Open Access
Fall 2012 An exact formula relating lattice points in symmetric spaces to the automorphic spectrum
Amy T. DeCelles
Illinois J. Math. 56(3): 805-823 (Fall 2012). DOI: 10.1215/ijm/1391178549

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 $(\Delta-\lambda_{z})^{\nu}$. On one hand, we form a Poincaré 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.

Citation

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Amy T. DeCelles. "An exact formula relating lattice points in symmetric spaces to the automorphic spectrum." Illinois J. Math. 56 (3) 805 - 823, Fall 2012. https://doi.org/10.1215/ijm/1391178549

Information

Published: Fall 2012
First available in Project Euclid: 31 January 2014

zbMATH: 1286.11083
MathSciNet: MR3161352
Digital Object Identifier: 10.1215/ijm/1391178549

Subjects:
Primary: 11F72
Secondary: 11F55 , 11M36 , 11P21

Rights: Copyright © 2012 University of Illinois at Urbana-Champaign

Vol.56 • No. 3 • Fall 2012
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