Automatic convergence and arithmeticity of modular forms on exceptional groups

2024/08/18 by Aaron Pollack, Pollack, Aaron
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2408.09519

Abstract

We prove that the space of cuspidal quaternionic modular forms on the groups of type F₄ and E_n have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal modular forms. The main component of the proof of the algebraic characterization is to show that certain infinite sums, which are potentially the Fourier expansion of a cuspidal modular form, converge absolutely. As a consequence of the algebraic characterization, we deduce that the cuspidal quaternionic modular forms have a basis consisting of forms all of whose Fourier coefficients are algebraic numbers.

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