Effective results for points on certain subvarieties of tori

Attila Bérczes, K. Györy, Jan Hendrik Evertse, Corentin Pontreau

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G MN(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of M N(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

Original languageEnglish
Pages (from-to)69-94
Number of pages26
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume147
Issue number1
DOIs
Publication statusPublished - Jul 2009

Fingerprint

Torus
Finite Rank
Number field
Estimate
Set of points
Logarithmic
Subgroup
Upper bound
Form
Language
Class

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Effective results for points on certain subvarieties of tori. / Bérczes, Attila; Györy, K.; Evertse, Jan Hendrik; Pontreau, Corentin.

In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 147, No. 1, 07.2009, p. 69-94.

Research output: Contribution to journalArticle

Bérczes, Attila ; Györy, K. ; Evertse, Jan Hendrik ; Pontreau, Corentin. / Effective results for points on certain subvarieties of tori. In: Mathematical Proceedings of the Cambridge Philosophical Society. 2009 ; Vol. 147, No. 1. pp. 69-94.
@article{e3c252470b6247d09caee54f0ff58ee6,
title = "Effective results for points on certain subvarieties of tori",
abstract = "The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G MN(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of M N(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.",
author = "Attila B{\'e}rczes and K. Gy{\"o}ry and Evertse, {Jan Hendrik} and Corentin Pontreau",
year = "2009",
month = "7",
doi = "10.1017/S030500410900231X",
language = "English",
volume = "147",
pages = "69--94",
journal = "Mathematical Proceedings of the Cambridge Philosophical Society",
issn = "0305-0041",
publisher = "Cambridge University Press",
number = "1",

}

TY - JOUR

T1 - Effective results for points on certain subvarieties of tori

AU - Bérczes, Attila

AU - Györy, K.

AU - Evertse, Jan Hendrik

AU - Pontreau, Corentin

PY - 2009/7

Y1 - 2009/7

N2 - The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G MN(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of M N(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

AB - The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G MN(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of M N(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

UR - http://www.scopus.com/inward/record.url?scp=68349135372&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68349135372&partnerID=8YFLogxK

U2 - 10.1017/S030500410900231X

DO - 10.1017/S030500410900231X

M3 - Article

VL - 147

SP - 69

EP - 94

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -