Del Pezzo surfaces over finite fields and their Frobenius traces

TL;DR

This paper classifies possible Frobenius trace values for smooth cubic and other del Pezzo surfaces over finite fields, answering Serre's question and correcting historical tables.

Contribution

It provides a complete characterization of Frobenius trace values for del Pezzo surfaces over finite fields, extending previous partial results and addressing the inverse Galois problem.

Findings

Complete classification of Frobenius traces for cubic surfaces

Extension of results to other del Pezzo surfaces

Correction of classical tables on cubic surfaces

Abstract

Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. 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.