Period Functions for Maass Wave Forms. I

Period functions for Maass wave forms

Contents Introduction Chapter I. The period correspondence via L-series 1. The correspondences u ↔ Lε ↔ f ↔ ψ 2. Periodicity, L-series, and the three-term functional equation 3. Even and odd 4. Relations between Mellin transforms; proof of Theorem 1 Chapter II. The period correspondence via integral transforms 1. The integral representation of ψ in terms of u 2. The period function as the integ...

Harmonic Weak Maass Forms, Automorphic Green Functions, and Period Integrals

Arakelov geometry, which is a mixture of algebraic geometry at finite primes (of a number field) and real analysis at infinite primes, was invented by Arakelov [Ar] in 70s to ‘compactify’ an arithmetic variety (see also [Fa2]). It has become a very important part of modern number theory after Faltings’ proof of the Mordell conjecture (see for example [Fa1], [So]) and the celebrated Gross-Zagier...

Finite Analogs of Maass Wave Forms

Orthogonality relations for finite Eisenstein series are obtained. It is shown that there exist finite analogs of Maass wave forms which are not Eisenstein series.

A thermodynamic formalism approach to period functions for Maass forms and generalisations

In this paper we study one-parameter families of signed transfer operators P± q associated to the Farey map, as well as two-parameter families of operators for the Gauss map, obtained from the Farey map by inducing. We characterise all the analytic eigenfunctions of P± q with eigenvalue λ not embedded in the continuous spectrum, which for λ = 1 are the period functions for Maass forms studied b...

Mock Theta Functions, Ranks, and Maass Forms