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Autour du problème de Lehmer relatif dans un tore

Abstract : Lehmer's problem consists in finding lower bounds for the Weil height of an algebraic number in terms of its degree over Q. Even if there is still no answer to Lehmer's original question, the sharpest corresponding conjecture has been proved up to an epsilon. Besides, there are several generalizations of this problem. On one hand, one can formulate the same kind of conjecture replacing the field of rationals by an abelian extension of a number field. On the other hand, one can generalize these statements in higher dimension. The point is to find lower bounds for the normalized height of a point or a subvariety of a torus; in this case, we substitute to the degree a more precise invariant: the obstruction index. It is then natural to try to combine these two generalizations: this is the relative Lehmer problem in a torus.

In this thesis, we first consider the one dimensional relative Lehmer problem. We give a lower bound for the height of an algebraic number in terms of its degree over an abelian extension of a number field. This is an improvement of a theorem of Amoroso and Zannier, obtained with a technically simpler proof. Furthermore, we precise the dependence of the lower bound on the ground field. Then, we focus on the Lehmer relative problem in higher dimension and find a lower bound for the normalized height of a hypersurface in terms of its obstruction index over an abelian extension of Q. Finally, we obtain an analogous result for a point, provided there's a technical hypothesis satisfied. Thus we show the sharpest conjectures up to an epsilon.
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https://tel.archives-ouvertes.fr/tel-00259956
Contributor : Emmanuel Delsinne <>
Submitted on : Friday, February 29, 2008 - 7:25:36 PM
Last modification on : Monday, April 27, 2020 - 4:14:03 PM
Long-term archiving on: : Thursday, May 20, 2010 - 11:47:55 PM

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  • HAL Id : tel-00259956, version 1

Citation

Emmanuel Delsinne. Autour du problème de Lehmer relatif dans un tore. Mathématiques [math]. Université de Caen, 2007. Français. ⟨tel-00259956⟩

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