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

In the 1940's,
Weil
proved that the number *N* of
**F**_{q}-rational points on a genus-*g* curve
satisfies *q*+1-2*gq*^{½} ≤ *N* ≤ *q*+1+2*gq*^{½}.
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 **F**_{q}
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 2*q*+2 points over
**F**_{q}, so for large genus these curves have few points.
For instance, any hyperelliptic curve over **F**_{2}
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 **F**_{2}-rational points.

Serre gave many examples of curves with many points constructed as abelian covers
of **P**^{1}. However,
Frey, Perret, and Stichtenoth
showed that, for fixed *q*, , the number of **F**_{q}-rational
points on any such curve is at most *O*(*g*/log *g*).
We prove that, conversely, there are abelian covers of **P**^{1} (over
**F**_{q})
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*(*g*^{1/3})
rational points; and conversely, there exist such curves of every
genus *g* having at
least *O*(*g*^{1/3}) 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.

*Michael Zieve*:
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