  Let $O_F$ be the ring of integers of a totally real field $F$ of degree $g$.
We study the reduction of the moduli space of separably polarized abelian
$O_F$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The
invariants and related conditions for the objects in the moduli space are
discussed. We construct a scheme-theoretic stratification by $a$-numbers on the
Rapoport locus and study the relation with the slope stratification. In
particular, we recover the main results of Goren and Oort [GO, J. Alg. Geom.
2000] on the stratifications when $p$ is unramified in $O_F$. We also prove the
strong Grothendieck conjecture for the moduli space in some restricted cases,
particularly when $p$ is totally ramified in $O_F$.
