#### Vol. 12, No. 7, 2018

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Density theorems for exceptional eigenvalues for congruence subgroups

### Peter Humphries

Vol. 12 (2018), No. 7, 1581–1610
##### 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 ${\Gamma }_{0}\left(q\right)$, ${\Gamma }_{1}\left(q\right)$, and $\Gamma \left(q\right)$. These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.

##### Keywords
Selberg eigenvalue conjecture, Ramanujan conjecture
Primary: 11F72
Secondary: 11F30
##### Milestones
Received: 30 January 2017
Revised: 2 April 2018
Accepted: 2 June 2018
Published: 27 October 2018
##### Authors
 Peter Humphries Department of Mathematics University College London United Kingdom