Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks,Dimitar Jetchev,Benjamin Wesolowski

doi:10.1007/s40993-017-0087-5

Ernest Hunter Brooks, Dimitar Jetchev + Show 1 more

Open Access

https://doi.org/10.1007/s40993-017-0087-5

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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, 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 (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.

Highlights

1.1 Isogeny graphs If A is a abelian variety over a finite field k, and B is an isogenous abelian variety, the discrete logarithm problem on A (k) may be transferred to a problem on B(k) assuming that one has an efficiently computable isogeny A → B
The real endomorphism ring of an absolutely simple ordinary abelian variety is the ring of totally real elements in its endomorphism ring
We say that the real endomorphism ring is maximal if it is integrally closed in its field of fractions

Summary

Introduction

1.1 Isogeny graphs If A is a abelian variety over a finite field k, and B is an isogenous abelian variety, the discrete logarithm problem on A (k) may be transferred to a problem on B(k) assuming that one has an efficiently computable isogeny A → B. (i) For each i ≥ 0, the varieties in Vipp share a common endomorphism ring Oi. The order O0 can be any order with locally maximal real multiplication at , whose conductor is not divisible by β; (ii) The level V0pp is isomorphic to the Cayley graph of the subgroup of C(O0) with generators (Li, β) where Li are the prime ideals in O0 above β; (iii) For any x ∈ V0pp, there are. We study the structure of the graph G , whose vertices are the isomorphism classes of principally polarizable surfaces A in the fixed isogeny class, which have maximal real multiplication locally at (i.e., oK+ ⊂ o(A )), with an edge of multiplicity m from such a vertex A to a vertex B if there are m distinct subgroups κ ⊂ A that are kernels of ( , )-isogenies such that A /κ ∼= B. If is inert in K +, if splits as l1l2 in K +, if ramifies as l2 in K +

The inert case

Motivation for a “going-up” algorithm

Prior work of Lauter–Robert