TY - MGZN
AU - Banwait, B.
AU - Fite, F.
AU - Loughran, D.
T2 - Mathematical proceedings of the Cambridge Philosophical Society
Y1 - 2019
VL - 167
IS - 1
SP - 35
EP - 60
DO - 10.1017/S0305004118000166
UR - https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/del-pezzo-surfaces-over-finite-fields-and-their-frobenius-traces/3D70794B5F27B0D655F34987CB2EB6F4
AB - Let S be a smooth cubic surface over a finite field q. It is known that #S( q) = 1 + aq + q2 for some a ¿ {-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.
TI - Del Pezzo surfaces over finite fields and their Frobenius traces
ER -