The solubility of diagonal cubic surfaces

Abstract Let F be an algebraic number field not containing the primitive cube roots of unity, and let a 1 X 1 3 +a 2 X 2 3 =a 3 X 3 3 +a 4 X 4 3 be a diagonal cubic surface defined over F and everywhere locally soluble. Subject to the assumption that the Tate–Safarevic group of every relevent elliptic curve is finite, the paper shows that under a very weak additional condition the surface contains points defined over F. Some condition (the Brauer–Manin obstruction) is known to be necessary, but the condition imposed in the paper (which is local) is slightly stronger. More remarkable is the condition on F, which seems to be an artefact of the proof and not intrinsic to the problem.