Density of rational points on certain surfaces

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

$$V_c:X_0^4+c{X}_1^4=X_2^4+c{X}_3^{4}forc=2,4\text{or }8;$$

then $V_c\left(\mathbb{Q}\right)$ is dense in $V_c\left({\mathbb{Q}}_2\right)$ for $c=2,4,8$.