Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

Hulek, Klaus; Kloosterman, Remke

Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1133-1179.

Dans cet article nous présentons une méthode pour calculer le rang d’une courbe elliptique générale sur le corps des fonctions rationnelles de deux variables. Nous réduisons ce problème au calcul de la cohomologie d’une hypersurface singulière dans un espace projectif pondéré de dimension quatre. Nous donnons alors une méthode de calcul de la cohomologie d’une certaine classe d’hypersurfaces singulières en étendant le travail de Dimca dans le cas des singularités isolées.

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective $4$-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Classification : 14J30, 14J70, 32S20, 32S35, 32S50
Keywords: Mordel-Weil group of Elliptic threefolds, Cohomology of singular varieties, Mixed Hodge structures
Mot clés : Groupe de Mordell-Weil des variétés elliptiques de dimension trois, cohomologie des variétés singulières, structures de Hodge mixtes

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