On exceptional eigenvalues of the Laplacian for congruence subgroups
by
Xian-Jin Li
Brigham Young University

For any congruence subgroup \Gamma₀(N) of square-free level N, let \lambda_j, j=1, 2, ... , h, be the exceptional eigenvalues of the Laplacian for the congruence subgroup. In 1965, Atle Selberg conjectured that there is no exceptional eigenvalues for congruence subgroups. In this talk, an explicit Dirichlet series L_N(s) will be given, which is analytic in the half-plane Re(s) > 1/2 except at s=1/2-i\kappa_j, j=1, 2, ... , h, where \kappa_j=\surd{\lambda_j-1/4}. The result is obtained by using Selberg's trace formula, Dirichlet's class number formula, and Siegel's theorem.

Date received: January 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadx-08.