All Siegel Hecke eigensystems (mod $p$) are cuspidal

Abstract

We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of fixed dimension g, fixed level N, and varying weight, are the same as the systems occurring in the spaces of Siegel cusp forms with the same parameters and varying weight. In particular, in the case g=1, this says that the Hecke eigensystems (mod p) coming from classical modular forms are the same as those coming from cusp forms. The proof uses restriction to the superspecial locus. We also give a comparison of cusp forms on the Satake compactification versus the toroidal compactifications of Siegel modular varieties.