Non-vanishing of geodesic periods of automorphic forms

TL;DR

This paper proves that all but finitely many closed geodesic periods of certain automorphic forms are non-zero, with implications for L-functions and distributional properties, using graph theory techniques.

Contribution

It introduces a novel approach linking geodesic periods and line integrals through graph theory, establishing non-vanishing results for automorphic forms.

Findings

100% non-vanishing of geodesic periods when ordered by length

Applications to non-vanishing of Rankin--Selberg L-values

Results towards normal distribution of periods

Abstract

We prove that one hundred percent of the closed geodesic periods of a Hecke--Maa{\ss} cusp form for the modular group are non-vanishing when ordered by length. We present applications to the non-vanishing of central values of Rankin--Selberg $L$-functions. Similar results for holomorphic forms for general Fuchsian groups of finite covolume with a cusp are also obtained, as well as results towards normal distribution. Our new key ingredient is to relate the distributions of closed geodesic periods and vertical line integrals via graph theory.