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

Abstract

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 c_q with the following property: for every integer g≥0, there is a genus-g curve over ${\mathbb{F}}_q$ with at least c_qg rational points over ${\mathbb{F}}_q$. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q=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 ${\mathbb{F}}_q$ that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to ${(\mathbb{Z}/n\mathbb{Z})}^r$ for some r>cg/n.