**Isogeny graphs of ordinary abelian varieties**

*Ernest Hunter Brooks and Dimitar Jetchev and Benjamin Wesolowski*

**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.

**Category / Keywords: **Jacobians of hyperelliptic curves, genus 2 cryptography, isogeny graphs, $(\ell,\ell)$-isogenies, principally polarised abelian varieties

**Date: **received 30 Sep 2016

**Contact author: **benjamin wesolowski at epfl ch

**Available format(s): **PDF | BibTeX Citation

**Version: **20161001:184635 (All versions of this report)

**Short URL: **ia.cr/2016/947

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