Open Access
2013 Density of rational points on certain surfaces
H. Peter Swinnerton-Dyer
Algebra Number Theory 7(4): 835-851 (2013). DOI: 10.2140/ant.2013.7.835


Let V 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 V is a K3 surface with these properties. We also assume that V() is not empty. The object of this paper is to prove, under a weak hypothesis, the Zariski density of V() and to study the closure of V() 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 and having at least two distinct elliptic fibrations. There is an explicitly computable Zariski closed proper subset X of V defined over such that if there is a point P0 of V() not in X then V() is Zariski dense in V.

The methods employed to study the closure of V() 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

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 Vc() is dense in Vc(2) for c=2,4,8.


Download Citation

H. Peter Swinnerton-Dyer. "Density of rational points on certain surfaces." Algebra Number Theory 7 (4) 835 - 851, 2013.


Received: 16 December 2010; Revised: 1 October 2012; Accepted: 10 December 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1277.11070
MathSciNet: MR3095228
Digital Object Identifier: 10.2140/ant.2013.7.835

Primary: 11G35

Keywords: K3 surfaces , rational points

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 4 • 2013
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