Computing square-free polarized abelian varieties over finite fields

TL;DR

This paper presents algorithms for classifying and computing properties of certain ordinary abelian varieties over finite fields with square-free Frobenius polynomials, including polarizations and automorphisms.

Contribution

It introduces new algorithms for computing isomorphism classes, polarizations, automorphism groups, and period matrices of square-free polarized abelian varieties over finite fields.

Findings

Algorithms for classifying ordinary abelian varieties over finite fields.

Methods to compute polarizations and automorphism groups.

Procedures to determine period matrices for principal polarizations.

Abstract

We give algorithms to compute isomorphism classes of ordinary abelian varieties defined over a finite field $\mathbb{F}_q$ whose characteristic polynomial (of Frobenius) is square-free and of abelian varieties defined over the prime field $\mathbb{F}_p$ whose characteristic polynomial is square-free and does not have real roots. In the ordinary case we are also able to compute the polarizations and the group of automorphisms (of the polarized variety) and, when the polarization is principal, the period matrix.