Formal Siegel modular forms for arithmetic subgroups

The notion of formal Siegel modular forms for an arithmetic subgroup of the symplectic group of genus n is a generalization of symmetric formal Fourier-Jacobi series. Assuming an upper bound on the affine covering number of the Siegel modular variety associated with, we prove that all formal Siegel modular forms are given by Fourier-Jacobi expansions of classical holomorphic Siegel modular forms. We also show that the required upper bound is always met if 2 n 4. As an application we consider the case of the paramodular group of squarefree level and genus 2.