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.