Cubic surfaces over finite fields
Peter Swinnerton‐Dyer · Mathematical Proceedings of the Cambridge Philosophical Society · 2010
Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions: