Curves of every genus with many points, II: Asymptotically good families

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant cq with the following property: for every integer g ≥ 0, there is a genus-g curve over Fq with at least cqg rational points over Fq. Moreover, we show that there exists a positive constant d such that for every q we can choose cq = d log q. We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over Fq that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ) for some r > cg/n.