Let be the fraction field of a two-dimensional, henselian, excellent local domain with finite residue field . When the characteristic of is not , we prove that every quadratic form of rank is isotropic over using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank with respect to discrete valuations of . The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field , 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 of prime order different from the characteristic of , if it is cyclic of degree over the completed field for every discrete valuation of , then the same holds over . This local-global principle for cyclicity is also established over function fields of -adic curves with the same method.
"Division algebras and quadratic forms over fraction fields of two-dimensional henselian domains." Algebra Number Theory 7 (8) 1919 - 1952, 2013. https://doi.org/10.2140/ant.2013.7.1919