Description of the strong approximation locus using Brauer-Manin obstruction for homogeneous spaces with commutative stabilizers

Abstract

For a homogeneous space $X$ over a number field $k$, the Brauer-Manin obstruction has been used to study strong approximation for $X$ away from a finite set $S$ of places, and known results state that $X(k)$ is dense in the omitting-$S$ projection of the Brauer-Manin set ${\mathrm{pr}}_S(X({\mathbb{A}}_k{)}^{\mathrm{br}})$, under certain assumptions. In order to completely understand the closure of $X(k)$ in the set of $S$-adelic points $X({\mathbb{A}}_k^S)$, we ask: (i) whether ${\mathrm{pr}}_S(X({\mathbb{A}}_k{)}^{\mathrm{br}})$ is closed in $X({\mathbb{A}}_k^S)$; (ii) whether $X(k)$ is dense in the closed subset of $X({\mathbb{A}}_k^S)$ cut out by elements in $\mathrm{br}X$ which induce zero evaluation maps at all the places in $S$. We also ask these questions considering only the algebraic Brauer group. We give answers to such questions for homogeneous spaces $X$ under semisimple simply connected groups with commutative stabilizers.

arXiv comment28 pages, comments welcome!