## Abstract

Let $V$ be a nonsingular projective surface defined over $\mathbb{Q}$ and having at least two elliptic fibrations defined over $\mathbb{Q}$; the most interesting case, though not the only one, is when $V$ is a K3 surface with these properties. We also assume that $V\left(\mathbb{Q}\right)$ is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of $V\left(\mathbb{Q}\right)$ and to study the closure of $V\left(\mathbb{Q}\right)$ under the real and the $p$-adic topologies. The first object is achieved by the following theorem:

Let $V$ be a nonsingular surface defined over $\mathbb{Q}$ and having at least two distinct elliptic fibrations. There is an explicitly computable Zariski closed proper subset $X$ of $V$ defined over $\mathbb{Q}$ such that if there is a point ${P}_{0}$ of $V\left(\mathbb{Q}\right)$ not in $X$ then $V\left(\mathbb{Q}\right)$ is Zariski dense in $V$.

The methods employed to study the closure of $V\left(\mathbb{Q}\right)$ in the real or $p$-adic topology demand an almost complete knowledge of $V$; a typical example of what they can achieve is as follows. Let ${V}_{c}$ be

$${V}_{c}:{X}_{0}^{4}+c{X}_{1}^{4}={X}_{2}^{4}+c{X}_{3}^{4}\phantom{\rule{1em}{0ex}}for\phantom{\rule{0.3em}{0ex}}c=2,4\text{or}8;$$

then ${V}_{c}\left(\mathbb{Q}\right)$ is dense in ${V}_{c}\left({\mathbb{Q}}_{2}\right)$ for $c=2,4,8$.

## Citation

H. Peter Swinnerton-Dyer. "Density of rational points on certain surfaces." Algebra Number Theory 7 (4) 835 - 851, 2013. https://doi.org/10.2140/ant.2013.7.835

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