Manin's and Peyre's conjectures on rational points and adelic mixing
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 41 (2008) no. 3, p. 385-437

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.

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.

@article{ASENS_2008_4_41_3_385_0,
     author = {Gorodnik, Alex and Maucourant, Fran\c cois 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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {3},
     year = {2008},
     pages = {385-437},
     doi = {10.24033/asens.2071},
     zbl = {1161.14015},
     mrnumber = {2482443},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2008_4_41_3_385_0}
}
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, Serie 4, Volume 41 (2008) no. 3, pp. 385-437. doi : 10.24033/asens.2071. http://www.numdam.org/item/ASENS_2008_4_41_3_385_0/

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