A New Method for Certain Diophantine Equations

1. Introduction. The original purpose of the research described in this article was to obtain results about Diophantine problems on rational surfaces — that is, surfaces defined over a field k which are birationally equivalent to P 2 over the algebraic closure ¯ k. (Throughout this article, K and k will always denote algebraic number fields, with respective rings of integers O and o. Except in the phrase 'rational surface' as defined above, 'rational' will always mean defined over k.) But as often happens, the research turns out to be also applicable to other problems: in this case, to certain K3 surfaces. This is significant, because Diophantine problems on K3 surfaces have hitherto been almost wholly intractable. Much of the research, which is still ongoing, is joint with one or both of Jean-Louis Colliot-Théì ene and Alexei Skorobogatov ; and I am grateful to both of them for their constructive comments.