Del Pezzo surfaces over finite fields and their Frobenius traces

Research output: Contribution to journalArticle

  • Authors:
  • Barinder Banwait
  • Francesc Fité
  • Daniel Loughran

Abstract

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.

Bibliographical metadata

Original languageEnglish
JournalMathematical Proceedings of the Cambridge Philosophical Society
Early online date10 Apr 2018
DOIs
Publication statusPublished - 2018