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

Abstract:

Let $k$ be a number field and $X$ a smooth, geometrically integral quasi-projective variety over $k$. For any linear algebraic group $G$ over $k$ and any $G$-torsor $g: Z \to X$, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places $S$ for all twists of $Z$ 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 $S$ for $X$. As an application, we show that any homogeneous space of the form $G/H$ with $G$ a connected linear algebraic group over $k$ satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when $k$ is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form $G/H$ with $G$ semisimple simply connected and $H$ finite, using the theory of torsors and descent.