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17
Small points on subvarieties of a torus
 Journal
, 2009
"... Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. Especially, we determine whether such a set is or not dense in V. We the ..."
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Abstract. Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. Especially, we determine whether such a set is or not dense in V. We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees. As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor. 1.
Measures Of Simultaneous Approximation For QuasiPeriods Of Abelian Varieties
 J. Number Theory, 94 (2002), N
, 2002
"... this paper, the functions # i will be assumed to be normalized as above, i.e. so that all secondorder derivatives of # 0 vanish at 0. 3.4. Conclusion ..."
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this paper, the functions # i will be assumed to be normalized as above, i.e. so that all secondorder derivatives of # 0 vanish at 0. 3.4. Conclusion
Special points on fibered powers of elliptic surfaces
 J.Reine Angew. Math (Crelle
"... Abstract. Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the ManinMumford and AndréOort Conjecture and is related to a ..."
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Abstract. Consider a fibered power of an elliptic surface. We characterize its subvarieties that contain a Zariski dense set of points that are torsion points in fibers with complex multiplication. This result can be viewed as a mix of the ManinMumford and AndréOort Conjecture and is related to a conjecture of Pink
SLOPES AND ABELIAN SUBVARIETY THEOREM
"... Abstract. In this article we show how to modify the proof of the Abelian Subvariety Theorem by Bost ([3] theorem 5.1) in order to improve the bounds in a quantitative respect and to extend the theorem to subspaces instead of hyperplanes. Given an abelian variety A defined over a number field κ and a ..."
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Abstract. In this article we show how to modify the proof of the Abelian Subvariety Theorem by Bost ([3] theorem 5.1) in order to improve the bounds in a quantitative respect and to extend the theorem to subspaces instead of hyperplanes. Given an abelian variety A defined over a number field κ and a nontrivial period γ in a subspace W of tA K with K a finite extension of κ, the Abelian Subvariety Theorem shows the existence of an abelian subvariety B of A Q, whose degree is bounded in terms of the height of W, the norm of γ, the degree of κ and the degree and dimension of A. If A is principally polarized
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),
Seshadri Constants and Interpolation on Commutative Algebraic Groups
, 2012
"... In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the ..."
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In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.