We determine in this paper the distribution of the number of points on the covers of ℙ1(Fq) such that K(C) is a Galois extension and Gal(K(C)/K) is abelian when q is fixed and the genus, g, tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over Fq. In all cases, the distribution is given by a sum of q + 1 random variables.
- finite fields
- function fields