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 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.

Final version. To appear on IMRN

References in corpus (9)

• Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
• Computing the ideal class monoid of an order
• Abelian varieties isogenous to a power of an elliptic curve
• Computing square-free polarized abelian varieties over finite fields
• Almost ordinary abelian varieties over finite fields
• On the nonexistence of certain curves of genus two
• Elliptic curves, random matrices and orbital integrals
• Serre's Tensor Construction and Moduli of Abelian Schemes
• Local heuristics and an exact formula for abelian surfaces over finite fields

Cited by in corpus (3)

• Polarized superspecial simple abelian surfaces with real Weil numbers
• Abelian varieties over finite fields and their groups of rational points
• Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields