Frobenius actions on Del Pezzo surfaces of degree 2

TL;DR

This paper counts degree 2 Del Pezzo surfaces over finite fields with specific Frobenius actions, addressing the inverse Galois problem and deriving results on point counts and related classifications.

Contribution

It explicitly determines the number of such surfaces with given Frobenius actions over finite fields, advancing understanding of their Galois representations.

Findings

Count of Del Pezzo surfaces with specified Frobenius action

Number of points on these surfaces over finite fields

Recovery of known classification results

Abstract

We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e. we solve the "quantitative inverse Galois problem". As applications we determine the number of Del Pezzo surfaces of degree 2 with a given number of points and recover results of Banwait-Fit\'e-Loughran and Loughran-Trepalin.