Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms

Abstract

In his letter (Israel J. Math. 95 (1996) 281), Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions ABx/Bx → F̄p, where B is the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving that the systems of Hecke eigenvalues given by Siegel modular forms (mod p) of genus g are the same as the ones given by algebraic modular forms (mod p) on the group GUg (B), as defined in Gross (Math. Res. Notices (16) (1998) 865; Israel J. Math. 113 (1999) 61). The correspondence is obtained by restricting to the superspecial locus of the moduli space of abelian..