Effective results for points on certain subvarieties of tori

Abstract

AbstractThe combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those pointsxof a given subvarietyof$\mathbb{G}_{\bf m}^N(\oQ )=(\oQ^*)^N$, that with respect to the height are “very close” to a given subgroup Γ of finite rank of$\mathbb{G}_{\bf m}^N(\oQ)$. 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 pointsxunder 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 varietywith very small height.