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Matching Witts with global fields. (English) Zbl 0807.11024
Jacob, William B. (ed.) et al., Recent advances in real algebraic geometry and quadratic forms. Proceedings of the RAGSQUAD year, Berkeley, CA, USA, 1990-1991. Providence, RI: American Mathematical Society. Contemp. Math. 155, 365-387 (1994).
The clever title of this paper refers to the problem of determining conditions under which two global fields are Witt equivalent, in the sense that they have isomorphic Witt rings of symmetric bilinear forms. In light of a result of R. Baeza and R. Moresi [J. Algebra 92, 446-453 (1985; Zbl 0553.10016)], it suffices to restrict to the case of fields of characteristic different from 2.
The authors define two global fields \(K\) and \(L\) to be reciprocity equivalent if there exist a group isomorphism \(t\) between the square- class groups of \(K\) and \(L\), and a bijection \(T\) between the sets of places on \(K\) and \(L\), such that Hilbert symbols are preserved; i.e., \((a,b)_ P= (ta, tb)_{TP}\) holds for all \(a\), \(b\) in \(K^*/K^{*^ 2}\) and all places \(P\) on \(K\). They then prove that \(K\) and \(L\) are Witt equivalent if and only if they are reciprocity equivalent. One consequence of this theorem is that Witt equivalent number fields must have the same degree over \(\mathbb{Q}\).
The authors then address the problem of constructing a reciprocity equivalence. For this purpose, they introduce the notion of small equivalence, and prove that any small equivalence (which is by definition a finite object) extends to a reciprocity equivalence. This is combined with results of J. Carpenter [Math. Z. 209, 153-166 (1992; Zbl 0736.11024)] to prove the following Hasse principle for Witt equivalence: Two global fields are Witt equivalent if and only if their places can be paired so that corresponding completions are Witt equivalent.
This paper, which is addressed to a general audience, also summarizes other recent work on the problem of Witt equivalence of global fields.
For the entire collection see [Zbl 0788.00051].

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E12 Quadratic forms over global rings and fields
11E08 Quadratic forms over local rings and fields