## Algebra & Number Theory

### Density theorems for exceptional eigenvalues for congruence subgroups

Peter Humphries

#### 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
Revised: 2 April 2018
Accepted: 2 June 2018
First available in Project Euclid: 9 November 2018

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

#### 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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