Density theorems for exceptional eigenvalues for 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 ${\mathrm{\Gamma }}_0\left(q\right)$, ${\mathrm{\Gamma }}_1\left(q\right)$, and $\mathrm{\Gamma }\left(q\right)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.