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2013 Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains
Yong Hu
Algebra Number Theory 7(8): 1919-1952 (2013). DOI: 10.2140/ant.2013.7.1919


Let K be the fraction field of a two-dimensional, henselian, excellent local domain with finite residue field k. When the characteristic of k is not 2, we prove that every quadratic form of rank 9 is isotropic over K using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank 5 with respect to discrete valuations of K. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field K, following Saltman’s methods. A key step is the proof of the following result, which answers a question of Colliot-Thélène, Ojanguren and Parimala: for a Brauer class over K of prime order q different from the characteristic of k, if it is cyclic of degree q over the completed field Kv for every discrete valuation v of K, then the same holds over K. This local-global principle for cyclicity is also established over function fields of p-adic curves with the same method.


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Yong Hu. "Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains." Algebra Number Theory 7 (8) 1919 - 1952, 2013.


Received: 31 May 2012; Revised: 9 September 2012; Accepted: 15 October 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1294.11035
MathSciNet: MR3134039
Digital Object Identifier: 10.2140/ant.2013.7.1919

Primary: 11E04
Secondary: 16K99

Keywords: division algebras , local-global principle , Quadratic forms

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.7 • No. 8 • 2013
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