Some observations about isogenies between $K3$ surfaces

TL;DR

This paper explores properties of elliptic fibrations on a specific singular K3 surface, revealing how certain 3-isogenies relate it to other K3 surfaces with different discriminants.

Contribution

It identifies how elliptic fibrations with 3-torsion sections induce isogenies connecting different K3 surfaces, highlighting new relationships among them.

Findings

Elliptic fibrations with 3-torsion induce 3-isogenies.

Connections between K3 surfaces of discriminant 72 and 8.

Identification of fibrations leading to specific K3 surfaces.

Abstract

Even if there are too many elliptic fibrations to investigate and describe on the singular $K3$ surface $Y_{10}$ of discriminant 72 and belonging to the Ap\'ery-Fermi pencil $(Y_k)$, we find on it many interesting properties. For example some of its elliptic fibrations with 3-torsion section induce by 3-isogeny either an elliptic fibration of $Y_2$, the unique $K3$ surface of discriminant 8, or an elliptic fibration of other $K3$ surfaces of discriminant 72.