Vol. 7, No. 4, 2013

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Density of rational points on certain surfaces

Sir Peter Swinnerton-Dyer

Vol. 7 (2013), No. 4, 835–851

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 V c be

V c : X04 + cX 14 = X 24 + cX 34for c = 2,4  or 8;

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

rational points, K3 surfaces
Mathematical Subject Classification 2010
Primary: 11G35
Received: 16 December 2010
Revised: 1 October 2012
Accepted: 10 December 2012
Published: 29 August 2013
Sir Peter Swinnerton-Dyer
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
United Kingdom