Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Advance publication (2019), 39 pages.
Apéry–Fermi pencil of $K3$-surfaces and 2-isogenies
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.
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
Permanent link to this document
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]
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. https://projecteuclid.org/euclid.jmsj/1571212902