Del Pezzo surfaces over finite fields and their Frobenius traces

Barinder Banwait, Francesc Fité, Daniel Loughran

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 wh…