Sir Peter Swinnerton-Dyer (University of Cambridge)
hpfs100@newton.cam.ac.uk
It is known that there are diagonal cubic surfaces defined over Q which do not contain rational points, although for each prime p they do contain p-adic points. The simplest example is
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Provided one assumes the finiteness of the Tate-Šafarevič groups of elliptic curves, I shall show that there is a sufficient condition for solubility which is only slightly stronger than the vanishing of the Brauer-Manin obstruction, and which can be expressed in very straightforward terms.