Illinois Journal of Mathematics

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

Amy T. DeCelles

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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

Subjects
Primary: 11F72: Spectral theory; Selberg trace formula
Secondary: 11P21: Lattice points in specified regions 11F55: Other groups and their modular and automorphic forms (several variables) 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas

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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