Koprowski, Przemysław Local-global principle for Witt equivalence of function fields over global fields. (English) Zbl 1030.11017 Colloq. Math. 91, No. 2, 293-302 (2002). Two fields (of characteristic \(\neq 2\)) are said to be Witt equivalent if their Witt rings are isomorphic. A well-known result by Harrison gives necessary and sufficient conditions for this to happen; see, for example, R. Perlis, K. Szymiczek, P. E. Conner and R. Litherland [Contemp. Math. 155, 365–387 (1994; Zbl 0807.11024)] where this criterion is then used in an essential manner to describe Witt equivalence of global fields. In [Math. Z. 242, No. 2, 323–345 (2002; Zbl 1067.11020)], the author determined when algebraic function fields over real closed fields are Witt equivalent, and in the present paper, he treats the case of algebraic function fields over fields belonging to a certain class which includes local and global fields. Let \(K\) and \(L\) be algebraic function fields over the fields of constants \(k\) and \(l\), respectively, and denote the set of all \(k\)-trivial (resp. \(l\)-trivial) places on \(K\) (resp. \(L\)) by \(\Omega (K)\) (resp. \(\Omega (L)\)). \(K\) and \(L\) are said to be quaternion-symbol equivalent if there exists a group isomorphism \(t : K^*/K^{*2}\to L^*/L^{*2}\) and a bijection \(T:\Omega (K)\to \Omega (L)\) such that for all \(f,g\in K^*/K^{*2}\) and all \({\mathfrak p}\in \Omega (K)\), the local quaternion symbol \((f,g)_{K_{\mathfrak p}}\) is trivial iff \((tf,tg)_{L_{T{\mathfrak p}}}\) is trivial. \(K\) and \(L\) are said to be uniformly locally Witt equivalent if there exists an isomorphism \(i:WK\to WL\), a bijection \(T:\Omega (K)\to \Omega (L)\) and isomorphisms \(i_{\mathfrak p}:WK_{\mathfrak p}\to WL_{T{\mathfrak p}}\) compatible with \(i\) under the natural epimorphisms \(WK\to WK_{\mathfrak p}\) and \(WL\to WL_{T{\mathfrak p}}\). The main result states that provided all finite extensions of \(k\) and \(l\) have at least four square classes, then \(K\) and \(L\) are Witt equivalent if and only if they are uniformly locally Witt equivalent, and that in the case of rational function fields \(K=k(X)\) and \(L=l(X)\), this is the same as \(K\) and \(L\) being quaternion-symbol equivalent. Reviewer: Detlev Hoffmann (Besançon) Cited in 1 ReviewCited in 6 Documents MSC: 11E81 Algebraic theory of quadratic forms; Witt groups and rings 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 11E04 Quadratic forms over general fields 14H05 Algebraic functions and function fields in algebraic geometry Keywords:isomorphism of Witt rings; Witt equivalence of fields; global field; algebraic function field Citations:Zbl 0807.11024; Zbl 1067.11020 PDFBibTeX XMLCite \textit{P. Koprowski}, Colloq. Math. 91, No. 2, 293--302 (2002; Zbl 1030.11017) Full Text: DOI