Automatic convergence for Siegel modular forms

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.