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

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 $N_q(C)\le (d-2)q+1$. Our result improves the previous bound $N_q(C)\le (d-1)q+1$ 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.