**Isogeny graphs with maximal real multiplication**

*Sorina Ionica and Emmanuel Thomé*

**Abstract: **An isogeny graph is a graph whose vertices are principally polarizable abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel describes
the structure of isogeny graphs for elliptic curves and shows that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth-first search (DFS) algorithm in the graph. In dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive. In this article, we show that, under certain circumstances, the problem of determining the endomorphism ring can also be solved in genus 2 with a DFS-based algorithm. We consider
the case of genus-2 Jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We describe the isogeny graphs in that case, locally at prime numbers which split in the real multiplication subfield. The resulting algorithm is implemented over finite fields, and examples are provided. To the best of our knowledge, this is the first DFS-based algorithm in genus 2.

**Category / Keywords: **isogeny graphs, abelian varieties, genus 2

**Date: **received 30 Mar 2014, last revised 17 Oct 2016

**Contact author: **sorina ionica at m4x org

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

**Version: **20161017:141946 (All versions of this report)

**Short URL: **ia.cr/2014/230

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