An exact formula relating lattice points in symmetric spaces to the automorphic spectrum
Abstract
We extract an exact formula relating the number of lattice points in an expanding region of a complex semi-simple symmetric space and the automorphic spectrum from a spectral identity, which is obtained by producing two expressions for the automorphic fundamental solution of the invariant differential operator (Delta - lambda_z)^N. On one hand, we form a Poincare series from the solution to the corresponding differential equation on the free space G/K, which is obtained using the harmonic analysis of bi-K-invariant functions. On the other hand, a suitable global automorphic Sobolev theory, developed in this paper, enables us to use the harmonic analysis of automorphic forms to produce a solution in terms of the automorphic spectrum.
Keywords
Cite
@article{arxiv.1104.5406,
title = {An exact formula relating lattice points in symmetric spaces to the automorphic spectrum},
author = {Amy DeCelles},
journal= {arXiv preprint arXiv:1104.5406},
year = {2011}
}
Comments
Improved normalization, resulting in a slightly different formula; 13 pages; results from the author's PhD thesis (University of Minnesota, 2011) under the direction of Paul Garrett