Matching Witts locally and globally. (English) Zbl 0766.11023

Let \(F\) and \(E\) be global fields of characteristic not 2, let \(W(F)\) and \(W(E)\) be their Witt rings of nondegenerate symmetric bilinear forms, and let \(\Omega(F)\) and \(\Omega(E)\) be their respective sets of primes (including infinite primes, if any). The fields \(F\) and \(E\) are said to be Witt equivalent if \(W(F)\) and \(W(E)\) are isomorphic. On the other hand, they are said to be reciprocity equivalent if there exist a bijective map \(T: \Omega(F)\to \Omega(E)\) and a group isomorphism \(t: \dot F/\dot F^ 2 \to \dot E/\dot E^ 2\) preserving Hilbert symbols; i.e., \((a,b)_ P=(ta,tb)_{TP}\) for all \(P\in\Omega(F)\) and all \(a,b\in \dot F/\dot F^ 2\). The fact that reciprocity equivalence implies Witt equivalence follows from the Hasse principle and a criterion for Witt equivalence involving the represented value sets of binary quadratic forms. Moreover, the existence of a reciprocity equivalence between \(F\) and \(E\) is equivalent to the existence of isomorphisms between the local Witt rings \(W(F_ P)\) and \(W(E_{TP})\) for all \(P\in\Omega(F)\).
The main purpose of the paper under review is to prove that, in fact, two global fields are Witt equivalent if and only if they are reciprocity equivalent. The proof that Witt equivalence implies reciprocity equivalence proceeds in two stages. First, valuation theory and rigidity properties of the fields are used to prove that the fields are what the author calls almost reciprocity equivalent, the definition for which is obtained by changing the sets \(\Omega(F)\) and \(\Omega(E)\) in the definition of reciprocity equivalence to the sets \(\Omega_ 1(F)\) and \(\Omega_ 1(E)\) consisting of finite non-dyadic primes and the infinite real primes of the respective fields. Then, by arguments attributed by the author to R. Perlis, it is shown that an almost reciprocity equivalence of two Witt equivalent fields can be extended to a reciprocity equivalence.
A consequence of the main theorem, and the elementary fact that reciprocity equivalence preserves degree over \(\mathbb{Q}\), is that Witt equivalent algebraic number fields have the same degree over \(\mathbb{Q}\). The main result of this paper has also been proved using different methods based on an analysis of the 2-torsion in the Brauer group by R. Perlis, the author, P. E. Conner and R. Litherland [“Matching Witts with global fields”, preprint, 1989; per ibid.].


11E12 Quadratic forms over global rings and fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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