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

[Fibrations elliptiques d’une certaine surface $K3$ du pinceau d’Apéry–Fermi]

Bertin, Marie José¹ ; Lecacheux, Odile¹

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 (VO)

We explain how to obtain from Kneser–Nishiyama’s method all the elliptic fibrations of the singular (i.e. of Picard number $20$) $K3$ 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 $K3$ surface $Y_2$ of discriminant $8$.

Résumé

On montre comment la méthode de Kneser–Nishiyama permet d’obtenir toutes les fibrations elliptiques de la surface $K3$ singulière (i.e. de nombre de Picard $20$) de discriminant $72$, notée $Y_{10}$, appartenant au pinceau ($Y_k$) de surfaces $K3$ 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 $8$.

Détail

Reçu le : 2020-07-12
Publié le : 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

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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