Rational points on K3 surfaces of degree 2

TL;DR

This paper investigates rational points on degree 2 K3 surfaces, providing bounds on field extensions and constructing explicit examples over $\,\mathbb{Q}$ with dense rational points.

Contribution

It establishes bounds for the degree of field extensions over which rational points are dense on degree 2 K3 surfaces and constructs explicit examples with dense rational points over $\,\mathbb{Q}$.

Findings

Bound on the degree of field extension for rational points on degree 2 K3 surfaces.

Explicit family of degree 2 K3 surfaces over $\,\mathbb{Q}$ with dense rational points.

Existence of K3 surfaces with geometric Picard number 1 and infinitely many rational points.

Abstract

A K3 surface over a number field has infinitely many rational points over a finite field extension. For K3 surfaces of degree 2, arising as double covers of $\mathbb{P}^2$ branched along a smooth sextic curve, we give a bound for the degree of such an extension. Moreover, using ideas of van Luijk and a surface constructed by Elsenhans and Jahnel, we give an explicit family of K3 surfaces of degree 2 defined over $\mathbb{Q}$ with geometric Picard number 1 and infinitely many $\mathbb{Q}$-rational points that is Zariski dense in the moduli space of K3 surfaces of degree 2.