**Isogeny graphs with maximal real multiplication**

*Sorina Ionica and Emmanuel Thomé*

**Abstract: **An isogeny graph is a graph whose vertices are principally polarized
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 19 Jun 2015

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

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

**Version: **20150619:094222 (All versions of this report)

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