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 Γ0(q), Γ1(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