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

Abstract

Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a numberfield 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 L2 (G (K) \ G (A)) which we obtain with a rate of convergence. © 2008 Société Mathématique de France. Tous droits réservés.