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

## TL;DR

This paper investigates the closure properties of rational points on homogeneous spaces over number fields, using the Brauer-Manin obstruction, specifically focusing on spaces with commutative stabilizers and semisimple simply connected groups.

## Contribution

It provides new results on the closure of rational points in the adelic space for homogeneous spaces with commutative stabilizers, addressing questions about the Brauer-Manin set's topological properties.

## Key findings

- Proves the closedness of the Brauer-Manin projection in certain cases.
- Establishes density of rational points in specific subsets of adelic points.
- Clarifies the role of algebraic Brauer group in strong approximation.

## 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.