Isogeny graphs of ordinary abelian varieties

Abstract

Fix a prime number . Graphs of isogenies of degree a power of 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 l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the "volcanoes" occurring as graphs of isogenies of elliptic curves. 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 ( , )-isogenies: those whose kernels are maximal isotropic subgroups of the -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.

Computing the endomorphism ring of an ordinary abelian surface over a finite field

“…Given an order O ⊆ K containing O F , the structure of C(O) can be seen through the following exact sequence Proof. This Proposition summarizes Proposition 5.4 and Lemma 5.5 of [8].…”

“…Following Brooks, Jetchev and Wesolowski [8], we will focus on l-isogenies, defined below. Assume that A has maximal RM and write v for the ideal f + (O F [π]).…”

“…The following lemma was stated without proof in the definition of l-isogeny in [8]. We record a proof here for completeness.…”

“…For this, we need the following additional result about isogeny graphs. More background may be found in [8]. This theorem implies that finding the power of each l | v which divides f + (A) is equivalent to finding the level of A in the l-isogeny graph.…”

Hilbert modular polynomials

“…There is a case for attempting to generalise these protocols to genus 2, especially given the recent research [RSSB16] suggesting that genus 2 arithmetic can be more efficient than elliptic curve arithmetic. The protocol presented in [DKS18] should generalise to genus 2 curves with maximal real multiplication directly-given that the structure of isogeny graphs for principally polarised simple ordinary abelian surfaces with maximal real multiplication is the same as for ordinary elliptic curves defined over Fq [Mar18;BJW17]. In order to actually implement such a protocol, there has to be a method of navigating the isogeny graphs, that is, of computing all the surfaces that are µ-isogenous to a given surface.…”