Density of rational points on certain surfaces

作者：Swinnerton Dyer Peter^*

来源：Algebra and Number Theory, 2013, 7(4): 835-851.

DOI：10.2140/ant.2013.7.835

• rational points
• K3 surfaces

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 P-0 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 V-c be
V-c : X-0(4) + cX(1)(4) = X-2(4) + C X-3(4) for c = 2, 4 or 8;
then V-c(Q) is dense in V-c(Q(2)) for c = 2, 4, 8.

• 出版日期2013