Density Theorems for Exceptional Eigenvalues for Congruence Subgroups

TL;DR

This paper employs the Kuznetsov formula to establish density theorems for exceptional eigenvalues of Maass cusp forms across various congruence subgroups, improving upon prior results obtained via the Selberg trace formula.

Contribution

It introduces new density theorems for exceptional eigenvalues using the Kuznetsov formula, extending and strengthening previous results by Sarnak and Huxley.

Findings

Density theorems for eigenvalues of Maass cusp forms

Improved bounds over previous results

Extension to multiple congruence subgroups

Abstract

Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maass cusp forms of weight 0 or 1 for the congruence subgroups $\Gamma_0(q)$, $\Gamma_1(q)$, and $\Gamma(q)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.