Witt equivalence of function fields of curves over local fields

TL;DR

This paper investigates Witt equivalence of function fields of curves over local fields, extending previous global field results to the local case, and explores implications for the underlying local fields.

Contribution

It extends the theory of Witt equivalence to function fields over local fields and establishes conditions linking the equivalence of these fields to their base local fields.

Findings

Witt equivalence of function fields over local fields implies equivalence of the base local fields under certain conditions.

The paper generalizes previous global field results to local fields.

It provides new insights into the structure of quadratic forms over function fields.

Abstract

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global fields, as well as function fields of curves defined over archimedean local fields under Witt equivalence is well-understood. Numbers of classes of Witt equivalent fields with finite numbers of square classes are also known in some cases. Witt equivalence of general function fields over global fields was studied in the earlier work [13] by the authors, and applied to study Witt equivalence of function fields of curves over global fields. In this paper we extend these results to local case, i.e. we discuss Witt equivalence of function fields of curves over local fields. As an application, we show that, modulo some additional assumptions, Witt equivalence of two such function fields implies Witt equivalence of underlying local fields.