Analytic Continuation of L-functions through Distributions

Stephen Miller
The analytic continuation of L-functions is a central goal in number theory, which usually comes through two approaches. The first, originally due to Riemann in his famous paper on the zeta function, expresses the L-function as an integral of an automorphic form. Properties of the form then translate to the L-function. Another approach, due to Langlands-Shahidi, finds L-functions in the Fourier expansions of exotic Eisenstein series, and obtains the analytic continuation from spectral theory. These methods have yielded outstanding successes, but still leave fundamental unsolved gaps between them. I will discuss a third method, developed in joint work with Wilfried Schmid, which uses pairings of automorphic distributions, to obtain the full analytic continuation and functional equation of some new examples of Langlands L-functions.