The Maximal Growth Of Toric Periods and Oscillatory Integrals for Maximal Flat Submanifolds

TL;DR

This paper establishes new bounds on toric periods and flat periods of automorphic forms on certain symmetric spaces, advancing understanding of their maximal growth and oscillatory behavior.

Contribution

It introduces a new omega result for toric periods of Hecke-Maass forms on PGL(3) and provides mean square asymptotics for flat periods in more general settings.

Findings

Proved a new omega result for toric periods.

Derived mean square asymptotics for flat periods.

Connected period bounds to orbital integral estimates.

Abstract

We prove a new omega result for toric periods of Hecke-Maass forms on compact locally symmetric spaces associated to forms of PGL(3). This is motivated by conjectures on the maximal growth of L-functions as well as by questions about the size of automorphic periods. We also prove a mean square asymptotic result for maximal flat periods on more general locally symmetric spaces of non-compact type, which takes as main input bounds for real relative orbital integrals.