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

Corvaja, Pietro, and Zannier, Umberto. "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 015.5 (2013): 1927-1942. <http://eudml.org/doc/277285>.

@article{Corvaja2013,
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.},
author = {Corvaja, Pietro, Zannier, Umberto},
journal = {Journal of the European Mathematical Society},
keywords = {diophantine approximation; curves over finite fields; Vojta's conjecture; Diophantine approximation; curves over finite fields; Vojta's conjecture},
language = {eng},
number = {5},
pages = {1927-1942},
publisher = {European Mathematical Society Publishing House},
title = {Greatest common divisors of $u-1$, $v-1$ in positive characteristic and rational points on curves over finite fields},
url = {http://eudml.org/doc/277285},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Corvaja, Pietro
AU - Zannier, Umberto
TI - Greatest common divisors of $u-1$, $v-1$ in positive characteristic and rational points on curves over finite fields
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 5
SP - 1927
EP - 1942
AB - 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.
LA - eng
KW - diophantine approximation; curves over finite fields; Vojta's conjecture; Diophantine approximation; curves over finite fields; Vojta's conjecture
UR - http://eudml.org/doc/277285
ER -