Witt equivalence of function fields of conics

TL;DR

This paper investigates Witt equivalence of function fields of conics, providing new theorems that quantify the number of Witt non-equivalent classes, extending previous work on related classes of fields.

Contribution

It extends prior research by analyzing Witt equivalence specifically for function fields of conics and offers quantitative results on the classification of such fields.

Findings

Derived new theorems quantifying Witt non-equivalent classes

Extended understanding of Witt equivalence in the context of conic function fields

Provided insights into the structure of Witt rings for these fields

Abstract

Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely function fields over global fields and function fields of curves over local fields, were investigated by the authors in their earlier works. In the present work, which can be viewed as a sequel to the earlier papers, we discuss the previously obtained results in the specific case of function fields of conic sections, and apply them to provide a few theorems of a somewhat quantitive flavour shedding some light on the question of numbers of Witt non-equivalent classes of such fields.