. Analytic and combinatorial torsion for congruence subgroups of Bianchi groups

From now on, we let , . We take to be the standard parabolic subgroup of consiting of all upper triangular matrices in . Then the unipotent radical of is given by all upper triangular matrices whose diagonal entries are one. Moreover, we let and denote the subgroups of defined by

 (3.1)