  In a letter to Tate (published in Israel J. Math. in 1996), J.-P. 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 on an adelic double
coset space constructed from 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) are the
same as the ones given by algebraic modular forms (mod p) on a quaternionic
unitary group, as defined by Gross in Israel J. Math. in 1999. The
correspondence is obtained by restricting to the superspecial locus of the
moduli space of abelian varieties.
