COUNTING POINTS ON CUBIC SURFACES II Sir

Let V be a non-singular surface defined over Q which is embedded in projective space P by means of anticanonical divisors, and let U be the open subset of V obtained by deleting the lines on V . For any point P in U(Q) denote by h(P ) the height of P . In this paper h will usually be the standard height h1(P ) = max(|x0|, . . . , |xn|) where P = (x0, . . . , xn) for integers xi with highest common factor 1; but this choice is in no way canonical, so that in a thorough investigation we should consider other heights also. Indeed, the eventual theory will almost certainly need the replacement of h by a generalized height — a concept which I am not able to make precise. Recall that for abelian varieties the canonical height function is actually a generalized height; and this is the only case in which a canonical height function is known with certainty, though the heights which have been defined for toric varieties and for certain toric bundles are probably also canonical. For any integer b > 0 write