Vol. 260, No. 2, 2012

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The functional equation and beyond endoscopy

P. Edward Herman

Vol. 260 (2012), No. 2, 497–513

In his paper “Beyond endoscopy,” Langlands tries to understand functoriality via poles of L-functions. This paper further investigates the analytic continuation of an L-function associated to a GL2 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 the perspective of analytic number theory, obtaining the functional equation from the trace formula implies that Voronoi summation should in general be also a consequence of the trace formula.

trace formula, L-functions, functional equation, automorphic forms
Mathematical Subject Classification 2010
Primary: 11F12, 11F66, 11F72, 11L05
Received: 1 August 2012
Accepted: 27 September 2012
Published: 30 November 2012
P. Edward Herman
Department of Mathematics
University of Chicago
5734 S. University Avenue
Chicago, IL 60637
United States