Designing Poincare Series for Number Theoretic Applications

TL;DR

This paper introduces a method for designing Poincaré series tailored to various number theoretic applications, extending previous work on $GL_2$ to higher rank groups and diverse problems.

Contribution

It generalizes the construction of Poincaré series to higher rank groups, enabling explicit formulas and applications in lattice point counting, L-function moments, and pseudo-Laplacian problems.

Findings

Constructed Poincaré series for lattice point counting in symmetric spaces.

Developed series for moments of $GL_n imes GL_n$ L-functions.

Designed series for applications involving pseudo-Laplacians.

Abstract

The $GL_2$ 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 $GL_n \times GL_n$ L-functions, and a Poincar\'{e} series designed for applications involving pseudo-Laplacians.