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

Bertin, Marie José; Lecacheux, Odile

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

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 $8$.

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 $8$.

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

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