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The cohomology of lattices in SL(2,C)

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the standard self-dual modules. In the case Gamma = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for some geometrically constructed and mostly non-arithmetic groups. The computations for SL(2,O) lead us to discover two instances with non-lifted classes in the cohomology. We also derive an upper bound of size O(n^2 / log n) for any fixed lattice Gamma in the general case. We discuss a number of new questions and conjectures suggested by our results and our experimental data.


Introduction
For a semisimple Lie group G and a lattice Γ in G (i.e. a discrete subgroup of finite covolume), it is natural to consider the cohomology groups H * (Γ, E) of Γ with coefficients in finite-dimensional representation spaces E of G. If K is a maximal compact subgroup of G and X = G/K the associated Riemannian symmetric space, these cohomology groups are canonically isomorphic to the cohomology groups of the quotient orbifold Γ\X with coefficients in the local system associated to E. By the main result of [22], at least for arithmetic Γ the cohomology of Γ can be described by automorphic forms. For the contribution of the cuspidal spectrum one has (for general Γ) a generalized Matsushima formula describing the so-called cuspidal cohomology in terms of the multiplicities of cohomological unitary representations of G in L 2 cusp (Γ\G), a well-known result of Borel [5] predating [22]. The closer study of the non-cuspidal part is the object of the theory of Eisenstein cohomology initiated by G. Harder (cf. [loc. cit.], [32]), and there are fairly complete results in many cases.
While there is therefore a complete correspondence between cohomology and representation theory for the cuspidal part, generalizing the classical Eichler-Shimura homomorphism for the case G = SL(2, R) to all G, the behavior of the dimensions of the cohomology spaces is understood only under the hypothesis that G has a compact Cartan subgroup. In this case, one can compute these dimensions using Euler-Poincaré characteristics [57] and the trace formula [1]. If the highest weight of the representation E is regular, Arthur obtained in [loc. cit.] an explicit formula for the dimension of the cohomology, which in this case is accounted for by the packet of discrete series representations with the same infinitesimal character as the dual of E. In particular, the leading term in his formula is a constant multiple of the dimension of E. The question of computing the cohomology for non-regular highest weights and of separating the individual representations in the packet is connected to the problem of stabilization and to Arthur's conjectures. In the easiest case, namely for lattices Γ in G = SL(2, R), it is well-known that the dimension of the cohomology with coefficients in the symmetric power representations of G can be explicitly computed in terms of the basic invariants of Γ (i.e. the covolume and the orders of the elliptic elements), and in fact such a dimension formula follows without difficulties from the description of the group-theoretic structure of Γ or from the Riemann-Roch theorem (see formula (1.9) below).
The situation is different if G has no compact Cartan subgroup and there are therefore no discrete series representations. No explicit dimension formulas are known in this case. In this paper, we consider the simplest case of this type, namely lattices in the Lie group G = SL(2, C). Although the structure of this Lie group is very simple, the study of the cohomology of lattices in G presents a number of deep problems. The irreducible finite-dimensional representations of G are given by the tensor products E n,m = Sym n ⊗ Sym m (n, m ∈ Z, n, m ≥ 0). (1.1) Here Sym n stands for the n-th symmetric power of the standard two-dimensional representation of G and Sym m for its complex conjugate. For a lattice Γ in G = SL(2, C) we consider therefore the finite-dimensional cohomology spaces H i (Γ, E n,m ).
The main problem studied in this paper is the behavior of the dimension of these spaces as a function of n and m for a fixed Γ. Another problem which we consider is the behavior of the dimensions when Γ ranges over the subgroups of finite index in a lattice Γ 0 . Since the virtual cohomological dimension of Γ is three in the cocompact case, and two otherwise, the only dimensions with interesting cohomology are i = 1 and i = 2. We now define the subspaces of cuspidal cohomology classes and give their description in terms of automorphic forms. As a consequence, it will turn out that we only have to consider the case n = m and may in addition restrict to the first cohomology. Consider the set of all proper parabolic subgroups P = M U of SL(2, C) with the property that Γ ∩ U is a lattice in U . Here U is the unipotent radical of P and M is a Levi subgroup of P . Let C be a system of representatives for the finitely many classes of such parabolics under Γ-conjugation and consider the direct sum of restriction maps H i (Γ, E n,m ) −→ U i (Γ, E n,m ) = P ∈C H i (Γ ∩ P, E n,m ). (1. 2) The kernel of this map is called the cuspidal cohomology of Γ and denoted by H i cusp (Γ, E n,m ) ⊆ H i (Γ, E n,m ). (1.3) If Γ is cocompact, the set C is empty and we have H i cusp (Γ, E n,m ) = H i (Γ, E n,m ). We can also describe this construction geometrically. The group of orientation preserving isometries of three dimensional hyperbolic space X = H 3 can be identified with PSL(2, C) = SL(2, C)/{±1}, and every lattice Γ of SL(2, C) gives rise to a quotient orbifold Γ\X. If this orbifold is not compact, it can be compactified by adding a boundary ∂(Γ\H 3 ), which consists of finitely many disjoint two-dimensional tori or spheres. The inclusion is a homotopy equivalence. The cohomology of Γ with coefficients in E n,m can be computed as the cohomology of a sheafÊ n,m on the compactified orbifold Γ\H 3 . The restriction map coincides with the restriction map (1.2) (see [30] for a more detailed account).  [30]) and we will therefore restrict to the case i = 1 in the following. Furthermore, by a result of Serre [56,Th. 8], for i = 1 the dimension of the image of the map in (1.2) is one half of the dimension of the target space U 1 (Γ, E n,m ). It is not difficult to calculate the latter dimension explicitly. For example, if Γ ∩ P ⊆ ±U for all P ∈ C and n + m is even, the dimension of the image is equal to the number of cusps (the number of elements of C) by [loc. cit., Cor. 1]. For i = 2 one sees immediately from the long exact cohomology sequence for the pair ( Γ\H 3 , ∂(Γ\H 3 )) that the image of (1.2) is the entire target space, except in the case n = m = 0, where it has codimension one. It therefore remains to study the space H 1 cusp . The theorem of Borel mentioned above yields an isomorphism H 1 cusp (Γ, E n,m ) ≃ H 1 (g, K; L 2 cusp (Γ\G) ∞ ⊗ E n,m ), (1.5) where the superscript ∞ denotes the subspace of smooth vectors. Since the space of cuspidal functions decomposes discretely as a representation of G, we can also write In the case n = m the module E n := E n,n = Sym n ⊗ Sym n (1.8) is self-dual in the terminology of [4], and the dimension of H 1 cusp (Γ, E n ) is equal to the multiplicity of the principal series representation π 2n+2,0 (the representation unitarily induced from the character z → (z/|z|) 2n+2 of the maximal torus T ≃ C × ) in the space L 2 cusp (Γ\G). We will use this connection extensively in the following. We would like to stress that it does not yield an explicit dimension formula.
In the following we study the behavior of the dimension of H 1 (Γ, E n ) both theoretically and numerically. We focus primarily on the following problems.
A: How does the dimension of H 1 (Γ, E n ) behave when Γ is fixed and n grows? B: Are there formulas for the dimension of H 1 (Γ, E n ) in terms of n at least for some groups Γ?
Are there formulas for the dimension valid for all n ≥ n 0 , where n 0 is allowed to depend on Γ?
C: Are there lattices Γ such that H 1 cusp (Γ, E n ) = 0 or H 1 (Γ, E n ) = 0 for all n (necessarily cocompact in the latter case)? D: How does the dimension of H 1 (Γ, E n ) and H 1 cusp (Γ, E n ) behave when n is fixed and Γ ranges over the subgroups of finite index of a fixed lattice in G? E: How does the asymptotic behavior of H 1 (Γ, E n ) and H 1 cusp (Γ, E n ) for n → ∞ change when Γ ranges over the subgroups of finite index of a fixed lattice in G?
While we do not know of previous work on problems A-C and E, problem D has been studied qualitatively in the context of limit multiplicities [15,54] and of the conjecture of Waldhausen and Thurston (cf. [17]) in three-manifold topology. The results and computations described below are of very preliminary nature, but we hope to provide at least some evidence on what might be true. In Section 1.1 below we summarize our theoretical results on upper and lower bounds for the dimension of H 1 (Γ, E n ). In Section 1.2 we shall formulate more specific questions about the behavior of the these dimensions and discuss the numerical evidence accumulated in the later sections.

Theoretical results
We now describe our theoretical results. Before we proceed, let us briefly comment on the situation for Fuchsian groups, i.e. discrete subgroups Γ of SL(2, R) of finite covolume. Let g be the genus of Γ, k the number of cusps and r 1 , . . . , r s be the orders of the elliptic elements in the image of Γ in PSL(2, R), considered up to conjugacy. For an integer n and a positive integer r let µ be the remainder of n after division by 2r and set Then a consideration of the group-theoretical structure of Γ shows that for all n > 0, where for −1 ∈ Γ one has in addition to assume n to be even (the cohomology spaces vanish for odd n in this case). So, the dimension of the cohomology is given by simple linear functions on congruence classes. Note also that the coefficient of n + 1 in (1.9) is equal to vol(Γ\H 2 )/2π. Our first theoretical result on lattices in SL(2, C) concerns a general upper bound for the dimension of the cohomology. We prove This result is obtained by an application of the Selberg trace formula in Section 5. Note that the module E n has dimension (n + 1) 2 . Since a group Γ as above is finitely presented, dim H 1 (Γ, E n ) = O(n 2 ) is the trivial upper bound (cf. Lemma 3.1 below). Non-trivial lower bounds are not known for general lattices Γ ⊆ SL(2, C). In fact, our examples in Sections 7.2 and 7.3 indicate that there are probably none.
Lattices in SL(2, C) can be classified into arithmetic and non-arithmetic ones. The arithmetic lattices arise from quaternion algebras over number fields with precisely one complex place, and are intimately connected to number theory. The primary examples are the Bianchi groups SL(2, O K ), where O K is the ring of integers of an imaginary quadratic field K. It is well known that every non-cocompact arithmetic lattice in SL(2, C) is commensurable to a Bianchi group. For arithmetic groups one can use Langlands functoriality (base change, automorphic induction, and the Jacquet-Langlands correspondence) to obtain lower bounds on the cohomology in certain cases (cf. [11,47,52]). Unfortunately, because base change is only available for solvable extensions of number fields, the results are not complete (cf. [48,Section 6]). The structure of the non-arithmetic lattices is even less well understood. The explicit examples considered in this paper are on the one side Bianchi groups for certain K of small discriminant, and on the other side certain geometrically constructed lattices and series of lattices, almost all of which are non-arithmetic.
A related (but in general not equivalent) method for obtaining lower bounds is based on studying the action of the complex conjugation automorphism c of SL(2, C) on the cohomology, if the lattice Γ is invariant under c (as the Bianchi groups are, for example). In this case, one can use the Lefschetz fixed point formula to compute the trace of this involution acting on H 1 cusp (Γ, E n ), and thereby obtain a lower bound for the dimension of this space. For the case of the Bianchi groups with trivial coefficients this approach was carried out in [45,53]. Here the results turn out to be equivalent to the ones given by theory of base change. While this method is not restricted to arithmetic groups, on the other hand it does not cover all lower bounds obtainable by Langlands functoriality for arithmetic groups.
We work out the consequences of base change and automorphic induction (CM automorphic forms) for the cohomology of the Bianchi groups in Section 4 below. The base change construction detailed there associates to holomorphic automorphic forms for certain congruence subgroups of SL(2, Z) elements of H 1 (SL(2, O K ), E n ). Let us write for the corresponding subspace. Our main result here is a precise formula for the dimension of H 1 bc (SL(2, O K ), E n ) in terms of the prime factorization of the discriminant of K. To state it, we need to introduce functions ε k and µ k defined for all integers k which depend only on the residue class of k modulo 4 and 3, respectively, and a function ν K,k depending on K and at most on k modulo 2 or 3 (see Sections 4.1 and 4.3 below for the precise definitions). Theorem 1.2. Let K be an imaginary quadratic extension of Q, R the set of primes ramified in K (the prime divisors of the discriminant of K), and for each p ∈ R let ν p be the exact power of p dividing the discriminant. Then there are non-negative constants c 2 , c 3 and c 4 (depending on K) such that for all n ≥ 0, where h K is the class number of K and δ n,0 denotes the Kronecker delta symbol.
Note that for every K the dimension of H 1 bc is given by linear functions of n spread out over the congruence classes modulo 12. If one takes the precise value of c 2 given in Section 4.3 into account, one sees that the coefficient of n in these linear functions is always positive, and that therefore the dimension of H 1 cusp (SL(2, O K ), E n ) grows at least linearly with n. Also, for a fixed n the dimension grows linearly in the (absolute value of) the discriminant. This implies non-vanishing results for the cuspidal cohomology. The first results of this nature (in a much more limited situation) are in [28,29]. Let us describe the special cases K = Q( √ d), d = −2, −7, −11, more explicitly. (1.12) A second construction of cohomology classes is via automorphic induction from Hecke characters of quadratic extensions of K (in fact necessarily biquadratic extensions of Q unramified over K). In Section 4.2 we describe the corresponding contribution H 1 CM to the cuspidal cohomology. In many cases, it is already contained in H 1 bc . The precise criterion for an additional CM contribution to the cohomology is as follows.
Proposition 1.4. Let K be an imaginary quadratic field. There is a CM contribution to a space if and only if for some real quadratic field L ′ such that KL ′ /K is unramified, the narrow class number h + L ′ is strictly bigger than the corresponding number of genera g + L ′ = 2 |R(L ′ )|−1 , where R(L ′ ) denotes the set of primes ramified in L ′ . See Section 4.2 for the smallest examples of real quadratic fields L ′ with this property. In any case, the additional CM contribution is always constant on residue classes modulo 12, and the total contribution from base change and automorphic induction is therefore again given by linear functions on residue classes mod 12.
The techniques of Langlands functoriality have been previously used to prove (a generalization of) the conjecture of Waldhausen and Thurston for some arithmetic groups Γ, i.e. to establish for fixed n the existence of a finite index subgroup ∆ of Γ with H 1 (∆, E n ) = 0. In fact, one can in all cases in the literature show that there exists such a ∆ with H 1 (∆, E n ) = 0 for all n. Building on the work of Labesse-Schwermer [47], Rajan considers in [52] arithmetic groups Γ associated to quaternion algebras defined over fields L such that the extension L/L tr , where L tr is the maximal totally real subfield of L, is solvable (but not necessarily Galois). His methods easily imply the following result on Problem E. Proposition 1.5. Let Γ be an arithmetic subgroup of SL(2, C) such that the field of definition L of the corresponding quaternion algebra is a solvable extension of its maximal totally real subfield L tr . Then for every c > 0 there exists a finite index subgroup ∆ of Γ such that If base change for SL (2) for arbitrary extensions of number fields was available, one could prove the corresponding result for all arithmetic lattices Γ.
We now turn to a result which gives an upper bound for special cases of problem D. We consider finite index subgroups of the Bianchi groups SL(2, O K ). For any non-zero ideal a of O K we have the classical congruence subgroup . (1.14) The following theorem gives a bound for the dimension of H 1 (Γ ∩ Γ 0 (a), E n ) for each subgroup Γ of finite index in SL 2 (O K ), which improves the trivial bound O(ι(a)) by a logarithm. Theorem 1.6. Let Γ be a subgroup of finite index in SL(2, O K ), K imaginary quadratic. Then for any fixed n ≥ 0 we have This theorem, which again results from an application of the trace formula, should be compared to the limit multiplicity results of de George-Wallach [15], Lück and Savin [54], which imply for fixed n and towers of normal subgroups Γ i of a fixed lattice Γ such that i Γ i = {1}. Note also that for a = ao K , a a positive integer, we can get by base change arguments a lower bound of the form Ca = CN(a) 1/2 . If a and its conjugate are relatively prime, there is no nontrivial lower bound known. On the other hand, it is certainly not possible to improve the trivial bound O([SL(2, O K ) : Γ]) on the dimension of H 1 (Γ, E n ) for finite index subgroups Γ of SL(2, O K ) without making any assumption on Γ. This follows easily from the fact that the Bianchi groups (and more generally all non-cocompact lattices in SL(2, C)) are large, i. e. contain a finite index subgroup surjecting onto a non-abelian free group. This property is in fact conjectured to be true for all lattices (cf. [48]).

Experimental results and questions
Here we formulate more specific versions of problems A-E from above. We shall also discuss the numerical results accumulated in the later sections.
Let us begin by reporting on our numerical calculations. In Section 3.1 we describe how the cohomology space H 1 (Γ, E n ) can be effectively computed from a presentation of Γ together with explicit matrices for the generators. We have developed computer codes for this task. The results of the computations are documented in Sections 6 and 7. Consider first the case of Bianchi groups explained in Section 6. We consider the fields K = Q( In both cases there is a uniquely determined two-dimensional complement invariant under the action of the Hecke algebra. We document the eigenvalues of the first Hecke operators on these subspaces in Section 6.2 below.
These computations suggest the following question: • For a given field K, is for all but finitely many n? In Section 7 we consider some examples of (mostly) non-arithmetic lattices. All examples are compatible with an affirmative answer to the following question: Question 1.9. For a given lattice Γ in SL(2, C), do there exist integers n 0 ≥ 0, N > 0 depending on Γ such that for each n ≥ n 0 and n in a fixed residue class modulo N the dimension dim H 1 (Γ, E n ) is given by a linear function in n?
As we have seen, an affirmative answer to the first part of Question 1.8 would imply that one might take N = 12 for the Bianchi groups. A weaker but still unresolved question is: Question 1.10. Do we have dim H 1 (Γ, E n ) = O(n) as n → ∞ for every lattice Γ or is it possible that these dimensions grow faster than linearly in n?
The computations in Section 7.2 and 7.3 suggest an affirmative answer to the following question: Question 1.11. Are there lattices Γ such that dim H 1 (Γ, E n ) remains bounded as n → ∞? Is it possible that H 1 cusp (Γ, E n ) = 0 or even H 1 (Γ, E n ) = 0 for all n (Γ being necessarily cocompact in the latter case)?
In Section 7.2 an infinite sequence of non-arithmetic groups with one cusp is considered, which provides candidates for lattices with H 1 cusp (Γ, E n ) = 0 for all n. In Section 7.3 we consider a cocompact non-arithmetic lattice and its finite index subgroups of low index and obtain many candidates for lattices with H 1 (Γ, E n ) = 0 for all n.
Concerning Problem D, we pose the following variant of the conjecture of Waldhausen and Thurston as a question: We have also made extensive computations of dim H 1 (Γ 0 (p), C) for the standard congruence subgroups Γ 0 (p) of SL(2, O −1 ) associated to degree one prime ideals p of O −1 . The results are documented in Section 6.3. The cohomology groups H 1 (Γ 0 (p), C) are particularly interesting for number theory since their non-vanishing is conjectured to be related to the existence of certain elliptic curves (or more generally abelian varieties) defined over K = Q(i) (cf. [14,25,27]). Also, the methods of Langlands functoriality do not provide any non-trivial lower bound for the dimension, and in fact there are many examples of prime ideals p with H 1 (Γ 0 (p), C) = 0. The analogy with distribution questions for elliptic curves (cf. [7]) suggests: Question 1.13. Is there a constant C such that the asymptotic relation holds as x tends to infinity, where the sum is to be extended over all degree one prime ideals p of O −1 of norm at most x?
The computational results in Section 6.3 are compatible with an affirmative answer to this question. But the range of our computations seems to be too small to allow a more detailed analysis (cf. [7]). The behavior of the dimensions dim H 1 (Γ 0 (p), E n ) seems to be quite different if n ≥ 1 is fixed and p varies (see Section 6.3).
Finally, we pose the following question regarding Problem E: Question 1.14. For a given lattice Γ, does there exist a subgroup ∆ of finite index such that While the theoretical evidence summarized in Section 1.1 above suggests that this question has an affirmative answer for arithmetic lattices Γ, our computations for non-arithmetic groups are inconclusive. Namely, for the groups considered in Sections 7.2 and 7.3 we were not able to find such a finite index subgroup ∆, but to search through all subgroups of a given index very quickly becomes prohibitive.

The Bianchi groups
This section contains some notation and preliminary material concerning the Bianchi groups, as well as the explicit finite presentations on which our computer calculations are based. The first subsection fixes notation which we will use throughout this paper. The results needed from algebraic number theory are contained in [49]. We also follow this book in our notational conventions.

The Bianchi groups and their congruence subgroups
The discriminant of the field K is We set R = R d for the set of rational primes p ramified in K. The set R d consists exactly of the prime divisors of D d .
We also fix the following notation concerning subgroups of SL(2, C) commensurable with the Bianchi groups SL(2, O). Let a ⊆ O be a non-zero ideal. The subgroup is called the full congruence subgroup of level a. It clearly has finite index in SL(2, O). A subgroup Γ ⊆ SL(2, K) is called a congruence subgroup if Γ∩SL(2, O) has finite index in both Γ and SL(2, O), and if Γ contains a full congruence subgroup Γ(a) for a non-zero ideal a of O. Let a ⊂ K now be a fractional ideal of O, that is a is a non-zero finitely generated O-submodule of K. We define Notice that SL(2, a) is a congruence subgroup of SL(2, K). It is equal to the stabilizer in SL(2, K) of the O-submodule O ⊕ a of K 2 . We write PSL(2, O) or PSL(2, a) for the images of the corresponding subgroups of SL(2, K) in PSL(2, C).
We write A K for the ring of adeles of K and A K,f for the ring of finite adeles. We view A K,f as the subring of A K consisting of those elements which are 0 at the infinite place of K. The adele rings A K , A K,f and their unit groups A * K , A * K,f are equipped with their standard topologies (see [49]). We also consider the profinite completion of the ring O, which we denote byÔ, to be embedded as a compact and open subring of A K,f in the usual way.
Recall the standard description of the adelic coset space GL(2, K)\GL(2, A K ) in terms of the coset spaces Γ\GL(2, C) for congruence subgroups Γ of GL(2, K). By the strong approximation theorem, for any compact open subgroup K of GL(2, A K,f ) the determinant map identifies the space of connected components of with the finite set A * K,f /K * det(K), and for a set S ⊂ GL(2, A K,f ) with the property that det(S) forms a system of representatives for To obtain the special case of the groups SL(2, a), let K 0 = GL(2,Ô) be the standard maximal compact subgroup of GL(2, A K,f ) and for each finite index subgroup ∆ ofÔ * set (2.6)

Presentations
This subsection contains explicit finite presentations for some of the Bianchi groups. We include them here, because some of them have not yet appeared in print. The presentations are taken from [21,55,62]. We use the standard notation for presentations of groups: G = g 1 , . . . , g n R 1 , . . . , R l means that the group G is generated by g 1 , . . . , g n and presented by the words R 1 , . . . , R l .
The following three matrices are in the set of generators in almost all cases: We first give the results for the cases d = −1, −2, −3, −7, −11, which are exactly the cases in which the ring of integers O d is euclidean.
Next we consider the case d = −19. In this case O d is a non-euclidean principal ideal ring. We have with the matrix In the cases d = −5, −6, −10 the class number of O d is equal to 2. We give presentations of both PSL(2, O d ) and of PSL(2, a) for a non-principal ideal a.
with the ideal a −5 = 2, 1 − √ −5 of O −5 and with the matrices with the matrices with the ideal a −6 = 2, √ −6 of O −6 and with the matrices with the matrices with the ideal a −10 = 2, √ −10 of O −10 and with the matrices The ideal class group of O −14 is cyclic of order 4. The ideal a −14 = 3, 1 + √ −14 is not a square in the ideal class group.
with the matrices with the matrices

Group cohomology
In this section we report basic definitions from the cohomology of groups. Section 3.1 reports a method to compute the first cohomology group for finitely presented groups. Our basic reference here is [6]. Let Γ be a group and M a RΓ-module for a commutative ring R.
is a derivation and is called the inner derivation corresponding to m. We write Der(Γ, M ) for the space of all derivations and IDer(Γ, M ) for its subspace consisting of inner derivations. If H 1 (Γ, M ) is the first cohomology group of Γ with coefficients in M , we have Here we are interested in the case when Γ ⊆ SL(2, L) ⊆ SL(2, C) where L ⊂ C is a number field. The modules we consider are derived from the symmetric powers of the standard representation of SL(2, C). So, let V be a two-dimensional L 1 -vector space with basis x, y where L 1 is a field between L and C invariant under complex conjugation. Let n be a non-negative integer. The symmetric power Sym n (L 1 ) has the L 1 -basis x n−i y i , 0 ≤ i ≤ n. The action of g ∈ SL(2, L 1 ) is given by The module Sym n (L 1 ) is equal to Sym n (L 1 ) as an L 1 -vector space, and the action is given by replacing g in (3.4) by its complex conjugate. We often use the follwing simple facts from group cohomology without further notice. First of all the spaces H 1 (Γ, Sym n (L 1 ) ⊗ Sym m (L 1 )) ⊗ C and H 1 (Γ, Sym n (C) ⊗ Sym m (C))) are isomorphic for all n, m ≥ 0. Secondly, if m + n is even, the action of Γ on Sym n (L 1 ) ⊗ Sym m (L 1 ) factors through an action of the imageΓ of Γ in PSL(2, C) and H 1 (Γ, Sym n (L 1 ) ⊗ Sym m (L 1 )) is isomorphic to H 1 (Γ, Sym n (L 1 ) ⊗ Sym m (L 1 )).

H 1 (Γ, M) for finitely presented groups
Here we explain how information about H 1 (Γ, M ) can be computed from equation (3.3). We assume here that R is an euclidean ring and M is a free R-module of finite rank in which a basis has been chosen. Let Γ be a finitely presented group given explicitly in the form Here we consider the relations R 1 , . . . , R t to be explicitly given words in the generators g 1 , . . . , g s of Γ and their inverses. Assume also that the matrices for the action of g 1 , . . . , g s on M are explicitly given. Consider now the R-linear map The image of Φ lies in the kernel of the linear map Λ : M s → M t , which is obtained by formally expanding the image of each of the relators R 1 , . . . , R t under a derivation f : Γ → M in terms of the values f (g 1 ), . . . , f (g s ). It is easily seen that Φ maps Der(Γ, M ) isomorphically to the kernel ker(Λ) of Λ. Since M s is a free R-module, a basis for the free module ker(Λ) can be computed. Consider now the linear map The image of µ may then be described as the linear span of the images of the basis elements of M . If we express these in terms of the previously computed basis of ker(Λ), we see that the effective version of the elementary divisor theorem can be used to compute the structure of If R is a field, the dimension of H 1 (Γ, M ) can be computed by this method. Apart from being important for the computation of cohomology spaces, (3.5) leads to the following (trivial) estimate. A typical problem encountered in our computations is that the module M can be a vector space of big dimension (up to around 50000) over an algebraic number field, and that the direct computation of the dimension of H 1 (Γ, M ) from (3.5) is not feasible. All discrete subgroups Γ ⊆ SL(2, C) considered in this paper have the property that they are contained in SL(2, R) for a finitely generated ring R inside an algebraic number field. Let O Γ be a ring containing R and its complex conjugate. Suppose p is a prime and O Γ → F p is a surjective ring homomorphism. Then E n (F p ) inherits the structure of a Γ-module. By the usual universal coefficient theorem we have A standard argument using Tchebotarev's density theorem shows that dim C H 1 (Γ, E n ) is equal to the minimum of the dimensions dim Fp H 1 (Γ, E n (F p )), where p ranges over all primes with the above compatibility property. For all real numbers x we define where p ranges over all primes with p ≤ x which admit a surjective ring homomorphism O Γ → F p . The numbers dim ≤x H 1 (Γ, E n ) are much cheaper to compute than the actual dimensions dim C H 1 (Γ, E n ). Of course, in the computations below we hope to have chosen the bound x to be large enough to capture dim C H 1 (Γ, E n ). Also, if a lower bound for this dimension is known beforehand, we can by this method verify that the actual dimension is equal to the bound.

Hecke operators
In this section we introduce the Hecke operators on cohomology spaces in a way suitable for explicit computations. We chose a treatment similar to [59,Section 8.5], see also [25]. Let now Γ be a congruence subgroup of SL(2, O), where O is the ring of integers in an imaginary quadratic number field K, and M be one of the GL(2, C)-modules E n,m . The groups Γ and δΓδ −1 are easily seen to be commensurable for every δ ∈ GL(2, K). Define the Hecke operator whereδ Is the isomorphism in cohomology induced by conjugation with δ. For a non-zero element a ∈ O we define The following properties of the linear maps T δ : [25]): • Each T δ is diagonalizable.
• The characteristic polynomial of T δ has integral coefficients and its zeroes are real numbers.
• T δ depends only on the double coset ΓδΓ.

The Eichler-Shimura isomorphism
In this subsection we briefly recall the generalized Eichler-Shimura isomorphism sketched already in the introduction, which will give us the possibility of using results from the theory of automorphic forms in our study of cohomology spaces. See also [32] and [64, Théorème 3.2] for the case of congruence subgroups of GL(2, K), K imaginary quadratic. From [4, Chapter II] we know that for any integer n ≥ 0, and any unitary representation π of G = SL(2, C), the (g, K)-cohomology space H 1 (g, K; H ∞ π ⊗ E n ) is non-trivial if and only if π is the principal series representation π 2n+2,0 (the representation unitarily induced from the character z → (z/|z|) 2n+2 of the maximal torus T ≃ C × , cf. Section 5), and one-dimensional in this case. Therefore, we can deduce from (1.6) the more explicit isomorphism for any lattice Γ of G.
For use in Section 4, we quickly rewrite this isomorphism in a form involving GL(2, C). Define a unitary character of C * by χ ∞ (x) = x/|x|, and for each integer n ≥ 0 consider the principal series be the center of GL(2, C). Then we have an isomorphism

Base change
This section contains our results on the construction of cohomology classes for the Bianchi groups by base change from classical modular forms for congruence subgroups of SL(2, Z) and automorphic induction from Hecke characters of quadratic extensions. In particular, we derive explicit dimension formulas for the corresponding subspaces of the cohomology. For this we fix an imaginary quadratic number field K = Q( √ d) and use the notation of Section 2.1.

General results on the base change construction
Here we present the consequences of the theory of base change and automorphic induction for the cohomology of the groups SL(2, a). We give a precise description of the base change process and the relevant spaces of holomorphic elliptic modular forms. The notation and concepts from the theory of automorphic forms are taken from [4]. For a quadratic extension L of Q denote by ω L the associated quadratic character of A * Q /Q * . Let A K be the set of all cuspidal automorphic representations of GL(2, A K ). See [50] for information on the base change map π → π K from GL(2, A Q ) to GL(2, A K ). We shall be interested in the following subset of A K .
Definition 4.1. The set A bc K of (twisted) base change representations is the set of all Π ∈ A K such that Π ≃ π K ⊗ χ for an automorphic representation π of GL(2, A Q ) and an idele class character χ of K.
Recall from Section 3.3 the definition of the representations ρ n ∞ of GL(2, C). For an integer n ≥ 0 and a finite index subgroup ∆ ofÔ * consider Recall that each representation in A K occurs with multiplicity one in L 2 cusp (GL(2, K)\GL(2, A K )). Furthermore, Π ∈ A 1 K (n) is equivalent to the condition that Π ∞ ≃ ρ n ∞ and that the local components Π p at the finite places p are twists of unramified principal series representations by characters. Therefore, for is actually one-dimensional. If we take a subgroup ∆ ofÔ * with the property ∆∩O * = {1}, we have by (2.5 where a ranges over a system of representatives for the ideal classes of K. Combining the Eichler-Shimura isomorphism from Section 3.3 with multiplicity one and the fact that dim Π It is not difficult to obtain also a finer description distinguishing between the individual cohomology spaces dim H 1 cusp (SL(2, a), E n ) for representatives a of different ideal classes. For this consider the action of the abelian group X(∆) on the space Hom(ρ ∞ n , L 2 cusp (GL(2, K)\GL(2, A K )/Z ∞ K(∆))) given by letting ξ ∈ X(∆) (see Section 2.1) act as multiplication of functions on [64,Section 5]) eigenfunctions for the Hecke algebra of K(∆) (which correspond to the representations in A 1 K (n, ∆)), one sees that the action of X(∆) induces a permutation of this basis, and therefore the trace of the action of a non-trivial element ξ ∈ X(∆) is equal to the number of elements Π ∈ A 1 K (n, ∆) with Π ⊗ ξ ≃ Π. It is clear that this number can only be non-zero if ξ is quadratic, and indeed unramified quadratic, i. e. necessarily of the form ω L • N K/Q for an imaginary quadratic field L = K such that LK/K is unramified (see Proposition 4.5 below). Therefore, we get where L(K) denotes the set of all imaginary quadratic fields L = K with LK/K unramified. Furthermore, if A ⊆ A 1 K (n, ∆) is any subset invariant under twisting by characters in X(∆), we can consider inside the space L 2 cusp (GL 2 (K)\GL(2, A K )/Z ∞ K(∆)) the subspace spanned by representations in A and apply the same arguments to see that it splits as a direct sum of spaces of functions supported on a single connected component. This implies that it makes sense to speak of the contribution of representations in A to each space H 1 cusp (SL(2, a), E n ) and that the dimension of the corresponding subspace is given by In particular, this dimension depends only on the genus of a and it assumes its maximum on the principal genus. We are especially interested in evaluating the contribution of twisted base change forms to the cohomology, i. e. in the case A = A 1,bc K (n, ∆).

Definition 4.2. For the set
Note that this definition makes sense, since the right-hand side is indeed independent of the subgroup ∆ with ∆ ∩ O * = {1}.
Our first goal is to describe the set A 1,bc K (n, ∆) in terms of holomorphic automorphic forms for GL(2, A Q ) fulfilling explicit local conditions. We also need to distinguish the automorphic representations of CM type. Recall that for any quadratic extension E/F of number fields there is a canonical map from Hecke characters of E to automorphic representations of GL(2, A F ) [37] which is called automorphic induction (notation: AI E/F ). The map is characterized by AI E/F (θ χ•N E/F ) = AI E/F (θ) ⊗ χ for Hecke characters χ of F , and the L-function identity L(s, AI E/F (θ)) = L(s, θ). The automorphically induced representation is cuspidal if and only if θ τ = θ, where τ is the automorphism of E/F , and the fibers of the automorphic induction map are precisely the orbits {θ, θ τ } of τ . There are compatible local induction maps, which we also denote by AI.
In the following, we fix once and for all for each p ∈ R a character θ p of K * p with θ p /θ c p unramified quadratic.
be the set of all cuspidal automorphic representations π of GL(2, A Q ) such that π ∞ is the holomorphic discrete series representation of weight n + 2, π p is unramified for p / ∈ R, and of one of the following three types for p ∈ R: 1. unramified principal series, For any (necessarily imaginary) quadratic extension L of Q let A 1 Q (n; L) be the subset of A 1 Q (n) consisting of representations automorphically induced from L. Recall that π ∈ A 1 Q (n; L) if and only if π ⊗ ω L ≃ π [46]. We will see that the set of possible extensions L is precisely L(K).
The basic classification statement is the following proposition. It shows that we obtain the representations in A 1,bc K (n, ∆) by base change and character twists from the elliptic modular forms satisfying the local conditions of Definition 4.3. Of course, the description depends on the choice of the local characters θ p for p ∈ R. 1. If Π ∈ A 1,bc K (n), one can find π ∈ A 1 Q (n)\A 1 Q (n; K) such that Π ≃ π K ⊗ χ for some finite order idele class character χ of K.

If
, for any π ∈ A 1 Q (n)\A 1 Q (n; K) there exists a finite order idele class character χ of K with π K ⊗ χ ∈ A 1 K (n, ∆). The set of all such characters χ is a principal homogeneous space for X(∆).
Proof. For an automorphic representation π of GL(2, A Q ) we have π K ∈ A 1 K (n) if and only if π is not automorphically induced from K, π ∞ is up to a character twist the holomorphic or antiholomorphic discrete series representation of weight n + 2, and each π p satisfies up to a character twist the local condition of Definition 4.3. It is not difficult to deduce from this the assertions of the Proposition.
We can also classify the CM representations of interest to us as follows.
1. If A 1 Q (n; L) is nonempty, L is an imaginary quadratic extension of Q such that for all primes p the character ω L,p is either unramified or the product of ω K,p and an unramified character.
2. If for Π ∈ A 1 K (n) there exists a character γ = 1 with Π ⊗ γ ≃ Π, the character γ is necessarily of the form ω KL/K = ω L • N K/Q for some quadratic extension L/Q as above.
The set of all imaginary quadratic number fields different from K and satisfying the conditions of Part 1 of Proposition 4.5 is precisely the set L(K) of imaginary quadratic fields different from K for which LK/K is unramified. Equivalently, it is the set of all imaginary quadratic fields L for which the discriminant d L is a proper divisor of the discriminant of K and the two factors d L and d K /d L are coprime.
consists of the twists π ⊗ γ for all characters γ such that γ p is unramified for all p where π p is unramified, and γ p is unramified or the product of ω K,p and an unramified character at the primes p where π p is ramified. In particular, it has cardinality 2 |R(π)| , where R(π) ⊆ R denotes the set of all primes p where π p is ramified.
Therefore, if we want to write the cardinality of A 1,bc K (n, ∆) as a sum over all representations π ∈ A 1 Q (n)\A 1 Q (n; K), each non-CM representation π ∈ A 1 Q (n)\ L A 1 Q (n; L) has to be weighted by the factor |X(∆)|2 −|R(π)| . Example 4.7. Consider the case where a single prime p is ramified in K. In this case, the set A 1 Q (n) consists of the automorphic representations associated to classical modular forms of weight n + 2 for SL(2, Z), for Γ 0 (p) with character ω K , or of p-power level with π p ≃ AI(θ p ) ⊗ γ p , γ p unramified. The CM forms for K have to be omitted. In this case, there are no other fields L to be considered. To obtain the dimension of H 1 bc (SL(2, O), E n ), the dimension of the corresponding spaces of modular forms has to be weighted by a factor 1/2 except in the SL(2, Z) case.
In the count for the representations in A 1,bc K (n, ∆), the main term is therefore given by |X(∆)| π∈A 1 Q (n) 2 −|R(π)| . The contributions from CM representations have to be modified by omitting the representations automorphically induced from the field K and weighting the contribution of the representations induced from quadratic fields L ∈ L(K) by an additional factor 1 2 . The reason for this is that for these representations there are more equivalences A 1 K (n, ∆; π ′ ) = A 1 K (n, ∆; π) than in the non-CM case.
To give some more details, we first explicate the local conditions on CM representations in A 1 Q (n). Recall the definition of the local character at infinity χ ∞ in Section 3.3. 1. If p ∈ R splits in L, ψ p is either unramified or of the form (α p , ω K,p β p ) or (ω K,p α p , β p ) for unramified characters α p and β p of Q × p .
2. If p ∈ R is inert in L, ψ p is either unramified or the product of θ p,Lp or θ c p,Lp and an unramified character.
3. If p ∈ R ramifies in L, ψ p is either unramified or the product of θ p,Lp and an unramified character.
Lemma 4.9. If A 1 K (n, ∆; π ′ ) = A 1 K (n, ∆; π) for π, π ′ ∈ A 1 Q (n) which are automorphically induced from quadratic extensions, they are necessarily induced from the same quadratic extension L. Furthermore, for π = AI L/Q (ψ) and π ′ = AI L/Q (ψ ′ ) with ψ and ψ ′ as above, the equivalence With these descriptions in hand, one can obtain a preliminary formula for the cardinality of A 1,bc K (n, ∆), which will in a second step be refined to a completely explicit expression. Using (4.4) we can then compute the contribution to the cohomology of each individual group SL(2, a).
To simplify the notation, we need the following definition: for an integer n and an imaginary quadratic field L define ν L,n ∈ {0, 1} as follows: 2. If L = Q(i), set ν L,n = 1, if n ≡ 1 (2), 0, otherwise.
3. If L is not one of the two exceptional fields, we simply set ν L,n = 1 for all n.
1. The cardinality of the set A 1,bc K (n, ∆) is given by

For L ∈ L(K) we have
Combining this Proposition with (4.4) and the fact that A 1 Q (n; K) = ν K,n 2 |R|−1 h K , we can immediately deduce:

CM classes
As a consequence we also obtain the following results on cohomology spaces associated to CM automorphic forms. We introduce the following notation. We can also consider all representations in A 1 K (n, ∆) automorphically induced from a fixed quadratic extension of K, necessarily of the form KL for an imaginary quadratic extension L as above. For this, let L ′ be the real quadratic subfield of LK and h + L ′ its narrow ideal class number. The total number of such representations is then equal to Consequently, we obtain: Proposition 4.14. For L ∈ L(K) the contribution of representations automorphically induced from KL to H 1 cusp (SL(2, a), E n ) has dimension Note that this is precisely the contribution of twisted base change representations of the corresponding type times a factor of h + L ′ /2 |R|−|RL|−1 , which is the number of narrow ideal classes in a narrow genus of L ′ .
The following relation between the dimension of the cohomology spaces for SL(2, a) and SL(2, O) follows immediately from (4.5) and (4.4), this time applied with A = A 1 K (n, ∆): The existence of such representations (i. e. the failure of the relation H 1 CM ⊆ H 1 bc for the field K) is equivalent to the existence of a real quadratic field L ′ with h + L ′ > g + L ′ and KL ′ /K unramified (equivalently, d L ′ divides the discriminant d K and d L ′ and d K /d L ′ are coprime).
In the following table we give the real quadratic fields L ′ = Q( √ D) with the five smallest discriminants, for which the criterion of Corollary 4.16 is satisfied.

Dimension formulas
We now deduce from the preliminary formula of Proposition 4.11 a completely explicit dimension formula for H 1 bc . For p ∈ R let ν p be the exact power of p dividing the discriminant of K. We have ν p = 1 for p = 2 and ν 2 = 2 or 3.
For any integer n set where δ n,0 stands for the Kronecker delta symbol. The constant c 2 is given by The constants c 4 and c 3 are given by It is not difficult to prove this Proposition using the description of SL(2, Z) as an amalgamated product of −S 3 and S 4 and the Eichler-Shimura isomorphism. Of course, it is also a consequence of the trace formula. By taking for σ a representation induced from the Borel subgroup, one recovers the classical dimension formulas for the group Γ 0 (N ) with nebentype (cf. [12]).
It remains to make explicit the representations of SL(2, Z/N Z) corresponding to the local conditions of Definition 4.3 and to compute the terms appearing in Proposition 4.18. Actually, we will consider irreducible representations σ of GL(2, Z/N Z) occurring in the automorphic representations in question with multiplicity one and use their restrictions to SL(2, Z/N Z). The representations σ can be written as tensor products of representations σ p of GL(2, Z/p νp Z) for p ∈ R(π). For the principal series representations of Definition 4.3 the necessary computation of dimensions and character values is standard and we refer to [12]. For the convenience of the reader we repeat the results here. The dimension of the corresponding representation σ p is p νp−1 (p + 1). The dimension of the space of U N -invariants equals 2. The character values are given by The parity of σ p is equal to ω K,p (−1). For the supercuspidal components we can use the constructions of [8].
Lemma 4.19. Let p ∈ R and Q p 2 be the unramified quadratic extension of Q p . We can write the representation The minimal conductor of such a character is p νp . Assume in the following that θ ′ p has this minimal conductor. Then π p contains (with multiplicity one) a unique representation of GL(2, Z p ), and this representation factors through a representation σ p of GL(2, Z/p νp Z). The dimension of σ p is p νp−1 (p−1). If p is odd, σ p is the cuspidal representation of GL(2, F p ) associated to the character of F × p 2 obtained by restricting θ ′ p . Furthermore, we have the following values for the traces at the torsion elements of SL(2, Z): The parity of σ p is −ω K,p (−1) except in the case p = 2, ν p = 2, where it is −1 while ω K,p (−1) = −1.

Proof.
We quickly sketch the ingredients of the proof. Everything is based on the Tame Parametrization Theorem of [8,Theorem 20.2]  We can now finish the proof of Theorem 4.17. For any tensor product of local representations σ p the dimensions and character values are obtained by multiplication. The space of U N -invariants is non-trivial only if all local components are principal series representations. The last term in Proposition 4.18 only appears for representations of level one (i. e. for σ the trivial representation). It remains to compute for each n the sum of the contributions in Proposition 4.11 for all possible combinations of local components with total parity (−1) n . This is a tedious but elementary computation which we omit here.

Bounds for the cohomology of Bianchi groups
In this subsection we use the results of Section 4.3 to give some bounds for the dimension of the cohomology spaces H 1 (SL(2, O d ), E n ) as |d| or n go to infinity. The first result is a more or less obvious consequence of Theorem 4.17.
Corollary 4.20. Let K be an imaginary quadratic number field with ring of integers O K . There is a bound C 1 > 0 such that For the proof we only have to show that the coefficient of n + 1 in the formula of Theorem 4.17 is non-negative.
The second result we see as a complement to the following theorem which is proved in [3]. To use this theorem note that where ζ K (s) is the Dedekind zeta function of K, see [20,Section 7]. It is easy to see that ζ K (2) is bounded between two positive real numbers for all imaginary quadratic fields K. In view of Lemma 3.1 we obtain

Corollary 4.22. Let n be a (fixed) non-negative integer. There is a constant
We remark that this result also follows from the trace formula methods of Section 5 below. Theorem 4.17 implies Proposition 4.23. Let n be a (fixed) non-negative integer. There is a constant C 4 > 0 such that

Base change and cocompact arithmetic groups
By the work of Labesse-Schwermer [47] and Rajan [52], it is possible to use base change and the Jacquet-Langlands correspondence to study the cohomology of the cocompact arithmetic groups Γ associated to quaternion algebras defined over fields L such that the extension L/L tr , where L tr is the maximal totally real subfield of L, is solvable (but not necessarily Galois). The resulting bound for dim H 1 (Γ, E n ) is determined by the dimension of certain spaces of Hilbert modular forms of weight (n + 2, 2, . . . , 2) for L tr , and will be again linear on congruence classes. We do not go into the details here. Note that, in contrast to the case of the Bianchi groups, for a particular group Γ the resulting bound will often be trivial. If we however consider the collection of all congruence subgroups, the conjecture of Waldhausen and Thurston has been verified for these groups by Rajan [52]. We can easily deduce from his arguments the following qualitative result: Proof. By [47,52], for a suitable ∆ a lower bound for the dimension of the cohomology is given by the dimension of the space of Hilbert modular newforms of weight (n + 2, 2, . . . , 2) for certain congruence subgroups of GL(2, L tr ). By adding additional local conditions, it is easily seen that it is possible to assume that the subgroups in question are torsion-free. Furthermore, their covolume can be made arbitrarily large by changing ∆. Shimizu's dimension formula [58] implies then that for any c > 0 we can find a subgroup ∆ such that the dimension of the corresponding space of Hilbert modular forms is ≥ c(n + 1) for all n ≥ 0.
If base change for SL (2) for arbitrary extensions of number fields was available, one could prove the corresponding result for all arithmetic lattices. This provides strong theoretical evidence for a positive answer to Question 1.14 for arithmetic lattices.

Upper bounds for the dimension of H 1
In this section we derive upper bounds for the dimension of the cohomology spaces H 1 (Γ, E n ) by using the generalized Eichler-Shimura isomorphism to transform the problem into a question on multiplicities of representations in L 2 (SL(2, C)/Γ) and then using the trace formula to get information on these multiplicities. We first set up the form of the trace formula we need by specializing the work of W. Hoffmann [35,36] to our situation. Then we consider the behavior of the dimension of H 1 (Γ, E n ) as a function of n and its behavior for fixed n as Γ varies over the standard congruence subgroups Γ 0 (a) of a Bianchi group (our result is in fact slightly more general, cf. Theorem 5.5 below).

Review of the invariant trace formula for SL(2, C)
Let Γ be a general discrete subgroup of G = SL(2, C) of finite covolume and consider the discrete part of L 2 (G/Γ), which is a Hilbert space direct sum of irreducible unitary representations π of G, each one occurring with a finite multiplicity m(π, Γ). The irreducible unitary representations of G most important to us are the principal series representations π m,iν for integers m and real parameters ν, which are obtained by unitary induction from the characters σ m,iν (e u+iθ ) = e i(νu+mθ) of the maximal torus T ≃ C × of G. The representations π m,iν and π −m,−iν are equivalent.
We are interested in bounding the multiplicities m(π m,0 ) from above. As explicated above, the dimension of H 1 cusp (Γ, E n ) is the same as the multiplicity m(π 2n+2,0 ). We first recall the trace formula for L 2 (G/Γ) in the form in which it has been explicitly worked out by Hoffmann for lattices of rank one [36]. We specialize his results to the simpler case of G = SL(2, C) and the trivial Hecke operator. As a preparation, we need to recall the basic relations between orbital integrals and principal series characters and the explicit form of the Plancherel formula, for which we use [43,Ch. XI] as a reference. We normalize measures as in [loc. cit.], i.e. we use the Haar measure on G given by the product measure dk dn da associated to the Iwasawa decomposition G = KN A, where dk gives K = SU(2) total measure one, dn is the standard measure on N ≃ C, the upper triangular unipotent subgroup, and the measure da on A ≃ R >0 is du in the parametrization u → diag (e u , e −u ). We consider compactly supported functions f ∈ C ∞ c (G). To such a function is associated the function ad (g)).
Then |D G (g)| 1/2 = |t − t −1 | 2 for a regular semisimple element g with eigenvalues t and t −1 , and D G (g) = 1 otherwise. For g ∈ G define the orbital integral Then J G (t, f ) = F T f (t) for all regular elements t ∈ T [loc. cit., (11.13), (11.14)]. From this one sees immediately that F T f (t) = F T f (t −1 ). It is also easy to see that F T f (±1) = 8πJ G (±n 1 , f ) for n 1 = 1 1 0 1 . The Fourier transform of F T f yields the characters of the principal series representations: The Plancherel formula for G is given by [loc. cit., Theorem 11.2] (up to a minor correction): For a discrete subgroup Γ of G of finite covolume let C be the set of all cuspidal parabolic subgroups of G, i. e. of all parabolic subgroups fixing a cusp of Γ. Let Γ( * ) be the set of all semisimple elements of Γ which do not fix a cusp together with the elements of Γ ∩ {±1}, and on the other hand Γ ce the set of semisimple elements of Γ different from ±1 and stabilizing a cusp. For ξ ∈ Γ ce let A(ξ) be the unique conjugate of the real torus A ⊆ T in the centralizer of ξ. For the definition of the weight factor v ξ for ξ ∈ Γ ce we refer to [36, p. 105]. For each P ∈ C let Γ M (P ) be the set of projections to a Levi component L of P of the elements of Γ ∩ P . We define constants C(P, η, Γ) (called C P (ηn 1 , χ Γ ) in [loc. cit., p. 106]) in terms of Epstein zeta functions associated to ηΓ ∩ N . Namely, C(P, η, Γ) is the constant term in the Laurent expansion at z = 1 of the function where we choose k P ∈ K such that k −1 P P k P is the standard upper triangular Borel subgroup P 0 and write The absolute value of u(ξ) does not depend on the choice of k P . For η = 1 we can write C(P, 1, Γ) = 2πκ Λ(P ) /|Γ M (P )| for the lattice Λ(P ) = u(Γ ∩ N ) in C, where κ Λ denotes the constant term in the expansion of |Λ| π λ∈Λ\{0} |λ| −2z at z = 1 (this notation agrees with [20, Lemma 6.5.2]). We also need distributions I L (η), which are Arthur's invariant modifications of weighted orbital integrals (cf. [loc. cit., Sect. 5]). Finally, let Φ(σ m,s ) be the scattering matrix of Γ defined in [loc. cit., p. 122] (and denoted by S(χ Γ ,w, σ Λ ) there) and φ = det Φ its determinant (with respect to a suitable identification of the vector spaces in question, the choice of which is unimportant). We can now quote [36,Theorem 6.4], specialized to our situation.
the trace of the corresponding convolution operator on the discrete part of L 2 (G/Γ) is given by The distributions I L (η) can be explicitly described in terms of the character values Θ m,iν (f ). We need here only the limiting case η = 1. Denote by ψ(s) = Γ ′ (s)/Γ(s) the logarithmic derivative of the gamma function. The following Proposition follows easily from Hoffmann's work in [35].
Proposition 5.2. For the trivial element of G the distribution I L (1) is given by with the function Note that although the function ψ has a simple pole at s = 0, the real part of ψ(iν/2) is continuous at ν = 0, and in fact Re ψ(iν/2) = Re ψ(1 + iν/2).
The reader may compare the resulting explicit trace formula, which involves only the function F T f on T and its Fourier transform, with the trace formula for K-biinvariant functions f given in [20, Theorem 6.5.1].

The dimension of H 1 : behavior with n
We now turn to the behavior of the multiplicities m(π m,0 , Γ) as m → ∞ for a fixed group Γ. The method extends to cohomological representations of real rank one groups which are not in the discrete series. It is an adaption of the method of [16,Sect. 9] for bounding the remainder term in Weyl's law. Our result is the following. As an immediate consequence we have: Proof of the Theorem. By passing to a finite index subgroup, we can assume that Γ is torsion-free and Γ M (P ) = {1} for all P .
and Θ n,iν (f ) = 0 for |n| = m. Insert f into the trace formula and note that because of our assumption on Γ, the sum in the second line is empty while the sums in the third and fourth line involve only η = 1. Also, the expression in the last line vanishes. Moving the integral involving the scattering matrix to the other side, we obtain an expression for as the sum of the remaining terms on the right hand side (i. e. the sum of the first, third and fourth line). As usual, we split the sum in the first line as By the Plancherel formula, we can express the first term as with constants C 1 and C 2 depending only on Γ and h 0 . By [16, pp. 90-91], since the absolute values of the orbital integrals J G (ξ, f ) are bounded independently of m, the second term can be estimated by C 3 e C4/ε . The third line of the trace formula is a constant multiple of F T f (1), and therefore C 5 ε. To estimate the weighted orbital integrals, we use the standard approximation ψ(s) = log s + O(1) for Re s ≥ δ > 0 to get From this one obtains Taking logarithmic derivatives, one sees that log q m − φ ′ (σ m,iν )/φ(σ m,iν ) is positive real for all ν. Since h 0 was assumed to be non-negative, (5.2) is therefore up to a term going to zero with m an upper bound for m(π m,0 )h 0 (0). The Theorem follows. We remark that for congruence subgroups of the Bianchi groups SL(2, O K ), K imaginary quadratic, standard estimates for the logarithmic derivatives of Hecke L-functions imply that the contribution from the continuous spectrum in (5.2) is O(ε log m) if ε goes to zero for m → ∞, and that it is therefore bounded with our choice of ε.

The dimension of H 1 : congruence subgroups of Bianchi groups
We now consider finite index subgroups of the Bianchi groups SL(2, O K ). For any non-zero ideal a of O K we have the classical congruence subgroup The index of Γ 0 (a) in SL(2, O K ) is given by the multiplicative function .
We also need the following subgroups closely related to the principal congruence subgroups: The following theorem gives a bound for the multiplicity of a representation π m,0 in L 2 (G/Γ∩Γ 0 (a)), Γ a subgroup of finite index in SL(2, O K ), which improves the trivial bound O(ι(a)) by a logarithm. This theorem can be regarded as a quantitative variant of the limit multiplicity results of de George-Wallach [15], Lück and Savin [54] (which however concern towers of normal subgroups). Note also that for a = aO K , a a positive integer, we can get by base change arguments a lower bound of the form Ca = CN(a) 1/2 . If a and its conjugate are relatively prime, there is no non-trivial lower bound known.
Corollary 5.6. Let Γ be a subgroup of finite index in SL(2, O K ). Then for any fixed n ≥ 0 we have , N(a) → ∞.
Proof. The corresponding assertion for the cuspidal part is an immediate consequence of Theorem 5.5. To bound the dimension of the non-cuspidal part use Lemma 5.7 below.
The proof of Theorem 5.5 is again based on the trace formula. By passing to a finite index subgroup, we can assume that Γ is torsion-free and Γ M (P ) = {1} for all P . Let ∆ ⊆ Γ be a subgroup of finite index. We may then write (cf. [13]) for every f ∈ C ∞ c (G) the spectral side We now need a sequence of elementary lemmas to deal with the parabolic and hyperbolic contributions. The following well-known lemma is used to bound the parabolic contribution. To deal with the hyperbolic contribution, we need to consider first the numbers c Γ∩Γ0(a) (ξ). The following lemma follows easily from the definitions.
Lemma 5.8. Let ξ ∈ Γ and b be the largest divisor of a such that ξ ∈Γ(b). Then where ν(a) denotes the number of prime divisors of a and c is defined by extending

multiplicatively.
For any semisimple element γ ∈ G let its norm N(γ) ≥ 1 be the maximum value of |t| 2 for the two eigenvalues t of γ. We need to estimate the number of Γ-conjugacy classes of bounded norm which are contained inΓ(b). Such an estimate can be deduced from the following well-known lemma.
Lemma 5.9. There is a constant B depending only on Γ such that every semisimple conjugacy class The crude estimate of the next lemma is an easy consequence. The reader may verify that a better estimate would not change the final result (apart from the constant implicit in the O). Proof. Apply Lemma 5.9 to see that each such conjugacy class has a representative γ = a b c d with |a| 2 , |b| 2 , |c| 2 , |d| 2 ≤ BT . Furthermore, bc = 0 since the conjugacy class was assumed to lie in Γ( * ). The number of possible pairs (b, c) corresponding to elements ofΓ(b) is therefore bounded by C ′ T 2 N(b) −2 with C ′ depending only on K. For each such pair the number of possible entries a and d with ad = 1 + bc = 0 is clearly bounded by O(T δ ) with a constant depending only on δ. This proves the assertion. Proof of Theorem 5.5. We take a test function f ∈ C ∞ c (G) depending on m ≥ 1 and ε as above. We fix m and assume at first only that ε is bounded. Then the identity contribution to the trace formula for ∆ = Γ ∩ Γ 0 (a) is bounded by C 1 ι(a)ε. The lattices Λ(P ) = u(∆ ∩ N ) appearing in the definition of C(P, 1, ∆) = 2πκ Λ(P ) are all invariant under a fixed order of the field K, and belong therefore to finitely many classes up to multiplication by elements of K * . Using that κ Λ + log |Λ| is invariant under such homotheties, this implies that the constants κ Λ(P ) are bounded by C + log N(a) for a constant C. By Lemma 5.7 the parabolic contribution is therefore bounded by C 2 ι(a)(log N(a))N(a) −1/2 ε.
As for the contribution of classes in Γ( * ), the estimates of Lemma 5.8 and Lemma 5.10 show that it is bounded by  for any µ > 0. Taking ε = c/ log N(a) with a suitable constant c, we see that the geometric side of the trace formula is indeed O(ι(a)/ log N(a)). A positivity argument as in the proof of Theorem 5.3 yields the result.

Computational results for Bianchi groups
This section contains computational results on the dimensions of the cohomology groups H 1 (Γ, E n ) where Γ = SL(2, a) is one of the Bianchi groups of Section 2.2 and n a non-negative integer. We also consider certain congruence subgroups of SL(2, O −1 ). The method of computation is explained in Section 3.1. Often the resulting systems of linear equations turned out to be far to big to do computations over the rational numbers. In these cases we were able to use the lower estimates of Section 4.3 to deduce the dimension of the solution space over the complex numbers from the dimension over various finite fields.

Dimensions of H 1 (SL(2, O), E n )
Let us start with a little table. In Table 1 we have listed the dimension of the cohomology spaces for all imaginary quadratic fields K, using the notation introduced in Section 4.1. We see that the cuspidal cohomology consists only of lifted forms except in the two cases marked in boldface. These two cases will be analyzed more closely below. As a result of some heavy computer calculations we can report the following results. It remains to report the results of the computations in the non-euclidean cases. We have found the following: 11  0  0  1  0  1  1  1  1  1  0  1  1  2  0  1  1  1  2  3  1  2  0  1  2  4  0  1  0  2  2  5  2  3  1  2  3  6  0  2  1  2  4  7  3  4  1  3

Hecke operators on non-lifted cohomology classes
In this subsection we give the numerical values of some of the Hecke operators on the two spaces of non-lifted cohomology classes exhibited in Section 6.1.

Hecke operators on
We consider the prime element π 11 = 2 + √ −7 of O −7 , which has degree one and norm 11. By the methods described in Section 3.2 the characteristic polynomial of the corresponding Hecke operator can easily be computed to be .
We know that there is a unique two-dimensional complement of the space of base change classes in the cohomology space H is this space of non-lifted classes. We write L π for the restriction of the Hecke operators T π , π a prime element of O −7 , to the space NL(−7, 12). The following properties of the linear maps L π hold for all prime elements π of O −7 .
P7.4 After a suitable choice of basis for NL(−7, 12), the matrices giving the action of the Hecke operators L π have integral entries.
Here A adj stands for the adjoint of a linear map A. If A is given by a two-by-two matrix, we have  Table 2. Examples of the integral matrices corresponding to the linear maps L π for primes π of degree one are contained in Tables 3, 4   can easily be computed to be We know that there is a unique two-dimensional complement of the space of base change classes in the cohomology space H 1 cusp (SL(2, O −11 ), E 10 ) invariant under the Hecke operators. We infer that the kernel NL(−11, 10) = Ker(T 2 π3 + 700 T π3 + 40671) (6.3) is equal to this space of non-lifted classes. We write L π for the restriction of the Hecke operator T π , π a prime element of O −11 , to the space NL(−11, 10). The following properties of the linear maps L π hold for all prime elements π of O −11 .
3 If π = p is a prime of degree two, then L π is an integer scalar denoted by µ p . The case d = −11, k = 10, differs from the case d = −7, k = 12, since we have L −π = L π for d = −11 and and L −π = −L π for d = −7. In the case d = −11 this leads directly to property P11.6.
Examples of the scalars µ p for primes of degree two are contained in Table 6. Examples of the integral matrices corresponding to the linear maps L π for primes π of degree one are contained in Tables 7,8,9.

Cohomology of congruence subgroups
In this subsection we give some computational results concerning the dimensions of the cohomology groups H 1 (Γ, E n ) where Γ ⊆ SL(2, O −1 ) is a congruence subgroup.

The case of trivial coefficients
Here we consider the congruence subgroups  Table 9: Hecke operators L π on NL(−11, 10) of SL(2, O −1 ), where p is a prime ideal of O = O −1 of degree one. Note that Γ 0 (p) is conjugate in SL(2, O) to the congruence subgroup Γ 0 (p) considered in Section 5.3. The norm of p is either equal to 2 or a rational prime p congruent to 1 modulo 4. The index of Γ 0 (p) in SL(2, O −1 ) is p + 1. The cohomology groups H 1 (Γ 0 (p), C) are particularly interesting for number theory since their non-vanishing is conjectured to be related to the existence of certain elliptic curves (or more generally abelian varieties) defined over K = Q(i) (cf. [14,25,27]). We shall report here on extensive computations of the dimensions of the spaces H 1 (Γ 0 (p), C). Note that we have H 1 (Γ 0 (p), C) = H 1 cusp (Γ 0 (p), C) and H 1 (SL(2, O), C) = 0, and that H 1 (Γ 0 (p), C) consists therefore entirely of new classes (cf. Section 6.3.2).
The elements A i , 0 ≤ i ≤ p − 1, and B (cf. Section 2.2) form a system of coset representatives for Γ 0 (p) in SL(2, O −1 ). From this we obtain the following generating system for Γ 0 (p): where ρ 2 + 1 = 0 in the field F p . From the presentation (2.7) we may compute a presentation of the finitely generated abelian group Γ 0 (p) ab and in particular the dimension of Γ 0 (p) ab ⊗C (which is the same as the dimension of H 1 (Γ 0 (p), C)). This computation may be speeded up in the following way, i. e. the presentation of Γ 0 (p) obtained from the Reidemeister-Schreier method can be simplified a lot. We shall describe a result contained in [25] which gives such a simplification. Let R be a commutative ring and let P 1 (R, p) be a (p + 1)-dimensional free R-module with basis u x indexed by the projective line P 1 (F p ). This module has rank p + 1 and is a PGL(2, F p )-module by the natural permutation action on the basis elements. Let U(R, p) be the submodule of P 1 (R, p) generated by u 0 and the elements x ∈ P 1 (F p ), (6.4) with the matrices The value 8 is attained for the prime 58313.
Let us now define for real numbers x the function where the sum is extended over all degree one prime ideals of O −1 . The function S(x) can be tabulated in the range x ≤ 60000 as follows: x

The case of non-trivial coefficients
We now report on some computational results on the cohomology spaces H 1 (Γ 0 (p), E n ) where p is a prime of O −1 of degree one and n ≥ 1. Let π be a generator of p and let δ π ∈ GL(2, Q( √ −1)) be defined as in (3.8). The two injective homomorphisms where ι 1 is just the injection and ι 2 is induced by conjugation with the element δ π , give rise to an injection ι : The image of ι is traditionally called the space of old classes. It is invariant under the Hecke operators and has an invariant complement H 1 new (Γ 0 (p), E n ). We have found: • H 1 new (Γ 0 (p), E 1 ) = 0 for all prime ideals p with N (p) ≤ 1000 except for the case N (p) = 41, where H 1 new (Γ 0 (p), E 1 ) has dimension 2.
• H 1 new (Γ 0 (p), E n ) = 0 for all prime ideals p with N (p) ≤ 90 and 3 ≤ n ≤ 10. Table 12 contains some examples of the characteristic polynomials of the Hecke operators on H 1 new (Γ 0 (p), E 1 ) for N (p) = 41. The Hecke operators on H 1 new (Γ 0 (p), E 1 ) satisfy T iπ = −T π for all prime elements π of degree one. There is no apparent connection between T π and Tπ. We thank Haluk Sengun for help with this computation.

Cocompact groups
As in the case considered before, we take from [41] (see also [26]) a series of explicit pairs of matrices generating a discrete subgroup Γ ⊆ PSL(2, C). In this case the groups Γ act on three-dimensional hyperbolic space with a compact quotient.
Let k ≥ 8 be an even integer. We define t = t k as in (7.1) and set Define GTet 1 [k, 3, 2] ⊆ PSL(2, C) to be the image in PSL(2, C) of the group generated by f and g.
The following facts are known.
Again these results are sufficient to compute cohomology spaces. We find
• Θ m has finite covolume and exactly one cusp.
• Θ m is defined by the relations (7.11).
The zero z in question is specified (up to complex conjugation) by the condition |z − 2| < 4 sin 2 π 2m for m ≥ 3 odd and Re (z) > 0, |z 2 − 4| < 4 sin 2 π m for m ≥ 4 even. For m = 1 or 2, z may be taken to be any zero of f m . Concerning the cohomology of the groups Θ m , we can report the following computations:
We have found that H 1 (Γ 26 , E n ) = 0 (7.17) for 0 ≤ n ≤ 90. The group Γ 26 has 222 conjugacy classes of subgroups of index less than or equal to 24. We have also determined the dimensions of some cohomology spaces of these subgroups. Of the 222 subgroups 191 satisfied H 1 (Γ, E n ) = 0 in the range 0 ≤ n ≤ 10. Thirty subgroups had dim ≤1000 H 1 (Γ, E n ) = 1 in the range 0 ≤ n ≤ 10. One of the 222 had dim ≤1000 H 1 (Γ, E n ) = 2, again in this range.