Noam D. Elkies, Everett W. Howe, Andrew Kresch, Bjorn Poonen, Joseph L. Wetherell, and Michael E. Zieve: Curves of every genus with many points, II: Asymptotically good families, Duke Math. J. 122 (2004) 399–422, MR 2005h:11123, Zbl 1072.11041.
(An official version and a preprint version are available online.)
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 positive constant c such that for every q and every positive integer n, 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)r for some r>cg/n.