Sturm type theorem for Siegel modular forms of genus 2 modulo $p$
Suppose that $f$ is an elliptic modular form with integral coefficients. Sturm obtained bounds for a nonnegative integer $n$ such that every Fourier coefficient of $f$ vanishes modulo a prime $p$ if the first $n$ Fourier coefficients of $f$ are zero modulo $p$. In the present note, we study analogues of Sturm's bounds for Siegel modular forms of genus 2. As an application, we study congruences involving an analogue of Atkin's $U(p)$-operator for the Fourier coefficients of Siegel modular forms of genus 2.