Arithmetic purity of strong approximation for homogeneous spaces

Cao, Yang; Liang, Yongqi; Xu, Fei

doi:10.1016/j.matpur.2019.02.018

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Arithmetic purity of strong approximation for homogeneous spaces

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Cao, Yang
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1016/j.matpur.2019.02.018
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arXiv:1701.07259.pdf
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Cao, Y., Liang, Y., & Xu, F. (2019). Arithmetic purity of strong approximation for homogeneous spaces. Journal de Mathématiques Pures et Appliquées, 132, 334-368. doi:10.1016/j.matpur.2019.02.018.

Cite as: https://hdl.handle.net/21.11116/0000-0005-6CB8-3

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.