Rational points on K3 surfaces of degree 2

TL;DR

The paper establishes a uniform bound for obtaining infinitely many rational points on K3 surfaces of degree $2$, showing that for any such surface over a number field $K$ there exists an extension $L/K$ with $[L:K]\le 12$ giving infinitely many $L$-rational points. It achieves this by first locating a genus-1 curve on an appropriate extension (degree at most $6$) and then proving that a quadratic extension renders its normalization into an elliptic curve with a non-torsion point, invoking Merel’s theorem to guarantee infinitely many points. The authors construct a family of degree-$2$ K3 surfaces with geometric Picard number $1$ and infinitely many rational points over $\mathbb{Q}$, and prove that this locus is Zariski dense in the moduli space $\mathcal{M}_2$ of K3 surfaces of degree $2$. The results extend the density phenomena known for degree-4 K3 surfaces to degree-2 surfaces, providing both explicit examples and a constructive method for realizing infinite rational points over controlled field extensions.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2: van Luijk, 2005 vluijk
• Theorem 1.3
• Proposition 3.1
• proof
• Lemma 4.1
• proof
• Proposition 4.2
• proof
• proof : Proof of Theorem \ref{['bound']}