Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains

Hu, Yong

doi:10.2140/ant.2013.7.1919

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Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains

Yong Hu

Vol. 7 (2013), No. 8, 1919–1952

DOI: 10.2140/ant.2013.7.1919

Abstract

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 $\geq 9$ 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 $K_{v}$ 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.

Keywords

quadratic forms, division algebras, local-global principle

Mathematical Subject Classification 2010

Primary: 11E04

Secondary: 16K99

Milestones

Received: 31 May 2012

Revised: 9 September 2012

Accepted: 15 October 2012

Published: 24 November 2013

Authors

Yong Hu
Université Paris-Sud 11 15 rue Georges Clemenceau Mathématiques Bâtiment 425 91405 Orsay Cedex France

Vol. 7, No. 8, 2013