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|
|Pages (from-to)||113 - 161|
|Number of pages||49|
|Journal||Journal für die reine und angewandte Mathematik|
|Publication status||Published - Jun 2007|