ON THE RATIONAL POINTS ON CUBIC SURFACES

Abstract

It is shewn that, if N(P) be the number of solutions of theindeterminate equation

ax^3+by^3+cz^3+dw^3 =0 \qquad (a,b,c,d \neq0)

for which \vert x \vert,\vert y\vert, \vert z\vert, \vert w\vert \leq P, then

N(P) = KP^2 + o(P^2),

where, to within a term O(P),KP^2 is the contribution to N(P) corresponding to therational lines in the projective surface defined by the equation. This proves a conjecture made byHeath-Brown, who has studied N(P) under the assumption of the Riemann Hypothesisfor certain Hasse-Weil L-functions. The remainder term o(P^2)in the formula represents O(P^{ {4 \over 3}+ϵ}),O(P^ {{5 \over 3}+ϵ}),or O(P^{2}/^3 \sqrt{ \log P}) according as the surface contains three, one, or norational lines.