Positive density of primes of ordinary reduction for abelian varieties of simple signature

Abstract

By a result of Serre, if $A$ is an elliptic curve without CM defined over a number field $L$, then the set of primes of $L$ for which $A$ has ordinary reduction has density $1$. Katz and Ogus proved the same is true when $A$ is an abelian surface, after possibly passing to a finite extension of $L$. More recently, Sawin computed the density of the set of primes of $L$ for which an abelian surface $A$ has ordinary reduction, depending on the endomorphism algebra of $A$. In this paper, we prove some generalizations of these results when $A$ is an absolutely simple abelian variety of arbitrary dimension whose endomorphism algebra is a CM field $F$, under specific conditions on the signature of the multiplication action of $F$ on $A$. We include explicit examples from Jacobians of curves of genus three through seven admitting cyclic covers to the projective line.