Polarizations of abelian varieties over finite fields via canonical liftings

TL;DR

This paper characterizes all polarizations of abelian varieties over finite fields within a fixed isogeny class, leveraging canonical liftings and categorical equivalences to explicitly compute isomorphism classes.

Contribution

It provides a complete description of polarizations for abelian varieties over finite fields with canonical liftings, including explicit algorithms for classification.

Findings

Explicit classification of polarizations in fixed isogeny classes

Development of algorithms for computing isomorphism classes

Application of categorical equivalences to abelian varieties

Abstract

We describe all polarizations for all abelian varieties over a finite field in a fixed isogeny class corresponding to a squarefree Weil polynomial, when one variety in the isogeny class admits a canonical liftings to characteristic zero, i.e., a lifting for which the reduction morphism induces an isomorphism of endomorphism rings. Categorical equivalences between abelian varieties over finite fields and fractional ideals in \'etale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.