Inose's construction and elliptic K3 surfaces with Mordell-Weil rank 15 revisited

TL;DR

This paper presents new explicit constructions of elliptic K3 surfaces with high Mordell-Weil rank, using Kummer surfaces and base-change methods, including an analogue of Inose fibration, over the rationals.

Contribution

It introduces two novel constructions of elliptic K3 surfaces with Mordell-Weil rank 15, expanding explicit examples in the field.

Findings

Constructed elliptic K3 surfaces with Mordell-Weil rank 15 over rationals.

Provided explicit examples using Kummer surfaces and base-change techniques.

Extended Inose fibration concepts to new classes of K3 surfaces.

Abstract

We describe two constructions of elliptic K3 surfaces starting from the Kummer surface of the Jacobian of a genus 2 curve. These parallel the base-change constructions of Kuwata for the Kummer surface of a product of two elliptic curves. One of these also involves the analogue of an Inose fibration. We use these methods to provide explicit examples of elliptic K3 surfaces over the rationals of geometric Mordell-Weil rank 15.