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

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.

Categories: math.AG