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Patching and weak approximation in isometry groups

Authors: Eva Bayer-Fluckiger and Uriya A. First
Journal: Trans. Amer. Math. Soc. 369 (2017), 7999-8035
MSC (2010): Primary 11E39, 11E41, 16H10
Published electronically: May 11, 2017
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Abstract: Let $ R$ be a semilocal principal ideal domain. Two algebraic objects over $ R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of $ R$ and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over (not necessarily commutative) $ R$-orders is always a finite power of $ 2$, and under further assumptions, e.g., that the order is hereditary, this number is $ 1$. The same result is also shown for related objects, e.g., systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings.

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Additional Information

Eva Bayer-Fluckiger
Affiliation: École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

Uriya A. First
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada

Keywords: Quadratic form, hermitian form, algebraic patching, weak approximation, genus, order, hereditary order, sesquilinear form, hermitian category
Received by editor(s): April 7, 2015
Received by editor(s) in revised form: December 15, 2015
Published electronically: May 11, 2017
Additional Notes: The second-named author performed this research at EPFL, the Hebrew University of Jerusalem and the University of British Columbia (in this order), where he was supported by an SNFS grant #IZK0Z2_151061, an ERC grant #226135, and the UBC Mathematics Department, respectively
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