Division Algebras and Quadratic Forms over Fraction Fields of Two-dimensional Henselian Domains

TL;DR

This paper proves new local-global principles for quadratic forms and division algebras over the fraction fields of 2-dimensional henselian domains, extending understanding of isotropy and cyclicity in algebraic structures.

Contribution

It establishes a local-global principle for cyclicity of Brauer classes over such fields, answering a question of Colliot-Thélène–Ojanguren–Parimala, and applies these results to quadratic forms.

Findings

Quadratic forms of rank ≥ 9 are isotropic over the field K.

A local-global principle for isotropy of rank 5 quadratic forms is proved.

A key result shows Brauer classes of prime order are cyclic if they are cyclic over completions.

Abstract

Let $K$ be the fraction field of a 2-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 $\ge 9$ 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-Th\'el\`ene--Ojanguren--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.