A bound for the number of points of space curves over finite fields

TL;DR

This paper establishes a new upper bound on the number of rational points of space curves over finite fields, improving previous bounds and proposing a conjecture that generalizes known results for plane curves.

Contribution

It provides a tighter bound for the number of points on space curves over finite fields and introduces a conjecture extending Sziklai's bound to higher-dimensional curves.

Findings

Proved that $N_q(C) \,\leq\, (d-2)q+1$ for space curves.

Improved upon the previous bound of $(d-1)q+1$ by Homma.

Formulated a conjecture generalizing Sziklai's bound for higher dimensions.

Abstract

For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves the previous bound $N_q(C) \leq (d-1)q+1$ obtained by Homma in 2012 and leads to a natural conjecture generalizing Sziklai's bound for the number of points of plane curves over finite fields.