Ordinary abelian varieties: isogeny graphs and polarizations

Abstract

Given an integer $D$ and an ordinary isogeny class of abelian varieties defined over a finite field ${\mathbb{F}}_q$ with commutative ${\mathbb{F}}_q$-endomorphism algebra, we provide algorithms for computing all isogenies of degree dividing $D$ and polarizations of degree dividing $D$. We discuss phenomena that arise for higher dimension abelian varieties but not elliptic curves, bounds on the diameter of the graph of minimal isogenies, and decompositions of isogeny graphs into orbits for the Picard group of the Frobenius order.