Abstract

The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety X of GNm(Q) = (Q)N, that with respect to the height are “very close ” to a given subgroup of finite rank of GNm(Q). 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 X, 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 X with very small height. 1.