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.
Translated title of the contribution | Numerical computations with the trace formula and the Selberg eigenvalue conjecture |
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Original language | English |
Pages (from-to) | 113 - 161 |
Number of pages | 49 |
Journal | Journal für die reine und angewandte Mathematik |
Volume | 2007 (607) |
DOIs | |
Publication status | Published - Jun 2007 |