Andrew Kresch, Joseph L. Wetherell, and Michael E. Zieve:
Curves of every genus with many points, I: Abelian and toric families,
J. Algebra 250 (2002), 353–370. MR 2003i:11084

(Both the published version and the arXiv version are available online.)

In the 1940's, Weil proved that the number  N  of  Fq-rational points on a genus-g  curve satisfies  q+1-2gq½ ≤ N ≤ q+1+2gq½.  When  g  is large compared to  q,  Weil's lower bound is negative, so it can be improved to the trivial bound  0≤N.  In the early 1980's, Manin and Ihara observed that in this situation one can also significantly improve Weil's upper bound. This observation, together with various applications, led many authors to search for large-genus curves with many points over small fields.

For fixed  q,  Weil's upper bound has the form  N < O(g).  Conversely, there are infinite sequences of curves over  Fq  having  O(g)  rational points. In 1983, Serre asked whether one could construct such a sequence including curves of every genus. This question is the focus of the present paper.

The simplest family of curves which attains every genus is the family of hyperelliptic curves; however, a hyperelliptic curve has at most  2q+2  points over  Fq,  so for large genus these curves have few points. For instance, any hyperelliptic curve over  F2  has at most six points. This bound can be improved to ten by allowing bielliptic curves, but before the present paper it was not known whether there are curves of every large genus which have more than ten  F2-rational points.

Serre gave many examples of curves with many points constructed as abelian covers of  P1.  However, Frey, Perret, and Stichtenoth showed that, for fixed  q, , the number of  Fq-rational points on any such curve is at most  O(g/log g). We prove that, conversely, there are abelian covers of  P1 (over Fq) having any prescribed genus  g  and having  O(g/log g) rational points.

We also analyze curves embedded in toric surfaces. We show that, for fixed  q,  any such curve has at most  O(g1/3)  rational points; and conversely, there exist such curves of every genus  g  having at least  O(g1/3)  rational points. Our proof of the upper bound uses a result of Arnol'd on the minimal area of a convex lattice n-gon.

Additional comment:  In subsequent work we answered Serre's question by constructing curves of every genus with  O(g)  rational points. However, the results of the present paper are still of interest for the following reason. We know very little about the behavior of the family of all high-genus curves over a small finite field, so in general it makes sense to study special families of curves, as we have done here.

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