3270 documents avec fichiers associés – 5421 références bibliographiques  [english version]
 HAL : hal-00426338, version 1
 arXiv : 0910.4668
 Journal of Algebra 343, 1 (2011) 248-277
 Versions disponibles : v1 (24-10-2009) v2 (25-11-2012)
 Computing modular correspondences for abelian varieties
 (2011)
 The aim of this paper is to give a higher dimensional equivalent of the classical modular polynomials $\Phi_\ell(X,Y)$. If $j$ is the $j$-invariant associated to an elliptic curve $E_k$ over a field $k$ then the roots of $\Phi_\ell(j,X)$ correspond to the $j$-invariants of the curves which are $\ell$-isogeneous to $E_k$. Denote by $X_0(N)$ the modular curve which parametrizes the set of elliptic curves together with a $N$-torsion subgroup. It is possible to interpret $\Phi_\ell(X,Y)$ as an equation cutting out the image of a certain modular correspondence $X_0(\ell) \rightarrow X_0(1) \times X_0(1)$ in the product $X_0(1) \times X_0(1)$. Let $g$ be a positive integer and $\overn \in \N^g$. We are interested in the moduli space that we denote by $\Mn$ of abelian varieties of dimension $g$ over a field $k$ together with an ample symmetric line bundle $\pol$ and a symmetric theta structure of type $\overn$. If $\ell$ is a prime and let $\overl=(\ell, \ldots , \ell)$, there exists a modular correspondence $\Mln \rightarrow \Mn \times \Mn$. We give a system of algebraic equations defining the image of this modular correspondence. We describe an algorithm to solve this system of algebraic equations which is much more efficient than a general purpose Gr¨obner basis algorithm. As an application, we explain how this algorithm can be used to speed up the initialisation phase of a point counting algorithm.
 1 : SALSA (INRIA Rocquencourt) INRIA – CNRS : UMR7606 – Université Pierre et Marie Curie [UPMC] - Paris VI 2 : Institut de Recherche Mathématique de Rennes (IRMAR) 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 3 : CACAO (INRIA Lorraine - LORIA) CNRS : UMR7503 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
 Domaine : Informatique/Calcul formel
Liste des fichiers attachés à ce document :
 PDF
 practical.pdf(468.1 KB)
 PS
 practical.ps(465.8 KB)
 hal-00426338, version 1 http://hal.archives-ouvertes.fr/hal-00426338 oai:hal.archives-ouvertes.fr:hal-00426338 Contributeur : Damien Robert <> Soumis le : Samedi 24 Octobre 2009, 16:45:21 Dernière modification le : Vendredi 9 Septembre 2011, 09:47:25