Witt Equivalence of Function Fields over Global Fields

Witt equivalent fields can be understood to be fields having the same symmetric bilinear form theory. Witt equivalence of finite fields, local fields and global fields is well understood. Witt equivalence of function fields of curves defined over archimedean local fields is also well understood. In the present paper, Witt equivalence of general function fields over global fields is studied. It is proved that for any two such fields K,L, any Witt equivalence K ∼ L induces a cannonical bijection v ↔ w between Abhyankar valuations v on K having residue field not finite of characteristic 2 and Abhyankar valuations w on L having residue field not finite of characteristic 2. The main tool used in the proof is a method for constructing valuations due to Arason, Elman and Jacob [1]. The method of proof does not extend to non-Abhyankar valuations. The result is applied to study Witt equivalence of function fields over number fields. It is proved, for example, that if k, l are number fields and k(x1, . . . , xn) ∼ l(x1, . . . , xn), n ≥ 1, then k ∼ l and the 2-ranks of the ideal class groups of k and l are equal.