Abstract
In studying rational points on elliptic K3 surfaces of the form $$\begin{equation*} f(t)y^2=g(x), \end{equation*}$$ 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, 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.
AbstractFollowing some remarks made by O'Grady and Oguiso, the potential density of rational points on the second punctual Hilbert scheme of certain K3 surfaces is proved.
Rational Points on K3 Surfaces and Derived Equivalence