Witt equivalence of function fields over global fields

TL;DR

This paper studies Witt equivalence of function fields over global fields, establishing a canonical correspondence between certain valuations and deriving implications for number fields and their class groups.

Contribution

It proves that Witt equivalence induces a bijection between specific valuations on function fields over global fields, with applications to number fields and class group properties.

Findings

Witt equivalence induces a canonical bijection between Abhyankar valuations with non-finite residue fields of characteristic 2.

Witt equivalence of function fields over number fields implies Witt equivalence of the base fields.

Number fields Witt equivalent to rational function fields have equal 2-ranks of their ideal class groups.

Abstract

In our work we investigate Witt equivalence of general function fields over global fields. It is proven that for any two such fields K and L the Witt equivalence induces a canonical bijection between Abhyankar valuations on K and L having residue fields not finite of characteristic 2. The main tool used in the proof is a method of constructing valuations due to Arason, Elman and Jacob. Numerous applications are provided, in particular to Witt equivalence of function fields over number fields: it is proven, among other things, that for two number fields k and l the Witt equivalence between the fields k(x_1,...,x_n) and l(x_1,...,x_n) implies that k and l are themselves Witt equivalent and have equal 2-ranks of their ideal class groups.