# Isogeny graphs with maximal real multiplication

Sorina Ionica

Date(s) : 05/02/2015   iCal
11 h 00 min - 12 h 00 min

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, David Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search 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. We fully describe isogeny graphs of genus 2 Jacobians, with maximal real multiplication. Over finite fields, we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers, if the real multiplication is maximal. To the best of our knowledge, this is the first DFS-based algorithm in genus 2. This is joint work with Emmanuel Thomé.

Sorina Ionica, Laboratoire MIS, Universite de Picardie, Amiens

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