Algebra & Number Theory

Density of rational points on certain surfaces

H. Peter Swinnerton-Dyer

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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.

Article information

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G35: Varieties over global fields [See also 14G25]

rational points K3 surfaces


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.

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