# Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields

### Umberto Zannier

Scuola Normale Superiore, Pisa, Italy### Pietro Corvaja

Università di Udine, Italy

## Abstract

In our previous work we proved a bound for the $\gcd(u-1,v-1)$, for $S$-units $u,v$ of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in [3]. 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 $u-1, v-1$ in positive characteristic and rational points on curves over finite fields. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1927–1942

DOI 10.4171/JEMS/409