ABSTRACT. To a finite quadratic module, that is, a finite abelian group D together with a non-singular quadratic form Q : D → Q/Z, it is possible to associate a representation of either the modular group, SL 2 (Z), or its metaplectic cover, Mp 2 (Z), on C [D], the group algebra of D. This representation is usually called the Weil representation associated to the finite quadratic module. The main result of this paper is a general explicit formula for the matrix coefficients of this representation. The formula, which involves the p-adic invariants of the quadratic module, is given in a way which is easy to implement on a computer. The result presented completes an earlier result by Scheithauer for the Weil representation associated to a discriminant form of even signature.

In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.

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In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups G_q, q=3,4,..., which are non-arithmetic for q \not= 3,4,6. For this we make use of a Poincare map for the geodesic flow on the corresponding Hecke surfaces which has been constructed in arXiv:0801.3951 and which is closely related to the natural extension of the generating map for the so called Hurwitz-Nakada continued fractions. We derive simple functional equations for the eigenfunctions of the transfer operator which for eigenvalues rho =1 are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.Comment: 30 pages; revised versio

The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke-Selberg and Phillips-Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of "new" discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) cerify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.