Ordinary abelian varieties: isogeny graphs and polarizations

TL;DR

This paper develops algorithms to compute isogenies and polarizations of abelian varieties over finite fields, exploring their graph structures and decompositions, with implications for higher-dimensional cases beyond elliptic curves.

Contribution

It introduces new algorithms for computing isogenies and polarizations of abelian varieties, and analyzes their isogeny graph structures and decompositions.

Findings

Algorithms for isogeny and polarization computation.

Bounds on the diameter of isogeny graphs.

Decomposition of isogeny graphs into Picard group orbits.

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.