Beyond Endoscopy via the trace formula - II: Asymptotic expansions of Fourier transforms and bounds towards the Ramanujan conjecture

TL;DR

This paper advances the understanding of the trace formula's elliptic part by deriving asymptotic expansions of Fourier transforms of orbital integrals, leading to bounds towards the Ramanujan conjecture.

Contribution

It provides detailed asymptotic expansions of Fourier transforms of orbital integrals and applies these results to establish bounds related to the Ramanujan conjecture.

Findings

Derived exact asymptotic expansions of Fourier transforms of orbital integrals.

Reproduced Kuznetsov's bound on Hecke operator traces using the trace formula.

Enhanced understanding of the elliptic part of the trace formula in the context of beyond endoscopy.

Abstract

We continue the analysis of the elliptic part of the trace formula initiated in \cite{Altug:2015aa}. In that reference Poisson summation was applied to the elliptic part and the dominant term was analyzed. The main aim of this paper is to study the remaining terms after Poisson summation. We analyze the the Fourier transforms of (smoothed) orbital integrals and obtain exact asymptotic expansions. As an application we recover, using the Arthur-Selberg trace formula, Kuznetsov's result (cf. \cite{Kuznetsov:1980aa}) that the trace of the $p$th Hecke operator on cuspidal automorphic representations is bounded by $p^{\frac14}$.