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

We resolve a 1983 question of
Serre's 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 integer

The strategy of our proof is to begin with some sequence of curves with
many points that achieve `enough' genera, and then to fill in the missing
genera via degree-2 covers. More specifically, we start with a sequence
of curves over **F**_{q} that have
many points and whose genera grow at most exponentially. Then we show
that for every curve *C* in this sequence, and for every
integer *h* greater than some constant multiple of the
genus of *C*, there exists a degree-2 cover
*B*→*C* over **F**_{q}
such that *B* has genus *h*. Then either
*B* or its quadratic twist will have at least as many
**F**_{q}-rational points as does *C*.

We produce the degree-2 covers by showing that, for any genus-*g*
curve *C* over **F**_{q},
there are degree-2 covers *B*→*C* over
**F**_{q} in which the genus of *B*
is any prescribed integer greater than 4*g*. We produce the
initial sequence of curves via class field towers.

If *q*
is a square then we can prove better results by starting with a sequence of
Shimura curves:
we show that for every *g* there is
a genus-*g* curve over **F**_{q}
having at least *g*(*q*^{½}-1+*o*(1))/3
rational points. Conversely,
Drinfeld
and Vladut have shown that any genus-*g* curve over **F**_{q}
has at most *g*(*q*^{½}-1+*o*(1))
rational points. Thus, over any prescribed field of square order,
every large genus *g* behaves in roughly the same way as
every other, in terms of the maximum value of
#*C*(**F**_{q})/*g*
where *C* varies over the genus-*g* curves
over **F**_{q}.

It is known that, over any square field, there are infinitely many curves
achieving equality in the Drinfeld–Vladut bound (up to
the *o*(1) error term). The known curves with this property
are the
modular curves and
the analogous
Shimura and
Drinfeld
modular curves,
which have many rational points because all supersingular points are defined
over a small square field. In particular, the
classical modular curve
*X*_{0}(*N*) achieves equality in the
Drinfeld–Vladut bound over **F**_{p2}
for any prime *p* coprime to *N*.
Since the genus of *X*_{0}(*N*)
is *N*/12 plus lower-order terms, and on average the
genus is less than *N*, it is natural to ask whether
one can achieve equality in the Drinfeld–Vladut bound in every
large genus by means of modular curves. However, it was shown by
Csirik, Wetherell, and Zieve that
the values occurring as genera of *X*_{0}(*N*)
tend to occur in this way for many values *N*, so
a random integer has probability zero of occurring as the genus of any
curve *X*_{0}(*N*). In light of this, our
approach via double covers does not seem farfetched.

The results of this paper improve the results of our previous paper, in which we used different methods to construct curves of every genus with many points.

*Michael Zieve*:
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