# Minoration de la hauteur de Néron-Tate pour les points et les sous-variétés : variations sur le problème de Lehmer

Abstract : This thesis is dedicated to the problems of lower bound for the normalised height of points and subvarieties. In the chapter 2, we prove a result of density of small points. This enables us to obtain for the height of subvarieties of Abelian varieties of C.M. type, a lower bound, optimal up to a power of a log'' of the degree. We prove in full generality that a good lower bound'' for the height of points implies the analogous lower bound for the subvarieties. This enables us in particular to prove that, on abelian varieties, the Lehmer's problem for points is equivalent to the Lehmer's problem for all the subvarieties. The chapter 3 is an amelioration in the case of hypersurfaces. The proof, introducing an auxiliary function, follows the classical scheme of transcendance's proofs. Using the slopes inequality of Bost, we reobtain then a celebrated result of Dobrowolski in the chapter 4. The chapter 5 extends a result of Amoroso and Zannier to the case of C.M. elliptic curves: we obtain of Lehmer's type lower bound, but with the degree of the extension generated by P over K replaced by the extension generated by P over the abelian extension of K. This enables us to simplify the proof of a result of Viada. Finally we explain the relations between different conjectures concerning the Lehmer's problem on Abelian varieties.
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Theses
Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2004. French
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https://tel.archives-ouvertes.fr/tel-00006163
Contributor : Nicolas Ratazzi <>
Submitted on : Friday, May 28, 2004 - 11:19:43 AM
Last modification on : Friday, May 28, 2004 - 11:19:43 AM

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• HAL Id : tel-00006163, version 1

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Nicolas Ratazzi. Minoration de la hauteur de Néron-Tate pour les points et les sous-variétés : variations sur le problème de Lehmer. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2004. French. <tel-00006163>

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