Ap\'ery-Fermi pencil of $K3$-surfaces and their $2$-isogenies

Abstract

Given a generic $K3$ surface $Y_k$ of the Ap\'ery-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 it 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.