# Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups

Balestrieri F. 2023. Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups. Proceedings of the American Mathematical Society. 151(3), 907–914.

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Abstract

Let k be a number field and X a smooth, geometrically integral quasi-projective variety over k. For any linear algebraic group G over k and any G-torsor g : Z → X, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for all twists of Z by elements in H^1(k, G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for X. As an application, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent.

Publishing Year

Date Published

2023-01-01

Journal Title

Proceedings of the American Mathematical Society

Volume

151

Issue

3

Page

907-914

ISSN

eISSN

IST-REx-ID

### Cite this

Balestrieri F. Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups.

*Proceedings of the American Mathematical Society*. 2023;151(3):907-914. doi:10.1090/proc/15239Balestrieri, F. (2023). Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups.

*Proceedings of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/proc/15239Balestrieri, Francesca. “Some Remarks on Strong Approximation and Applications to Homogeneous Spaces of Linear Algebraic Groups.”

*Proceedings of the American Mathematical Society*. American Mathematical Society, 2023. https://doi.org/10.1090/proc/15239.F. Balestrieri, “Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups,”

*Proceedings of the American Mathematical Society*, vol. 151, no. 3. American Mathematical Society, pp. 907–914, 2023.Balestrieri, Francesca. “Some Remarks on Strong Approximation and Applications to Homogeneous Spaces of Linear Algebraic Groups.”

*Proceedings of the American Mathematical Society*, vol. 151, no. 3, American Mathematical Society, 2023, pp. 907–14, doi:10.1090/proc/15239.**All files available under the following license(s):**

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