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Computing modular correspondences for abelian varieties

hal.structure.identifierSolvers for Algebraic Systems and Applications [SALSA]
dc.contributor.authorFAUGÈRE, Jean-Charles
hal.structure.identifierInstitut de Recherche Mathématique de Rennes [IRMAR]
dc.contributor.authorLUBICZ, David
hal.structure.identifierCurves, Algebra, Computer Arithmetic, and so On [CACAO]
hal.structure.identifierLithe and fast algorithmic number theory [LFANT]
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorROBERT, Damien
dc.date.accessioned2024-04-04T02:24:45Z
dc.date.available2024-04-04T02:24:45Z
dc.date.created2009-05
dc.date.issued2011-10
dc.identifier.issn0021-8693
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/189835
dc.description.abstractEnThe 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.sponsorshipCourbes Hyperelliptiques : Isogénies et Comptage - ANR-09-BLAN-0020
dc.language.isoen
dc.publisherElsevier
dc.title.enComputing modular correspondences for abelian varieties
dc.typeArticle de revue
dc.identifier.doi10.1016/j.jalgebra.2011.06.031
dc.subject.halInformatique [cs]/Calcul formel [cs.SC]
bordeaux.journalJournal of Algebra
bordeaux.page248-277
bordeaux.volume343
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue1
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-00426338
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00426338v1
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