Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks, Dimitar Jetchev, Benjamin Wesolowski
2017 Research in Number Theory  
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 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
more » ... 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.
doi:10.1007/s40993-017-0087-5 fatcat:mehwdw2wpfco7fkhjnw4cpeaga