Beyond Endoscopy via the Trace Formula-I: Poisson Summation and Contributions of Special Representations

TL;DR

This paper develops a refined analysis of the elliptic part of the trace formula using Poisson summation and the approximate functional equation, facilitating applications in Beyond Endoscopy and isolating special representations.

Contribution

It introduces a method to control elliptic trace formula components and isolates contributions of special representations, advancing analytic techniques for automorphic forms.

Findings

Controlled the analytic behavior of volumes of tori.

Smoothed out singularities of orbital integrals.

Rewrote the elliptic part for analytic applications.

Abstract

With analytic applications in mind, in particular Beyond Endoscopy ([13]), we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation to the elliptic part we control the analytic behavior of the volumes of tori that appear in the elliptic part. Furthermore by carefully choosing the truncation parameter in the approximate functional equation we smooth-out the singularities of orbital integrals. Finally by an application of Poisson summation we rewrite the elliptic part so that it is ready to be used in analytic applications, and in particular in Beyond Endoscopy. As a by product we also isolate the contributions of special representations as pointed out in [13].