Computing modular correspondences for abelian varieties
hal.structure.identifier | Solvers for Algebraic Systems and Applications [SALSA] | |
dc.contributor.author | FAUGÈRE, Jean-Charles | |
hal.structure.identifier | Institut de Recherche Mathématique de Rennes [IRMAR] | |
dc.contributor.author | LUBICZ, David | |
hal.structure.identifier | Curves, Algebra, Computer Arithmetic, and so On [CACAO] | |
hal.structure.identifier | Lithe and fast algorithmic number theory [LFANT] | |
hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
dc.contributor.author | ROBERT, Damien | |
dc.date.accessioned | 2024-04-04T02:24:45Z | |
dc.date.available | 2024-04-04T02:24:45Z | |
dc.date.created | 2009-05 | |
dc.date.issued | 2011-10 | |
dc.identifier.issn | 0021-8693 | |
dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/189835 | |
dc.description.abstractEn | 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. | |
dc.description.sponsorship | Courbes Hyperelliptiques : Isogénies et Comptage - ANR-09-BLAN-0020 | |
dc.language.iso | en | |
dc.publisher | Elsevier | |
dc.title.en | Computing modular correspondences for abelian varieties | |
dc.type | Article de revue | |
dc.identifier.doi | 10.1016/j.jalgebra.2011.06.031 | |
dc.subject.hal | Informatique [cs]/Calcul formel [cs.SC] | |
bordeaux.journal | Journal of Algebra | |
bordeaux.page | 248-277 | |
bordeaux.volume | 343 | |
bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
bordeaux.issue | 1 | |
bordeaux.institution | Université de Bordeaux | |
bordeaux.institution | Bordeaux INP | |
bordeaux.institution | CNRS | |
bordeaux.peerReviewed | oui | |
hal.identifier | hal-00426338 | |
hal.version | 1 | |
hal.popular | non | |
hal.audience | Internationale | |
hal.origin.link | https://hal.archives-ouvertes.fr//hal-00426338v1 | |
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