In our previous work we proved a bound for the , for -units of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in . As pointed out by Silverman, the exact analogue does not work for function fields in positive characteristic. In the present work, we investigate possible extensions in that direction; it turns out that under suitable assumptions some of the results still hold. For instance we prove Theorems 2 and 3 below, from which we deduce in particular a new proof of Weil's bound for the number of rational points on a curve over finite fields. When the genus of the curve is large compared to the characteristic, we can even go beyond it. What seems a new feature is the analogy with the characteristic zero case, which admitted applications to apparently distant problems.
Cite this article
Umberto Zannier, Pietro Corvaja, Greatest common divisors of in positive characteristic and rational points on curves over finite fields. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1927–1942