Formal Siegel modular forms for arithmetic subgroups

The notion of formal Siegel modular forms for an arithmetic subgroup $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$, 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 $2\le n\le 4$. As an application we consider the case of the paramodular group of squarefree level and genus $2$.