Measures, modular forms, and summation formulas of Poisson type
Number Theory
2025-10-22 v1 Mathematical Physics
Classical Analysis and ODEs
math.MP
Abstract
In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call -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 -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 -spherical measures.
Cite
@article{arxiv.2405.15620,
title = {Measures, modular forms, and summation formulas of Poisson type},
author = {Claudia Alfes and Paul Kiefer and Jan Mazáč},
journal= {arXiv preprint arXiv:2405.15620},
year = {2025}
}