Manchester Geometry Seminar 1999/2000

Special session. First talk
Thursday, 4 May. Room 9.05, Mathematics Building, University of Manchester. 3 p.m.

Rational Points on Diagonal Cubic Surfaces

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

3X03+4X13+10X23+15X33 = 0.
It is natural to ask what condition additional to solubility in each p-adic field is needed to ensure solubility in Q; and there is strong numerical evidence that this condition is that there is no Brauer-Manin obstruction. However, one does not know how to exploit such a condition.

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.

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