  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.
