The circle method and diagonal cubic forms

We use the Hardy-Littlewood circle method, in the form developed by Heath-Brown in 1996, to investigate the number of integer zeros of diagonal cubic forms. The results are subject to unproved hypotheses concerning certain Hasse-Weil L-functions. For six variables we show that there are O(P3+epsilon) zeros up to height P, for any epsilon > 0. For four variables we show that there are O(P3/2+epsilon) such zeros, excluding any that lie on rational lines in the corresponding surface.