Illinois Journal of Mathematics

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

Amy T. DeCelles

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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.

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

First available in Project Euclid: 31 January 2014

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Zentralblatt MATH identifier

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


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.

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