Arithmetic purity of strong approximation for homogeneous spaces

Abstract

We prove that any open subset U of a semi-simple simply connected quasi-split linear algebraic group G with codim(G∖U,G)≥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.