Geodesic
Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields
Journal of the European Mathematical Society, Tome 15 (2013) no. 5, pp. 1927-1942

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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.
DOI : 10.4171/jems/409
Classification : 11-XX, 14-XX, 00-XX
Keywords: Diophantine approximation, curves over finite fields, Vojta's conjecture 
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     title = {Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields},
     journal = {Journal of the European Mathematical Society},
     pages = {1927--1942},
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Pietro Corvaja; Umberto Zannier. Greatest common divisors of $u-1, v-1$ in positive characteristic and rational points on curves over finite fields. Journal of the European Mathematical Society, Tome 15 (2013) no. 5, pp. 1927-1942. doi: 10.4171/jems/409

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