Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The Brauer group of cubic surfaces

Sir Peter Swinnerton-Dyera1

a1 Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection

S0305004100076106_eqnU1

is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic S0305004100076106_inline1.

(Received June 08 1992)

(Revised September 25 1992)