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1. Introduction. Let Q be a positive definite n × n matrix with integral entries and even diagonal entries. It is well known that the theta function ${\vartheta }_Q\left(z\right):={\sum }_{g\in {\mathbb{Z}}^n}exp\pi i{}^{t}gQgz$, Im z > 0, is a modular form of weight n/2 on ${\Gamma }_0\left(N\right)$, where N is the level of Q, i.e. $N{Q}^{-1}$ is integral and $N{Q}^{-1}$ has even diagonal entries. This was proved by Schoeneberg [5] for even n and by Pfetzer [3] for odd n. Shimura [6] uses the Poisson summation formula to generalize their results for arbitrary n and he also computes the theta multiplier explicitly....

We employ recent results on Jacobi forms to investigate congruences and filtrations of Siegel modular forms of degree $2$. In particular, we determine when an analog of Atkin’s $U\left(p\right)$-operator applied to a Siegel modular form of degree $2$ is nonzero modulo a prime $p$. Furthermore, we discuss explicit examples to illustrate our results.