Automorphic partial differential equations and spectral theory with applications to number theory
While proofs of the Riemann hypothesis and the Lindelöf hypothesis remain elusive, for some number-theoretic applications any bound that surpasses the “trivial” or “convex” bound for the growth of an L-function, i.e. any subconvex bound, suffices. In this paper, we construct a Poincaré series suitable for proving a subconvex bound for Rankin-Selberg convolutions for GL n × GLn over totally complex number fields. The Poincaré series, with transparent spectral expansion, is obtained by winding-up a free space fundamental solution for the operator (Delta - lambdaz)ν on the free space G/K. As a sample application, not obviously related to subconvexity, a Perron transform extracts, from the Poincaré series, information about the number of lattice points in an expanding region in G/K, and from the spectral expansion, terms corresponding to the automorphic spectrum of the Laplacian. The result is an explicit formula relating the automorphic spectrum to the number of lattice points in an expanding region. A global automorphic Sobolev theory as well as a zonal spherical Sobolev theory legitimize derivations and manipulations of spectral expansions. This line of inquiry is relevant not only to the hoped-for subconvexity result but also to the development of techniques applicable to harmonic analysis of automorphic forms on higher rank groups.