We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
Let $pi_1(C)$ be the algebraic fundamental group of a smooth connected affine curve, defined over an algebraically closed field of characteristic $p>0$ of countable cardinality. Let $N$ be a normal (resp. characteristic) subgroup of $pi_1(C)$. Under
the hypothesis that the quotient $pi_1(C)/N$ admits an infinitely generated Sylow $p$-subgroup, we prove that $N$ is indeed isomorphic to a normal (resp. characteristic) subgroup of a free profinite group of countable cardinality. As a consequence, every proper open subgroup of $N$ is a free profinite group of countable cardinality.
Let $k$ be an algebraically closed field of characteristic $p>0$ and let $C/k$ be a smooth connected affine curve. Denote by $pi_1(C)$ its algebraic fundamental group. The goal of this paper is to characterize a certain subset of closed normal subgro
ups $N$ of $pi_1(C)$. In Normal subgroups of fundamental groups of affine curves in positive characteristic we proved the same result under the additional hypothesis that $k$ had countable cardinality.
