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arXiv · 2405.15620

Measures, modular forms, and summation formulas of Poisson type

Claudia Alfes·Paul Kiefer·Jan Mazáč
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Abstract

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call $k$-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct $k$-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn-Gon\c{c}alves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional $k$-spherical measures.

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Claudia Alfes, Paul Kiefer, Jan Mazáč. 2024-05-24. Measures, modular forms, and summation formulas of Poisson type. https://doi.org/10.1007/s00220-025-05313-6

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