Algebra & Number Theory

Density theorems for exceptional eigenvalues for congruence subgroups

Peter Humphries

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Selberg eigenvalue conjecture Ramanujan conjecture


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.

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