Ordinary Primes for Abelian Surfaces

TL;DR

This paper determines the density of ordinary primes for abelian surfaces over number fields, showing it can be 1, 1/2, or 1/4 based on the l-adic monodromy group classification.

Contribution

It explicitly computes the density of ordinary primes for abelian surfaces using the classification of their l-adic monodromy groups.

Findings

Density of ordinary primes is 1, 1/2, or 1/4.

Density depends on the classification of l-adic monodromy groups.

Provides a clear link between monodromy groups and prime densities.

Abstract

We compute the density of the set of ordinary primes of an abelian surface over a number field in terms of the l-adic monodromy group. Using the classification of l-adic monodromy groups of abelian surfaces by Fite, Kedlaya, Rotger, and Sutherland, we show the density is 1, 1/2, or 1/4.