Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields

TL;DR

This paper presents algorithms for classifying conjugacy classes of semisimple integral matrices and for enumerating isomorphism classes of certain abelian varieties over finite fields, advancing computational methods in algebraic number theory and algebraic geometry.

Contribution

It introduces new algorithms for computing conjugacy class representatives of semisimple matrices and for classifying abelian varieties over finite fields based on their Frobenius characteristic polynomial.

Findings

Algorithm for conjugacy class representatives of integral matrices.

Algorithm for classifying abelian varieties over finite fields.

Effective enumeration of isomorphism classes in specified cases.

Abstract

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes of abelian varieties over a finite field $\mathbb{F}_q$ which belong to an isogeny class determined by a characteristic polynomial $h$ of Frobenius when $h$ is ordinary, or $q$ is prime and $h$ has no real roots.