Upper bounds for geodesic periods over rank one locally symmetric spaces

TL;DR

This paper establishes upper bounds for geodesic periods of automorphic forms on rank one locally symmetric spaces, extending previous hyperbolic results to a broader class of symmetric spaces.

Contribution

It generalizes upper bounds for geodesic periods from hyperbolic manifolds to all rank one locally symmetric spaces, incorporating twisted automorphic forms.

Findings

Upper bounds depend on Laplace eigenvalues of automorphic forms

Results extend previous hyperbolic space bounds to all rank one symmetric spaces

Provides a unified framework for geodesic period estimates

Abstract

We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.