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 .
H. Peter Swinnerton-Dyer. "Density of rational points on certain surfaces." Algebra Number Theory 7 (4) 835 - 851, 2013. https://doi.org/10.2140/ant.2013.7.835