On the distribution of the eigenvalues of the hyperbolic Laplacian for PSL(2, Z)
by
C. J. Mozzochi
Princeton, N.J.

Let H be the upper half plane and X = SL(2, Z)\H the corresponding modular surface. Theory and experiment suggest that the eigenvalues of the hyperbolic Laplacian, \Delta, on X, denoted by \lambda_j = 1/4 + t²_j, behave in many ways like a random sequence. In particular for any A > 0 the numbers A\lambda_j, j = 1, 2, 3, ... should be well distributed modulo 1 (that is to say there should be square root cancellation in the corresponding Weyl sums). In this paper we show in sharp contrast to the above that the sequence 2t_j log (t_j/\pie) is not well distributed modulo 1. This reflects a certain structure that the closed geodesics on X carry, precisely that the norms of the hyperbolic conjugacy classes (which correspond to closed geodesics) of \Gamma are very close to being squares of integers. This phenomenon no doubt occurs for all arithmetic quotients X of H but not for the generic hyperbolic surface.

Date received: November 22, 1999

Copyright © 1999 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-03.