Approximation forte pour les variétés avec une action d’un groupe linéaire

Abstract

Let $G$ be a connected linear algebraic group over a number field $k$. Let $U{\hookrightarrow}X$ be a $G$-equivariant open embedding of a $G$-homogeneous space $U$ with connected stabilizers into a smooth $G$-variety $X$. We prove that $X$ satisfies strong approximation with Brauer–Manin condition off a set $S$ of places of $k$ under either of the following hypotheses:(i)$S$ is the set of archimedean places;(ii)$S$ is a non-empty finite set and $\bar{k}^{\times }=\bar{k}[X]^{\times }$.The proof builds upon the case $X=U$, which has been the object of several works.