Del Pezzo surfaces over finite fields and their Frobenius traces

Let S be a smooth cubic surface over a finite field F-q. It is known that #S(F-q) = 1 + aq + q(2) for some a is an element of{- 2,- 1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton-Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton-Dyer's tables on cubic surfaces over finite fields.