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

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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 non-negative integer g, there is a genus-g curve over Fq with at least cq * g 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 c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)ʳ for some r > c*g/n.