Curves with a prescribed number of rational points (Q650831)

Let \(q\) be a prime power. Let \(g\) and \(N\) be non-negative integers. Here a curve refers to an algebraic curve over \(\mathbb{F}_q\) which is smooth, projective and absolutely irreducible. The author introduces the interesting concept \[ \mathcal{G}(q,N):= \{ g\mid\text{there exists a curve over \(\mathbb{F}_q\) of genus \(g\) having exactly \(N\) rational points} \}. \] The main result says: For all \(q\) and \(N\), the set \(\mathbb{N} \setminus \mathcal{G}(q,N)\) is finite. He also comments that determining \(\mathcal{G}(q,N)\) seems to be impossible in general. He determines \(\mathcal{G}(q,N)\) for some specific values of \(q\) and \(N\) as well.