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.

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