Curves of Every Genus with Many Points, I: Abelian and Toric Families,

Let Nq(g) denote the maximal number of q-rational points on any curve of genus g over q. Ihara (for square q) and Serre (for general q) proved that lim supg → ∞Nq(g)/g > 0 for any fixed q. Here we prove limg → ∞Nq(g) = ∞. More precisely, we use abelian covers of 1 to prove lim infg → ∞Nq(g)/(g/log g) > 0, and we use curves on toric surfaces to prove lim infg → ∞Nq(g)/g1/3 > 0; we also show that these results are the best possible that can be proved using these families of curves.