Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups

Abstract

Let k k be a number field and X X a smooth, geometrically integral quasi-projective variety over k k . For any linear algebraic group G G over k k and any G G -torsor g : Z → X g: Z \to X , we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S S for all twists of Z Z by elements in H^1_{\text {\'{e}t}}(k,G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S S for X X . As an application, we show that any homogeneous space of the form G / H G/H with G G a connected linear algebraic group over k k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G / H G/H with G G semisimple simply connected and H H finite, using the theory of torsors and descent.