Vendredi 14 juin 2013 à 17h00 en salle C48
Sorina Ionica (ENS de Paris)

Titre : Isogeny graphs with real multiplication

Résumé :
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 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. For 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 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.