Computing square-free polarized abelian varieties over finite fields

Abstract

We give algorithms to compute isomorphism classes of ordinary abelian varieties defined over a finite field ${\mathbb{F}}_q$ whose characteristic polynomial (of Frobenius) is square-free and of abelian varieties defined over the prime field ${\mathbb{F}}_p$ whose characteristic polynomial is square-free and does not have real roots. In the ordinary case we are also able to compute the polarizations and the group of automorphisms (of the polarized variety) and, when the polarization is principal, the period matrix.

arXiv commentaccepted by Math. Comp. major revision: added computation of the group of points; examples have been exported on the repo

References in corpus (3)

• Abelian varieties isogenous to a power of an elliptic curve
• Almost ordinary abelian varieties over finite fields
• Elliptic curves, random matrices and orbital integrals

Cited by in corpus (13)

• Almost ordinary abelian varieties over finite fields
• Polarizations of abelian varieties over finite fields via canonical liftings
• Counting abelian varieties over finite fields via Frobenius densities
• Cohen-Macaulay type of orders, generators and ideal classes
• Every finite abelian group is the group of rational points of an ordinary abelian variety over ${\mathbb{F}}_2$, ${\mathbb{F}}_3$ and ${\mathbb{F}}_5$
• Class numbers of CM algebraic tori, CM abelian varieties and components of unitary Shimura varieties
• Polarized superspecial simple abelian surfaces with real Weil numbers
• Local heuristics and an exact formula for abelian varieties of odd prime dimension over finite fields
• Computing base extensions of ordinary abelian varieties over finite fields
• Products and Polarizations of Super-Isolated Abelian Varieties
• Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields
• Ideal classes of orders in quaternion algebras
• Abelian varieties over finite fields and their groups of rational points