## Illinois Journal of Mathematics

### An exact formula relating lattice points in symmetric spaces to the automorphic spectrum

Amy T. DeCelles

#### 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.

#### Article information

Source
Illinois J. Math., Volume 56, Number 3 (2012), 805-823.

Dates
First available in Project Euclid: 31 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1391178549

Digital Object Identifier
doi:10.1215/ijm/1391178549

Mathematical Reviews number (MathSciNet)
MR3161352

Zentralblatt MATH identifier
1286.11083

#### Citation

DeCelles, Amy T. An exact formula relating lattice points in symmetric spaces to the automorphic spectrum. Illinois J. Math. 56 (2012), no. 3, 805--823. doi:10.1215/ijm/1391178549. https://projecteuclid.org/euclid.ijm/1391178549

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