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 with at least cqg rational points over . 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 that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to for some r>cg/n.
Noam D. Elkies. Everett W. Howe. Andrew Kresch. Bjorn Poonen. Joseph L. Wetherell. Michael E. Zieve. "Curves of every genus with many points, II: Asymptotically good families." Duke Math. J. 122 (2) 399 - 422, 1 April 2004. https://doi.org/10.1215/S0012-7094-04-12224-9