Density theorems for exceptional eigenvalues for congruence subgroups

Humphries, Peter

doi:10.2140/ant.2018.12.1581

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Density theorems for exceptional eigenvalues for congruence subgroups

Peter Humphries

Vol. 12 (2018), No. 7, 1581–1610

DOI: 10.2140/ant.2018.12.1581

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 $Γ_{0} (q)$ , $Γ_{1} (q)$ , and $Γ (q)$ . These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.

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Keywords

Selberg eigenvalue conjecture, Ramanujan conjecture

Mathematical Subject Classification 2010

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

Vol. 12, No. 7, 2018