##
**Local-global principle for Witt equivalence of function fields over global fields.**
*(English)*
Zbl 1030.11017

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.

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)

### 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 |