Curves of every genus with a prescribed number of rational points

Abstract

A fundamental problem in the theory of curves over finite fields is to determine the sets M q (g):= {N ∈ ℕ / there is a curve over $$ \mathbb{F}_q $$ of genus g with exactly N rational points}. A complete description of M q (g) is out of reach. So far, mostly bounds for the numbers N q (g):= maxM q (g) have been studied. In particular, Elkies et al. proved that there is a constant γ q > 0 such that for any g ≥ 0 there is some N ∈ M q (g) with N ≥ γ q g. This implies that lim inf g→∞ N q (g)/g > 0, and solves a long-standing problem by Serre. We extend the result of Elkies et al. substantially and show that there are constants α q , β q > 0 such that for all g ≥ 0, the whole interval [0, α q g − β q ] ∩ ℕ is contained in M q (g).