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Bogomolov on tori revisited
"... Let V ⊆ Gnm ⊆ Pn be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ> 0 let V (θ) be the set of α ∈ V (Q) of Weil’s height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang), ..."
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Let V ⊆ Gnm ⊆ Pn be a geometrically irreducible variety which is not torsion (i. e. not a translate of a subtorus by a torsion point). For θ> 0 let V (θ) be the set of α ∈ V (Q) of Weil’s height h(α) ≤ θ. By the toric case of Bogomolov conjecture (which is a theorem of Zhang),
Effective results for points on certain subvarieties of tori
, 2008
"... The combined conjecture of LangBogomolov 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, t ..."
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The combined conjecture of LangBogomolov 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 LangBogomolov 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 Bakertype logarithmic forms estimates and Bogomolovtype estimates for the number of points on the variety X with very small height. 1.