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- W2000693464 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." @default.
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- W2000693464 date "2013-01-01" @default.
- W2000693464 modified "2023-10-13" @default.
- W2000693464 title "Greatest Common Divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields" @default.
- W2000693464 doi "https://doi.org/10.4171/jems/409" @default.
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