Algebra & Number Theory

Density theorems for exceptional eigenvalues for congruence subgroups

Peter Humphries

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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.

Article information

Source
Algebra Number Theory, Volume 12, Number 7 (2018), 1581-1610.

Dates
Received: 30 January 2017
Revised: 2 April 2018
Accepted: 2 June 2018
First available in Project Euclid: 9 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1541732434

Digital Object Identifier
doi:10.2140/ant.2018.12.1581

Mathematical Reviews number (MathSciNet)
MR3871503

Zentralblatt MATH identifier
06976295

Subjects
Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11F30: Fourier coefficients of automorphic forms

Keywords
Selberg eigenvalue conjecture Ramanujan conjecture

Citation

Humphries, Peter. Density theorems for exceptional eigenvalues for congruence subgroups. Algebra Number Theory 12 (2018), no. 7, 1581--1610. doi:10.2140/ant.2018.12.1581. https://projecteuclid.org/euclid.ant/1541732434


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