Spectral correspondences for Maass waveforms on quaternion groups (original) (raw)
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Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quaternion division algebra H, we define several Kleinian discrete subgroups of P SL(2, H). We define first a Kleinian subgroup P SL(2, L) of P SL(2, H(Z)). This group is a generalization of the modular group P SL(2, Z). Next we define a discrete subgroup P SL(2, H) of P SL(2, H) which is obtained by using Hurwitz integers and in particular the subgroup of order 24 consisting of Hurwitz units. It contains as a subgroup P SL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic half space H 1 H := {q ∈ H : (q) > 0}. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients H 1 H /P SL(2, L) and H 1 H /P SL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. We give a thorough study of the Iwasawa decompositions, affine subgroups, and their descriptions by Lorentz transformations in the Lorentz-Minkowski model of hyperbolic 4-space. We give abstract finite presentations of these modular groups in terms of generators and relations via the Cayley graphs associated to the fundamental domains. We also describe a set of Selberg covers (corresponding to finite-index subgroups acting freely) which are quaternionic hyperbolic manifolds of finite volume with cusps whose sections are 3-tori. These hyperbolic arithmetic 4-manifolds are topologically the complement of linked 2-tori in the 4-sphere, in analogy with the complement in the 3-sphere of the Borromean rings and are related to the ubiquitous hyperbolic 24-cell. Finally we study the Poincaré extensions of these Kleinian groups to arithmetic Kleinian groups acting on hyperbolic 5-space and described in the quaternionic setting. In particular P SL(2, H(Z)) and P SL(2, Hur(Z)) are discrete subgroups of isometries of H 5 R and H 5 R /P SL(2, H(Z)), H 5 R /P SL(2, Hur(Z)) are examples of arithmetic 5-dimensional hyperbolic orbifolds of finite volume.
1987
In the present paper the elementary divisor theory over the Hurwitz order of integral quaternions is applied in order to determine the structure of the Hecke-algebras related to the attached unimodular and modular group of degree n. In the case n = 1 the Hecke-algebras fail to be commutative. If n > 1 the Hecke-algebras prove to be commutative and coincide with the tensor product of their primary components. Each primary component turns out to be a polynomial ring in n resp. n + 1 resp. 2n resp. 2n + 1 algebraically independent elements. In the case of the modular group of degree n, the law of interchange with the Siegel ~b-operator is described. The induced homomorphism of the Hecke-algebras is surjective except for the weights r = 4n -4 and r = 4n -2.