Elliptic Fibrations of a certain $K3$ surface of the Apéry–Fermi pencil

Marie José Bertin; Odile Lecacheux

doi:10.5802/pmb.44

Elliptic Fibrations of a certain

K 3

surface of the Apéry–Fermi pencil

Marie José Bertin ¹; Odile Lecacheux ¹

¹ Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 4 Place Jussieu, 75252 PARIS, Cedex 85, France

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2022), pp. 5-36.

Abstract
Résumé

We explain how to obtain from Kneser–Nishiyama’s method all the elliptic fibrations of the singular (i.e. of Picard number $20$ ) $K 3$ surface $Y_{10}$ of discriminant $72$ and belonging to the Apéry–Fermi pencil ( $Y_{k}$ ). The case of its extremal elliptic fibrations is developped together with Weierstrass equations, noticing that two of them are obtained by $3$ -isogeny from extremal fibrations of the $K 3$ surface $Y_{2}$ of discriminant $8$ .

On montre comment la méthode de Kneser–Nishiyama permet d’obtenir toutes les fibrations elliptiques de la surface $K 3$ singulière (i.e. de nombre de Picard $20$ ) de discriminant $72$ , notée $Y_{10}$ , appartenant au pinceau ( $Y_{k}$ ) de surfaces $K 3$ d’Apéry–Fermi. Les fibrations elliptiques extrémales sont en outre données avec des équations de Weierstrass. On remarque que deux d’entre elles sont obtenues par $3$ -isogénie à partir de fibrations extrémales de la surface $Y_{2}$ de discriminant $8$ .

Received: 2020-07-12
Published online: 2022-08-18

DOI: 10.5802/pmb.44

Classification: 11F23, 11G05, 14J28, 14J27
Keywords: Niemeier Lattices, Kneser–Nishiyama Method for Elliptic Fibrations of $K3$ Surfaces, Elkies r-neighbor Method for Weierstrass Equations

Author's affiliations:

Marie José Bertin ¹; Odile Lecacheux ¹

¹ Sorbonne Université, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 4 Place Jussieu, 75252 PARIS, Cedex 85, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Elliptic {Fibrations} of a certain $K3$ surface of the {Ap\'ery{\textendash}Fermi} pencil},
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Marie José Bertin; Odile Lecacheux. Elliptic Fibrations of a certain $K3$ surface of the Apéry–Fermi pencil. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2022), pp. 5-36. doi : 10.5802/pmb.44. https://pmb.centre-mersenne.org/articles/10.5802/pmb.44/

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