One construction of a K3 surface with a dense set of rational points

TL;DR

This paper constructs a specific K3 surface over a number field with geometric Picard number one, demonstrating that it has a Zariski dense set of rational points, advancing understanding of rational points on K3 surfaces.

Contribution

It provides a new explicit example of a K3 surface with minimal Picard number and dense rational points over a number field, which was previously unknown.

Findings

Existence of a K3 surface over a number field with Picard number 1

The rational points on this surface are Zariski dense

The construction advances understanding of rational points on K3 surfaces

Abstract

We prove that there exists a number field $\fie$ and a smooth projective $\mathrm{K3}$ surface $S_{22}$ (of genus $12$) over $\fie$ such that the geometric Picard number of $S_{22}$ is equal to $1$ and the $\fie$-rational points of $S_{22}$ are Zariski dense.