Isogeny graphs with maximal real multiplication

Sorina Ionica 1, 2 Emmanuel Thomé 3
1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
3 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
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 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. Our setting considers genus 2 jacobians with complex multiplication, with the assumptions that the real multiplication subring is maximal and has class number one. We fully describe the isogeny graphs in that case. 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.
Type de document :
Pré-publication, Document de travail
2015
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https://hal.archives-ouvertes.fr/hal-00967742
Contributeur : Sorina Ionica <>
Soumis le : lundi 17 octobre 2016 - 16:28:30
Dernière modification le : mercredi 19 octobre 2016 - 01:23:43

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  • HAL Id : hal-00967742, version 3
  • ARXIV : 1407.6672

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Sorina Ionica, Emmanuel Thomé. Isogeny graphs with maximal real multiplication. 2015. <hal-00967742v3>

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