Effective equidistribution of shears and applications

TL;DR

This paper proves effective equidistribution of shears, which are unipotent translates of cuspidal geodesic rays, with applications to automorphic L-functions and lattice point counting.

Contribution

It establishes power-saving rates for the equidistribution of shears and applies these results to problems in automorphic forms and lattice point enumeration.

Findings

Proves regularized equidistribution of shears with effective rates

Provides bounds for weighted second moments of automorphic L-functions

Counts lattice points on locally affine symmetric spaces

Abstract

A "shear" is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of PSL(2,R), possibly of infinite co-volume. We prove the regularized equidistribution of shears under large translates with effective (that is, power saving) rates. We also give applications to weighted second moments of GL(2) automorphic L-functions, and to counting lattice points on locally affine symmetric spaces.