@article{2009b02dce4b4d1a9b9d667d23f032a8,
title = "Polarizations of abelian varieties over finite fields via canonical liftings",
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 lifting to characteristic zero, that is, 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 {\'e}tale algebras enable us to explicitly compute isomorphism classes of polarized abelian varieties satisfying some mild conditions. We also implement algorithms to perform these computations.",
keywords = "Construction, Curves, Isogeny classes, Number, Surfaces",
author = "V.Z. Karemaker and S. Marseglia and Jonas Bergstr{\"o}m",
note = "Funding Information: The authors thank Knut och Alice Wallenbergs Stiftelse for financial support through grants MG2018-0044 and 2017.0418. The second author is supported by the Dutch Research Council (NWO) through grant VI.Veni.192.038. The third author thanks the Max Planck Institute for Mathematics, and NWO grants 613.001.651 and VI.Veni.202.107 for support. The third author would like to express his gratitude to Frans Oort for helpful discussions and to the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation for granting access to their computational facilities. The authors thank Frans Oort, Christophe Ritzenthaler, Andrew Sutherland, and John Voight for helpful comments on an earlier draft, and the anonymous referee for careful reading and useful suggestions. Publisher Copyright: {\textcopyright} The Author(s) 2021. Published by Oxford University Press.",
year = "2023",
month = feb,
day = "1",
doi = "10.1093/imrn/rnab333",
language = "English",
volume = "2023",
pages = "3194–3248",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",
number = "4",
}