#### Vol. 7, No. 4, 2013

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Density of rational points on certain surfaces

### Sir Peter Swinnerton-Dyer

Vol. 7 (2013), No. 4, 835–851
##### Abstract

Let $V$ be a nonsingular projective surface defined over $ℚ$ and having at least two elliptic fibrations defined over $ℚ$; the most interesting case, though not the only one, is when $V$ is a K3 surface with these properties. We also assume that $V\left(ℚ\right)$ is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of $V\left(ℚ\right)$ and to study the closure of $V\left(ℚ\right)$ 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 $ℚ$ and having at least two distinct elliptic fibrations. There is an explicitly computable Zariski closed proper subset $X$ of $V$ defined over $ℚ$ such that if there is a point ${P}_{0}$ of $V\left(ℚ\right)$ not in $X$ then $V\left(ℚ\right)$ is Zariski dense in $V$.

The methods employed to study the closure of $V\left(ℚ\right)$ 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

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

##### Keywords
rational points, K3 surfaces
Primary: 11G35