Density of rational points on certain surfaces

Swinnerton-Dyer, H. Peter

doi:10.2140/ant.2013.7.835

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Density of rational points on certain surfaces

Sir Peter Swinnerton-Dyer

Vol. 7 (2013), No. 4, 835–851

DOI: 10.2140/ant.2013.7.835

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 $V (ℚ)$ is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of $V (ℚ)$ and to study the closure of $V (ℚ)$ 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 $V (ℚ)$ not in $X$ then $V (ℚ)$ is Zariski dense in $V$ .

The methods employed to study the closure of $V (ℚ)$ 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} f o r c = 2, 4 or 8;

then $V_{c} (ℚ)$ is dense in $V_{c} (ℚ_{2})$ for $c = 2, 4, 8$ .

Keywords

rational points, K3 surfaces

Mathematical Subject Classification 2010

Primary: 11G35

Milestones

Received: 16 December 2010

Revised: 1 October 2012

Accepted: 10 December 2012

Published: 29 August 2013

Authors

Sir Peter Swinnerton-Dyer
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge CB3 0WB United Kingdom

Vol. 7, No. 4, 2013