Sturm type theorem for Siegel modular forms of degree 2

TL;DR

This paper extends Sturm's theorem to Siegel modular forms of degree 2, providing explicit bounds for Fourier coefficients to determine congruences, with applications to Jacobi forms and prime level cases.

Contribution

It generalizes Sturm's congruence criterion to Siegel modular forms of degree 2, including explicit bounds and applications to prime level forms.

Findings

Explicit bounds for Fourier coefficients to determine congruences.

Analog of Sturm's theorem established for Jacobi forms.

Verification of congruences with finitely many Fourier coefficients.

Abstract

We attempt to generalize a congruence property of elliptic modular forms proved by Sturm to that of Haupttypus of Siegel modular forms of degree 2 with level. Namely, we give an explicit bound of Fourier coefficients required to determine the congruence of modular forms. We give the analog of Sturm's theorem for Jacobi forms, which is required in the proof. In the case that Nebentypus of Siegel modular forms with prime level, in order to prove the congruence between two modular forms, we show that it suffices to check the congruence of the finitely many Fourier coefficients. Finally, we give two examples of our main theorem.