Computing abelian varieties over finite fields isogenous to a power

TL;DR

This paper provides a module-theoretic classification of abelian varieties over finite fields that are isogenous to a power of a given variety, with explicit descriptions under certain conditions, including polarizations in the ordinary case.

Contribution

It introduces a new module-theoretic framework for classifying isomorphism classes of abelian varieties isogenous to a power, with explicit computational methods for certain polynomials.

Findings

Classifies isomorphism classes using fractional ideals in number fields.

Provides explicit descriptions under specific polynomial conditions.

Includes a module-theoretic approach to polarizations in the ordinary case.

Abstract

In this paper we give a module-theoretic description of the isomorphism classes of abelian varieties $A$ isogenous to $B^r$, where the characteristic polynomial $g$ of Frobenius of $B$ is an ordinary square-free $q$-Weil polynomial, for a power $q$ of a prime $p$, or a square-free $p$-Weil polynomial with no real roots. Under some extra assumptions on the polynomial $g$ we give an explicit description of all the isomorphism classes which can be computed in terms of fractional ideals of an order in a finite product of number fields. In the ordinary case, we also give a module-theoretic description of the polarizations of $A$.