# Arithmetic purity of strong approximation for homogeneous spaces

## TL;DR

This paper establishes strong approximation properties for certain algebraic groups and homogeneous spaces over number fields, using fibrations over toric varieties and analyzing Brauer-Manin obstructions.

## Contribution

It proves strong approximation for open subsets of semi-simple simply connected groups and for certain compactifications, extending previous results with new techniques.

## Key findings

- Strong approximation holds for open subsets with codimension ≥ 2.
- Strong approximation with Brauer-Manin obstruction is established for specific compactifications.
- Counterexamples are provided for some semi-abelian varieties.

## Abstract

We prove that any open subset $U$ of a semi-simple simply connected quasi-split linear algebraic group $G$ with ${codim} (G\setminus U, G)\geq 2$ over a number field satisfies strong approximation by establishing a fibration of $G$ over a toric variety. We also prove a similar result of strong approximation with Brauer-Manin obstruction for a partial equivariant smooth compactification of a homogeneous space where all invertible functions are constant and the semi-simple part of the linear algebraic group is quasi-split. Some semi-abelian varieties of any given dimension where the complements of a rational point do not satisfy strong approximation with Brauer-Manin obstruction are given.