Large Sieve Inequalities for periods of Maass forms

TL;DR

This paper develops large sieve inequalities for hyperbolic periods of Maass forms on Fuchsian groups, extending previous work and enabling applications in counting problems involving hyperbolic subgroups.

Contribution

It introduces new large sieve inequalities with weights based on hyperbolic periods of Maass forms for Fuchsian groups, generalizing prior results for weight zero forms.

Findings

Established large sieve inequalities for hyperbolic periods of Maass forms

Extended previous inequalities to forms of even weight

Enabled applications in counting problems in hyperbolic geometry

Abstract

For $\Gamma$ a Fuchsian Group of the first kind, we obtain large sieve inequalities with weights the hyperbolic periods of Maass forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights values of Maass forms of weight $0$. The motivation is applications in counting problems in $\Gamma_1 \backslash \Gamma / \Gamma_2$, where $\Gamma_1$, $\Gamma_2$ are hyperbolic subgroups of $\Gamma$.