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

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.

### MSC:

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

Zbl 1206.11082
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