  We study the special fibers of a certain class of absolutely simple abelian
varieties over number fields with endomorphism rings $\bz$ and possessing
$l$-adic monodromy groups of the least possible rank. We also study the
Dirichlet density of the places at which the possible reductions occur and
confirm a special case of a broader conjecture for the splitting of reductions
of abelian varieties over number fields.
