Curves of every genus with many points. I: Abelian and toric families

Abstract

Let Nq(g) denote the maximal number of Fq-rational points on any curve of genus g over Fq. 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 ℙ1 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. © 2002 Elsevier Science (USA).