Greatest common divisors of \(u-1, v-1\) in positive characteristic and rational points on curves over finite fields (Q363241)

Summary: 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 [the authors, Monatsh. Math. 144, No. 3, 203--224 (2005; Zbl 1086.11035)]. 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 (see \S4). 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.