All Siegel Hecke eigensystems (MOD p) are cuspidal

Abstract

Fix integers g ≥ 1 and N ≥ 3, and a prime p not dividing N. We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of dimension g, 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 both the main theorem of [Ghi04] and a modification of the techniques used there, namely restriction to the superspecial locus. © International Press 2006.