Elements of large order on varieties over prime finite fields

Mei-Chu Chang[1]; Bryce Kerr[2]; Igor E. Shparlinski[3]; Umberto Zannier[4]

  • [1] Department of Mathematics University of California Riverside, CA 92521, USA
  • [2] Department of Pure Mathematics University of New South Wales Sydney, NSW 2052, Australia
  • [3] Department of Computing Macquarie University Sydney, NSW 2109, Australia
  • [4] Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, Italy

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 3, page 579-593
  • ISSN: 1246-7405

Abstract

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Let 𝒱 be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C ( 𝒱 ) such that for almost all primes p for all but at most C ( 𝒱 ) points on the reduction of 𝒱 modulo p at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

How to cite

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Chang, Mei-Chu, et al. "Elements of large order on varieties over prime finite fields." Journal de Théorie des Nombres de Bordeaux 26.3 (2014): 579-593. <http://eudml.org/doc/275730>.

@article{Chang2014,
abstract = {Let $\{\mathcal\{V\}\}$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C(\{\mathcal\{V\}\})$ such that for almost all primes $p$ for all but at most $C(\{\mathcal\{V\}\})$ points on the reduction of $\{\mathcal\{V\}\}$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.},
affiliation = {Department of Mathematics University of California Riverside, CA 92521, USA; Department of Pure Mathematics University of New South Wales Sydney, NSW 2052, Australia; Department of Computing Macquarie University Sydney, NSW 2109, Australia; Scuola Normale Superiore Piazza dei Cavalieri, 7 56126 Pisa, Italy},
author = {Chang, Mei-Chu, Kerr, Bryce, Shparlinski, Igor E., Zannier, Umberto},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {12},
number = {3},
pages = {579-593},
publisher = {Société Arithmétique de Bordeaux},
title = {Elements of large order on varieties over prime finite fields},
url = {http://eudml.org/doc/275730},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Chang, Mei-Chu
AU - Kerr, Bryce
AU - Shparlinski, Igor E.
AU - Zannier, Umberto
TI - Elements of large order on varieties over prime finite fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/12//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 3
SP - 579
EP - 593
AB - Let ${\mathcal{V}}$ be a fixed algebraic variety defined by $m$ polynomials in $n$ variables with integer coefficients. We show that there exists a constant $C({\mathcal{V}})$ such that for almost all primes $p$ for all but at most $C({\mathcal{V}})$ points on the reduction of ${\mathcal{V}}$ modulo $p$ at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.
LA - eng
UR - http://eudml.org/doc/275730
ER -

References

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