We provide an upper bound for the number of rational points on a nonisotrivial curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, and not on the Jacobian variety of the curve.

The first author wishes to thank the Institut de Mathématiques de Jussieu, Paris, for its support during his sabbatical. This stay had also the support of CNPq (Brazil), CNRS (France) and Paris Science Foundation. Both authors would like to thank Université Bordeaux 1, ANR-10-BLAN-0115 Hamot and ANR-10-JCJC-0107 Arivaf (France).