Formal Siegel modular forms for arithmetic subgroups

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The notion of formal Siegel modular forms for an arithmetic subgroup Γ \Gamma of the symplectic group of genus n 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 Γ \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 ≤ n ≤ 4 2\leq n \leq 4 . As an application we consider the case of the paramodular group of squarefree level and genus 2 2 .