Isogeny graphs of ordinary abelian varieties

TL;DR

This paper investigates the structure of isogeny graphs of ordinary abelian varieties, revealing volcano-like structures and developing algorithms for navigating these graphs to aid cryptographic and number-theoretic applications.

Contribution

It characterizes the structure of isogeny graphs for higher-dimensional abelian varieties and provides algorithms for computing paths to varieties with maximal endomorphism rings.

Findings

Isogeny graphs form volcanoes in any dimension.

Algorithms for finding isogeny paths to maximal endomorphism rings.

Applications to genus 2 cryptography and CM-methods.

Abstract

Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $\mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(\ell, \ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $\ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.