Efficiently generated spaces of classical Siegel modular forms and the Boecherer conjecture

TL;DR

This paper constructs and verifies a generating set for classical Siegel modular forms up to weight 172, enabling new computational approaches and confirming the Boecherer conjecture for non-rational eigenforms.

Contribution

It introduces a new generating set for Siegel modular forms, verified up to weight 172, facilitating Fourier expansion calculations and confirming key conjectures.

Findings

Verified the Boecherer conjecture for non-rational eigenforms.

Confirmed other conjectures for weights up to 150.

Described arithmetic properties of the constructed basis.

Abstract

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set we verify the Boecherer conjec- ture for non-rational eigenforms. As one further application we verify another conjectures for weights up to 150 and investigate an analogue of the Victor-Miller basis. Additionally, we describe some arithmetic properties of the basis we found.