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]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 385-437.

Soit X la compactification merveilleuse d’un groupe semi-simple 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+, et construisons de manière explicite une mesure sur X(𝔸), généralisant celle de Peyre, qui décrit la répartition asymptotique des points rationnels 𝐆(K) sur X(𝔸). Ce travail repose sur la propriété de mélange de L 2 (𝐆(K)𝐆(𝔸)), qui est démontrée avec une estimée de vitesse.

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) of the number of K-rational points of X of height less than T, and give an explicit construction of a measure on X(𝔸), generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points 𝐆(K) on X(𝔸). Our approach is based on the mixing property of L 2 (𝐆(K)𝐆(𝔸)) which we obtain with a rate of convergence.

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     author = {Gorodnik, Alex and Maucourant, Fran\c{c}ois and Oh, Hee},
     title = {Manin's and {Peyre's} conjectures on rational points and adelic mixing},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {385--437},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {3},
     year = {2008},
     doi = {10.24033/asens.2071},
     mrnumber = {2482443},
     zbl = {1161.14015},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2071/}
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Gorodnik, Alex; Maucourant, François; Oh, Hee. Manin's and Peyre's conjectures on rational points and adelic mixing. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 385-437. doi : 10.24033/asens.2071. http://www.numdam.org/articles/10.24033/asens.2071/

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