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

Pietro Corvaja; Umberto Zannier

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 5, page 1927-1942
- ISSN: 1435-9855

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topCorvaja, 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 -

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