Vol. 7, No. 8, 2013

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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
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 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.

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