Polarizations of abelian varieties over finite fields via canonical liftings
Jonas Bergström, Valentijn Karemaker, Stefano Marseglia
TL;DR
The paper addresses the problem of describing polarizations, especially principal polarizations, of abelian varieties over finite fields within a fixed isogeny class determined by a squarefree Weil polynomial $h$, under the presence of a canonical CM-lifting. It develops an explicit, computable bridge between characteristic $p$ and characteristic zero using CM theory and the Centeleghe-Stix and Mar-AbVar equivalences, encoding abelian varieties as fractional ideals in an étale algebra and reducing polarization questions to ideal-theoretic data. A key theoretical contribution is a polarization criterion: for a CM-lifting with CM-type $\Phi$ and a totally real unit $\alpha$ arising from reduction, a polarization corresponds to an element $\gamma=\alpha^{-1}\lambda$ that is totally imaginary and $\Phi$-positive, with $\lambda$ encoding the polarization; this yields a practical algorithm to enumerate principal polarizations within the isogeny class. The paper also provides algorithms to verify the generalized residual reflex condition (RRC) for canonical liftability and to compute the principal polarizations, along with explicit examples and open-source code for reproducible computations. The results significantly expand the ability to classify polarized abelian varieties over finite fields and demonstrate the feasibility of explicit polarization counts across a broad range of squarefree isogeny classes, clarifying when the obstruction from $\alpha$ can be neglected and when it must be accounted for.
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 étale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.