Algebra & Number Theory

Density of rational points on certain surfaces

H. Peter Swinnerton-Dyer

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 $P0$ 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 $Vc$ be

then $Vc(ℚ)$ is dense in $Vc(ℚ2)$ for $c=2,4,8$.

Article information

Source
Algebra Number Theory, Volume 7, Number 4 (2013), 835-851.

Dates
Revised: 1 October 2012
Accepted: 10 December 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729983

Digital Object Identifier
doi:10.2140/ant.2013.7.835

Mathematical Reviews number (MathSciNet)
MR3095228

Zentralblatt MATH identifier
1277.11070

Subjects

Keywords
rational points K3 surfaces

Citation

Swinnerton-Dyer, H. Peter. Density of rational points on certain surfaces. Algebra Number Theory 7 (2013), no. 4, 835--851. doi:10.2140/ant.2013.7.835. https://projecteuclid.org/euclid.ant/1513729983

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