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Witt equivalence of function fields over global fields


Authors: Paweł Gładki and Murray Marshall
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11E81, 12J20; Secondary 11E04, 11E12
DOI: https://doi.org/10.1090/tran/6898
Published electronically: April 11, 2017
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Abstract: Witt equivalent fields can be understood to be fields having the same symmetric bilinear form theory. Witt equivalence of finite fields, local fields and global fields is well understood. Witt equivalence of function fields of curves defined over archimedean local fields is also well understood. In the present paper, Witt equivalence of general function fields over global fields is studied. It is proved that for any two such fields $ K,L$, any Witt equivalence $ K \sim L$ induces a cannonical bijection $ v \leftrightarrow w$ between Abhyankar valuations $ v$ on $ K$ having residue field not finite of characteristic $ 2$ and Abhyankar valuations $ w$ on $ L$ having residue field not finite of characteristic $ 2$. The main tool used in the proof is a method for constructing valuations due to Arason, Elman and Jacob [J. Algebra 110 (1987), 449-467]. The method of proof does not extend to non-Abhyankar valuations. The result is applied to study Witt equivalence of function fields over number fields. It is proved, for example, that if $ k$, $ \ell $ are number fields and $ k(x_1,\dots ,x_n) \sim \ell (x_1,\dots ,x_n)$, $ n\ge 1$.


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Additional Information

Paweł Gładki
Affiliation: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland – and – Department of Computer Science, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
Email: pawel.gladki@us.edu.pl

Murray Marshall
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan S7N 5E6, Canada

DOI: https://doi.org/10.1090/tran/6898
Keywords: Symmetric bilinear forms, quadratic forms, Witt equivalence of fields, function fields, global fields, valuations, Abhyankar valuations
Received by editor(s): April 21, 2015
Received by editor(s) in revised form: November 28, 2015
Published electronically: April 11, 2017
Additional Notes: The research of the second author was supported in part by NSERC of Canada.
Article copyright: © Copyright 2017 American Mathematical Society