Density of rational points on certain surfaces. (English) Zbl 1277.11070

Swinnerton-Dyer, H. P. F.

Algebra Number Theory 7, No. 4, 835-851 (2013).

Let \(V\) be a nonsingular surface defined over \(\mathbb{Q}\) having at least two elliptic fibrations defined over \(\mathbb{Q}\). There are many K3 surfaces with this property, for example. The first result gives an effective criterion to decide whether \(V(\mathbb{Q})\) is Zariski dense in \(V\). A Zariski closed proper subset \(X\) of \(V\) is constructed, in terms of the two fibrations, such that \(X\) is defined over \(\mathbb{Q}\), and with the property that \(V(\mathbb{Q})\) is Zariski dense in \(V\) provided only that \(V-X\) contains at least one rational point.
The paper goes on to consider the density of rational points under the real topology. Let \(X\) be as above, and let \(R\) be the closure of \((V-X)(\mathbb{Q})\) in \(V\) under the real topology. A second Zariski closed proper subset \(X'\) of \(V\) is constructed such that the boundary of \(R\) must lie in \(X'\). Of course it is quite possible that \(R\) is the whole of \(V\).
As examples the author considers surfaces \[ V:\; a_0x_0^4+a_1x_1^4+a_2x_2^4+a_3x_3^4=0 \] with rational coefficients for which the product \(a_0a_1a_2a_3\) is a square. In this case \(V(\mathbb{Q})\) is Zariski dense in \(V\), and dense under the real topology in \(V(\mathbb{R})\), provided only that \(V\) has a rational point with \(x_0x_1x_2x_3\not=0\) and not lying on any line in \(V\).
There are also results for the \(p\)-adic topology, and as an example it is shown that for the surfaces \[ V_c:\; x_0^4+cx_1^4=x_2^4+cx_3^4\quad(c=2,4\text{ or } 8) \] \(V_c(\mathbb{Q})\) is dense in \(V_c(\mathbb{Q}_2)\).
These results build on ideas of A. Logan, D. McKinnon and R. van Luijk [Algebra Number Theory 4, No. 1, 1–20 (2010; Zbl 1206.11082)], who considered a particular family of surfaces.

Reviewer: D. R. Heath-Brown (Oxford)

MSC:

11G35	Varieties over global fields
14G05	Rational points
11D25	Cubic and quartic Diophantine equations

Keywords:

algebraic surface; K3 surface; rational points; Zariski dense; elliptic fibrations; real topology

Citations:

Zbl 1206.11082