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

TL;DR

This paper classifies elliptic fibrations of a specific family of K3 surfaces, identifying those with 2-torsion sections and exploring their relation to 2-isogenies and Shioda-Inose structures.

Contribution

It applies the Kneser-Nishiyama technique to classify all elliptic fibrations of Apéry-Fermi K3 surfaces and analyzes their 2-torsion sections and isogeny relations.

Findings

Classified all elliptic fibrations of the Apéry-Fermi K3 family.

Identified fibrations with 2-torsion sections and their associated structures.

Compared generic and singular members of the family regarding their lattice properties.

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.