Vol. 7, No. 4, 2013

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Author Index
To Appear
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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