Numerical computations with the trace formula and the Selberg eigenvalue conjecture

Andrew R Booker; Andreas Strömbergsson

doi:10.1515/CRELLE.2007.047

Published by De Gruyter July 22, 2007

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

Andrew R Booker and Andreas Strömbergsson

From the journal Journal für die reine und angewandte Mathematik

https://doi.org/10.1515/CRELLE.2007.047

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Abstract

We verify the Selberg eigenvalue conjecture for congruence groups of small squarefree conductor, improving on a result of Huxley [M. N. Huxley, Introduction to Kloostermania, in: Elementary and analytic theory of numbers, Banach Center Publ. 17, Warsaw (1985), 217–306.]. The main tool is the Selberg trace formula which, unlike previous geometric methods, allows for treatment of cases where the eigenvalue 1/4 is present. We present a few other sample applications, including the classification of even 2-dimensional Galois representations of small squarefree conductor.

Received: 2005-10-20

Revised: 2006-03-14

Published Online: 2007-07-22

Published in Print: 2007-06-27

Numerical computations with the trace formula and the Selberg eigenvalue conjecture

Abstract

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