Manin's and Peyre's conjectures on rational points and adelic mixing

[Conjectures de Manin et de Peyre sur des points rationnels et mélange adélique]

Gorodnik, Alex ; Maucourant, François ; Oh, Hee

Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008), p. 385-437 / Harvested from Numdam

Résumé

Soit $X$ la compactification merveilleuse d’un groupe semi-simple $\mathbf {G}$, connexe, de type adjoint, algébrique défini sur un corps de nombre $K$. Nous démontrons l’asymptotique conjecturée par Manin du nombre de points $K$-rationnels sur $X$ de hauteur plus petite que $T$, lorsque $T\rightarrow +\infty $, et construisons de manière explicite une mesure sur $X(\mathbb {A})$, généralisant celle de Peyre, qui décrit la répartition asymptotique des points rationnels $\mathbf {G}(K)$ sur $X(\mathbb {A})$. Ce travail repose sur la propriété de mélange de $L^2(\mathbf {G}(K)\backslash \mathbf {G}(\mathbb {A}))$, qui est démontrée avec une estimée de vitesse.

Abstract

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$. We prove Manin’s conjecture on the asymptotic (as $T\rightarrow \infty $) of the number of $K$-rational points of $X$ of height less than $T$, and give an explicit construction of a measure on $X(\mathbb {A})$, generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $\mathbf {G}(K)$ on $X(\mathbb {A})$. Our approach is based on the mixing property of ${L}^2(\mathbf {G}(K)\backslash \mathbf {G}(\mathbb {A}))$ which we obtain with a rate of convergence.

Détail

Publié le : 2008-01-01

MR 2482443 | Zbl 1161.14015 | 2 citations dans Numdam

DOI : https://doi.org/10.24033/asens.2071

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