Serre’s ordinary-reduction density conjecture for abelian varieties

Establish that, after passing to a suitable finite extension of the base number field, every abelian variety has ordinary reduction at a set of primes of natural density 1.

Background

The paper situates its study of ordinary and μ-ordinary reduction for K3 surfaces within a broader conjecture concerning abelian varieties. For an abelian variety over a number field, ordinary reduction is defined by the Newton polygon having only slopes 0 and 1. The conjecture asserts that a finite extension of the base field suffices to make ordinary reduction occur at density 1.

The authors note that the conjecture is known for elliptic curves and abelian surfaces but remains unresolved for abelian varieties in general. This is a background open problem rather than one of the paper’s own unresolved questions; the paper proves an analogous density result for certain K3 surfaces under explicit hypotheses.