The Functional Equation and Beyond Endoscopy

TL;DR

This paper demonstrates that the functional equation of an L-function associated with a $GL_2$ automorphic form can be derived from the trace formula, advancing understanding of analytic continuation and functoriality in number theory.

Contribution

It shows that the functional equation of L-functions can be recovered solely from the trace formula, providing new insights into their analytic continuation and functoriality.

Findings

Functional equation can be derived from the trace formula.

Trace formula implies Voronoi summation as a consequence.

Advances understanding of L-function analytic continuation.

Abstract

In his paper "Beyond Endoscopy," Langlands tries to understand functoriality via poles of L-functions. The following paper further investigates the analytic continuation of a L-function associated to a $GL_2$ automorphic form through the trace formula. Though the usual way to obtain the analytic continuation of an L-function is through its functional equation, this paper shows that by simply assuming the trace formula, the functional equation of the L-function may be recovered. This paper is a step towards understanding the analytic continuation of the L-function at the same time as capturing information about functoriality. From an analytic number theory perspective, obtaining the functional equation from the trace formula implies that Voronoi summation should in general be also a consequence of the trace formula.