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

TL;DR

This paper generalizes known results on the density of primes of ordinary reduction for abelian varieties, focusing on absolutely simple varieties with CM endomorphism algebras and specific signatures, including explicit examples from Jacobians.

Contribution

It extends the density results to higher-dimensional abelian varieties with CM endomorphism algebras under certain signature conditions, with explicit examples.

Findings

Proves positive density of primes with ordinary reduction for certain CM abelian varieties.

Generalizes previous results from elliptic curves and surfaces to higher dimensions.

Includes explicit examples from Jacobians of curves of genus three to seven.

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.