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Sur quelques questions d'équidistribution en géométrie arithmétique

Abstract : We prove some equidistribution result on the modular curves: Galois orbits of modular invariants within a same non CM isogeny class distribute along the Poincaré measure on the modular curve. As a corollary, the height of the considered points diverges, recovering a result of Szpiro and Ullmo. To prove such a statement, we combine galois properties (Serre's theorem on the Galois action on division points) and ergodic properties (Ratner's theorem on unipotent flows, or rather the equidistribution of Hecke points). We generalise our method in the setup of Shimura varieties. But in the latter setup, on of our ingredients rely on some variant of Mumford-Tate conjecture. This brings us to study, in a second part, some refinements of the equidistribution of Hecke points. Some issues of non-divergences in lattices spaces has then to be dealt with. Dani-Margulis linearization tecnhique reduces this to some geometric statement. We provide an answer to this question. In the real case, it is a collaboration with Nimish Shah. In the p-adic case, we are lead to use the ultrametric geometry recently developped by Berkovich, together with Bruhat-Tits theory, and more particularly the recent work of B. Remy, A. Thuillier et A. Werner. We precisely prove - some decomposition properties in buildings inspired by Mostow decomposition theorem for symmetric spaces; - some convexity properties, on buildings, for analytic functions, in the ultramétric sense, on the associated group. We finally illustrate how our results, in combination with D. Kleinbock and G. Tomanov work, and Ratner's theorem, applies to the study of some S-arithmetic problems in lattices spaces.
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Contributor : Richard Rodolphe <>
Submitted on : Tuesday, January 12, 2010 - 4:44:01 PM
Last modification on : Thursday, October 29, 2020 - 3:01:40 PM
Long-term archiving on: : Wednesday, November 30, 2016 - 9:55:46 AM


  • HAL Id : tel-00438515, version 1


Rodolphe Richard. Sur quelques questions d'équidistribution en géométrie arithmétique. Mathématiques [math]. Université Rennes 1, 2009. Français. ⟨tel-00438515⟩



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