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Distribution asymptotique fine des points de hauteur bornée sur les variétés algébriques

Abstract : The study of the distribution of rational points on algebraic varieties is a classic subject of Diophantine geometry. The program proposed by V. Batyrev and Y. Manin in the 1990s gives a prediction on the order of growth whereas its later version due to E. Peyre conjectures the existence of a global distribution. In this thesis we propose a study of the local distribution of rational points of bounded height on algebraic manifolds. This aims at giving a description finer than the global one by counting the points closest to a fixed point. We set ourselves on the recent framework of the work of D. McKinnon and M. Roth who prefers that the geometry of the variety governs the Diophantine approximation on it and we take up the results of S. Pagelot. The expected order of growth and the existence of an asymptotic measure on some toric surfaces are demonstrated, while we demonstrate a totally different result for another surface on which there is no asymptotic measure and the best generic approximates are obtained on nodal rational curves. These two phenomena are of a radically different nature from the point of view of the Diophantine approximation.
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Submitted on : Friday, January 12, 2018 - 4:17:06 PM
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Zhizhong Huang. Distribution asymptotique fine des points de hauteur bornée sur les variétés algébriques. Géométrie algébrique [math.AG]. Université Grenoble Alpes, 2017. Français. ⟨NNT : 2017GREAM036⟩. ⟨tel-01682961⟩



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