Designing Poincare Series for Number Theoretic Applications

Abstract

The $G{L}_2$ Poincaré series giving the subconvexity results of Diaconu and Garrett is the solution to an automorphic partial differential equation, constructed by winding-up the solution to the corresponding differential equation on the free space. Generalizing this approach allows design of higher rank Poincaré series with specific number theoretic applications in mind: a Poincaré series for producing an explicit formula for the number of lattice points in an expanding region in a symmetric space, a Poincaré series producing moments of $G{L}_n\times G{L}_n$ L-functions, and a Poincaré series designed for applications involving pseudo-Laplacians.

17 pages, supercedes arxiv:1104.4313. arXiv admin note: substantial text overlap with arXiv:1104.5406