Sturm Bounds for Siegel Modular Forms

TL;DR

This paper establishes Sturm bounds for degree g Siegel modular forms modulo a prime p, using an inductive approach based on Fourier-Jacobi expansions and properties of Jacobi forms, extending previous results to general g.

Contribution

The paper introduces a new inductive proof method for Sturm bounds applicable to all degrees g, unlike previous proofs limited to specific cases.

Findings

Established Sturm bounds for all degrees g of Siegel modular forms

Developed an inductive proof leveraging Fourier-Jacobi expansions

Extended the applicability of Sturm bounds beyond known cases

Abstract

We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g=1,2, which do not extend to the case of general g.