The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume
aa r X i v : . [ m a t h . SP ] J u l THE ANALYTIC TORSION AND ITS ASYMPTOTIC BEHAVIOURFOR SEQUENCES OF HYPERBOLIC MANIFOLDS OF FINITEVOLUME
WERNER M ¨ULLER AND JONATHAN PFAFF
Abstract.
In this paper we study the regularized analytic torsion of finite volume hyper-bolic manifolds. We consider sequences of coverings X i of a fixed hyperbolic orbifold X .Our main result is that for certain sequences of coverings and strongly acyclic flat bun-dles, the analytic torsion divided by the index of the covering, converges to the L -torsion.Our results apply to certain sequences of arithmetic groups, in particular to sequences ofprincipal congruence subgroups of SO ( d, Z ) and to sequences of principal congruencesubgroups or Hecke subgroups of Bianchi groups. Introduction
The aim of this paper is to extend the results of Bergeron and Venkatesh [BV] on theasymptotic equality of analytic and L -torsion for strongly acyclic representations from thecompact to the finite volume case.Therefore, we shall first recall the results of Bergeron and Venkatesh about the compactcase. Let G be a semisimple Lie group of non-compact type. Let K be a maximal compactsubgroup of G and let e X = G/K be the associated Riemannian symmetric space endowedwith a G -invariant metric. Let Γ ⊂ G be a co-compact discrete subgroup. For simplicitywe assume that Γ is torsion free. Let X := Γ \ e X . Then X is a compact locally symmetricmanifold of non-positive curvature. Let τ be an irreducible finite dimensional complexrepresentation of G . Let E τ → X be the flat vector bundle associated to the restrictionof τ to Γ. By [MtM], E τ can be equipped with a canonical Hermitian fibre metric, calledadmissible, which is unique up to scaling. Let ∆ P ( τ ) be the Laplace operator on E τ -valued p -forms with respect to the metric on X and in E τ . Let ζ p ( s ; τ ) be the zeta function of∆ p ( τ ) (see [Sh]). Then the analytic torsion T X ( τ ) ∈ R + is defined by(1.1) T X ( τ ) := exp d X p =1 ( − p p dds ζ p ( s ; τ ) (cid:12)(cid:12) s =0 ! . Date : September 19, 2018.1991
Mathematics Subject Classification.
Primary: 58J52, Secondary: 53C53.
Key words and phrases. analytic torsion, locally symmetric manifolds.
On the other hand there is the L -torsion T (2) X ( τ ) (see [Lo]). Since the heat kernels on e X are G -invariant, one has(1.2) log T (2) X ( τ ) = vol( X ) t (2) e X ( τ ) , where t (2) e X ( τ ) is a constant that depends only on e X and τ . It is an interesting problemto see if the L -torsion can be approximated by the torsion of finite coverings X i → X .This problem has been studied by Bergeron and Venkatesh [BV] under a certain non-degeneracy condition on τ . Representations which satisfy this condition are called stronglyacyclic . One of the main results of [BV] is as follows. Let X i → X , i ∈ N , be a sequence offinite coverings of X . Let τ be strongly acyclic. Let inj( X i ) denote the injectivety radiusof X i and assume that inj( X i ) → ∞ as i → ∞ . Then by [BV, Theorem 4.5] one has(1.3) lim i →∞ log T X i ( τ )vol( X i ) = t (2) e X ( τ ) . If rk C ( G ) − rk C ( K ) = 1, one can show that t (2) e X ( τ ) = 0. Using the equality of analytictorsion and Reidemeister torsion [Mu2], Bergeron and Venkatesh [BV] used this result tostudy the growth of torsion in the cohomology of cocompact arithmetic groups. Further-more, recently P. Scholze [Sch] has shown the existence of Galois representations associatedwith mod p cohomology of locally symmetric spaces for GL n over a totally real or CM field.This makes it desirable to extend these results in various directions. Especially, one wouldlike to extend (1.3) to the finite volume case. However, due to the presence of the contin-uous spectrum of the Laplace operators in the non-compact case, one encounters serioustechnical difficulties in attempting to generalize (1.3) to the finite volume case. In [Ra1]J. Raimbault has dealt with finite volume hyperbolic 3-manifolds. In [Ra2] he appliedthis to study the growth of torsion in the cohomology for certain sequences of congruencesubgroups of Bianchi groups. His result generalized the exponential growth of torsion, ob-tained in [Pf2] for local systems induced from the even symmetric powers of the standardrepresentation of SL ( C ), to all strongly acyclic local systems and furthermore they impliedthat the limit of the normalized torsion size exists. The main purpose of the present paperis to extend (1.3) to hyperbolic manifolds of finite volume and arbitrary dimension.So from now on we let G = Spin( d, K = Spin( d ) or G = SO ( d,
1) and K = SO( d ) for d >
1. Then K is a maximal compact subgroup of G . Let e X = G/K . Choose an invariantRiemannian metric on e X . If the metric is suitably normalized, e X is isometric to the d -dimensional hyperbolic space H d . Let Γ ⊂ G be a torsion free lattice, i.e., Γ is a discrete,torsion free subgroup with vol(Γ \ G ) < ∞ . Let X = Γ \ e X . Then X is an oriented d -dimensional hyperbolic manifold of finite volume. Let τ be an irreducible finite dimensionalcomplex representation of G and let E τ → X be the flat vector bundle associated to τ asabove, endowed with an admissible Hermitian fibre metric. The first problem is to definethe analytic torsion if X is non-compact, which is the case we are interested in. Then theLaplace operator ∆ p ( τ ) has a non-empty continuous spectrum and hence, the zeta function ζ p ( s ; τ ) can not be defined in the usual way. It requires an additional regularization. We use the method introduced in [MP2]. One uses an appropriate height function to truncate X at sufficiently high level Y > Y to get a compact submanifold X ( Y ) ⊂ X with boundary ∂X ( Y ). Let K p,τ ( t, x, y ) be the kernel of the heat operator exp( − t ∆ p ( τ )). Then it followsthat there exists α ( t ) ∈ R such that R X ( Y ) tr K p,τ ( x, x, t ) dx − α ( t ) log Y has a limit as Y → ∞ . Then we put(1.4) Tr reg (cid:0) e − t ∆ p ( τ ) (cid:1) := lim Y →∞ (cid:18)Z X ( Y ) tr K p,τ ( t, x, x ) dx − α ( t ) log Y (cid:19) . As pointed out in [MP2, Remark 5.4], the regularized trace is not uniquely defined. Itdepends on the choice of truncation parameters on the manifold X . However, if a locallysymmetric space X = Γ \ e X of finite volume is given and if truncation parameters on X are fixed, then every locally symmetric manifold X which is a finite covering of X iscanonically equipped with truncation parameters: One simply pulls back the truncation on X to a truncation on X via the covering map. This will be explained in detail in section6 of the present paper.We remark that we do not assume that the group Γ is torsion-free. In fact, the typicalexample for Γ in the arithmetic case will be Γ = SO ( d, Z ) or Γ = SL ( O D ), where O D is the ring of integers of an imaginary quadratic number field Q ( √− D ), D ∈ N beingsquare-free. Then Γ will denote, for example, a principal congruence subgroup. However,we assume that Γ is not only a torsion-free lattice but also that Γ satisfies the followingcondition: For each Γ-cuspidal parabolic subgroup P ′ of G one has(1.5) Γ ∩ P ′ = Γ ∩ N P ′ , where N P ′ denotes the nilpotent radical of P ′ . This condition holds naturally, for example,for all principal congruence subgroups of sufficiently high level.Let θ be the Cartan involution of G with respect to our choice of K . Let τ θ = τ ◦ θ . If τ = τ θ , it can be shown that Tr reg (cid:0) e − t ∆ p ( τ ) (cid:1) is exponentially decreasing as t → ∞ andadmits an asymptotic expansion as t →
0. Therefore, the regularized zeta function ζ p ( s ; τ )of ∆ p ( τ ) can be defined as in the compact case by(1.6) ζ p ( s ; τ ) := 1Γ( s ) Z ∞ Tr reg (cid:0) e − t ∆ p ( τ ) (cid:1) t s − dt. The integral converges absolutely and uniformly on compact subsets of the half-planeRe( s ) > d/ s = 0. So in analogy with the compact case, the analytic torsion T X ( τ ) ∈ R + can be defined by the same formula (1.1).In even dimensions, T X ( τ ) is rather trivial (see [MP2]). So we assume that d = 2 n + 1, n ∈ N . To formulate our main result, we need to introduce some notation. We let Γ be afixed lattice in G and we let X := Γ \ e X . We let Γ i , i ∈ N be a sequence of finite indextorsion-free subgroups of Γ . Then following Raimbault [Ra1], in definition 8.2 we definethe condition on the sequence Γ i to be cusp-uniform. This condition is, roughly spoken,a condition on the shape of the 2n-tori which form the cross-sections of the cusps of themanifolds X i := Γ i \ e X . For more details, we refer to section 8. We let ℓ (Γ i ) be the length of WERNER M ¨ULLER AND JONATHAN PFAFF the shortest closed geodesic on X i . We assume that truncation parameters on the orbifold X are fixed and for each i and τ with τ = τ θ we define the analytic torsion with respectto the induced truncation parameters on X i as above. Then our main result can be statedas the following theorem. Theorem 1.1.
Let Γ be a lattice in G . Let Γ i , i ∈ N be a sequence of finite-index subgroupsof Γ which is cusp-uniform. Assume that for i ≥ the group Γ i is torsion free and satisfies (1.5) . Let P Γ i = { P i,j , j = 1 , . . . , κ (Γ i ) } be a set of representatives of Γ i -conjugacy classesof Γ i -cuspidal parabolic subgroups of G and let N P i,j denote the nilpotent radical of P i,j .Assume that lim i →∞ ℓ (Γ i ) = ∞ and that lim i →∞ : Γ i ] (cid:0) κ (Γ i ) + κ (Γ i ) X j =1 log[Γ ∩ N P i,j : Γ i ∩ N P i,j ] (cid:1) = 0 . (1.7) Then for X i := Γ i \ e X and every τ with τ = τ θ one has lim i →∞ log T X i ( τ )[Γ : Γ i ] = t (2) e X ( τ ) vol( X ) . We remark that the condition (1.7) is independent of the choice of P Γ i . Furthermore,one immediately sees that it is satisfied, for example, iflim i →∞ κ (Γ i ) log[Γ : Γ i ][Γ : Γ i ] = 0 . (1.8)For hyperbolic 3-manifolds, Theorem 1.1 was proved by J. Raimbault [Ra1] under addi-tional assumptions on the intertwining operators. We emphasize that we don’t need thisassumption.For sequences of cusp uniform normal subgroups Γ i of Γ which exhaust Γ , the as-sumption (1.7) is easily verified and we have the following theorem for the case of normalsubgroups. Theorem 1.2.
Let Γ be a lattice in G and let Γ i , i ∈ N , be a sequence of finite-indexnormal subgroups which is cusp uniform and such that each Γ i , i ≥ , is torsion-free andsatisfies (1.5) . If lim i →∞ [Γ : Γ i ] = ∞ and if each γ ∈ Γ − { } only belongs to finitelymany Γ i , then for each τ with τ = τ θ one has lim i →∞ log T X i ( τ )[Γ : Γ i ] = t (2) e X ( τ ) vol( X ) . (1.9) In particular, if under the same assumptions Γ i is a tower of normal subgroups, i.e. Γ i +1 ⊂ Γ i for each i and ∩ i Γ i = { } , then (1.9) holds. We shall now give applications of our main results to the case of arithmetic groups.Firstly let Γ := SO ( d, Z ). Then Γ is a lattice in SO ( d, q ∈ N let Γ( q ) bethe principal congruence subgroup of level q (see section 10). Using a result of Deitmarand Hoffmann [DH], it follows that the family of principal congruence subgroups is cuspuniform (see Lemma 10.1). Thus, Theorem 1.2 implies the following corollary. Corollary 1.3.
For any finite-dimensional irreducible representation τ of SO ( d, with τ = τ θ the principal congruence subgroups Γ( q ) , q ≥ , of Γ := SO ( d, Z ) satisfy lim q →∞ log T X q ( τ )[Γ : Γ( q )] = t (2) e X ( τ ) vol( X ) , where X q := Γ( q ) \ H d and X := Γ \ H d . Secondly, we give some specific applications in the 3-dimensional case. There is a naturalisomorphism Spin(3 , ∼ = SL ( C ). If ρ is the standard-representation of SL ( C ) on C , thenthe finite-dimensional irreducible representations of SL ( C ) are given as Sym m ρ ⊗ Sym n ρ , m, n ∈ N . Here Sym k denotes the k -th symmetric power and ρ denotes the complex-conjugate representation of ρ . One has (Sym m ρ ⊗ Sym n ρ ) θ = Sym n ρ ⊗ Sym m ρ . For D ∈ N square-free let O D be the ring of integers of the imaginary quadratic number field Q ( √− D ) and let Γ( D ) := SL ( O D ). Then Γ( D ) is a lattice in SL ( C ). If a is a non-zeroideal in O D , let Γ( a ) be the associated principal congruence subgroup of level a (see section11). Then Theorem 1.2 implies the following corollary. Corollary 1.4. If a i is a sequence of non-zero ideals in O D such that each N ( a i ) issufficiently large and such that lim i →∞ N ( a i ) = ∞ , then for any representation τ =Sym n ρ ⊗ Sym m ¯ ρ with m = n and for X D := Γ( D ) \ H and X i := Γ( a i ) \ H one has lim i →∞ log T X i ( τ )[Γ( D ) : Γ( a i )] = t (2) e X ( τ ) vol( X D ) . (1.10)Finally, due to their arithmetic significance, in the 3-dimensional case we also wantto treat Hecke subgroups of the Bianchi groups. These groups do not fall directly in theframework of our two main theorems, since their systole does not necessarily tend to infinityif their index in the Bianchi groups does. However, a slight modification of the proof of ourmain results will also give the corresponding statement for these groups. More precisely,for a non-zero ideal a of O D let Γ ( a ) be the corresponding Hecke subgroup. Actually,since these groups are not torsion-free, we have to take a fixed torsion-free subgroup Γ ′ ofΓ( D ) of finite index which satisfies assumption (1.5), for example a principal congruencesubgroup of sufficiently high level, and consider the intersections Γ ′ ( a ) := Γ ( a ) ∩ Γ ′ . Thenwe have the following theorem: Theorem 1.5. If a i is a sequence of non-zero ideals in O D such that lim i →∞ N ( a i ) = ∞ ,then for any representation τ = Sym n ρ ⊗ Sym m ¯ ρ with m = n and for X D := Γ( D ) \ H , X ′ i := Γ ′ ( a i ) \ H one has lim i →∞ log T X ′ i ( τ )[Γ( D ) : Γ ′ ( a i )] = t (2) e X ( τ ) vol( X D ) . (1.11)We shall now outline our method to prove our main results. Let d = 2 n + 1. Weassume that the representation τ is not invariant under the Cartan involution. To indicatethe dependence of the heat operator, the regularized trace and other quantities on the WERNER M ¨ULLER AND JONATHAN PFAFF covering X i , we use the subscript X i . Let(1.12) K X i ( t, τ ) := 12 d X p =1 ( − p p Tr reg; X i (cid:0) e − t ∆ Xi,p ( τ ) (cid:1) . As observed above, K X i ( t, τ ) is exponentially decreasing as t → ∞ and admits an as-ymptotic expansion as t →
0. Thus the analytic torsion T X i ( τ ) ∈ R + can be definedby(1.13) log T X i ( τ ) = dds (cid:18) s ) Z ∞ K X i ( t, τ ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 . The integral converges for Re( s ) > d/ s = 0 is defined by analyticcontinuation. For T > T X i ( τ ) = dds (cid:18) s ) Z T K X i ( t, τ ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + Z ∞ T K X i ( t, τ ) t − dt. Now we study the behaviour as i → ∞ of the terms on the right hand side. We start withthe second term. Our assumption about τ implies that the spectrum of the Laplacians∆ X i ,p , i ∈ N , have a uniform positive lower bound. Using the definition (6.12) of theregularized trace, it follows that there exist constants C i , c > t ≥
10 wehave | K X i ( t, τ ) | ≤ C i e − ct The problem is to estimate C i . In Proposition 7.2, we will show that there exists a constant C such that for each i and each t ≥
10 one has an estimation | Tr reg; X i (cid:0) e − t ∆ Xi,p ( τ ) (cid:1) | ≤ Ce − ct (cid:0) Tr reg; X i (cid:0) e − ∆ Xi,p ( τ ) (cid:1) + vol( X i ) (cid:1) (1.15)for each p = 1 , . . . , d . This estimate is easy to prove in the compact case and one does notneed the term vol( X i ) here. More precisely, if X i is compact and if λ ( i ) ≤ λ ( i ) ≤ · · · arethe eigenvalues of ∆ X i ,p ( τ ), counted with multiplicity, then for t ≥ (cid:0) e − t ∆ Xi,p ( τ ) (cid:1) = ∞ X j =1 e − tλ j ( i ) ≤ e − tλ ( i ) / ∞ X j =1 e − λ j ( i ) = e − tλ ( i ) / Tr (cid:0) e − ∆ Xi,p ( τ ) (cid:1) , and the assumption on τ implies that there is c > λ ( i ) ≥ c for all i ∈ N .In the non-compact case, the proof of equation (1.15) is more difficult since one alsohas to deal with the contribution of the continuous spectrum to the regularized trace,which is given by the logarithmic derivative of certain intertwining operators. The keyingredient of our approach to treat the terms involving the intertwining operators is thefactorization of the determinant of the intertwining operators, which we will study carefullyunder coverings in section 4. Our main result is Theorem 4.6.To estimate Tr reg; X i (cid:0) e − ∆ Xi,p (cid:1) we use that the regularized trace of the heat operator,up to a minor term, is equal to the spectral side of the Selberg trace formula applied tothe heat operator (see [MP2]). Then we apply the Selberg trace formula to express theregularized trace through the geometric side of the trace formula. More precisely, let e E τ be the homogeneous vector bundle over e X = G/K associated to τ | K and let e ∆ p ( τ ) be theLaplacian on e E τ -valued p -forms on e X . The heat operator e − t e ∆ p ( τ ) is a convolution operatorwith kernel H ν p ( τ ) t : G → End(Λ p p ∗ ⊗ V τ ). Let h ν p ( τ ) t ( g ) = tr H ν p ( τ ) t ( g ), g ∈ G . Then by thetrace formula we get(1.16) Tr reg; X i (cid:0) e − t ∆ Xi,p ( τ ) (cid:1) = I X ( h τ,pt ) + H X ( h τ,pt ) + T ′ X ( h τ,pt ) + S X ( h τ,pt ) , where I X i , H X i , T ′ X i , and S X i are distributions on G associated to the identity, the hy-perbolic and the parabolic conjugacy classes of Γ i , respectively. The distributions aredescribed in section 8. For example, the identity contribution is given by I X i ( h τ,pt ) = vol( X i ) h τ,pt (1) . Now we put t = 1 and estimate each term on the right hand side of (1.16). In this way wecan conclude that there exist C, c > t ≥
10 and all i ∈ N we have | K X i ( t, τ ) | ≤ C (vol( X i ) + κ ( X i ) + α ( X i )) e − ct , where α ( X i ) is defined in terms of the lattices associated to the cross sections of the cuspsof X i (see (8.11)). Using the assumptions of Theorem 1.1, we finally get that there exist C, c > X i ) (cid:12)(cid:12)(cid:12)(cid:12) Z ∞ T K X i ( t, τ ) t − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − cT for all i ∈ N .To deal with the first term on the right hand side of (1.14), put(1.18) k τt := 12 d X p =1 ( − p ph τ,pt . Then by (1.12) and (1.16) we get(1.19) K X i ( t, τ ) = I X ( k τt ) + H X ( k τt ) + T ′ X ( k τt ) + S X ( k τt ) . Now we take the partial Mellin transform of each term on the right hand side, take itsderivative at s = 0, and study its behaviour as i → ∞ . For the contribution of the identitywe get vol( X i )( t (2) e X ( τ ) + O ( e − cT )). Using the assumptions of Theorem 1.1, it follows thatthe other terms, divided by [Γ : Γ i ], converge to 0. Thus we get(1.20) lim i →∞ : Γ i ] dds (cid:18) s ) Z T K X i ( t, τ ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = vol( X )( t (2) e X ( τ ) + O ( e − cT )) . Combining (1.20), (1.14) and (1.17), and using that
T > C -matrix. WERNER M ¨ULLER AND JONATHAN PFAFF
The main result is Theorem 4.6. In section 5 we consider Bochner-Laplace operators andestablish some properties of their spectrum. In section 6 we introduce the regularized traceof the heat operator using the truncated heat kernel and express it in terms of spectraldata of the corresponding Laplace operator. Section 7 deals with the estimation of theregularized trace of the heat operator for large time. The bound obtained in Proposition7.2 involves the regularized trace of the heat operator at time t = 1. In section 8 we use thegeometric side of the trace formula to study this term in detail. Of particular importanceare the constants obtained from the contribution of the parabolic conjugacy classes whichwe need to estimate uniformly with respect to the covering. In section 9 we prove our maintheorems. In the final sections 10 and 11 we apply our results to derive the corollaries. Acknowledgement.
We would like to thank Tobias Finis for several very helpful expla-nations concerning the Hecke subgroups of the Bianchi groups. In particular, Proposition11.2 and its proof are due to Tobias Finis.2.
Preliminaries
We let d = 2 n + 1, n ∈ N and we let either G = SO ( d, K = SO( d ) or G = Spin( d, K = Spin( d ). Then K is a maximal compact subgroup of G and if the quotient e X := G/K is equipped with the G -invariant metric defined by (2.3), then e X is isometric to the d -dimensional hyperbolic space. Let G = N AK be the Iwasawa decomposition of G as in[MP2, section 2] and let M be the centralizer of A in K . Let g , n , a , k , m denote the Liealgebras of G , N , A K and M . Fix a Cartan subalgebra b of m . Then h := a ⊕ b is a Cartan subalgebra of g . We can identify g C ∼ = so ( d + 1 , C ). Let e ∈ a ∗ be thepositive restricted root defining n . Then we fix e , . . . , e n +1 ∈ i b ∗ such that the positiveroots ∆ + ( g C , h C ) are chosen as in [Kn2, page 684-685] for the root system D n +1 . We let∆ + ( g C , a C ) be the set of roots of ∆ + ( g C , h C ) which do not vanish on a C . The positive roots∆ + ( m C , b C ) are chosen such that they are restrictions of elements from ∆ + ( g C , h C ). For j = 1 , . . . , n + 1 let(2.1) ρ j := n + 1 − j. Then the half-sums of positive roots ρ G and ρ M , respectively, are given by ρ G := 12 X α ∈ ∆ + ( g C , h C ) α = n +1 X j =1 ρ j e j ; ρ M := 12 X α ∈ ∆ + ( m C , b C ) α = n +1 X j =2 ρ j e j . (2.2)Put h X, Y i θ := − d − B ( X, θ ( Y )) , X, Y ∈ g . (2.3)Let Z (cid:2) (cid:3) j be the set of all ( k , . . . , k j ) ∈ Q j such that either all k i are integers or all k i arehalf integers. Let Rep( G ) denote the set of finite dimensional irreducible representations τ of G . These are parametrized by their highest weights(2.4) Λ( τ ) = k ( τ ) e + · · · + k n +1 ( τ ) e n +1 ; k ( τ ) ≥ k ( τ ) ≥ · · · ≥ k n ( τ ) ≥ | k n +1 ( τ ) | , where ( k ( τ ) , . . . , k n +1 ( τ )) belongs to Z (cid:2) (cid:3) n +1 if G = Spin( d,
1) and to Z n +1 if G =SO ( d, ν ∈ ˆ K of K areparametrized by their highest weights(2.5) Λ( ν ) = k ( ν ) e + · · · + k n +1 ( ν ) e n +1 ; k ( ν ) ≥ k ( ν ) ≥ · · · ≥ k n ( ν ) ≥ k n +1 ( ν ) ≥ , where ( k ( ν ) , . . . , k n +1 ( ν )) belongs to Z (cid:2) (cid:3) n if G = Spin( d,
1) and to Z n if G = SO ( d, σ ∈ ˆ M of M are parametrizedby their highest weights(2.6) Λ( σ ) = k ( σ ) e + · · · + k n +1 ( σ ) e n +1 ; k ( σ ) ≥ k ( σ ) ≥ · · · ≥ k n ( σ ) ≥ | k n +1 ( σ ) | , where ( k ( σ ) , . . . , k n +1 ( σ )) belongs to Z (cid:2) (cid:3) n , if G = Spin( d, Z n , if G = SO ( d, ν ∈ ˆ K and σ ∈ ˆ M we denote by [ ν : σ ] the multiplicity of σ in the restriction of ν to M .Let Ω, Ω K and Ω M be the Casimir elements of G , K and M , respectively, with respectto the inner product (2.3). Then by a standard computation one hasΩ = H − nH + Ω M mod n U ( g ) . (2.7)Let M ′ be the normalizer of A in K and let W ( A ) = M ′ /M be the restricted Weyl-group.It has order two and it acts on the finite-dimensional representations of M as follows. Let w ∈ W ( A ) be the non-trivial element and let m ∈ M ′ be a representative of w . Given σ ∈ ˆ M , the representation w σ ∈ ˆ M is defined by w σ ( m ) = σ ( m mm − ) , m ∈ M. Let Λ( σ ) = k ( σ ) e + · · · + k n +1 ( σ ) e n +1 be the highest weight of σ as in (2.6). Then thehighest weight Λ( w σ ) of w σ is given by(2.8) Λ( w σ ) = k ( σ ) e + · · · + k n ( σ ) e n − k n +1 ( σ ) e n +1 . Let P := N AM . We equip a with the norm induced from the restriction of the normal-ized Killing form on g . Let H ∈ a be the unique vector which is of norm one and such thatthe positive restricted root, implicit in the choice of N , is positive on H . Let exp : a → A be the exponential map. Every a ∈ A can be written as a = exp log a , where log a ∈ a isunique. For t ∈ R , we let a ( t ) := exp ( tH ). If g ∈ G , we define n ( g ) ∈ N , H ( g ) ∈ R and κ ( g ) ∈ K by g = n ( g ) a ( H ( g )) κ ( g ) . Now let P ′ be any parabolic subgroup. Then there exists a k P ′ ∈ K such that P ′ = N P ′ A P ′ M P ′ with N P ′ = k P ′ N k − P ′ , A P ′ = k P ′ Ak − P ′ , M P ′ = k P ′ M k − P ′ . We choose a set of k P ′ ’s, which will be fixed from now on. Let k P = 1. We let a P ′ ( t ) := k P ′ a ( t ) k − P ′ . If g ∈ G ,we define n P ′ ( g ) ∈ N P ′ , H P ′ ( g ) ∈ R and κ P ′ ( g ) ∈ K by g = n P ′ ( g ) a P ′ ( H P ′ ( g )) κ P ′ ( g )(2.9) and we define an identification ι P ′ of (0 , ∞ ) with A P ′ by ι P ′ ( t ) := a P ′ (log( t )). For Y >
0, let A P ′ [ Y ] := ι P ′ ( Y, ∞ ) and A P ′ [ Y ] := ι P ′ [ Y, ∞ ). For g ∈ G as in (2.9) we let y P ′ ( g ) := e H P ′ ( g ) .Let Γ be a discrete subgroup of G such that vol(Γ \ G ) < ∞ . We do not assume atthe moment that Γ is torsion-free. Let X := Γ \ e X . Let pr X : G → X be the projection.A parabolic subgroup P ′ of G is called a Γ-cuspidal parabolic subgroup if Γ ∩ N P ′ is alattice in N P ′ . Let P Γ = { P , . . . , P κ (Γ) } be a set of representatives of Γ-conjugacy-classesof Γ-cuspidal parabolic subgroups of G . Then for each P ′ ∈ P Γ one hasΓ ∩ P ′ = Γ ∩ ( M P ′ N P ′ ) . (2.10)The number(2.11) κ ( X ) := κ (Γ) = P Γ is finite and equals the number of cusps of X . More precisely, for each P i ∈ P Γ there existsa Y P i > C = C ( Y P , . . . , Y P κ (Γ) ) of G suchthat in the sense of a disjoint union one has G = Γ · C ⊔ κ ( X ) G i =1 Γ · N P i A P i [ Y P i ] K (2.12)and such that γ · N P i A P i [ Y P i ] K ∩ N P i A P i [ Y P i ] K = ∅ ⇔ γ ∈ Γ ∩ P i . (2.13)For each P i ∈ P Γ let(2.14) Y P i (Γ) = inf { Y P i : Y P i ∈ R + satisfies (2.13) } . Moreover, we define the height-function y Γ ,P i on X by(2.15) y Γ ,P i ( x ) := sup { y P i ( g ) : g ∈ G, pr X ( g ) = x } . By (2.12) and (2.13) the supremum is finite. For Y , . . . , Y κ ( X ) ∈ (0 , ∞ ) we let(2.16) X ( P , . . . , P κ ( X ) ; Y , . . . , Y κ ( X ) ) := { x ∈ X : y Γ ,P i ( x ) ≤ Y i , i = 1 , . . . , κ ( X ) } . If Y ∈ (0 , ∞ ), we write X P Γ ( Y ) or X ( P , . . . , P κ ( X ) ; Y ) for X ( P , . . . , P κ ( X ) ; Y, . . . , Y ), i.e. X P Γ ( Y ) := X ( P , . . . , P κ ( X ) ; Y ) := { x ∈ X : y Γ ,P i ( x ) ≤ Y, i = 1 , . . . , κ ( X ) } . (2.17)For later purposes we now recall the interpretation of the semisimple elements in termsof closed geodesics. For further details we refer, for example, to [Pf1, section 3]. We let Γ s denote the semisimple elements of Γ which are not G -conjugate to an element of K . ByC(Γ) s we denote the set of Γ-conjugacy classes of elements of Γ s . Then for each γ ∈ Γ s there exists a unique geodesic e c γ in e X which is stabilized by γ . If one lets ℓ ( γ ) = inf x ∈ e X d ( x, γx ) , (2.18)then ℓ ( γ ) > e X lying on e c γ . Let C ( X ) denote the set of closed geodesics of X . For γ ∈ Γ s let c γ be the projection of thesegment of e c γ from x to γx , x a point on e c γ , to X . Then one can show that c γ depends only on the Γ-conjugacy class of γ and that the assignment γ c γ induces a bijectionbetween C(Γ) s and C ( X ). For c ∈ C ( X ) let ℓ ( c ) denote its length. Then there exists aconstant C X such that for each R one can estimate { c ∈ C ( X ) : ℓ ( c ) ≤ R } ≤ C X e nR . (2.19)In particular, if one sets ℓ (Γ) := ℓ ( X ) := inf { ℓ ( c ) : c ∈ C ( X ) } , (2.20)then ℓ (Γ) > K such that K has volume 1. We fix an isometric identification of R n with n with respect to the innerproduct h· , ·i θ . We give n the measure, induced from the Lebesgue measure under thisidentification. Moreover, we identify n and N by the exponential map and we will denoteby dn the Haar measure on N , induced from the measure on n under this identification.We normalize the Haar measure on G by setting Z G f ( g ) dg = Z N Z R Z K e − nt f ( na ( t ) k ) dkdtdn. (2.21)If P ′ is a parabolic subgroup of G , the measures on N P ′ and A P ′ will be the measuresinduced from N and A via the conjugation with k P ′ . Let f be integrable over Γ \ G . Thenidentifying f with a measurable function on G it follows from (2.21), (2.12) and (2.13) thatfor every Y ≥ Y one has Z Γ \ G f ( x ) dx = Z C ( Y ) f ( g ) dg + κ (Γ) X i =1 Z Γ ∩ N Pi \ N Pi Z ∞ log Y Z K e − nt f ( n P i a P i ( t ) k ) dn P i dtdk (2.22)For σ ∈ ˆ M and λ ∈ C let π σ,λ be the principal series representation of G parametrizedas in [MP2, section 2.7]. In particular, the representations π σ,λ are unitary iff λ ∈ R . Wedenote by Θ σ,λ the global character of π σ,λ . For σ ∈ ˆ M with highest weight Λ( σ ) as in(2.6) let σ (Ω M ) be the Casimir eigenvalue of σ and let c ( σ ) := σ (Ω M ) − n = n +1 X j =2 ( k j ( σ ) + ρ j ) − n +1 X j =1 ρ j , (2.23)where the second equality follows from a standard computation.3. Eisenstein series
In this section we recall the definition and some basic properties of the Eisenstein series.Let Γ be a discrete subgroup of G such that vol(Γ \ G ) is finite. Furthermore, for conveniencewe assume in this section that Γ is torsion-free and that for each Γ-cuspidal parabolicsubgroup P ′ of G one has Γ ∩ P ′ = Γ ∩ N P ′ . (3.1) Let P Γ be a fixed set of representatives of Γ-conjugacy classes of Γ-cuspidal parabolicsubgroups of G . Let P i ∈ P Γ . For σ ∈ ˆ M we define a representation σ P i of M P i by(3.2) σ P i ( m P i ) := σ ( k − P i m P i k P i ) , m P i ∈ M P i . Now let ν ∈ ˆ K and σ P ∈ ˆ M such that [ ν : σ ] = 0. Then we let E P i ( σ, ν ) be the set of allcontinuous functions Φ on G which are left-invariant under N P i A P i such that for all x ∈ G the function m Φ P i ( mx ) belongs to L ( M P i , σ P i ), the σ P i -isotypical component of theright regular representation of M P i , and such that for all x ∈ G the function k Φ P i ( xk )belongs to the ν -isotypical component of the right regular representation of K . The space E P i ( σ, ν ) is finite dimensional and in fact one hasdim( E P i ( σ, ν )) = dim( σ ) dim( ν ) . (3.3)We define an inner product on E P i ( σ, ν ) as follows. Any element of E P i ( σ, ν ) can beidentified canonically with a function on K . For Φ , Ψ ∈ E P i ( σ, ν ) put h Φ , Ψ i := vol(Γ ∩ N P i \ N P i ) Z K Φ( k ) ¯Ψ( k ) dk. (3.4)Define the Hilbert space E P i ( σ ) by E P i ( σ ) := M ν ∈ ˆ K [ ν : σ ] =0 E P i ( σ, ν ) . For Φ P i ∈ E P i ( σ, ν ) and λ ∈ C letΦ P i ,λ ( g ) := e ( λ + n )( H Pi ( x )) Φ P i ( g ) . (3.5)Let x ∈ Γ \ G , x = Γ g . Then the Eisenstein series E (Φ P i : λ : x ) is defined by E (Φ P i : λ : x ) := X γ ∈ (Γ ∩ N Pi ) \ Γ Φ P i ,λ ( γg ) . (3.6)On Γ \ G × { λ ∈ C : Re( λ ) > n } the series (3.6) is absolutely and locally uniformly conver-gent. As a function of λ , it has a meromorphic continuation to C with only finitely manypoles in the strip 0 < Re( λ ) ≤ n which are located on (0 , n ] and it has no poles on the lineRe( λ ) = 0. By (2.7), for σ ∈ ˆ M with [ ν : σ ] = 0 and Φ P i ∈ E ( σ, ν ) one hasΩΦ P i ,λ = ( λ + c ( σ ))Φ P i ,λ , (3.7)where c ( σ ) is as in (2.23). Since Ω is G -invariant it follows thatΩ E (Φ P i : λ : x ) = ( λ + c ( σ )) E (Φ P i : λ : x ) . (3.8)Let E ( σ, ν ) := M P i ∈ P Γ E P i ( σ, ν ); E ( σ ) := M P i ∈ P Γ E P i ( σ ) . By (3.3) one has dim E ( σ, ν ) = κ (Γ) dim( σ ) dim( ν ) . (3.9) Let P i , P j ∈ P Γ and let σ ∈ ˆ M . For Φ P i ∈ E P i ( σ, ν ), i = 1 ,
2, and g ∈ G let E P j (Φ P i : g : λ ) := 1vol (cid:0) Γ ∩ N P j \ N P j (cid:1) Z Γ ∩ N Pj \ N Pj E (Φ P i : ng : λ ) dn be the constant term of E (Φ P i : − : λ ) along P j . Then there exists a meromorphic function C P i | P j ( σ : ν : λ ) : E P i ( σ, ν ) −→ E P j ( w σ, ν ) , such that for P i , P j ∈ P Γ one has E P j (Φ P i : g : λ ) = δ i,j Φ P i ,λ ( g ) + ( C P i | P j ( σ : ν : λ )Φ P i ) − λ ( g ) . (3.10)Now we let C P i | P j ( σ P i , λ ) := M ν ∈ ˆ K [ ν : σ ] =0 C P i | P j ( σ, ν, λ ) , where σ P i is defined by (3.2). Furthermore, let C ( σ, λ ) : E ( σ ) → E ( w σ ); C ( σ, ν, λ ) : E ( σ, ν ) → E ( w σ, ν )be the maps built from the maps C P i | P j ( σ, λ ), resp. C P i | P j ( σ, ν, λ ). Then one has C ( w σ, λ ) C ( σ, − λ ) = Id; C ( σ, λ ) ∗ = C ( w σ, ¯ λ ) . (3.11)Let σ ∈ ˆ M and ν ∈ ˆ K . If σ = w σ , let E P i ( σ, ν ) := E P i ( σ, ν ), E ( σ, ν ) := E ( σ, ν ), C ( σ : ν : s ) := C ( σ : ν : s ). If σ = w σ , let E P i ( σ, ν ) := E P i ( σ, ν ) ⊕ E P i ( w σ, ν ) E ( σ, ν ) := E ( σ, ν ) ⊕ E ( w σ , ν ) and(3.12) C ( σ, ν, s ) : E ( σ, ν ) → E ( σ, ν ); C ( σ, ν, s ) := (cid:18) C ( w σ, ν, s ) C ( σ, ν, s ) 0 (cid:19) . Let R σ (resp. R w σ ) denote the right regular representation of K on E ( σ ) (resp. E ( w σ )).Then C ( σ, s ) is an intertwining operator between R σ and R w σ . Thus if ν is a finite-dimensional representation of K on V ν , we can define e C ( σ, ν, s ) as the restriction of( C ( σ, s ) ⊗ Id) to a map from ( E ( σ ) ⊗ V ν ) K to ( E ( w σ ) ⊗ V ν ) K . For later purpose weneed the following Lemma. Lemma 3.1.
In the sense of meromorphic functions one has Tr (cid:18) e C ( σ, ν, s ) − dds e C ( σ, ν, s ) (cid:19) = 1dim( ν ) Tr (cid:18) C ( σ, ν, s ) − dds C ( σ, ν, s ) (cid:19) for each σ ∈ ˆ M , ν ∈ ˆ K with [ ν : σ ] = 0 .Proof. Let P be the projection form E ( σ ) to E ( σ, ν ) and let P be the projection from( E ( σ ) ⊗ V ν ) to ( E ( σ ) ⊗ V ν ) K . Then using that ˇ ν ∼ = ν we have P = dim( ν ) Z K χ ν ( k ) R σ ( k ); P = Z K R σ ( k ) ⊗ ν ( k ) dk, where χ ν is the character of ν . Thus one hasTr (cid:18) e C ( σ, ν, s ) − dds e C ( σ, ν, s ) (cid:19) = Tr (cid:18) C ( σ, s ) − dds C ( σ, s ) ⊗ Id ◦ P (cid:19) = Tr (cid:18)Z K C ( σ, s ) − dds C ( σ, s ) ◦ R σ ( k ) ⊗ ν ( k ) dk (cid:19) = Tr (cid:18)Z K C ( σ, s ) − dds C ( σ, s ) ◦ χ ν ( k ) R σ ( k ) dk (cid:19) = 1dim( ν ) Tr (cid:18) C ( σ, s ) − dds C ( σ, s ) ◦ P (cid:19) = 1dim( ν ) Tr (cid:18) C ( σ, ν, s ) − dds C ( σ, ν, s ) (cid:19) , which concludes the proof of the proposition. (cid:3) Factorization of the C-matrix
We let Γ be a discrete subgroup of G satisfying (3.1) and we keep the notations ofthe previous section. By the results of M¨uller, in particular [Mu1, equation (6.8)], thedeterminant of the matrix C ( σ, ν, λ ) factorizes into a product of an exponential factor andan infinite Weierstrass product involving its zeroes and poles. For the case of a hyperbolicsurface, this factorization was first established by Selberg (see[Se, page 656]).While the poles and zeroes of the C -matrices are easily seen to be independent of thechoice of P Γ , the exponential factor depends on P Γ or, equivalently, on the choice oftruncation parameters. This fact will become particularly crucial if one lets the manifold X vary. In [Mu1], the manifold X and the set P Γ were fixed. Therefore, for the purposesof the present article we have to go through the arguments of the paper [Mu1] whichled to equation (6.8) in this paper and to keep track of the precise choices of truncationparameters.Let R Γ be the right regular representation of G on L (Γ \ G ). If ν is a finite dimensionalrepresentation of K , let L (Γ \ G ) ν denote the ν -isotypical component of the restriction of R Γ to K . Let C ∞ c (Γ \ G ) ν := C ∞ c (Γ \ G ) ∩ L (Γ \ G ) ν . Then it is easy to see that C ∞ c (Γ \ G ) ν is dense in L (Γ \ G ) ν .Now let ∆ ν be the differential operator in C ∞ (Γ \ G ) ν , which is induced by − R Γ (Ω). If weregard it as an operator in L (Γ \ G ) ν with domain C ∞ c (Γ \ G ) ν , it is symmetric, essentiallyselfadjoint and satisfies ∆ ν ≥ − ν (Ω K ), where ν (Ω K ) ∈ R + is the Casimir eigenvalue of ν .This follows easily from the considerations in the next section 5. The closure of ∆ ν will bedenoted by ∆ ν . One has σ (∆ ν ) ⊂ ( − ν (Ω K ) , ∞ ) . (4.1)We fix a smooth function φ on R with values in [0 ,
1] such that φ ( t ) = 0 for t ≤ φ ( t ) = 1 for t ≥
1. If P i ∈ P Γ , then for Y ∈ (0 , ∞ ) we let ψ P i ,Y ( n P i a P i ( t ) k ) := φ ( t − log Y ) , n P i ∈ N P i , t ∈ R . Now let Y P i ∈ (0 , ∞ ), i = 1 , . . . , κ (Γ), such that Y P i ≥ Y P i (Γ), where Y P i (Γ) is definedby (2.14). For Φ P i ∈ E ( σ P i , ν ) we define a function θ (Φ P i : Y P i : λ : x ) on Γ \ G by θ (Φ P i : Y P i : λ : x ) := X γ ∈ Γ ∩ N Pi \ Γ ψ P i ,Y Pi ( γg )Φ P i ,λ ( γg ); x = Γ g. (4.2)By (2.13) at most one summand in this sum can be non-zero . We let H (Φ P i : Y P i : λ : x ) := (∆ ν + c ( σ P i ) + λ ) θ (Φ P i : Y P i : λ : x ) . Then by (3.7) one has H (Φ P i : Y P i : λ : x ) ∈ C ∞ c (Γ \ G ) ν . Moreover, the Eisenstein seriescan be characterized by the following Proposition, which for dim X = 2 is due to Colin deVerdi`ere [CV]. Proposition 4.1.
For P i ∈ P Γ , Y P i ≥ Y P i (Γ) and λ ∈ C with λ + c ( σ ) / ∈ ( −∞ , ν (Ω K )) and Re( λ ) > one has E (Φ P i : λ : x ) = θ (Φ P i : Y P i : λ : x ) − (∆ ν + λ + c ( σ )) − ( H (Φ P i : Y P i : λ : x )) . Proof.
This was proved in general in [Mu1, Proposition 4.7]. For the convenience of thereader we recall the proof. Denote the right hand side by e E (Φ P i : λ : x ). By definitionit satisfies (∆ ν + λ + c ( σ )) e E (Φ P i : λ : x ) = 0. By (3.8), E (Φ P i : λ : x ) satisfies thesame differential equation. By [Mu1, Lemma 4.5], E (Φ P i : λ ) − θ (Φ P i : Y P i : λ ) is squareintegrable for Re( λ ) > n . Hence, u := E ((Φ P i : λ ) − e E (Φ P i : λ ) is square integrable forRe( λ ) > n and satisfies (∆ ν + λ + c ( σ )) u = 0. Since ∆ ν is essentially self-adjoint, it followsthat E ((Φ P i : λ ) = e E (Φ P i : λ ) for Re( λ ) > n . The proposition follows by the uniquenessthe analytic continuation. (cid:3) Lemma 4.2.
There exists a constant C which is independent of Γ and P Γ such thatfor all λ ∈ C with λ + c ( σ ) / ∈ ( −∞ , ν (Ω K )) and Re( λ ) > , all Y P i ≥ Y P i (Γ) , and all Φ P i ∈ E P i ( σ, ν ) , P i ∈ P Γ , one has k H (Φ P i : Y P i : λ : x ) k L (Γ \ G ) ≤ C e Re( λ )(log Y Pi +2) k Φ P i k E Pi ( σ,ν ) . Proof.
There exists a unique Φ P ∈ E P ( σ, ν ) such that Φ P i ,λ ( g ) = Φ P,λ ( κ − P i gκ P i ). Since ∆ ν commutes with the right-action of G on Γ \ G , it follows from (2.22) that Z Γ \ G | H (Φ P i : Y P i : λ : x ) | dx = vol(Γ ∩ N P i \ N P i ) Z log Y Pi +1log Y Pi Z K e − nt | (∆ ν + c ( σ ) + λ ) ψ P i ,Y Pi ( a P i ( t ))Φ P i ,λ ( a P i ( t ) k ) | dkdt = vol(Γ ∩ N P i \ N P i ) Z log Y Pi +1log Y Pi Z K e − nt | (∆ ν + c ( σ ) + λ ) ψ P,Y Pi ( a ( t ))Φ P,λ ( a ( t ) k ) | dkdt. Now using (2.7) and (3.8) one obtains(∆ ν + c ( σ ) + λ )( ψ P,Y Pi ( a ( t ))Φ P,λ ( a ( t ) k ))= − e ( λ + n ) t Φ P ( k )( φ ′′ ( t − log Y P i ) + 2 λφ ′ ( t − log Y P i )) . This proves the proposition. (cid:3)
Corollary 4.3.
There exists a constant C which is independent of Γ and P Γ such thatfor all λ ∈ C with Re( λ ) + c ( σ ) ≥ ν (Ω K ) + 1 and Re( λ ) > , all Y P i ≥ Y P i (Γ) and all Φ P i ∈ E P i ( σ, ν ) , P i ∈ P Γ , one has k (∆ ν + λ + c ( σ )) − H (Φ P i : Y P i : λ : x ) k L (Γ \ G ) ≤ C e Re( λ )(log Y Pi +2) k Φ P i k E Pi ( σ,ν ) Proof.
By [Ka, V, § k (∆ ν + λ + c ( σ )) − k ≤ − λ − c ( σ ) , spec(∆ ν )) , where the estimate holds without any constant. Applying the previous Lemma and (4.1),the corollary follows. (cid:3) In the following proposition we estimate the coefficients of the C -matrix. Proposition 4.4.
There exists a constant C , which is independent of Γ and P Γ suchthat for all P i , P j ∈ P Γ , all Y P i , Y P j ∈ (0 , ∞ ) with Y P i ≥ Y P i (Γ) , Y P j ≥ Y P j (Γ) , all Φ P i ∈E P i ( σ, ν ) , Φ P j ∈ E P j ( σ, ν ) and all λ ∈ C with Re( λ ) + c ( σ ) ≥ ν (Ω K ) + 1 and Re( λ ) > ,one has | (cid:10) C P i | P j ( σ, ν, λ )(Φ P i ) , Φ P j (cid:11) E Pj ( σ,ν ) | ≤ C e Re( λ )(log Y Pi +log Y Pj +4) k Φ P i k E Pi ( σ,ν ) · k Φ P j k E Pj ( σ,ν ) . Proof.
By the definition (3.10) of the constant term it follows that for each t ∈ R and each k ∈ K one has C P i | P j ( σ, ν, λ )(Φ P i )( k ) = e ( λ − n ) t ( C P i | P j ( σ, ν, λ )(Φ P i )) − λ ( a P j ( t ) k )= e ( λ − n ) t (cid:0) E P j (Φ P i : a P j ( t ) k : λ ) − δ i,j Φ P i ,λ ( a P j ( t ) k ) (cid:1) . Moreover, by (2.12) and (2.13), for t ≥ log Y P j + 1 one has θ (Φ P i : Y P i : λ : a P j ( t ) k ) = δ i,j Φ P i ,λ ( a P j ( t ) k ) . Thus by Proposition 4.1 for t ≥ log Y P j + 1 one has E P j (Φ P i : a P j ( t ) k : λ ) − δ i,j Φ P i ,λ ( a P j ( t ) k )= − (cid:0) Γ ∩ N P j \ N P j (cid:1) Z Γ ∩ N Pj \ N Pj (∆ ν + c ( σ ) + λ ) − ( H (Φ P i : Y P i : λ : n P j a P j ( t ) k )) dn P j . Combining these equations, it follows that for each t ≥ log Y P j + 1 one has (cid:10) C P i | P j ( σ, ν, λ )(Φ P i ) , Φ P j (cid:11) E Pj ( σ,ν ) = vol(Γ ∩ N P j \ N P j ) Z K Φ P j ( k ) C P i | P j ( σ, ν, λ )(Φ P i )( k ) dk = − e ( λ − n ) t × Z K Φ P j ( k ) Z Γ ∩ N Pj \ N Pj (∆ ν + c ( σ ) + λ ) − ( H (Φ P i : Y P i : λ : n P j a P j ( t ) k )) dn P j dk. (4.3) Now we define a function e f P j ,λ on G by e f P j ,λ ( n P j a P j ( t ) k ) = e ( n + λ ) t χ [log Y Pj , log Y Pj +1] ( t )Φ P j ( k ) , where χ [log Y Pj , log Y Pj +1] ( t ) denotes the characteristic function of the interval [log Y P j , log Y P j +1]. Then we define a function f P j ,λ on Γ \ G by f P j ,λ ( x ) = X γ ∈ Γ ∩ P j \ Γ e f P j ,λ ( γg ) , x = Γ g. By (2.13), at most one summand in this sum can be nonzero. Integrating equation (4.3)over t in the interval [log Y P j , log Y P j + 1] and using (2.22), we obtain (cid:12)(cid:12) (cid:10) C P i | P j ( σ, ν, λ )(Φ P i ) , Φ P j (cid:11) E Pj ( σ,ν ) (cid:12)(cid:12) = (cid:12)(cid:12) (cid:10) (∆ ν + c ( σ ) + λ ) − ( H (Φ P i : Y P i : λ )) , f P j ,λ (cid:11) L (Γ \ G ) (cid:12)(cid:12) . Now observe that k f P j ,λ k L (Γ \ G ) ≤ e Re( λ )(log Y Pj +1) k Φ P j k E Pj ( σ,ν ) . Applying Corollary 4.3, the Proposition follows. (cid:3)
Summarizing our results, we obtain the following refinement of [Mu1, Lemma 6.1].
Corollary 4.5.
Let ¯ d ( σ, ν ) := dim E P ( σ, ν ) . For each P i ∈ P Γ let Y P i ≥ Y P i (Γ) be given.Put q := κ (Γ) Y i =1 e Y Pi +2) ¯ d ( σ,ν ) . There exists a constant
C > which is independent of Γ , P Γ , and Y P i , i = 1 , ..., κ (Γ) , suchthat for all λ ∈ C satisfying Re( λ ) + c ( σ ) ≥ ν (Ω K ) + 1 and Re( λ ) > , one has | det( C ( σ, ν, λ )) | ≤ Cq Re( λ )1 . Proof.
If one chooses for each i = 1 , . . . , κ (Γ) an orthonormal base of E P i ( σ, ν ) resp. E P i ( w σ, ν ) and applies the preceding proposition, the corollary follows immediately fromthe Leibniz formula for the determinant. (cid:3) Applying the previous Corollary we can restate the factorization of the C -matrix, [Mu1,equation 6.8] with an expression for the exponential factor in terms of the truncationparameters that will be sufficient for our later considerations. Theorem 4.6.
Let σ j , j = 1 , . . . , l denote the poles of det( C ( σ, ν, λ )) in the interval (0 , n ] and let η run through the poles of det( C ( σ, ν, λ )) in the half-plane Re( λ ) ≤ , both countedwith multiplicity. Then one has det( C ( σ, ν, λ )) = det( C ( σ, ν, q λ l Y j =1 λ + σ j λ − σ j Y η λ + ¯ ηλ − η . Moreover, if for each P i ∈ P Γ a Y P i ∈ (0 , ∞ ) with Y P i ≥ Y P i (Γ) is given, then q can bewritten as q = e a κ (Γ) Y i =1 e Y Pi +2) ¯ d ( σ,ν ) , (4.4) where a ∈ R , a ≤ .Proof. Using the previous Corollary instead of [Mu1, Lemma 6.1], one can proceed exactlyas in [Mu1, section 6] to obtain the Theorem. (cid:3) Twisted Laplace operators
Let ν be a finite dimensional unitary representation of K on ( V ν , h· , ·i ν ). Let˜ E ν := G × ν V ν be the associated homogeneous vector bundle over ˜ X . Then h· , ·i ν induces a G -invariantmetric ˜ B ν on ˜ E ν . Let E ν := Γ \ ( G × ν V ν )be the associated locally homogeneous bundle over X . Since ˜ B ν is G -invariant, it can bepushed down to a fiber metric B ν on E ν . Let C ∞ ( G, ν ) := { f : G → V ν : f ∈ C ∞ , f ( gk ) = ν ( k − ) f ( g ) , ∀ g ∈ G, ∀ k ∈ K } . (5.1)Let C ∞ (Γ \ G, ν ) := { f ∈ C ∞ ( G, ν ) : f ( γg ) = f ( g ) , ∀ g ∈ G, ∀ γ ∈ Γ } . (5.2)Let C ∞ ( X, E ν ) denote the space of smooth sections of E ν . Then there is a canonicalisomorphism A : C ∞ ( X, E ν ) ∼ = C ∞ (Γ \ G, ν )(see [Mi1, p. 4]). There is also a corresponding isometry for the space L ( X, E ν ) of L -sections of E ν .Let τ be an irreducible finite dimensional representation of G on V τ . Let E τ be the flatvector bundle associated to the restriction of τ to Γ. Let e E τ → e X be the homogeneousvector bundle associated to τ | K . Then by [MtM] there is canonical isomorphism E τ ∼ = Γ \ e E τ . By [MtM], there exists an inner product h· , ·i on V τ such that(1) h τ ( Y ) u, v i = − h u, τ ( Y ) v i for all Y ∈ k , u, v ∈ V τ (2) h τ ( Y ) u, v i = h u, τ ( Y ) v i for all Y ∈ p , u, v ∈ V τ . Such an inner product is called admissible. It is unique up to scaling. Fix an admissibleinner product. Since τ | K is unitary with respect to this inner product, it induces a fibermetric on e E τ , and hence on E τ . This fiber metric will also be called admissible. LetΛ p ( X, E τ ) be the space of E τ -valued p -forms. This is the space of smooth sections of thevector bundle Λ p ( E τ ) := Λ p T ∗ X ⊗ E τ . Let(5.3) d p ( τ ) : Λ p ( X, E τ ) → Λ p +1 ( X, E τ )be the exterior derivative and let(5.4) ∆ p ( τ ) = d p ( τ ) ∗ d p ( τ ) + d p − ( τ ) d p − ( τ ) ∗ be the Laplace operator on E τ -valued p -forms. This operator can be expressed in thelocally homogeneous setting as follows. Let ν p ( τ ) be the representation of K defined by(5.5) ν p ( τ ) := Λ p Ad ∗ ⊗ τ : K → GL(Λ p p ∗ ⊗ V τ ) . There is a canonical isomorphismΛ p ( E τ ) ∼ = Γ \ ( G × ν p ( τ ) (Λ p p ∗ ⊗ V τ )) , (5.6)which induces an isomorphismΛ p ( X, E τ ) ∼ = C ∞ (Γ \ G, ν p ( τ )) . (5.7)There is a corresponding isometry of the L -spaces. Let τ (Ω) be the Casimir eigenvalue of τ . With respect to the isomorphism (5.7) on has∆ p ( τ ) = − R Γ (Ω) + τ (Ω) Id(5.8)(see [MtM, (6.9)]). Next we want to show that the discrete spectrum of the operators ∆ p ( τ )is greater or equal than 1 / p and each τ ∈ Rep( G ) satisfying τ = τ θ . This wasalready stated in [MP2, Lemma 7.3]. However, as it was kindly brought to our attention byMartin Olbrich, the parametrization of the complementary series used in the proof of thatLemma was incorrect. Therefore we shall now correct the part of the argument leadingto the proof of [MP2, Lemma 7.3] which involved the complementary series. We let ˆ G un denote the unitary dual of G . Lemma 5.1.
Let τ ∈ Rep( G ) such that τ = τ θ . Let π ∈ ˆ G un belong to the complementaryseries. Let p ∈ { , . . . , d } . Then if [ π : ν p ( τ )] = 0 one has − π (Ω) + τ (Ω) ≥ .Proof. Let τ be a finite-dimensional irreducible representation of G of highest weightΛ( τ ) = τ e + · · · + τ n +1 e n +1 as in (2.4) and assume that τ = τ θ . Let p ∈ { , . . . , d } and let σ ∈ ˆ M such that [ ν p ( τ ) : σ ] = 0. Assume that σ = w σ . Let Λ( σ ) = k ( σ ) e + · · · + k n +1 ( σ ) e n +1 be the highest weight of σ as in (2.6). It was shown in theproof of [MP2, Lemma 7.1] that τ j − + 1 ≥ | k j ( σ ) | for every j ∈ { , . . . , n + 1 } . Let c ( σ ) beas in (2.23) and let l ∈ { , . . . , n } be minimal with the property that k l +1 ( σ ) = 0. Using ρ j − = ρ j + 1 and [MP1, equation 2.20], it follows that one can estimate c ( σ ) = l X j =2 ( k j ( σ ) + ρ j ) − n +1 X j =1 ρ j ≤ l X j =2 ( τ j − + ρ j − ) − n +1 X j =1 ρ j = τ (Ω) − n +1 X j = l ( τ j + ρ j ) . (5.9)We parametrize the principal series representations as above. Then if π belongs to thecomplementary series, by [KS, Proposition 49, Proposition 53] and our parametrizationthere exists a σ ∈ ˆ M , σ = w σ and a λ ∈ (0 , n − l + 1), where l is minimal with theproperty that k l +1 ( σ ) = 0, such that π σ,iλ is unitarizable with unitarization π . We write π = π cσ,iλ . If [ π cσ,iλ : ν p ( τ )] = 0, then by Frobenius reciprocity [Kn1, page 208] onehas [ ν p ( τ ) : σ ] = 0. Thus, since σ = w σ , it follows easily from the branching lawsfor restrictions of representations from G to K and from K to M , [GW][Theorem 8.1.3,Theorem 8.1.4] that all k j ( τ ) defined as in (2.4) are integral. By [MP1, Corollary 2.4] onehas − π cσ,iλ (Ω) + τ (Ω) = − λ − c ( σ ) + τ (Ω)(5.10)and if we apply equation (5.9) and the condition | τ n +1 | ≥
1, it follows that − π cσ,iλ (Ω) + τ (Ω) ≥ n +1 X j = l ( τ j + ρ j ) − ( n − l + 1) = n +1 X j = l ( τ j + ρ j ) − ρ l ≥ τ n +1 ≥ (cid:3) Corollary 5.2.
Let τ ∈ Rep( G ) , τ = τ θ . For p ∈ { , . . . , d } let λ be an eigenvalue of ∆ p ( τ ) . Then one has λ ≥ .Proof. Using the preceding Lemma, one can proceed exactly as in the proof of [MP2,Lemma 7.3] to establish the corollary. (cid:3) The regularized trace under coverings
Let X = Γ \ H d be a finite-volume hyperbolic manifold. For τ a finite-dimensional ir-reducible representation of G let e − t ∆ p ( τ ) be the heat operator associated to the Laplaceoperator (5.4) acting on the locally homogeneous vector-bundle E τ over X . To begin withrecall the definition of the regularized trace of the heat operators e − t ∆ p ( τ ) introduced in[MP2]. Let K τ,pX ( t ; x, y ) ∈ C ∞ ( X × X , E τ ⊠ E ∗ τ )be the kernel of e − t ∆ p ( τ ) . If a set P Γ of representatives of Γ-cuspidal parabolic subgroupsof X is fixed, then according to (2.17), one obtains compact smooth manifolds X P Γ ( Y ) with boundary which exhaust X . Using the Maass-Selberg relations, one can show thatthere is an asymptotic expansion Z X P Γ ( Y ) Tr K τ,pX ( t ; x, x ) dx = α − ( t ) log Y + α ( t ) + o (1) , (6.1)as Y → ∞ . Now recall that on a compact manifold the trace of the heat operator is given bythe integral of the pointwise trace of the heat kernel. Based on this observation one definesthe regularized trace Tr reg ( e − t ∆ p ( τ ) ) as the constant term of the asymptotic expansion (6.1).However, this definition depends on the choice of the set P Γ of representatives of Γ-cuspidalparabolic subgroups of G or equivalently on the choice of a truncation parameter on themanifold X , see [MP2, Remark 5.4]. Therefore, this definition is not suitable if one wantsto study the regularized trace for families of hyperbolic manifolds.To overcome this problem, we remark that if π : X → X is a finite covering of X and if truncation parameters on the manifold X are given, then there is a canonical wayto truncate the manifold X , putting X ( Y ) := π − ( X ( Y )). Thus one only has to fixtruncation parameters for the manifold X or equivalently a set P Γ of representatives ofΓ -cuspidal parabolic subgroups of G . To make this approach rigorous, we first need todiscuss some facts about height functions.Let Γ be a discrete subgroup of G of finite covolume. We emphasize that we do notassume that Γ is torsion-free. Let P Γ := { P , , . . . , P ,κ ( X ) } be a fixed set of Γ -cuspidalparabolic subgroups of G . Each P ,l , l = 1 , . . . , κ ( X ), has a Langlands decomposition P ,l = N ,l A ,l M ,l . If P ′ is any Γ -cuspidal parabolic subgroup of G , there exists γ ′ ∈ Γ and a unique l ′ ∈ { , . . . , κ (Γ ) } such that γ ′ P ′ γ ′− = P ,l ′ . Write γ ′ = n ,l ′ ι P ,l ′ ( t P ′ ) k ,l ′ , (6.2) n ,l ′ ∈ N P ,l ′ , t P ′ ∈ (0 , ∞ ), ι P ,l ′ ( t P ′ ) ∈ A P ,l ( P ′ ) as above, and k ,l ′ ∈ K . Since P ,l ′ equalsits normalizer in G , the projection of the element γ ′ to (Γ ∩ P ,l ′ ) \ Γ is unique. Moreover,since P ,l ′ is Γ -cuspidal, one has Γ ∩ P ,l ′ = Γ ∩ N P ,l ′ M P ,l ′ . Thus t P ′ depends only on P Γ and P ′ .Now we let Γ ⊂ Γ be a subgroup of finite index. Then a parabolic subgroup P ′ of G is Γ -cuspidal iff it is Γ -cuspidal. We assume for simplicity that Γ satisfies (3.1). Let X = Γ \ e X , X = Γ \ e X . Let π : X → X be the covering map and let pr X : G → X and pr X : G → X be the corresponding projections. Let P Γ = { P , . . . , P κ ( X ) } be a setof representatives of Γ -cuspidal parabolic subgroups. Then for each j ∈ { , . . . , κ ( X ) } let l ( j ) ∈ { , · · · , κ (Γ ) } , γ j ∈ Γ , and t j := t P j be as in (6.2) with respect to P j . Fix Y (Γ ) ∈ (0 , ∞ ) such that for each P ,l ∈ P Γ , l = 1 , . . . , κ (Γ ), one has Y (Γ ) ≥ Y ( P ,l ) , (6.3)where Y ( P ,l ) is defined by (2.14). Then the following Lemma holds. Lemma 6.1.
For each P j ∈ P Γ let Y P j (Γ ) be defined by (2.14) . Then one has Y P j (Γ ) ≤ t − j Y (Γ ) . (6.4) Let X ( Y ) := X ( P , , . . . , P ,κ ( X ) , Y ) . Then for Y sufficiently large one has π − ( X ( Y )) = X ( P , . . . , P κ ( X ) ; t − Y, . . . , t − κ ( X ) Y ) . Proof.
Since P ,l ( j ) = k ,l ( j ) P j k − ,l ( j ) , and since the adjoint action by k ,l ( j ) is an isometry formthe Lie-algebra of A P j to the Lie-algebra of A P ,l ( j ) , it follows that for every j = 1 , . . . , κ ( X )and every g ∈ G one has y P ,l ( j ) ( γ j gk − ,l ( j ) ) = t j y P j ( g ) . (6.5)This implies (6.4). Indeed, if g ∈ G and γ ∈ Γ satisfy y P j ( g ) > t − j Y (Γ ) and y P j ( γg ) >t − j Y (Γ ), then by (6.5) and the choice of Y (Γ ) one has γ ∈ γ − j (Γ ∩ P l ( j ) ) γ j = Γ ∩ P j .To prove the second part of the lemma, let x ∈ X − X ( P , . . . , P κ ( X ) ; t − Y, . . . , t − κ ( X ) Y ).By (2.16) there exists j ∈ { , · · · , κ (Γ ) } such that y Γ ,P j ( x ) > t − j Y . Then by (2.15) thereexists g ∈ G satisfying pr X ( g ) = x and y P j ( g ) > t − j Y . Now observe that pr X ( γ j gk − ,l ( j ) ) = x . Using (6.5) and (2.15), it follows that y Γ ,P ,l ( j ) ( π ( x )) > Y , i.e., x ∈ π − ( X − X ( Y )).Thus we have shown that X − X ( P , . . . , P κ ( X ) ; t − Y, . . . , t − κ ( X ) Y ) ⊆ π − ( X − X ( Y )) . (6.6)It remains to prove the opposite inclusion. Fix l ∈ { , . . . , κ ( X ) } . Since P ,l equals itsnormalizer in G , it follows that { P j ∈ P Γ : γ j P j γ − j = P ,l } = \ Γ / Γ ∩ P ,l )(6.7)and the γ j with γ j P j γ − j = P ,l form a set of representatives of equivalence classes in thedouble coset (6.7). For each γ j with γ j P j γ − j = P ,l let µ i,j ∈ Γ \ Γ , i = 1 , . . . , r ( j ), be suchthat the orbit of Γ γ j under the action of Γ ∩ P ,l is given by the Γ µ i,j , i = 1 , . . . , r ( j ).Then [Γ : Γ ] = X j ∈{ ,...,κ (Γ ) } γ j P j γ − j = P ,l r ( j ) . (6.8)Write µ i,j = γ j p i,j with p i,j ∈ Γ ∩ P ,l . Choose Y P j ∈ (0 , ∞ ), j = 1 , . . . , κ ( X ), suchthat (2.12) and (2.13) hold for Γ . Let Y > max { t − j Y P j : j = 1 , . . . , κ ( X ) } . Let x ∈ X − X ( Y ). Then there exists a P ,l ∈ P Γ such that y Γ ,P ,l ( x ) > Y . Thus there exists g ∈ G such that x = pr X ( g ) and y P ,l ( g ) > Y . By (2.10) one has y P ,l ( p i,j g ) > Y . Weclaim that π − ( x ) = (cid:8) pr X ( γ − j p i,j g ) : γ j P j γ − j = P ,l , : i = 1 , . . . , r ( j ) (cid:9) . (6.9)Obviously, each pr X ( γ − j p i,j g ) is contained in π − ( x ). On the other hand, assume thatpr X ( γ − j p i,j g ) = pr X ( γ − j ′ p i ′ ,j ′ g ) =: x , where γ j P j γ − j = P ,l = γ j ′ P j ′ γ − j ′ . By (2.10) and(6.5) one obtains y Γ ,P j ( x ) > t − j Y > Y P j , y Γ ,P ′ j ( x ) > t − j ′ Y > Y P j ′ . (6.10) Applying (2.12), (2.13) one obtains j = j ′ and hence i = i ′ . Thus, since { π − ( x ) } =[Γ : Γ ], (6.9) follows from (6.8). Applying (6.9) and (6.10) one obtains π − ( X − X ( Y )) ⊆ X − X ( P , . . . , P κ ( X ) ; t − Y, . . . , t − κ ( X ) Y ) . and together with (6.6) the lemma follows. (cid:3) Let ∆ X ,p ( τ ) be the Laplace operator on E τ -valued p -forms on X . Using the precedingLemma, we can give an invariant definition of the regularized trace of e − t ∆ X ,p ( τ ) providedthe set P Γ is fixed. We fix a set P Γ of representatives of Γ -cuspidal parabolic subgroupsof G . Then by Lemma 6.1 we have Z π − ( X ( Y )) Tr K τ,pX ( t ; x, x ) dx = Z X ( P ,...,P κ ( X ; t − Y,...,t − κ ( X Y ) Tr K τ,pX ( t ; x, x ) dx. (6.11)Now arguing exactly as in [MP2, section 5] and applying Lemma 6.1, we obtain Z π − X ( Y ) Tr K τ,pX ( t ; x, x ) dx dx = X σ ∈ ˆ M [ ν p ( τ ): σ ] =0 X P j ∈ P Γ1 e − t ( τ (Ω) − c ( σ )) dim( σ ) log ( t − j Y ) √ πt + X j e − tλ j + X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) Tr( e C ( σ, ν, − π X σ ∈ ˆ M [ ν : σ ] =0 Z R e − t ( λ + τ (Ω) − c ( σ ) ) Tr (cid:18) e C ( σ, ν, − iλ ) ddz e C ( σ, ν, iλ ) (cid:19) dλ + o (1) , as Y → ∞ . Here the λ j in the first row are the eigenvalues of ∆ X ,p ( τ ), counted withmultiplicity. It follows that the integral on the left-hand side of (6.11) admits an asymptoticexpansion in Y as Y goes to infinity. Note that, since the factor factor τ (Ω) comes fromequation (5.8), the last equation coincides with [MP2, equation 5.7] up to the occurrenceof the t j ’s in the first sum. This occurrence is caused by the different choices of truncationparameters. The appearance of the t j ’s is exactly the reason why the above integral isindependent of the choice of P Γ and depends only on the choice of P Γ .We assume from now on that the set P Γ is fixed. By the above considerations we are letto the following definition of the regularized trace of the heat operator for finite coveringsof X . Definition 6.2.
Let X = Γ \ e X be a finite covering of X and assume that Γ is torsionfree and satisfies (3.1). Let ∆ X ,p ( τ ) be the Laplace operator on E τ -valued p -forms on X . For any choice of a set P Γ of representatives of Γ -cuspidal parabolic subgroups we put(6.12) Tr reg; X ( e − t ∆ X ,p ( τ ) ) := − X σ ∈ ˆ M [ ν p ( τ ): σ ] =0 X P j ∈ P Γ1 e − t ( τ (Ω) − c ( σ )) dim( σ ) log ( t j ) √ πt + X j e − tλ j + X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) Tr( e C ( σ, ν, − π X σ ∈ ˆ M [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) Z R e − tλ Tr (cid:18) e C ( σ, ν p ( τ ) , − iλ ) ddz e C ( σ, ν, iλ ) (cid:19) dλ, where the notation is as above.If one expresses Tr reg; X ( e − t ∆ X ,p ( τ ) ) using the geometric side of the trace formula, thenit becomes again transparent that the summands log t j compensate the ambiguity causedby the choice of P Γ so that Tr reg; X ( e − t ∆ X ,p ( τ ) ) depends only on the choice of P Γ . Forfurther details we refer the reader to section 8, in particular to equations (8.9) and (8.12).7. Exponential decay of the regularized trace for large time
In this section we estimate the regularized trace for large time and with respect tocoverings. Let Γ be a lattice in G and put X = Γ \ e X . Let X = Γ \ e X be a finitecovering of X such that Γ is torsion-free and satisfies (3.1). We assume that a set P Γ of representatives of Γ -cuspidal parabolic subgroups is fixed. We define the regularizedtrace according to Definition 6.2. To begin with we establish the following lemma. Lemma 7.1.
For every σ ∈ (0 , ∞ ) one has Z R σσ + λ e − tλ dλ = √ πt e tσ Z ∞ σ e − tu du. Proof.
Put f ( σ ) := Z R σσ + λ e − tλ dλ = Z R e − tσ λ
11 + λ dλ. Then f ′ ( σ ) = − tσ Z R e − tσ λ λ λ dλ = − tσ (cid:18)Z R e − tσ λ dλ − Z R e − tσ λ
11 + λ dλ (cid:19) = −√ πt + 2 tσf ( σ ) . The general solution of this differential equation on (0 , ∞ ) is given by y ( σ ) = e tσ (cid:18) √ πt Z ∞ σ e − tu du + C (cid:19) and since f satisfies lim σ →∞ f ( σ ) = 0, the Lemma follows. (cid:3) The following proposition is our main result concerning the large time estimation of theregularized trace of the heat kernel.
Proposition 7.2.
Let τ be such that τ θ = τ . There exist constants C, c > such that forall finite covers X of X one has (cid:12)(cid:12) Tr reg; X (cid:0) e − t ∆ X ,p ( τ ) (cid:1)(cid:12)(cid:12) ≤ Ce − ct (Tr reg; X ( e − ∆ X ,p ( τ ) ) + vol( X )) for t ≥ .Proof. Let C ( σ, ν p ( τ ) , s ) be as in (3.12). For each σ ∈ ˆ M one has c ( σ ) = c ( w σ ). Thus byLemma 3.1 the last line of 6.12 can be rewritten as − π dim( ν p ( τ )) X σ ∈ ˆ M/W ( A )[ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) Z R e − tλ Tr (cid:18) C ( σ, ν p ( τ ) , − iλ ) ddz C ( σ, ν, iλ ) (cid:19) dλ. We have Tr (cid:18) C ( σ, ν p ( τ ) , − iλ ) ddz C ( σ, ν p ( τ ) , iλ ) (cid:19) = ddz log det C ( σ, ν p ( τ ) , iλ ) . Let σ , . . . , σ l ∈ (0 , n ], n = ( d − /
2, be the poles of det C ( σ, ν p ( τ ) , s ) in the half-planeRe( s ) ≥
0. Poles occur only if σ = w σ . Let η run over the poles of det C ( σ, ν p ( τ ) , s ) inthe half-plane Re( s ) <
0, both counted with multiplicity. For σ ∈ ˆ M , put(7.1) σ = ( σ, σ = w σ ; σ ⊕ w σ, σ = w σ. Let Y (Γ ) be as in (6.3). By Lemma 6.1 we have t − j Y (Γ ) ≥ Y P j (Γ ) for j = 1 , · · · , κ (Γ ).Using Theorem 4.6 and (3.3) we get1dim( ν p ( τ )) Tr (cid:18) C ( σ, ν p ( τ ) , − iλ ) ddz C ( σ, ν p ( τ ) , iλ ) (cid:19) =2 dim( σ ) − κ (Γ ) X j =1 log t j + ( Y (Γ ) + 2) κ (Γ ) + a ( σ, ν ) + 1dim( ν p ( τ )) l X j =1 σ j λ + σ j + X η η )( λ − Im( η )) + Re( η ) ! , where a ( σ, ν ) ∈ R , a ( σ, ν ) ≤
0. Let σ pp (∆ X ,p ( τ )) denote the pure point spectrum of∆ X ,p ( τ ). Then σ pp (∆ X ,p ( τ )) is the union of the cuspidal spectrum σ cusp (∆ X ,p ( τ )) andthe residual spectrum σ res (∆ X ,p ( τ )). For a given eigenvalue λ ∈ σ pp (∆ X ,p ( τ )), let m ( λ )denote its multiplicity. Put I ( t, ν p ( τ )) := X λ ∈ σ cusp (∆ X ,p ( τ )) m ( λ ) e − tλ , I ( t, ν p ( τ )) := X λ ∈ σ res (∆ X ,p ( τ )) m ( λ ) e − tλ − π dim ν p ( τ ) X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 l X j =1 e − t ( τ (Ω − c ( σ ))) Z R e − tλ σ j λ + σ j dλ,I ( t, ν p ( τ )) := − X σ ∈ ˆ M/W ( A )[ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) (cid:18) a ( σ, ν )4 √ πt + 1 √ πt κ (Γ ) dim(¯ σ )( Y (Γ ) + 2)+ 12 π dim( ν p ( τ )) Z R e − tλ X η Re( η )Re( η ) + ( λ − Im( η )) dλ (cid:19) , and I ( t, ν p ( τ )) := X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) Tr( e C ( σ, ν, . Then it follows from (6.12) that we haveTr reg; X ( e − t ∆ X ,p ( τ ) ) = I ( t, ν p ( τ )) + I ( t, ν p ( τ )) + I ( t, ν p ( τ )) + I ( t, ν p ( τ )) . (7.2)To estimate I ( t, ν p ( τ )) we apply Corollary 5.2. It follows that for t ≥ | I ( t, ν p ( τ )) | ≤ e − t I (1 , ν p ( τ )) . (7.3)To deal with I ( t, ν p ( τ )) observe that to each λ j ∈ σ res (∆ p ( τ )) there correspond a σ ∈ ˆ M satisfying σ = w σ and [ ν p ( τ ) : σ ] = 0, and a pole σ j of det C ( σ, ν p ( τ ) , s ) in (0 , n ] such that λ j = − σ j + τ (Ω) − c ( σ ) . (7.4)Moreover, the multiplicity of σ j divided by dim( ν p ( τ )) equals the multiplicity of the eigen-value λ j . Let µ j be the sequence of the σ j ’s, where the multiplicity of each µ j is themultiplicity of σ j divided by dim( ν p ( τ )). Put h µ j ( t ) := 1 − √ t √ π Z ∞ µ j e − tu du = 1 − √ π Z ∞√ tµ j e − u du. Using Lemma 7.1, we get I ( t, ν p ( τ )) = X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) X j (cid:18) e tµ j − π Z R e − tλ µ j λ + µ j dλ (cid:19) = X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 X j e − t ( τ (Ω) − c ( σ ) − µ j ) h µ j ( t ) . Now observe that 1 ≥ h µ j ( t ) ≥ . Moreover, by Corollary 5.2 it follows that for every µ j we have − µ j + τ (Ω) − c ( σ ) ≥ . (7.5)Thus for each t ≥
10 we get | I ( t, ν p ( τ )) | ≤ e − t X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 X j e − t ( τ (Ω) − c ( σ ) − µ j )2 h µ j ( t ) ≤ e − t X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 X j e − ( τ (Ω) − c ( σ ) − µ j ) e − h µ j ( t ) ≤ e − t X σ ∈ ˆ M ; σ = w σ [ ν p ( τ ): σ ] =0 X j e − ( τ (Ω) − c ( σ ) − µ j ) h µ j (1) = e − t I (1 , ν p ( τ )) . (7.6)Next we deal with I ( t, ν p ( τ )). By [MP2, Lemma 7.1] we have τ (Ω) − c ( σ ) ≥ σ ∈ ˆ M with [ ν p ( τ ) : σ ] = 0. Then since a ( σ, ν ) ≤
0, Re( η ) <
0, for each t ≥ | I ( t, ν p ( τ )) | ≤ e − t X σ ∈ ˆ M/W ( A )[ ν p ( τ ): σ ] =0 e − ( τ (Ω) − c ( σ )) dim(¯ σ ) κ (Γ )( Y (Γ ) + 2) 1 √ π − e − t X σ ∈ ˆ M/W ( A )[ ν p ( τ ): σ ] =0 e − ( τ (Ω) − c ( σ )) (cid:18) a ( σ, ν )4 √ π + 12 π dim( ν p ( τ )) Z R e − λ X η Re( η )Re( η ) + ( λ − Im( η )) dλ (cid:19) = 2 e − t X σ ∈ ˆ M/W ( A )[ ν p ( τ ): σ ] =0 e − ( τ (Ω) − c ( σ )) dim(¯ σ )( Y (Γ ) + 2) κ (Γ ) 1 √ π + e − t I (1 , ν p ( τ )) . By [Ke, Proposition 3.3] there exists C ( d ) > κ ( X ) ≤ C ( d ) vol( X )for all complete hyperbolic manifolds of finite volume and dimension d . Thus for each t ≥ | I ( t, ν p ( τ )) | ≤ e − t ( I (1 , ν p ( τ )) + C vol( X )) , (7.9)where C depends only on Γ and P Γ . To estimate I ( t, ν p ( τ )) we recall that e C ( σ, ν, =Id. Hence there exist natural numbers c (Γ , σ, ν ), c (Γ , σ, ν ) such that c (Γ , σ, ν ) + c (Γ , σ, ν ) = dim ( E ( σ, ν ) ⊗ V ν ) K = κ ( X ) dim( σ ) , and Tr( e C ( σ, ν, c (Γ , σ, ν ) − c (Γ , σ, ν ) . Using (7.7) and (7.8) we obtain for t ≥ | I ( t, ν p ( τ )) |≤ e − t ( I (1 , ν p ( τ )) + 2 c (Γ , σ, ν )) ≤ e − t ( I (1 , ν p ( τ )) + 2 C ( d ) dim( σ ) vol( X )) . (7.10)Combining (7.2), (7.3), (7.6), (7.9) and (7.10), the proof of the proposition is complete. (cid:3) Geometric side of the trace formula
To study the behaviour of the analytic torsion under coverings we will apply the traceformula to the regularized trace of the heat operator. In this section we recall the structureof the geometric side of the trace formula and study the parabolic contribution.Let the assumptions be the same as at the beginning of the previous section. Let τ ∈ Rep( G ) and assume that τ = τ θ . Let e E τ be the homogeneous vector bundle over e X = G/K , associated to τ | K , equipped with an admissible Hermitian metric (see section5). Let e ∆ p ( τ ) be the Laplace operator on e E τ -valued p -forms. The on C ∞ ( G, ν p ( τ )) onehas e ∆ p ( τ ) = − Ω + τ (Ω) , (8.1)see [MtM, (6.9)] Let(8.2) H τ,pt : G → End(Λ p p ∗ ⊗ V τ )be the kernel of the heat operator e − t e ∆ p ( τ ) . Let(8.3) h τ,pt = tr H τ,pt . We apply the trace formulas in [MP2, section 6] to express the regularized trace as a sumof distributions evaluated at h τ,pt . The terms appearing on the geometric side of the traceformula are associated to the different types of Γ-conjugacy classes. We briefly recall theirdefinition. For further details, we refer the reader to [MP2, section 6] and the referencestherein. In order to indicate the dependence of the distributions on the manifold X , we shall use X as a subscript. The contribution of the identity to the trace formula is givenby(8.4) I X ( h τ,pt ) := vol( X ) h τ,pt (1) . The hyperbolic contribution is given by H X ( h τ,pt ) := Z Γ \ G X γ ∈ Γ , s −{ } h τ,pt ( x − γx ) dx, (8.5)where Γ , s are the semisimple elements of Γ . By [Wa, Lemma 8.1] the integral convergesabsolutely. Moreover, arguing as in the cocompact case [Wal], if G γ resp. (Γ ) γ denote thecentralizers of γ in G resp. Γ , one has H X ( h τ,pt ) = X [ γ ] ∈ C(Γ ) s − [1] vol((Γ ) γ \ G γ ) Z G γ \ G h τ,pt ( x − γx ) dx, where C(Γ ) s are the Γ -conjugacy classes of semisimple elements of Γ . Now the lattersum can also be written as a sum over the set C(Γ ) s of non elliptic semisimple conjugacyclasses of the group Γ as follows. For each γ ∈ Γ let c Γ ( γ ) be the number of fixed pointsof γ on Γ / Γ . This number clearly depends only on the Γ -conjugacy class of γ . Then ifΓ γ is the centralizer of γ in Γ , one has H X ( h τ,pt ) = X [ γ ] ∈ C(Γ ) s − [1] c Γ ( γ ) vol(Γ γ \ G γ ) Z G γ \ G h τ,pt ( x − γx ) dx, (8.6)see [Co, page 152-153]. This expression will be used when we treat the Hecke subgroupsof the Bianchi groups.Next we describe the distributions associated to the parabolic conjugacy classes. Firstlylet(8.7) T ′ X ( h τ,pt ) := κ ( X ) Z K Z N h τ,pt ( knk − ) log k log n k dkdn. We note that T is a non-invariant distribution which depends on X only via the numberof cusps of X . Now let P ′ be any Γ -cuspidal parabolic subgroup of G , or equivalentlya Γ -cuspidal parabolic subgroup of G . Let n P ′ denote the Lie algebra of N P ′ . Thenexp : n P ′ → N P ′ is an isomorphism and we denote its inverse by log. We equip n P ′ withthe inner product obtained by restriction of the inner product in (2.3). By k·k we denotethe corresponding norm. LetΛ P ′ (Γ ) := log(Γ ∩ N P ′ ); Λ P ′ (Γ ) := vol(Λ P ′ (Γ )) − n Λ P ′ (Γ ) . Then Λ P ′ (Γ ) and Λ P ′ (Γ ) are lattices in n P ′ and Λ P ′ (Γ ) is unimodular. Then for Re( s ) > ζ P ′ ;Γ , defined by ζ P ′ ;Γ ( s ) := X η ∈ Λ P ′ (Γ ) −{ } k η k − n (1+ s ) , (8.8) converges and ζ P ′ ;Γ has a meromorphic continuation to C with a simple pole at 0. Let C (Λ P ′ (Γ )) be the constant term of ζ P ′ ;Γ at s = 0. Now as before let P Γ be a set ofrepresentatives of Γ -cuspidal parabolic subgroups and for each P j ∈ P Γ let t j be as inthe previous sections. Then put S X ( h τ,pt ) := X P j ∈ P Γ1 (cid:18) C (Λ P j (Γ )) vol(Λ P j (Γ ))vol( S n − ) X σ ∈ ˆ M dim( σ )2 π Z R Θ σ,λ ( h τ,pt ) dλ − X σ ∈ ˆ M [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) dim( σ ) log ( t j ) √ πt (cid:19) . Comparing the Definition 6.2 and [MP2, Definition 5.1], it follows from [MP2, Theorem6.1] that Tr reg; X ( e − t ∆ X ,p ( τ ) ) = I X ( h τ,pt ) + H X ( h τ,pt ) + T ′ X ( h τ,pt ) + S X ( h τ,pt ) . (8.9)We now study the distribution S X ( h τ,pt ) in more detail. By [MP2, Proposition 4.1] wehave Θ σ,λ ( h τ,pt ) = e − t ( λ + τ (Ω) − c ( σ )) for [ ν p ( τ ) : σ ] = 0, and Θ σ,λ ( h νt ) = 0 otherwise. Thus we can rewrite S X ( h τ,pt ) := X σ ∈ ˆ M [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) dim( σ ) √ πt X P j ∈ P Γ1 C (Λ P j (Γ )) vol(Λ P j (Γ ))vol( S n − ) − log ( t j ) . Let Λ be a lattice in R n . The associated Epstein zeta function ζ Λ ( s ) := X λ ∈ Λ −{ } k λ k − n (1+ s ) (8.10)converges for Re( s ) > C . Let C (Λ) denote theconstant term of the Laurent expansion of ζ Λ ( s ) at s = 0. The following lemma describesthe behaviour of C (Λ) under scaling. Lemma 8.1.
Let Λ be a lattice in R n . Let µ ∈ (0 , ∞ ) and put Λ ′ := µ Λ . Then one has C (Λ ′ ) = µ − n (cid:18) C (Λ) − vol( S n − ) log µ vol(Λ) (cid:19) . Proof.
Let R (Λ) be the residue of ζ Λ at 0. Then one has C (Λ ′ ) = µ − n ( C (Λ) − R (Λ)2 n log µ ) . Moreover, by [Ter, Chapter 1.4, Theorem 1] one has R (Λ) = vol( S n − )2 n vol(Λ)and the lemma follows. (cid:3) Now we let P ′ be any Γ -cuspidal parabolic subgroup of G . Following section 6, we let l ′ ∈ { , . . . , κ ( X ) } such that there exists γ ′ ∈ Γ with γ ′ P ′ γ ′− = P ,l ′ . As in (6.2) wewrite γ ′ = n ,l ′ a ,l ′ (log t P ′ ) k ,l ′ . If Γ is a finite index subgroup of Γ , we define a lattice˜Λ P ′ (Γ ) in n P ,l ( P ′ ) as ˜Λ P ′ (Γ ) := log( γ ′ (Γ ∩ N P ′ ) γ ′− ) . If Γ is normal in Γ , one has ˜Λ P ′ (Γ ) = Λ P ,l ′ (Γ ). Since γ ′ is unique in Γ / (Γ ∩ P ′ ) andΓ ∩ P ′ = Γ ∩ ( M P ′ N P ′ ), the isometry class of ˜Λ P ′ (Γ ) is independent of the choice of γ ′ having the required property. Let ˆΛ P ′ (Γ ) be the unimodular lattice corresponding to˜Λ P ′ (Γ ), i.e. ˆΛ P ′ (Γ ) := (vol( ˜Λ P ′ (Γ )) − n · ˜Λ P ′ (Γ ) . With respect to the norms induced by the Killing form, the lattice Λ P ′ (Γ ) in n P ′ isisometric to the lattice t − P ′ ˜Λ P ′ (Γ ) in n P ,l ( P ′ ) . Thus the preceding Lemma implies that C (Λ P j (Γ )) vol(Λ P j (Γ ))vol( S n − ) = C ( ˜Λ P j (Γ )) vol( ˜Λ P j (Γ ))vol( S n − ) + log t j . Now define α ( X ) := α (Γ ) := κ ( X ) X j =1 C ( ˜Λ P j (Γ )) vol( ˜Λ P j (Γ ))vol( S n − ) . (8.11)Then, putting everything together, we can write S X ( h τ,pt ) = α ( X ) X σ ∈ ˆ M [ ν p ( τ ): σ ] =0 e − t ( τ (Ω) − c ( σ )) dim( σ ) √ πt . (8.12)Finally, for each l = 1 , . . . , κ (Γ ), we let P ( n P ,l ) be the set of isometry classes of unimod-ular lattices in n P ,l equipped with the standard topology, i.e., with the topology inducedby identification of P ( n P ,l ) with SO(2 n ) \ SL n ( R ) / SL n ( Z ). Now in order to control theconstant α (Γ i ) for sequences of finite coverings, we make the following definition. Definition 8.2.
Let Γ i be a sequence of finite index subgroups of Γ . Let P Γ be a fixedset of representatives Γ -cuspidal parabolic subgroups of Γ . Then the sequence Γ i is calledcusp uniform if for each l = 1 , . . . , κ (Γ ) there exists a compact set K l in P ( n P ,l ) such thatfor each Γ -cuspidal parabolic P ′ the lattices ˆΛ P ′ (Γ i ), i ∈ N , belong to K l .We can reformulate the condition of cusp-uniformity in a simpler way as follows. Welet P ( n ) be the space of isometry classes of unimodular lattices in n , equipped with thetopology as above. For each parabolic subgroup P ′ of G there exists a g P ′ ∈ G with g P ′ P ′ g − P ′ = P . Let Γ be a discrete subgroup of G of finite covolume. If P ′ is Γ-cuspidal,we let Λ P | P ′ (Γ) := vol (cid:0) log( g P ′ (Γ ∩ N P ′ ) g − P ′ ) (cid:1) n log( g P ′ (Γ ∩ N P ′ ) g − P ′ ) . (8.13) This a unimodular lattice in n and since the image of g P ′ in P \ G is unique, the isometryclass of Λ P ′ (Γ) is independent of the choice of g P ′ with g P ′ P ′ g − P ′ = P . Lemma 8.3.
The following conditions are equivalent:(1) The sequence Γ i is cusp-uniform.(2) For each Γ -cuspidal parabolic subgroup P ′ of G there exists a compact set K P ′ in P ( n P ′ ) such that Λ P ′ (Γ i ) ∈ K P ′ for every i .(3) There exists a compact set K P in P ( n P ) such that for each Γ -cuspidal parabolicsubgroup P ′ of G one has Λ P | P ′ (Γ i ) ∈ K P for each i ∈ N .Proof. By the preceding arguments all lattices are isometric. (cid:3)
Lemma 8.4.
Let K be a compact set of unimodular lattices in R n . Then the constantterm of the Laurent expansion of the Epstein zeta functions ζ Λ ( s ) at s = 0 is bounded on K .Proof. By [Ter, Chapt.I, § ζ Λ ( s ) is given by π − s Γ( s ) ζ Λ ( s ) = 2 ns − n (1 + s ) + Z ∞ ( t n (1+ s ) − + t − n s − ) X λ ∈ Λ −{ } e − tπ k λ k dt . (8.14)Now for a lattice Λ in R n , let λ (Λ) denote the smallest norm of a non-zero vector inΛ. Let B ( R ) denote the ball in R n around the origin of radius R . Then it follows from[BHW, Theorem 2.1] that for each R > { B ( R ) ∩ Λ } ≤ (cid:18) Rλ (Λ) + 1 (cid:19) n . If K is a compact set of unimodular lattices in R n , then by Mahler’s criterion there existsa constant µ such that λ (Λ) ≥ µ for each Λ ∈ K . Thus for each Λ ∈ K and for each t ∈ [1 , ∞ ) we have X λ ∈ Λ −{ } e − tπ k λ k ≤ e − tπµ X λ ∈ Λ −{ } e − π k λ k ≤ e − tπµ ∞ X k =1 e − π ( µk )22 { B ( µ ( k + 1)) ∩ Λ }≤ e − tπµ ∞ X k =1 e − π ( µk )22 (2 k + 3) n =: C e − tπµ , where C is a constant which is independent of Λ. Applying (8.14), the Lemma follows. (cid:3) Now we can control the behaviour of the constants, appearing in the definitions of theterms T ′ X i ( h τ,pt ) and S X i ( h τ,pt ), under sequences of coverings X i = Γ i \ e X of X . As always weassume that a set P Γ of representatives of Γ -cuspidal parabolic subgroups of G is fixed.For each i we let P Γ i = { P i,j , j = 1 , . . . , κ (Γ i ) } be a set of representatives of Γ i -conjugacyclasses of Γ i -cuspidal parabolic subgroups. We can estimate α (Γ i ) as follows. Proposition 8.5.
Let Γ i be cusp-uniform sequence of finite index subgroups of Γ . Thenthere exists a constant c (Γ ) such that | α (Γ i ) | ≤ c (Γ ) κ (Γ i ) + c (Γ ) κ (Γ i ) X j =1 log[Γ ∩ N P i,j : Γ i ∩ N P i,j ] . In particular, there exists a constant c (Γ ) such that we have | α (Γ i ) | ≤ c (Γ ) κ (Γ i ) log [Γ : Γ i ] . Proof.
By Lemma 8.1, for each P i,j ∈ P Γ i one has C ( ˜Λ P i,j (Γ i )) vol( ˜Λ P i,j (Γ i )) = C ( ˆΛ P i,j (Γ i )) − vol( S n − ) log vol( ˜Λ P i,j (Γ i )2 n . By assumption the lattices ˆΛ P i,j (Γ i ), i ∈ N , lie in a compact subset of P ( n P ,l ( j ) ). Thusby Lemma 8.4 there exists a constant c ′ (Γ ) such that for each i one has | C ( ˆΛ P i,j (Γ i )) | ≤ c ′ (Γ ). Since ˜Λ P i,j (Γ ) = Λ P ,l ( j ) (Γ ), the lattice ˜Λ P i,j (Γ i ) is a sublattice of Λ P ,l ( j ) (Γ ) ofindex [Γ ∩ N P i,j : Γ i ∩ N P i,j ]. Therefore one hasvol( ˜Λ P i,j (Γ i )) = vol(Λ P ,l ( j ) (Γ ))[Γ ∩ N P i,j : Γ i ∩ N P i,j ] ≤ c ′′ (Γ )[Γ ∩ N P i,j : Γ i ∩ N P i,j ] , where c ′′ (Γ ) is a constant which is independent of i . This proves the first estimate. Thesecond estimate follows immediately from the first one. (cid:3) In the next proposition we estimate the number of cusps and the behaviour of theconstant α (Γ i ) under sequences of normal coverings. Proposition 8.6.
Let Γ i be a sequence of normal subgroups of Γ of finite index [Γ : Γ i ] such that [Γ : Γ i ] → ∞ as i → ∞ and such that each γ ∈ Γ , γ = 1 , belongs only tofinitely many Γ i . Assume that each Γ i satisfies assumption (3.1) . Then one has lim i →∞ κ (Γ i )[Γ : Γ i ] = 0 . If in addition the sequence Γ i is cusp-uniform, then one has lim i →∞ | α (Γ i ) | [Γ : Γ i ] = 0 . Proof.
Using that each Γ i , i ≥
1, satisfies (3.1) and Γ i is normal in Γ , one has { Γ i \ Γ / Γ ∩ P ,l } = [Γ : Γ i ][Γ ∩ P ,l : Γ i ∩ P ,l ] ≤ [Γ : Γ i ][Γ ∩ N P ,l : Γ i ∩ N P ,l ]for each l = 1 , . . . , κ (Γ ). Thus using (6.7), one can estimate κ (Γ i )[Γ : Γ i ] = P P ,l ∈ P Γ0 { Γ i \ Γ / Γ ∩ P ,l } [Γ : Γ i ] ≤ X P ∈ P Γ0 ∩ N P ,l : Γ i ∩ N P ,l ] . Moreover, for each l = 1 , . . . , κ (Γ ) and each j = 1 , . . . , κ (Γ i ) one hasΓ ∩ N P i,j = γ j (Γ ∩ N P ,l ( j ) ) γ − j , Γ i ∩ N P i,j = γ j (Γ i ∩ N P ,l ( j ) ) γ − j , where the second equality is due to the assumption that Γ i is normal in Γ . Thus applying(6.7), one can estimate1[Γ : Γ i ] κ (Γ i ) X j =1 log[Γ ∩ N P i,j : Γ i ∩ N P i,j ]= P P ,l ∈ P Γ0 { Γ i \ Γ / Γ ∩ P ,l } log [Γ ∩ N P ,l : Γ i ∩ N P ,l ][Γ : Γ i ] ≤ X P ∈ P Γ0 log [Γ ∩ N P ,l : Γ i ∩ N P ,l ][Γ ∩ N P ,l : Γ i ∩ N P ,l ] . The condition that each γ ∈ Γ − { } , γ = 1, belongs only to finitely many Γ i impliesthat [Γ ∩ N P ,l : Γ i ∩ N P ,l ] goes to ∞ as i → ∞ . Thus the first statement and togetherwith the previous proposition also the second one are proved. (cid:3) Proof of the main results
We keep the assumptions of the previous sections. So Γ is a lattice in G and Γ isa torsion-free subgroup of finite index of Γ , which satisfies (3.1). We let X := Γ \ e X and X i := Γ i \ e X . We assume that a set P Γ of representatives of Γ -conjugacy classesof Γ -cuspidal parabolic subgroups of G is fixed. Then for each τ ∈ Rep( G ), τ = τ θ , letTr reg; X ( e − t ∆ X ,p ( τ ) ) be the the regularized trace of e − t ∆ X ,p ( τ ) , as defined by 6.2. It followsfrom Proposition (7.2) that there exist constants C, c > (cid:12)(cid:12) Tr reg; X (cid:0) e − t ∆ X ,p ( τ ) (cid:1) (cid:12)(cid:12) ≤ Ce − ct , (9.1)for t ≥
1. Applying [Proposition 6.9][MP2], it follows immediately from the definition ofTr reg; X ( e − t ∆ X ,p ( τ ) ) that there is an asymptotic expansionTr reg; X ( e − t ∆ X ,p ( τ ) ) ∼ ∞ X j =0 a j t j − d + ∞ X j =0 b j t j − log t + ∞ X j =0 c j t j as t → +0. Put K X ( t, τ ) := 12 d X p =1 ( − p p Tr reg; X (cid:0) e − t ∆ X ,p ( τ ) (cid:1) . Then it follows that we can define the analytic torsion T X ( τ ) bylog T X ( τ ) = dds (cid:18) s ) Z ∞ K X ( t, τ ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s =0 , where the integral converges in the half-plane Re( s ) > d/ s = 0 byanalytic continuation. Let T >
0. Then it follows from (9.1) that R ∞ T K X ( t, τ ) t s − dt isan entire function of s . Therefore we have(9.2) log T X ( τ ) = dds (cid:18) s ) Z T K X ( t, τ ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + Z ∞ T K X ( t, τ ) t − dt. To proceed further, we first need to estimate the integrand of the hyperbolic term (8.5).Recall that for a lattice Γ in G we denote by ℓ (Γ) the length of the shortest closed geodesicof Γ \ e X . Lemma 9.1.
Let h τ.pt ∈ C ∞ ( G ) be defined by (8.3) . For each T ∈ (0 , ∞ ) there existsa constant C > , depending on T and X only, such that for all hyperbolic manifolds X = Γ \ e X , which are finite coverings of X , and all g ∈ G one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X γ ∈ Γ , s −{ } h τ,pt ( g − γg ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − ℓ (Γ0)232 t e − ℓ (Γ1)28 t for all t ∈ (0 , T ] .Proof. Let ν p ( τ ) be the representation of K defined by (5.5). Let e E ν p ( τ ) be the associatedhomogeneous vector bundle over e X equipped with the canonical metric connection [MP2,section 4]. Let e ∆ ν p ( τ ) be the Bochner-Laplace operator acting on C ∞ ( e X, e E ν p ( τ ) ). Then on C ∞ ( G, ν p ( τ )), the action of this operator is given by e ∆ ν p ( τ ) = − R (Ω) + ν p ( τ )(Ω K ) , where Ω K is the Casimir eigenvalue of k with respect to the restriction of the normalizedKilling form g to k , see [Mi1, Proposition 1.1]. Thus by (8.1) there exists an endomorphism E p ( τ ) of Λ p p ∗ ⊗ V τ such that e ∆ p ( τ ) = e ∆ ν p ( τ ) + E p ( τ ) . Moreover E p ( τ ) commutes with e ∆ ν p ( τ ) . Let H ν p ( τ ) t : G → End(Λ p p ∗ ⊗ V τ )be the kernel of the heat operator e − t e ∆ νp ( τ ) . Then it follows that(9.3) H τ,pt = e − tE p ( τ ) ◦ H ν p ( τ ) t . Let H t ( g ) be the heat kernel for the Laplacian on functions on e X . Using (9.3) and [MP1,Proposition 3.1] it follows that there exist constants C > c ∈ R such that k H τ,pt ( g ) k ≤ Ce ct H t ( g ) , g ∈ G, t > . Hence we get | h τ,pt ( g ) | ≤ C dim( τ ) e ct H t ( g ) , g ∈ G, t > . By [Do1] there exists C > T such that for each t ∈ (0 , T ] onehas H t ( g ) ≤ C ′ t − d/ exp (cid:18) − d ( gK, K t (cid:19) for 0 < t ≤ T . The constant C ′ depends only on T . Thus we get X γ ∈ Γ , s −{ } | h τ,pt ( g − γg ) | ≤ C t − d/ e cT X γ ∈ Γ , s −{ } e − d ( γgK,gK ) / (4 t ) ≤ C e − ℓ (Γ i ) / (8 t ) e − ℓ (Γ ) / (32 t ) X γ ∈ Γ , s −{ } e − d ( γgK,gK ) / (16 T ) , (9.4)where C , C are constants which depend only on T . It remains to show that the last sumconverges and can be estimated independently of g . For r ∈ (0 , ∞ ) and x ∈ e X we let B r ( x ) be the metric ball of radius r around x . There exists a constant C > B r ( x )) ≤ Ce nr (9.5)for all r ∈ (0 , ∞ ). It easily follows from (2.12) and (2.13) that there exists an ǫ > x ∈ e X and all γ ∈ Γ , s , γ = 1 one has B ǫ ( x ) ∩ γB ǫ ( x ) = ∅ . Thus for each x ∈ e X the union G γ ∈ Γ , s : d ( x,γx ) ≤ R γB ǫ ( x )is disjoint and contained in B ǫ + R ( x ). Using (9.5) it follows that there exists a constant C X >
0, depending on X , such that for all R ∈ (0 , ∞ ) and all x ∈ e X one has { γ ∈ Γ , s : d ( x, γx ) ≤ R } ≤ C X e nR . Applying (9.4) the Lemma follows. (cid:3)
Applying the preceding lemma we obtain the following estimate for the regularized tracewhich is uniform with respect to coverings.
Proposition 9.2.
There exists a constant
C > such that for each hyperbolic manifold X = Γ \ e X , which is a finite covering of X , and for which Γ satisfies (3.1) , one has | Tr reg; X (cid:0) e − ∆ X ,p ( τ ) (cid:1) | ≤ C (vol( X ) + κ ( X ) + α ( X )) , where κ ( X ) is the number of cusps of X and α ( X ) is as in (8.11) .Proof. We put t = 1 in (8.9) and estimate the terms on the right hand side. The identitycontribution (8.4) can be estimated by C vol( X ). By (8.7), the third term can be esti-mated by C κ ( X ). Using (8.12), it follows that the forth term is bounded by C α ( X ).Finally, (8.5) and Lemma 9.1 imply that the hyperbolic term is bounded by C vol( X ).The constants C i > i = 1 , · · · ,
4, are all independent of X . This finishes the proof. (cid:3) Now we can deal with the second integral in (9.2). Using Proposition 7.2, Proposition9.2, assumption (1.7) and Proposition 8.5, it follows that there exists a
C, c > π : X → X as above we have(9.6) 1vol( X ) (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ T K X ( t, τ ) t − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − cT for all T ≥ k τt := 12 d X p =1 ( − p ph τ,pt . It follows from [MP2, section 9] that the Mellin transform R ∞ k τt (1) t s − dt converges ab-solutely and uniformly on compact subsets of Re( s ) > d/
2, and admits a meromorphicextension to C , which is holomorphic at s = 0. Let(9.8) t (2) e X ( τ ) := dds (cid:18) s ) Z ∞ k τt (1) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 . Then in analogy to the compact case (1.2), the L -torsion T (2) X ( τ ) ∈ R + is given bylog T (2) X ( τ ) = vol( X ) t (2) e X ( τ ) . For details we refer to [MP2, section 9]. Furthermore, it follows from [MP2, equation 9.4]that there exist
C, c > (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ T k τt (1) t − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ce − cT for T >
0. Hence we get(9.9) dds (cid:18) s ) Z T I X ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = vol( X ) · ( t (2) e X ( τ ) + O (cid:0) e − cT (cid:1) ) . Now let Γ i , i ∈ N , be a sequence of torsion-free subgroups of finite index of Γ , whichsatisfy the assumptions of Theorem 1.1. Firstly, by (9.9) we have(9.10) lim i →∞ : Γ i ] dds (cid:18) s ) Z T I X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = vol( X ) · ( t (2) e X ( τ ) + O (cid:0) e − cT (cid:1) ) . Let (Γ i ) s be the set of semi-simple elements in Γ i . By (8.5) the hyperbolic contribution isgiven by H X i ( k τt ) = Z Γ i \ G X γ ∈ (Γ i ) s −{ } k τt ( g − γg ) d ˙ g. It follows from Lemma 9.1 that dds (cid:18) s ) Z T H X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = Z T H X i ( k τt ) t − dt and that there exists a constant C , depending on T , such that (cid:12)(cid:12)(cid:12)(cid:12)Z T H X i ( k τt ) t − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C vol( X i ) e − ℓ (Γ i )28 T . Hence if ℓ (Γ i ) → ∞ as i → ∞ , one has(9.11) lim i →∞ : Γ i ] dds (cid:18) s ) Z T H X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 0 . Next we study the term associated to T ′ X i ( k τt ), defined in (8.7). We let J X i ( k τt ) and I X i ( k τt ) be defined according to [MP2, (6.13), (6.15)], where the subindex X i indicatesthat these distributions depend on the manifold X i . Then by definition we have T ′ X i ( k τt ) = κ ( X i ) I X i ( k τt ) + J X i ( k τt ) . Using the results of [MP2, section 6], it follows that there is an asymptotic expansion T ′ X i ( k τt ) ∼ ∞ X k =0 a k t k − ( d − / + ∞ X k =0 b k t k − / log t + c as t →
0. Thus for Re( s ) > ( d − /
2, the integral Z T T ′ X i ( k τt ) t s − dt converges and has a meromorphic extension to C , which at s = 0 has at most a simplepole. Applying the definition of T ′ X i it follows that there exists a function φ ( T, τ ) such that dds (cid:18) s ) Z T T ′ X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = φ ( T, τ ) · κ ( X i ) . Thus if lim i →∞ κ ( X i ) / [Γ : Γ i ] = 0, we obtain(9.12) lim i →∞ : Γ i ] dds (cid:18) s ) Z T T ′ X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 0 . Finally, by (8.12) the integral Z T t s − S X i ( k τt ) dt converges absolutely for s ∈ C with Re( s ) > and has a meromorphic extension to C withan at most a simple pole at s = 0. Moreover, it follows from (8.12) that there exists afunction ψ ( T, τ ) such that dds (cid:18) s ) Z T S X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = ψ ( T, τ ) · α (Γ i ) , where α (Γ i ) is as in (8.11). By assumption (1.7) and Proposition 8.5 it follows thatlim i →∞ : Γ i ] dds (cid:18) s ) Z T S X i ( k τt ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = 0 . Combined with (9.10), (9.11), and (9.12) we get(9.13) lim i →∞ : Γ i ] dds (cid:18) s ) Z T K X ( t, τ ) t s − dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) s =0 = vol( X ) · ( t (2) e X ( τ ) + O ( e − cT )) . Finally, combining (9.13), (9.2) and (9.6), and using that
T > (cid:3)
Now assume that Γ i is normal in Γ and each γ ∈ Γ belongs only to finitely manyΓ i . Note that ℓ ( γ ) depends only on the Γ -conjugacy class. Since by (2.19), for each R > γ ] ∈ C(Γ , s ) with ℓ ( γ ) ≤ R , onehas lim i →∞ ℓ (Γ i ) = ∞ . Thus, if one applies Proposition 8.6 and the preceding arguments,Theorem 1.2 follows.10. Principal congruence subgroups of SO ( d, ( d,
1) and prove Corollary 1.3. Therefore, throughout this section we let G := SO ( d, d odd, d = 2 n + 1. Let K = SO( d ), regarded as a subgroup of G . Then K is a maximalcompact subgroup of G .We realize the standard parabolic subalgebra p of g as follows. Denote by E i,j the matrixin g whose entry at the i-th row and j-th column is equal to 1 and all of whose other entriesare equal to 0 and let H := E , + E , . Let a := R H and let n = X ( v ) := v t v t v − v , v ∈ R d − . (10.1)Then for the standard ordering of the restricted roots of a in g , n is the direct sum of thepositive restricted root spaces. We let g = n ⊕ a ⊕ k be the associated Iwasawa decomposition. Let N be the connected Lie group with Liealgebra n and let A := exp( a ). Let M be the centralizer of A in K . Then P = M AN is a parabolic subgroup of G .For v ∈ R d − one hasexp( X ( v )) = 1 + X ( v ) + X ( v )2 = k v k / −k v k / v t k v k / − k v k / v t v − v I d − , (10.2)where I d − denotes the unit-matrix and where k · k denotes the Euclidean norm on R d − .We have N = exp( n ). The group G is an algebraic group defined over Q and we let Γ := G ( Z ) be its integralpoints. By [BoHa], Γ is a lattice in G . It follows from (10.2) thatlog (Γ ∩ N ) = v t v t v − v , v ∈ Z d − , k v k ∈ Z . (10.3)In particular, P is a Γ -cuspidal parabolic subgroup of G .Now for q ∈ N we let Γ( q ) be the principal congruence subgroup of level q , i.e.Γ( q ) = { A ∈ Γ : A ≡ I mod( q ) } . Then Γ( q ) coincides with the kernel of the canonical map Γ → G ( Z /q Z ). In particular,Γ( q ) is a normal subgroup of Γ . If q ≥
3, then the group Γ( q ) is neat in the sense of Borel,see [Bo, 17.4]. In particular, Γ( q ) is torsion free and satisfies (3.1).In the following Lemma we verify the cusp-uniformity of the groups Γ( q ). The Lemma isjust a special case of Lemma 4 of the paper [DH] of Deitmar and Hoffmann who treated themore general case of families of strictly bounded depth in algebraic Q -groups of arbitraryreal rank. However, for the convenience of the reader we shall now recall the proof ofDeitmar and Hoffmann in our situation. Lemma 10.1.
Let P ′ be a Γ -cuspidal parabolic subgroup defined over Q with nilpotentradical N P ′ . Let n P ′ be the Lie-algebra of N P ′ . Then there exists a lattice Λ + n P ′ in n P ′ suchthat q Λ + n P ′ ⊆ log (Γ( q ) ∩ N P ′ ) ⊆ q + n P ′ for each q ∈ N . In particular, the sequence Γ( q ) , q ∈ N , is cusp-uniform.Proof. Let Mat ( d +1) × ( d +1) ( Z ) be the integral ( d + 1) × ( d + 1)-matrices. Then by (10.1) n ∩ Mat ( d +1) × ( d +1) ( Z ) is a lattice in n . We choose g ∈ G ( Q ) such that P ′ = gP g − . Then n P ′ = g n g − and thus Λ + n P ′ := 2( n P ′ ∩ Mat ( d +1) × ( d +1) ( Z ))is a lattice in n P ′ . By (10.2), one has exp( Y ) = 1 + Y + Y for each Y ∈ n P ′ and thus thefirst inclusion is clear. Moreover, by (10.1), if k ≥ Y k = 0 for each Y ∈ n P ′ andthus for each n P ′ ∈ N P ′ one haslog n P ′ = ( n P ′ − −
12 ( n P ′ − and this gives the second inclusion. The second statement follows from Mahler’s criterionand Lemma 8.3. (cid:3) It is obvious that every γ ∈ Γ belongs only to finitely many Γ( q ). If we use equation(10.3), we easily see that [Γ ∩ N : Γ( q ) ∩ N ] goes to infinity if q does and so [Γ : Γ( q )]goes to infinity if q → ∞ . Thus applying Lemma 10.1, Corollary 1.3 follows from Theorem1.2. Principal congruence subgroups and Hecke subgroups of Bianchigroups
We finally turn to the proofs of Corollary 1.4 and Theorem 1.5. We let F := Q ( √− D ), D ∈ N square-free, be an imaginary quadratic number field. Let O D be the ring of integersof F , i.e. O D = Z + √− D Z if D ≡ , O D = Z + √− D Z if D ≡ D ) := SL ( O D ) be the associated Bianchi-group. Then X D := Γ( D ) \ H is offinite volume. More precisely, one hasvol( X D ) = | δ F | ζ F (2)4 π , where ζ F is the Dedekind zeta function of F and δ F is is the discriminant of F , see [Hu],[Sa, Proposition 2.1]. Let a be any nonzero ideal in O D and let N ( a ) denote its norm.Then the associated principal congruence subgroup Γ( a ) is defined asΓ( a ) := (cid:26)(cid:18) a bc d (cid:19) ∈ Γ( D ) : a − ∈ a ; d − ∈ a ; b, c ∈ a (cid:27) . Moreover, the associated Hecke subgroup Γ ( a ) is defined asΓ ( a ) := (cid:26)(cid:18) a bc d (cid:19) ∈ Γ( D ) : c ∈ a (cid:27) . Let P be the parabolic subgroup given by the upper triangular matrices in SL ( C ). Thenthe Langlands decomposition P = M AN is given by M = (cid:26)(cid:18) e iθ e − iθ (cid:19) , θ ∈ [0 , π ) (cid:27) and A = (cid:26)(cid:18) λ λ − (cid:19) , λ ∈ R , λ > (cid:27) ; N = (cid:26)(cid:18) b (cid:19) , b ∈ C (cid:27) . We recall that by [Ba, Corollary 5.2] the canonical map from SL ( O D ) to SL ( O D / a ) issurjective. Thus the sequence1 → Γ( a ) → Γ( D ) → SL ( O D / a ) → a it follows as in [Sh, Chapter 1.6] for theSL ( R )-case that [Γ( D ) : Γ( a )] = N ( a ) Y p | a (cid:18) − N ( p ) (cid:19) . (11.1)It also follows that the sequence1 → Γ( a ) → Γ ( a ) → P ( O D / a ) → is exact. Moreover the order of P ( O D / a ) is N ( a ) φ ( a ), where φ ( a ) := { ( O D / a ) ∗ } = N ( a ) Y p | a (cid:0) − N ( p ) − (cid:1) . (11.2)Thus one obtains [Γ( D ) : Γ ( a )] = N ( a ) Y p | a (1 + N ( p ) − ) . (11.3)Here the products in (11.1), (11.2) and (11.3) are taken over all prime ideals p in O D dividing a .Let P ( F ) be the one-dimensional projective space over F . As usual, we write ∞ for theelement [1 , ∈ P ( F ). Then SL ( F ) acts naturally on P ( F ) and by [EGM, Chapter 7.2,Proposition 2.2] one has κ (Γ( D )) = (cid:0) Γ( D ) \ P ( F ) (cid:1) . Using [EGM, Chapter 7.2, Theorem 2.4], it follows that κ (Γ( D )) = d F , where d F is theclass number of F . The group P is the stabilizer of ∞ in SL ( C ). For each η ∈ P ( F ) wefix a B η ∈ SL ( F ) with B η η = ∞ . We let B ∞ = Id. Then P η := B − η P B η is the stabilizerof η in SL ( C ) and the Γ( D )-cuspidal parabolic subgroups of G are given as P η . We let N η := B − η N B η . If η ∈ P ( F ), we let Γ( D ) η , Γ( a ) η , Γ ( a ) η be the stabilizers of η in Γ( D )resp. Γ( a ) resp. Γ ( a ).The following Proposition is an immediate consequence of the finiteness of the classnumber. Proposition 11.1.
The set of all principal congruence subgroups Γ( a ) and all Hecke sub-groups Γ ( a ) , a a non-zero ideal in O D , is cusp-uniform.Proof. Let J F be the ideal group of F , i.e. the group of all finitely generated non-zero O D -modules in F . We regard F ∗ as a subgroup of J F by identifying F ∗ with the group offractional principal ideals. Let I F := J F /F ∗ be the ideal class group. Then I F = d F < ∞ , see [Ne, chapter I.6]. Now for η ∈ P ( F ), B η as above, write B η = (cid:18) α βγ δ (cid:19) ∈ SL ( F )and let u be the O D -module generated by γ and δ and let b := u − ∩ γ − a . It is easy tosee that b = 0. Then proceeding as in [EGM, Chapter 8.2, Lemma 2.2], one obtains B η Γ( a ) η B − η ∩ N = (cid:26)(cid:18) ω ′ (cid:19) ; ω ′ ∈ au − (cid:27) ; B η Γ ( a ) η B − η ∩ N = (cid:26)(cid:18) ω ′′ (cid:19) ; ω ′′ ∈ b (cid:27) . Let P ′ be a Γ( D )-cuspidal parabolic subgroup of G and let Λ P | P ′ (Γ( a )) and Λ P | P ′ (Γ ( a ))denote the set of lattices defined as in (8.13). Since au − and b belong to J F , and I F is finite, it follows that Λ P | P ′ (Γ( a )), and Λ P | P ′ (Γ ( a )) are finite sets. Applying the thirdcriterion of Lemma 8.3, the proposition follows. (cid:3) The groups Γ( a ) are torsion-free and satisfy (3.1) for N ( a ) sufficiently large. This wasshown for example in the proof of Lemma 4.1 in [Pf2]. Since [Γ( D ) : Γ( a )] tends to ∞ if N ( a ) tends to ∞ and since each γ ∈ Γ( D ), γ = 1, is contained in only finitely many Γ( a ), Corollary 1.4 follows from Proposition 11.1 and Theorem 1.2.We finally turn to Theorem 1.5. The Hecke groups Γ ( a ) are never torsion-free and neversatisfy (1.5). However, we may take a finite index subgroup Γ ′ of Γ D , for example a fixedprincipal congruence subgroup of sufficiently high level, which is torsion free and satisfiesassumption (1.5). Then for each non zero ideal a of O D we letΓ ′ ( a ) := Γ ( a ) ∩ Γ ′ . This group satisfies now the required assumptions and if n := [Γ( D ) : Γ ′ ], then[Γ ( a ) : Γ ′ ( a )] ≤ n (11.4)for each non-zero ideal a . Thus since the set of all Γ ( a ) is cusp uniform by the precedinglemma, also the set of all Γ ′ ( a ), a a non-zero ideal in O D , is cusp uniform. Now, as in [AC,page 15], for an ideal b of O D we let φ u ( b ) := O D / b ) ∗ / O ∗ D ) . Then by [AC, Theorem 7] one has κ (Γ ( a )) = d F X b | a φ u ( b + b − a ) . (11.5)Now as in [FGT, Lemma 5.7], on the set of ideals in O D , we introduce the multiplicativefunction κ given by κ ( p k ) := ( N ( p ) k + N ( p ) k − k ≡ , N ( p ) k − k ≡ , where p is a prime ideal of O D . Using (11.5), it easily follows that κ (Γ ( a )) ≤ d F κ ( a ) , where one has equality if one replaces φ u by φ in (11.5). Now observe that κ ( a ) ≤ N ( a ) / Y p | a (cid:0) N ( p ) − (cid:1) . Using (11.3), we obtain κ (Γ ( a ))[Γ( D ) : Γ ( a )] ≤ d F p N ( a ) . Now by (11.3) we have the trivial bound [Γ( D ) : Γ ( a )] ≤ N ( a ) . It follows thatlim N ( a ) →∞ κ (Γ ( a )) log[Γ( D ) : Γ ( a )][Γ( D ) : Γ ( a )] = 0 . Thus every sequence Γ ( a ) satisfies assumption (1.8) for N ( a ) → ∞ . As above, if P , , . . . , P ,d F are fixed representatives of Γ( D )-cuspidal parabolic subgroups of SL ( C ), then κ (Γ ′ ( a )) = d F X j =1 { Γ ( a ) ′ \ Γ( D ) / Γ( D ) ∩ P ,j } and there is a similar formula for κ (Γ ( a )). Thus one has κ (Γ ′ ( a )) ≤ n κ (Γ ( a )) andputting everything together, it follows that the sequence Γ ′ ( a ) satisfies condition (1.8).It remains to prove that the contribution of the semisimple conjugacy classes to theanalytic torsion goes to zero for towers of Hecke subgroups. In order to prove this, weconsider the formula (8.6). According to section 8, for γ ∈ Γ( D ) we let c Γ ( a ) ( γ ) be thenumber of fixed points of γ on Γ( D ) / Γ ( a ). To begin with, as in [FGT] we let˜Γ( a ) := (cid:26)(cid:18) a bc d (cid:19) : a − d ∈ a : b, c ∈ a (cid:27) . Now we define a multiplicative function c ( · , · ) on the ideals of O D by putting c ( p k , p r ) := N ( p ) ( k + r ) / , k − r odd, k − r > N ( p ) ( k + r − / , k − r even, k − r > N ( p ) k + N ( p ) k − , k ≤ r, if p is a prime ideal and k, r ∈ N . Then the following proposition and its proof were kindlyprovided by Tobias Finis. Proposition 11.2.
Let γ ∈ Γ( D ) and let b be the largest divisor of a such that γ ∈ ˜Γ( b ) .Then one has c Γ ( a ) ( γ ) ≤ c ( a , b ) . In particular, if ν ( a ) denotes the number of prime divisors of a , one can estimate c Γ ( a ) ( γ ) ≤ ν ( a ) p N ( a ) N ( b ) . Proof.
We can identify the quotient Γ( D ) / Γ ( a ) with the projective line P ( O D / a ) and fora given γ ∈ Γ( D ) we have to estimate the number of its fixed points N ( γ, a ) on P ( O D / a ).By the strong approximation theorem we have N ( γ, a ) = Y p N ( γ, p ν p ( a ) ) , a = Y p p ν p ( a ) . So it suffices to study N ( γ, p k ) for a prime ideal p of O D . First assume that γ is scalarmodulo p k . Then every point of P ( O D / p k ) is a fixed point of γ . The number of elementsof the projective line P ( O D / p k ) equals N ( p ) k + N ( p ) k − . Thus in this case the lemma isproved. Next assume that γ is not scalar modulo p k . Let r < k be the maximal integersuch that γ is scalar modulo p r . We work over the completion O p of O at p . Let π be thecorresponding prime element. Then we have O p /π l ∼ = O / p l for every l . Over O p we havethe decomposition γ = a + π r η, where a is a scalar matrix and η is not scalar modulo π . A vector v ∈ O p which is notdivisible by π is an eigenvector of γ modulo π k if and only if it is an eigenvector of η modulo π k − r . If we consider the canonical map P ( O / p k ) → P ( O / p k − r ), then the preimage of eachelement in P ( O / p k − r ) has N ( p ) r elements. Thus if n denotes the number of eigenvaluesof η in P ( O / p k − r ), we have N ( γ, p k ) = N ( p ) r n . It remains to estimate n .To this end, we may assume that η has an eigenvalue. Otherwise there is nothing toprove. Then adding a scalar matrix and performing a base change over O p , which does notchange the number n , we may assume that η has the eigenvalue 0 with eigenvector (1 , t .Since we assumed that η is not scalar modulo π , after a base change we may assume that η is of the form η = (cid:18) d (cid:19) , where d ∈ O p . Now a set of representatives of eigenvectors in P ( O / p k − r ) of this matrixis given by all classes of vectors represented by (1 , y ), where y is chosen modulo p k − r andsatisfies y − dy ≡ p k − r . Thus n is the number of solutions of the quadraticcongruence for y ∈ O / p k − r . Let ν p be the valuation corresponding to p . Then this congru-ence is equivalent to ν p ( y ) + ν p ( y − d ) ≥ k − r . This implies that at least one summandis ≥ ( k − r ) /
2. We distinguish two cases. First, we assume that ν p ( d ) < ( k − r ) /
2. Thenexactly one summand is ≥ ( k − r ) / ν p ( d ). Thus in thiscase n is 2 times the number of all representatives whose valuation is ≥ k − r − ν p ( d ),i.e. n = 2 N ( p ) ν p ( d ) . Secondly, we assume that ν p ( d ) ≥ ( k − r ) /
2. Then the congruence isequivalent to ν p ( y ) ≥ ( k − r ) /
2. Thus in this case one has n = N ( p ) ⌊ k − r ⌋ . In all cases weobtain n ≤ N ( p ) ( k − r ) / if k − r is even and n ≤ N ( p ) ( k − r − / if k − r is odd. Putting ev-erything together, the first estimate follows. This estimate immediately implies the secondone. (cid:3) Remark . Proposition 11.2 also follows from more general estimates which are thecontent of a paper of Tobias Finis and Erez Lapid that is in preparation. Related resultsare also obtained in [A++].The following Lemma is due to Finis, Grunewald and Tirao.
Lemma 11.4.
For every δ > there is a constant C > such that for all non zero ideals b of O D and all R > the number of elements in [ γ ] ∈ C(Γ( D )) s which satisfy ℓ ( γ ) ≤ R and which belong to ˜Γ( b ) is bounded by N ( b ) − e (2+ δ ) R .Proof. This follows directly from [FGT, Lemma 5.10]. (cid:3)
Now we take a sequence a i of ideals such that N ( a i ) tends to infinity with i and welet Γ i := Γ ′ ( a i ), X i := Γ i \ H . We need to estimate the hyperbolic contribution H X i ( h τt ).We use formula (8.6), and apply the Fourier inversion formulas of Harish-Chandra to theinvariant orbital integrals using that the Fourier transform of h τt can be computed explicitly.This was carried out in [MP2]. If we combine [MP2, (10.4)] for the special case of dimension H X i ( h τt ) = X k =0 ( − k +1 e − tλ τ,k X [ γ ] ∈ C(Γ( D )) s − [1] c Γ i ( γ ) ℓ ( γ ) n Γ ( γ ) L sym ( γ ; σ τ,k ) e − ℓ ( γ ) / t (4 πt ) . (11.6)Here the λ τ,k ∈ (0 , ∞ ) are as in [MP2, (8.4)] and the σ τ,k ∈ ˆ M are determined by theirhighest weight Λ σ τ,k given as in [MP2, (8.5)]. Moreover, n Γ ( γ ) is the period of the closedgeodesic corresponding to γ and L sym ( γ ; σ τ,k ) is as in [MP2, (6.2), (10.3)]. By [MP2,(10.11)] and the definition of L sym ( γ ; σ τ,k ), there exists a constant C such that for all γ ∈ Γ( D ) s − { } one has ℓ ( γ ) n Γ ( γ ) | L sym ( γ ; σ τ,k ) | ≤ C . Thus together with equation (11.4), Proposition 11.2 and Lemma 11.4, it follows that thereexist constants C , C such that for each i we can estimate H X i ( h τt ) ≤ C ν ( a ) X b | a p N ( b ) N ( a ) X [ γ ] ∈ C(Γ( D )) s − [1] γ ∈ ˜Γ( b ) e − ℓ ( γ )24 t (4 πt ) ≤ C ν ( a ) X b | a p N ( ab ) ∞ X k =1 (cid:18) e − ( kℓ (Γ( D )))24 t (4 πt ) × { [ γ ] ∈ C(Γ( D )) s : γ ∈ ˜Γ( b ) : kℓ (Γ( D )) ≤ ℓ ( γ ) ≤ ( k + 1) ℓ (Γ( D )) } (cid:19) ≤ C ν ( a ) p N ( a ) X b | a N ( b ) − ∞ X k =1 ke − ( kℓ (Γ( D )))24 t (4 πt ) e (2+ δ ) kℓ (Γ( D )) . Let a = p k · · · · · p k ν ( a ) ν ( a ) be the prime ideal decomposition of a . Then we have2 ν ( a ) X b | a N ( b ) − ≤ ν ( a ) ν ( a ) Y j =1 − N ( p j ) − ≤ ν ( a ) . Now note that there are only finitely many prime ideals with a given norm. This impliesthat for every ǫ > C ( ǫ ) > a we have 2 ν ( a ) ≤ C ( ǫ ) N ( a ) ǫ .Hence the right hand side is O ( N ( a ) ǫ ) as N ( a ) → ∞ for any ǫ >
0, where the impliedconstant depends on ǫ . Thus there exist constants c, C , C > H X i ( h τt ) ≤ C ν ( a ) p N ( a ) X b | a N ( b ) − e − ct ≤ C N ( a ) e − ct . (11.7)Applying (11.3), it follows that for ever T ∈ (0 , ∞ ) one haslim i →∞ D ) : Γ i ] Z T t − H X i ( h τt ) dt = 0 . (11.8) Thus the analog of equation (9.11) is also verified for the present sequence Γ i of subgroupsderived from Hecke subgroups. Since it was shown above that this sequence is cusp uniformand satisfies condition 1.8, the proof of Theorem 1.1 given in section 9 can be carried overto the present case. Thus also Theorem 1.5 is proved. References [A++] M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault, I. Samet,
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