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Graded quaternion symbol equivalence of function fields. (English) Zbl 1190.11029

Let \(K\), \(L\) be two function fields and let \(A\), \(B\) be fixed sets of points of \(K\) and \(L\). A pair of maps \((t,T)\) in which \(t\) is an isomorphism of the square-class group of \(K\) to the square-class group of \(L\) and \(T:A\to B\) is a bijection is called a graded quaternion-symbol equivalence of the fields \(K\) and \(L\) with respect to the sets \(A\) and \(B\) if the following two conditions are satisfied:
(i) \(t(-1)=-1\);
(ii) The pair \((t,T)\) preserves the vanishing of local Clifford invariants in the sense that \[ \left<\frac{f,g}{K_{\mathfrak p}}\right>=1\in \text{BW}(K_{\mathfrak p})\Leftrightarrow \left<\frac{tf,tg}{L_{T\mathfrak p}}\right>=1\in \text{BW}(L_{T\mathfrak p}) \] for all square classes \(f,g\) of \(K\) and all points \({\mathfrak p}\in A\), where \(\text{BW}(K_{\mathfrak p})\) is the Brauer-Wall group of \(K_{\mathfrak p}\).
The author proves:
Proposition 1. Let \(K,L\) be two global function fields of characteristics\(\neq 2\), let \(t\) be an isomorphism of their square-class groups and \(T\) a bijection of their sets of points. The following conditions are equivalent:
(1) The pair \((t,T)\) is a graded quaternion-symbol equivalence.
(2) The pair \((t,T)\) preserves Hilbert-symbols.
(3) The pair \((t,T)\) preserves \(-1\) and induces isomorphisms of subgroups of local Brauer-Wall groups generated by graded quaternion algebras given by \(\left<\frac{f,g}{K_{\mathfrak p}}\right>\mapsto\left<\frac{tf,tg}{L_{T\mathfrak p}}\right>.\)
Proposition 2. Let \(K,L\) be two formally real algebraic function fields over a fixed real closed field \(\mathbb k\). Let \(t\) be an isomorphism of their square-class groups such that \(t(-1)=-1\) and let \(T\) be a bijection of their sets of points. The following three conditions are equivalent:
(1) The pair \((t,T)\) is a graded quaternion-symbol equivalence.
(2) The pair \((t,T)\) preserves quaternion-symbols.
(3) The pair \((t,T)\) induces isomorphisms of subgroups of local Brauer-Wall groups generated by graded quaternion algebras given by \(\left<\frac{f,g}{K_{\mathfrak p}}\right>\mapsto\left<\frac{tf,tg}{L_{T\mathfrak p}}\right>.\)

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E10 Forms over real fields
16K50 Brauer groups (algebraic aspects)
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References:

[1] A. Czogała: Równowa\.zność Hilberta ciał globalnych. Volume 1969 of Prace Naukowe Uniwersytetu Ślaskiego w Katowicach [Scientific Publications of the University of Silesia]. Wydawnictwo Uniwersytetu Ślaskiego, Katowice, 2001.
[2] M. Knebusch: On algebraic curves over real closed fields. II. Math. Z. 151 (1976), 189–205. · doi:10.1007/BF01213994
[3] P. Koprowski: Local-global principle for Witt equivalence of function fields over global fields. Colloq. Math. 91 (2002), 293–302. · Zbl 1030.11017 · doi:10.4064/cm91-2-8
[4] P. Koprowski: Witt equivalence of algebraic function fields over real closed fields. Math. Z. 242 (2002), 323–345. · Zbl 1067.11020 · doi:10.1007/s002090100336
[5] P. Koprowski: Integral equivalence of real algebraic function fields. Tatra Mt. Math. Publ. 32 (2005), 53–61. · Zbl 1150.11420
[6] T. Y. Lam: Introduction to quadratic forms over fields. Volume 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2005. · Zbl 1068.11023
[7] R. Perlis, K. Szymiczek, P. E. Conner and R. Litherland: Matching Witts with global fields. In Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991), volume 155 of Contemp. Math., pages 365–387. Amer. Math. Soc., Providence, RI, 1994. · Zbl 0807.11024
[8] K. Szymiczek: Matching Witts locally and globally. Math. Slovaca 41 (1991), 315–330. · Zbl 0766.11023
[9] K. Szymiczek: Witt equivalence of global fields. Comm. Algebra 19 (1991), 1125–1149. · Zbl 0724.11020 · doi:10.1080/00927879108824194
[10] K. Szymiczek: Hilbert-symbol equivalence of number fields. Tatra Mt. Math. Publ. 11 (1997), 7–16. · Zbl 0978.11012
[11] K. Szymiczek: A characterization of tame Hilbert-symbol equivalence. Acta Math. Inform. Univ. Ostraviensis 6 (1998), 191–201. · Zbl 1024.11022
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