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arXiv · 1806.07869

Rational points on elliptic K3 surfaces of quadratic twist type

Zhizhong Huang
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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.

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Zhizhong Huang. 2018-06-20. Rational points on elliptic K3 surfaces of quadratic twist type. https://doi.org/10.1093/qmath/haaa044

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