Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces
[Calcul du rang de Mordell-Weil des variétés elliptiques de dimension trois et de la cohomologie des hypersurfaces singulières]
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.

Reçu le :
Accepté le :
DOI : https://doi.org/10.5802/aif.2637
Classification : 14J30,  14J70,  32S20,  32S35,  32S50
Mots clés : Groupe de Mordell-Weil des variétés elliptiques de dimension trois, cohomologie des variétés singulières, structures de Hodge mixtes
@article{AIF_2011__61_3_1133_0,
     author = {Hulek, Klaus and Kloosterman, Remke},
     title = {Calculating the {Mordell-Weil} rank of elliptic threefolds and the cohomology of singular hypersurfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {1133--1179},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     doi = {10.5802/aif.2637},
     mrnumber = {2918726},
     zbl = {1246.14057},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2637/}
}
Hulek, Klaus; Kloosterman, Remke. Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces. Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 1133-1179. doi : 10.5802/aif.2637. https://aif.centre-mersenne.org/articles/10.5802/aif.2637/

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