Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type

authored by

Zhizhong Huang

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. 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.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

Quarterly Journal of Mathematics

Volume

72

Pages

755-772

No. of pages

18

ISSN

0033-5606

Publication date

09.2021

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Mathematics(all)

Electronic version(s)

https://arxiv.org/abs/1806.07869 (Access: Open)
https://doi.org/10.1093/qmath/haaa044 (Access: Closed)