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

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