Period Functions for Maass Wave Forms and Cohomology

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...

Mock Theta Functions, Ranks, and Maass Forms

Maass Forms and Their L-functions

We present examples of Maass forms on Hecke congruence groups, giving low eigenvalues on Γ0(p) for small prime p, and the first 1000 eigenvalues for Γ0(11). We also present calculations of the L-functions associated to the Maass forms and make comparisons to the predictions from random matrix theory.

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.