Maass Waveforms on (Γ0(N), χ) (Computational Aspects)

Abstract

Summary. The main topic of this paper is computational aspects of the theory of Maass waveforms, i.e. square-integrable eigenfunctions of the Laplace-Beltrami operator on certain Riemann surfaces with constant negative curvature and finite area. The surfaces under consideration correspond to quotients of the upper half-plane by certain discrete groups of isometries, the so called Hecke congruence subgroups. It is known that such functions can also be regarded as wavefunctions corresponding to a quantum-mechanical system describing a particle moving freely on the surface. Since the classical counterpart of this motion is chaotic, the study of these wavefunctions is closely related to the study of quantum chaos on the surface. The presentation here, however, will be purely from a mathematical viewpoint. Today, our best knowledge of generic Maass waveforms comes from numerical experiments, and these have previously been limited to the modular group, PSL (2, ℤ), and certain triangle groups (cf. [13, 14, 16] and [39]). One of the primary goals of this lecture is to give the necessary theoretical background to generalize these numerical experiments to Hecke congruence subgroups and non-trivial characters. We will describe algorithms that can be used to locate eigenvalues and eigenfunctions, and we will also present some of the results obtained by those algorithms. There are four parts, first elementary notations and definitions, mostly from the study of Fuchsian groups and hyperbolic geometry. Then more of the theoretical background needed to understand the rich structure of the space of Maass waveforms will be introduced. The third chapter deals with the computational aspects, and the final chapter contains some numerical results.