Designing Poincare Series for Number Theoretic Applications
Abstract
The GL2 Poincar\'e 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 Poincar\'e series with specific number theoretic applications in mind: a Poincar\'e series for producing an explicit formula for the number of lattice points in an expanding region in a symmetric space, a Poincar\'e series producing moments of GLn × GLn L-functions, and a Poincar\'e series designed for applications involving pseudo-Laplacians.
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