Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

TL;DR

This paper investigates elliptic fibrations on K3 surfaces that are double covers of extremal rational elliptic surfaces, focusing on their fields of definition and the influence of involutions.

Contribution

It provides a comprehensive list of elliptic fibrations on specific K3 surfaces and analyzes how involutions affect their field of definition and sections.

Findings

Identified all elliptic fibrations on certain K3 surfaces.

Determined the degrees of field extensions for fibrations to be defined.

Showed the action of involution influences the fibrations' fields of definition.

Abstract

We study K3 surfaces over a number field $k$ which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus 1 fibration.