# Hauteurs pour les sous-schémas et exemples d'utilisation de méthodes arakeloviennes en théorie de l'approximation diophantienne

Abstract : In this thesis we define and study some notions in the context of Arakelov geometry that have an intrinsic interest and should find applications in diophantine approximation theory.

Most of the text is devoted to the elaboration of a theory of heights for subschemes and to the proof of Hilbert-Samuel formulae'' for these heights. For two important classes of subschemes (integral subschemes and smooth with multiplicities'' subschemes) we show that the height of the subscheme relative to a high power of a positive line bundle is asymptotically determined by the height of the associated cycle. The proof essentially uses the arithmetic Hilbert-Samuel theorem'' of Gillet and Soulé, and reduces to it using techniques from hermitian analytic geometry. Then we give a finer analysis of the asymptotic expansion of heights of certain particular subschemes. Notably, in the case of relative dimension zero, we express the constant term of the asymptotic expansion by means of the ramification of the subscheme, which solves a question of Michel Laurent concerning heights of interpolation matrices.

Finally, as an independant part, we show some applications of arakelovian methods to diophantine approximation problems. In particular, we give a new proof of a classical criterion for algebraic independance. The originality of this proof is that it does not use any elimination theory but only arguments from arithmetic intersection theory.
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https://tel.archives-ouvertes.fr/tel-00359859
Contributor : Hugues Randriambololona Connect in order to contact the contributor
Submitted on : Monday, February 9, 2009 - 3:22:47 PM
Last modification on : Friday, October 7, 2022 - 3:49:07 AM
Long-term archiving on: : Tuesday, June 8, 2010 - 7:17:03 PM

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• HAL Id : tel-00359859, version 1

### Citation

Hugues Randriambololona. Hauteurs pour les sous-schémas et exemples d'utilisation de méthodes arakeloviennes en théorie de l'approximation diophantienne. Mathématiques [math]. Université Paris Sud - Paris XI, 2002. Français. ⟨tel-00359859⟩

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