Siegel modular forms mod $p$ and their $U(p)$ congruences

We discuss Jacobi coefficients with $p$-integral coefficients, and device them to study Siegel modular forms for the full Siegel modular group of arbitrary genus.

We revisit classical results on Jacobi forms due to Eichler and Zagier, and Sofer. Extensions of them to Jacobi forms of matrix index can be obtained by the recently much refined method of restriction. In particular, we will show that the module of Jacobi forms of fixed index with $p$-integral coefficients is free for all $p$ greater than 3.

As for Siegel modular forms, we give a complete description of algebras of Siegel modular forms mod $p$. We then turn to $U(p)$ congruences, which we characterize under a mild technical hypothesis. As an example, we give a complete discussion of $U(p)$ congruence of the Schottky form in genus 4.