Algebra & Number Theory
- Algebra Number Theory
- Volume 7, Number 4 (2013), 835-851.
Density of rational points on certain surfaces
Let be a nonsingular projective surface defined over and having at least two elliptic fibrations defined over ; the most interesting case, though not the only one, is when is a K3 surface with these properties. We also assume that is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of and to study the closure of under the real and the -adic topologies. The first object is achieved by the following theorem:
Let be a nonsingular surface defined over and having at least two distinct elliptic fibrations. There is an explicitly computable Zariski closed proper subset of defined over such that if there is a point of not in then is Zariski dense in .
The methods employed to study the closure of in the real or -adic topology demand an almost complete knowledge of ; a typical example of what they can achieve is as follows. Let be
then is dense in for .
Algebra Number Theory, Volume 7, Number 4 (2013), 835-851.
Received: 16 December 2010
Revised: 1 October 2012
Accepted: 10 December 2012
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11G35: Varieties over global fields [See also 14G25]
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