# Minoration de la hauteur normalisée en petite codimension

Abstract : The starting point of this thesis is the study of Lehmer's problem in dimension greater than two. It aims at finding, in the more general context of the multiplicative group $G_m^n$, lower bounds for the height of small dimension subvarieties, or rather small codimension ones.

First we resume some more or less known results about algebraic subgroups and the behavior of algebraic subvarieties under multiplication in $G_m^n$. Afterwards, we give arithmetic and geometric lower bounds for the height of subvarieties of codimension 1 and 2 in $G_m^2$ and $G_m^3$ respectively. Unlike what it can be done in previous works of F. Amoroso et S. David, for subvarieties of codimension different from 1, we do not use, to conclude our proofs, a final descent argument, but a new geometric one. This considerably simplifies the process, and leads to quantitative improvements in the considered cases.

Finally, we study the distribution of small points of a subvariety. For some geometrically irreducible surface $V$ of $G_m^3$, we show that outside a finite number of exceptional translated subtori included in $V$, of which we give an upper bound for the sum of their degrees, all the points have height bounded by below by a quasi optimal quantity $\epsilon(V)>0$, essentially linear in the inverse of the degree of $V$, which is not proved in the general case.
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Cited literature [15 references]

https://tel.archives-ouvertes.fr/tel-00011840
Contributor : Corentin Pontreau <>
Submitted on : Wednesday, March 8, 2006 - 5:27:39 PM
Last modification on : Monday, April 27, 2020 - 4:14:03 PM
Long-term archiving on: : Saturday, April 3, 2010 - 10:45:44 PM

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• HAL Id : tel-00011840, version 1

### Citation

Corentin Pontreau. Minoration de la hauteur normalisée en petite codimension. Mathématiques [math]. Université de Caen, 2005. Français. ⟨tel-00011840⟩

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