  Let A be an abelian variety defined over a number field and of dimension g.
When g<3, by the recent work of Sawin, we know the exact (nonzero) value of the
density of the set of primes which are ordinary for A. In higher dimension very
little is known. We show that if g=3 and A has multiplication by an imaginary
quadratic field E, then there exists a nonzero density set of ordinary primes
for A. We reach the same conclusion if g=4 and the pair (A,E) has signature
(2,2). We also obtain partial results when g=3 and A has multiplication by a
totally real cubic field. We show that our methods also apply to certain
abelian varieties of Albert type IV of higher dimension. These results are
derived from an extended version of the l-adic methods of Katz, Ogus, and Serre
in the presence of extra endomorphisms.
