The Functional Equation and Beyond Endoscopy

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.