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 *c*_{q} with the
following property: for every non-negative integer *g*,
there is a genus-*g* curve over **F**_{q}
with at least *c*_{q} g rational points over
**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 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 **F**_{q}
that has at least *cg/n* rational points and whose
Jacobian contains a subgroup of rational points isomorphic
to (**Z**/*n***Z**)^{r} for some
*r>cg/n*.