Rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields

TL;DR

This paper improves bounds on the number of rational points on smooth surfaces in projective three-space over finite fields and analyzes families of surfaces that reach or nearly reach these bounds, providing exact point counts.

Contribution

It refines existing bounds on rational points and explicitly computes point counts for extremal surface families, offering new insights and potential applications.

Findings

Improved upper bounds on rational points on smooth surfaces in $\\mathbb{P}^3$ over finite fields.

Explicit point counts for families of surfaces near the bounds.

Potentially useful computations with independent interest.

Abstract

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact number of rational points. These computations may have independent interest.