Witt equivalence of function fields over global fields

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.