Automatic convergence for Siegel modular forms

TL;DR

This paper provides a new proof for the automatic convergence of formal Siegel modular forms, confirming they are genuine holomorphic forms, and extends similar ideas to quaternionic modular forms on exceptional groups.

Contribution

It offers a novel proof of the cuspidal case of the automatic convergence theorem for formal Siegel modular forms, and applies these ideas to quaternionic modular forms.

Findings

Formal Siegel modular forms converge absolutely on the Siegel half-space.

Confirmed formal Siegel modular forms are actual holomorphic Siegel modular forms.

Extended proof techniques to quaternionic modular forms on exceptional groups.

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.