Bounds for Siegel Modular Forms of genus 2 modulo $p$

TL;DR

This paper extends Sturm's bounds from elliptic to genus 2 Siegel modular forms, establishing sharp bounds and exploring congruences involving Atkin's $U(p)$-operator for their Fourier coefficients.

Contribution

It provides the first sharp bounds for Siegel modular forms of genus 2 modulo a prime and investigates related congruences with Atkin's $U(p)$-operator.

Findings

The bounds for genus 2 Siegel modular forms are sharp.

Established congruences involving Atkin's $U(p)$-operator.

Extended Sturm's classical results to higher genus forms.

Abstract

Sturm obtained the bounds for the number of the first Fourier coefficients of elliptic modular form $f$ to determine vanishing of $f$ modulo a prime $p$. In this paper, we study analogues of Sturm's bound for Siegel modular forms of genus 2. We show the resulting bound is sharp. As an application, we study congruences involving Atkin's $U(p)$-operator for the Fourier coefficients of Siegel mdoular forms of genus 2.