Abstract Let be a ‐dimensional abelian variety defined over a number field . It is conjectured that the set of ordinary primes of over has positive density, and this is known to be true when , or for certain abelian varieties with extra endomorphisms. In this paper, we extend the family of abelian varieties whose sets of ordinary primes have positive density. Specifically, we show that if the endomorphism algebra of contains a number field of degree , then under certain conditions on the fields and , the set of ordinary primes of over has positive density. This includes ‐type abelian varieties over (resp., quadratic number fields) of dimension or (resp., ) for any rational prime . The proof is carried out in the general setting of compatible systems of Galois representations, and as a consequence, it also implies a positive density result for the sets of ordinary primes of certain modular forms of weight 2.
Wang et al. (Fri,) studied this question.