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

(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  cq  with the following property: for every integer  g≥0,  there is a genus-g  curve over  Fq  with at least  cq g  rational points over  Fq.  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  Fq  that has at least  (c/n)g  rational points and whose Jacobian contains a subgroup of rational points isomorphic to  (Z/nZ)r  for some  r > (c/n)g.

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  Fq  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  BC  over  Fq  such that  B  has genus  h.  Then either  B  or its quadratic twist will have at least as many  Fq-rational points as does  C.

We produce the degree-2 covers by showing that, for any genus-g  curve  C  over  Fq,  there are degree-2 covers  BC  over  Fq  in which the genus of  B  is any prescribed integer greater than  4g.  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  Fq  having at least  g(q½-1+o(1))/3  rational points. Conversely, Drinfeld and Vladut have shown that any genus-g  curve over  Fq  has at most  g(q½-1+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(Fq)/g  where  C  varies over the genus-g  curves over  Fq.

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  X0(N)  achieves equality in the Drinfeld–Vladut bound over  Fp2  for any prime  p  coprime to  N.  Since the genus of  X0(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  X0(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  X0(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.

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