Journal of the Mathematical Society of Japan

Apéry–Fermi pencil of $K3$-surfaces and 2-isogenies

Marie José BERTIN and Odile LECACHEUX

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Given a generic $K3$-surface $Y_k$ of the Apéry–Fermi pencil, we use the Kneser–Nishiyama technique to determine all its non isomorphic elliptic fibrations. These computations lead to determine those fibrations with 2-torsion sections T. We classify the fibrations such that the translation by T gives a Shioda–Inose structure. The other fibrations correspond to a $K3$-surface identified by its transcendental lattice. The same problem is solved for a singular member $Y_2$ of the family showing the differences with the generic case. In conclusion we put our results in the context of relations between 2-isogenies and isometries on the singular surfaces of the family.

Article information

J. Math. Soc. Japan, Advance publication (2019), 39 pages.

Received: 26 May 2018
Revised: 17 October 2018
First available in Project Euclid: 16 October 2019

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Digital Object Identifier

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 14J27: Elliptic surfaces 14J50: Automorphisms of surfaces and higher-dimensional varieties 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

elliptic fibrations of $K3$-surfaces Morrison–Nikulin involutions isogenies


BERTIN, Marie José; LECACHEUX, Odile. Apéry–Fermi pencil of $K3$-surfaces and 2-isogenies. J. Math. Soc. Japan, advance publication, 16 October 2019. doi:10.2969/jmsj/80638063.

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