Galois groups of low dimensional abelian varieties over finite fields

TL;DR

This paper introduces a new invariant called the weighted permutation representation to analyze relationships between Galois groups, Newton polygons, and angle ranks of abelian varieties over finite fields, classifying possible invariant triples.

Contribution

It defines the weighted permutation representation and uses it to classify invariant triples for abelian surfaces and simple abelian threefolds, advancing understanding of their Galois groups.

Findings

Classified invariant triples for abelian surfaces.

Classified invariant triples for simple abelian threefolds.

Established relationships between Galois groups, Newton polygons, and angle ranks.

Abstract

We consider three isogeny invariants of abelian varieties over finite fields: the Galois group, Newton polygon, and the angle rank. Motivated by work of Dupuy, Kedlaya, and Zureick-Brown, we define a new invariant called the weighted permutation representation which encompasses all three of these invariants and use it to study the subtle relationships between them. We use this permutation representation to classify the triples of invariants that occur for abelian surfaces and simple abelian threefolds.