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Spinor groups with good reduction

Part of: Forms and linear algebraic groups Algebraic number theory: global fields

Published online by Cambridge University Press:  07 March 2019

Vladimir I. Chernousov ,
Andrei S. Rapinchuk  and
Igor A. Rapinchuk
Vladimir I. Chernousov
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada email vladimir@ualberta.ca
Andrei S. Rapinchuk
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA email asr3x@virginia.edu
Igor A. Rapinchuk
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA email rapinchu@msu.edu
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Abstract

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$. We show that for a given $n\geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\mathsf{G}_{2}$.

Keywords

algebraic groupsGalois cohomologygood reductionHasse principlesunramified cohomology

MSC classification

Primary: 11E72: Galois cohomology of linear algebraic groups
Secondary: 11R34: Galois cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 3 , March 2019 , pp. 484 - 527
Copyright
© The Authors 2019 

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Footnotes

The first author was supported by the Canada Research Chairs Program and by an NSERC research grant. The second author was partially supported by the Simons Foundation. During the preparation of the final version of the paper, he visited Princeton University and the Institute for Advanced Study, and would like to thankfully acknowledge the hospitality of both institutions. The third author was partially supported by an AMS-Simons Travel Grant. We are grateful to P. Gille and M. Rapoport for useful conversations. Last but not least, we would like to express our gratitude to all anonymous referees whose comments, insights and suggestions helped to improve the original paper significantly.

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