Exponents for the Equidistribution of Shears and Applications

TL;DR

This paper enhances the understanding of equidistribution of shears in hyperbolic surfaces by employing spectral theory to obtain explicit exponents and uniform results, with applications to counting quadratic form values.

Contribution

It introduces a spectral theory-based approach to achieve explicit exponents and uniformity in the equidistribution of shears, improving upon previous soft methods.

Findings

Explicit exponents for equidistribution of shears

Uniform bounds in parameters

Applications to counting quadratic form values

Abstract

In previous work, the authors introduced "soft" methods to prove the effective (i.e. with power savings error) equidistribution of "shears" in cusped hyperbolic surfaces. In this paper, we study the same problem but now allow full use of the spectral theory of automorphic forms to produce explicit exponents, and uniformity in parameters. We give applications to counting square values of quadratic forms.