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.
Keywords
trace formula, L-functions,
functional equation, automorphic forms
