The hyperbolic lattice point problem in conjugacy classes

TL;DR

This paper investigates the hyperbolic lattice point problem within conjugacy classes for Fuchsian groups, providing new average error bounds and connecting results to quadratic forms, advancing understanding of geometric and number-theoretic properties.

Contribution

It introduces a novel approach using large sieve inequalities to analyze the problem and offers a new proof of the classical error bound, extending to cases like SL(2,Z).

Findings

Established average error bounds for the problem.

Provided a new proof of the classical $O(X^{2/3})$ error bound.

Connected results to indefinite quadratic forms for SL(2,Z).

Abstract

For $\Gamma$ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces $\Gamma\backslash \mathbb H$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound $O(X^{2/3})$, due to A. Good. For $\hbox{SL}(2,{\mathbb Z})$ we interpret our results in terms of indefinite quadratic forms.