Numbers of points of surfaces in the projective $3$-space over finite fields

TL;DR

This paper classifies all algebraic surfaces in projective 3-space over finite fields that reach a known upper bound on the number of rational points, identifying specific types such as hyperbolic, Hermitian, and minimal degree surfaces.

Contribution

It provides a complete classification of surfaces attaining the elementary bound on the number of points over finite fields in projective 3-space.

Findings

Hyperbolic surface attains the bound

Hermitian surface attains the bound

Surface of minimum degree contains all ${f F}_q$-points

Abstract

In the previous paper, we established an elementary bound for numbers of points of surfaces in the projective $3$-space over ${\Bbb F}_q$. In this paper, we give the complete list of surfaces that attain the elementary bound. Precisely those surfaces are the hyperbolic surface, the nonsingular Hermitian surface, and the surface of minimum degree containing all ${\Bbb F}_q$-points of the $3$-space.