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| HAL : hal-00446062, version 3 |
| DOI : 10.1112/S0010437X12000243 |
| Fiche détaillée |  Récupérer au format |
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| Compositio Mathematica 148, 05 (2012) 1483--1515 |
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| Versions disponibles : | v1 (13-01-2010) | v2 (13-01-2010) | v3 (22-09-2012) |
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| Computing isogenies between Abelian Varieties |
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| David Lubicz 1Damien Robert 2, 3 |
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| (01/09/2012) |
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| We describe an efficient algorithm for the computation of isogenies between abelian varieties represented in the coordinate system provided by algebraic theta functions. We explain how to compute all the isogenies from an abelian variety whose kernel is isomorphic to a given abstract group. We also describe an analog of Vélu's formulas to compute an isogenis with prescribed kernels. All our algorithms rely in an essential manner on a generalization of the Riemann formulas. In order to improve the efficiency of our algorithms, we introduce a point compression algorithm that represents a point of level $4\ell$ of a $g$ dimensional abelian variety using only $g(g+1)/2\cdot 4^g$ coordinates. We also give formulas to compute the Weil and commutator pairing given input points in theta coordinates. All the algorithms presented in this paper work in general for any abelian variety defined over a field of odd characteristic. |
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| 1 : | Institut de Recherche Mathématique de Rennes (IRMAR) |
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CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne |
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| 2 : | CARAMEL (INRIA Nancy - Grand Est / LORIA) |
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INRIA – CNRS : UMR7503 – Université de Lorraine |
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| 3 : | LFANT (INRIA Bordeaux - Sud-Ouest) |
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INRIA – Université de Bordeaux – CNRS : UMR5251 |
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| Géométrie algébrique réelle |
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| Domaine | : | Informatique/Calcul formel |
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| Isogeny |
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| Liste des fichiers attachés à ce document : | |||||
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| hal-00446062, version 3 | |
| http://hal.archives-ouvertes.fr/hal-00446062 | |
| oai:hal.archives-ouvertes.fr:hal-00446062 | |
| Contributeur : Damien Robert | |
| Soumis le : Vendredi 21 Septembre 2012, 19:15:22 | |
| Dernière modification le : Dimanche 25 Novembre 2012, 14:26:21 | |
