Points algébriques de hauteur bornée

Abstract : The study of the distribution of rational or algebraic points of an algebraic variety according to their height is a classic problem in Diophantine geometry. In this thesis, we will be interested in the asymptotic cardinality of the set of algebraic points of fixed degree and bounded height of a smooth Fano variety defined over a number field, when the bound on the height tends to infinity. In particular, we show that this can be connected to the Batyrev-Manin-Peyre conjecture, i.e. the case of rational points, on some ponctual Hilbert scheme. We thus deduce the distribution of algebraic points of fixed degree on a rational curve. When the variety is a smooth Fano surface, our study shows that the associated Hilbert schemes provide, under certain conditions, new counterexamples to the Batyrev-Manin-Peyre conjecture. However, in two cases detailed in this thesis, the associated Hilbert schemes satisfie a slightly weaker version of the Batyrev-Manin-Peyre conjecture.
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Submitted on : Friday, November 14, 2014 - 10:31:44 AM
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  • HAL Id : tel-01082702, version 1

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Cécile Le Rudulier. Points algébriques de hauteur bornée. Géométrie algébrique [math.AG]. Université Rennes 1, 2014. Français. ⟨NNT : 2014REN1S073⟩. ⟨tel-01082702⟩

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