Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type

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.

Root number of twists of an elliptic curve

“…Observe that the first example is a K3 surface and that the second is a rational elliptic surface. These specific surfaces have however a Zariski-dense set of rational points -the proof can be found respectively in [10] and in [15,Example 7.1]. By the time the present article was published, the author and B. Naskrȩcki [7] released a preprint presenting a simple algorithm to find the generic rank of any elliptic surface of the form y 2 = x 3 + AT 6 + B, which generalizes Várilly-Alvarado's result.…”

“…Monodromy of (v 2 (a) + 4, v 2 (b) + 5, v 2 (∆)) Are triples in Table 3.1 (0, 0, 0) ←→ (2, 3, 6) yes, yes (0, 0, > 0) ←→ (2, 3, > 6) no, no (3, 3, 0) ←→ (5,5,6) no, no (≥ 4, 3, 0) ←→ (≥ 6, 6, 6) yes, yes (2, 4, 0) ←→ (4,7,6) no, no (2, ≥ 5, 0) ←→ (4, ≥ 8, 6) no, no (2, 3, 1) ←→ (4,6,7) no, no (2, 3, 2) ←→ (4,6,8) no, no (2, 3, 3) ←→ (4,6,7) no, no (2, 3, ≥ 4) ←→ (4, 6, ≥ 8) yes, no (3, 4, 2) ←→ (5,7,8) no, no (3, 5, 3) ←→ (5,8,9) yes, yes (4, 4, 2) ←→ (6,7,8) yes, no (≥ 5, 4, 2) ←→ (≥ 7, 7, 8) yes, no (4, 5, 4) ←→ (6,8,10) no, yes (≥ 5, 5, 4) ←→ (≥ 7, 8, 10) no, yes…”

“…In [10] Huang presents a geometric method that proves the density of rational points for elliptic surfaces defined by the equation y 2 = x 3 − d 2 (1 + t 4 ) 2 x, for infinitely many squarefree values of d including d = 1 and 7.…”