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Effective results for points on certain subvarieties of tori

Published online by Cambridge University Press:  01 July 2009

ATTILA BÉRCZES ,
KÁLMÁN GYŐRY ,
JAN-HENDRIK EVERTSE  and
CORENTIN PONTREAU
ATTILA BÉRCZES
Affiliation:
Institute of Mathematics, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary. e-mail: berczesa@math.klte.hu, gyory@math.klte.hu
KÁLMÁN GYŐRY
Affiliation:
Institute of Mathematics, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences and University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary. e-mail: berczesa@math.klte.hu, gyory@math.klte.hu
JAN-HENDRIK EVERTSE
Affiliation:
Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden, The Netherlands. e-mail: evertse@math.leidenuniv.nl
CORENTIN PONTREAU
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme CNRS UMR 6139, Université de Caen, 14032 Caen cedex, France. e-mail: pontreau@math.unicaen.fr
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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 , that with respect to the height are “very close” to a given subgroup Γ of finite rank of . 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 69 - 94
Copyright
Copyright © Cambridge Philosophical Society 2009

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