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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Numerical computations with the trace formula and the Selberg eigenvalue conjecture

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2007, Issue 607, Pages 113–161, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2007.047, July 2007

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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.

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