Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups
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- by Francesca Balestrieri
- Proc. Amer. Math. Soc. 151 (2023), 907-914
- DOI: https://doi.org/10.1090/proc/15239
- Published electronically: December 15, 2022
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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 $g: Z \to X$, 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 $H^1_{\text {\'{e}t}}(k,G)$, 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.References
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Bibliographic Information
- Francesca Balestrieri
- Affiliation: IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
- MR Author ID: 1161310
- Email: fbalestr@ist.ac.at
- Received by editor(s): April 10, 2020
- Received by editor(s) in revised form: June 16, 2020
- Published electronically: December 15, 2022
- Additional Notes: This project received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska Curie grant agreement 840684.
- Communicated by: Rachel Pries
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 907-914
- MSC (2020): Primary 14G12; Secondary 11G35, 14M17, 20G10
- DOI: https://doi.org/10.1090/proc/15239
- MathSciNet review: 4531627