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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The étale Brauer-Manin obstruction to strong approximation on homogeneous spaces
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by Julian L. Demeio PDF
Trans. Amer. Math. Soc. 375 (2022), 8581-8634

Abstract:

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces $X$ under a linear group $G$ (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of $X$ are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilizers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients $G/H$, where $G$ and $H$ are connected algebraic groups and $H$ is linear, and certain abelianization morphisms associated with these spaces.
References
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Additional Information
  • Julian L. Demeio
  • Affiliation: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy; and Faculté des Sciences d’Orsay, Université Paris-Saclay, Bâtiment 307, rue Michel Magat, F-91405 Orsay Cedex, France
  • Address at time of publication: Max Planck Institute for Mathematics, Vivatgasse 7, 53111 Bonn, Germany
  • MR Author ID: 1378320
  • Email: julian.demeio@sns.it, demeio@mpim-bonn.mpg.de
  • Received by editor(s): August 25, 2020
  • Received by editor(s) in revised form: June 13, 2021, November 8, 2021, and February 2, 2022
  • Published electronically: September 28, 2022
  • Additional Notes: The final revisions of this manuscript happened while the author was a guest at the Max Planck Institute for Mathematics, which provided hospitality and financial support
  • © Copyright 2022 Julian L. Demeio
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8581-8634
  • MSC (2020): Primary 14G12; Secondary 14G05, 14M17
  • DOI: https://doi.org/10.1090/tran/8705