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

摘要

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 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.