Maass forms and Fourrier decomposition

Abstract

We develop a theory of Fourier analysis along closed geodesics of Maass automorphic forms. Maass automorphic forms are (skew-)invariant functions with respect to a Fuchsian group on the upper half-plane that are eigenfunctions of the Laplace-Beltrami operator. These functions arise in the spectral theory of (skew-)invariant functions on the upper half-plane or Riemann surfaces. The Fourier analysis of automorphic forms at interior points and cusps is well-documented, and with a nice result: Any Fourier term of an automorphic form at an interior point or cusp is the multiple of a unique normalized Fourier function, this is called a 'dimension one theorem'. However, this is not possible when working along closed geodesics.

In this talk we show that using the reflection in a geodesic a local dimension one theorem can be proven: 'If an automorphic form is (anti-)symmetric with respect to the reflection of a geodesic, then the Fourier term along this geodesic is a multiple of the unique normalized (anti-)symmetric Fourier function'.