Witt equivalence of function fields of curves over local 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.

KEYWORDS:

• Abhyankar valuations
• function fields
• local fields
• quadratic forms
• symmetric bilinear forms
• valuations
• Witt equivalence of fields

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 11E81
• 12J20 Secondary: 11E04
• 11E12

Acknowledgement

Murray Marshall passed away in May 2015. Our community lost a brilliant mathematician and a wonderful man. We sorely miss him.