Automatic convergence for Siegel modular forms

2024/08/29 by Aaron Pollack, Pollack, Aaron · 1 citation
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.16392

openalex publication_date 2024/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Bruinier and Raum, building on work of Ibukiyama-Poor-Yuen, have studied a notion of ``formal Siegel modular forms". These objects are formal sums that have the symmetry properties of the Fourier expansion of a holomorphic Siegel modular form. These authors proved that formal Siegel modular forms necessarily converge absolutely on the Siegel half-space, and thus are the Fourier expansion of an honest Siegel modular form. The purpose of this note is to give a new proof of the cuspidal case of this ``automatic convergence" theorem of Bruinier-Raum. We use the same basic ideas in a separate paper to prove an automatic convergence theorem for cuspidal quaternionic modular forms on exceptional groups.

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