inria-00071967, version 2
The arithmetic of Jacobian groups of superelliptic cubics
Abdolali Basiri 1, 2Andreas Enge
3, 4Jean-Charles Faugère 1, 2Nicolas Gürel 3, 4
Mathematics of Computation 74, 249 (2005) 389-410
Résumé : We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data.
- 1 : SPACES (INRIA Lorraine - LORIA)
- INRIA – CNRS : UMR7503 – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine
- 2 : Laboratoire d'Informatique de Paris 6 (LIP6)
- CNRS : UMR7606 – Université Paris VI - Pierre et Marie Curie
- 3 : TANC (INRIA Futurs)
- CNRS : UMR7161 – INRIA – Polytechnique - X
- 4 : Laboratoire d'informatique de l'école polytechnique (LIX)
- CNRS : UMR7161 – Polytechnique - X
- Domaine : Informatique/Autre
- Mots-clés : superelliptic curve – $C_{ab}$ curve – Jacobian – arithmetic – Gröbner basis
- Référence interne : RR-4618
- Versions disponibles : v1 (31-05-2006) v2 (31-10-2009)
- inria-00071967, version 2
- http://hal.inria.fr/inria-00071967
- oai:hal.inria.fr:inria-00071967
- Contributeur : Andreas Enge
- Soumis le : Vendredi 30 Octobre 2009, 18:07:10
- Dernière modification le : Samedi 31 Octobre 2009, 09:49:42
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