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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 thedensity of the set of primes which are ordinary for A. In higher dimension verylittle is known. We show that if g=3 and A has multiplication by an imaginaryquadratic field E, then there exists a nonzero density set of ordinary primesfor 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 atotally real cubic field. We show that our methods also apply to certainabelian varieties of Albert type IV of higher dimension. These results arederived from an extended version of the l-adic methods of Katz, Ogus, and Serrein the presence of extra endomorphisms.

Ordinary primes for some varieties with extra endomorphisms