Rational points on elliptic K3 surfaces of quadratic twist type

Abstract

We propose a double covering method to study the density of rational points and density of fibres of prescribed rank on quadratic twist type elliptic surfaces f (t)y 2 = g(x), where f, g are cubic or quartic polynomials (without repeated roots). We apply it to certain generic Mordell-Weil rank 0 cases such as the example of Cassels and Schinzel and prove unconditionally that rational points are dense both in Zariski topology and in real topology. Contents 1. Introduction 1 2. Preliminaries 5 3. Kummer type elliptic surfaces 8 4. The double covering method 11 5. The Diophantine equation X 4 + Y 4 = CZ 2 13 6. Demonstrations of the main theorem and particular examples 16 7. Applications to quadratic twist type elliptic surfaces of lower degree 18 References 19

Mazur's conjecture and an unexpected rational curve on Kummer surfaces and their superelliptic generalisations

“…Theorem 4.4 complements Kuwata and Wang's quartic example (t 4 + 1)y 2 = x 3 − 4x [KW93, p. 121] which they derived from the work by Elkies mentioned in the introduction. A recent preprint by Huang [Hua18] deals with d(t 4 + 1)y 2 = x 3 − x for some d. By entirely different methods and under some additional assumption, [HS16, Prop. 1.1] proves Mazur's conjecture for the Kummer quotient associated to the product of non-trivial 2-coverings of elliptic curves.…”