Rational points on elliptic K3 surfaces of quadratic twist type

TL;DR

This paper establishes criteria for the density of rational points on certain elliptic K3 surfaces of quadratic twist type, linking the problem to the ranks of related elliptic curves and proving density results unconditionally for Cassels-Schinzel type surfaces.

Contribution

It introduces a new condition on quadratic twists of elliptic curves that determines the Zariski density of rational points on these K3 surfaces.

Findings

Necessary and sufficient condition for Zariski density of rational points.

Proved density of rational points on Cassels-Schinzel type surfaces unconditionally.

Connected the density problem to the positive Mordell-Weil rank of related elliptic curves.

Abstract

In studying rational points on elliptic K3 surfaces of the form $f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated roots), we introduce a condition on the quadratic twists of two elliptic curves having simultaneously positive Mordell-Weil rank. We prove a necessary and sufficient condition for the Zariski density of rational points by using this condition, and we relate it to the Hilbert property. Applying to surfaces of Cassels-Schinzel type, we prove unconditionally that rational points are dense both in Zariski topology and in real topology.