In FrenchInternational audienceThe proof by Ullmo and Zhang of Bogomolov's conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for ``small points'' in the associated complex abelian variety. We study the analogous equidistribution property at $p$-adic places. Our results can be conveniently stated within the framework of the analytic spaces defined by Berkovich. The first one is valid in any dimension but is restricted to ``algebraic metrics'', the second one is valid for curves, but allows for more general metrics, in particular to the normalized heights with respect to dynamical systems
We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.Résumé. -Nousétablissons un développement asymptotique du nombre de points rationnels de hauteur bornée sur certaines compactificationséquivariantes du plan affine.
We establish asymptotic formulas for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures. Résumé. -Nous établissons un développement asymptotique du volume des boules de hauteur dans des variétés analytiques sur des corps locaux et sur les points adéliques de variétés algébriques sur des corps de nombres. Pour cela, nous relions les transformées de Mellin des fonctions hauteur à des intégrales de type Igusa et à des invariants géométriques globaux de la variété sous-jacente. Dans le cas adélique, nous construisons des mesures de Tamagawa dans un cadre général.
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