  In a letter to Tate, 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 (mod p), proving that the systems of Hecke eigenvalues obtained
from them are the same as the ones given by algebraic modular forms (mod p) on
a quaternionic unitary group, as defined by Gross. The correspondence is
obtained by restricting to the superspecial locus of the moduli space of
abelian varieties.
