Witt equivalence of algebraic function fields over real closed fields
type: Item
instance of: scholarly article
title: Witt equivalence of algebraic function fields over real closed fields
zbMATH Open document ID: 1067.11020
author: Przemysław Koprowski
published in: Mathematische Zeitschrift
publication date: 2004-06-17T00:00:00Z
review text: Two fields \(K\) and \(L\) are called Witt equivalent if their Witt rings are isomorphic. Harrison's criterion states that two fields are Witt equivalent iff there exists an isomorphism \(t\) between the respective square class groups sending \(-1\) to \(-1\) and such that the quadratic form \(\langle a,b\rangle\) over \(K\) represents \(1\) iff the form \(\langle t(a),t(b)\rangle\) over \(L\) represents \(1\). \(t\) is also called a Harrison map. Witt equivalence of global fields has been studied earlier by \textit{R. Perlis} et al. [Contemp. Math. 155, 365--387 (1994; Zbl 0807.11024)]. In the present paper, the author studies the case of function fields of transcendence degree one over a fixed real closed field. The first main result is that two such function fields \(K\) and \(L\) are Witt equivalent iff they are both real or both nonreal. In the real case, the author considers a refined version of Witt equivalence which he calls tame equivalence. A tame equivalence is a Harrison isomorphism which induces a map from so-called \(1\)-pt fans of \(K\) onto \(1\)-pt fans of \(L\). If \(\gamma (K)\) (resp. \(\gamma (L)\)) denotes the set of all real \(K\)-places (resp. all real \(L\)-places), equipped with the so-called strong topology, then the author shows that \(K\) and \(L\) are tamely equivalent iff \(\gamma (K)\) and \(\gamma (L)\) have the same number of semi-algebraically connected components. As a corollary, he obtains that there are at most \(g+1\) tame classes of such real function fields of transcendence degree \(1\) and genus not exceeding \(g\) over a fixed real closed field of constants. As an example, the author gives representatives for all tame equivalence classes for genus \(0\) and \(1\) for the field of constants \({\mathbb R}\).
reviewed by: Detlev Hoffmann
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 14H05
zbMATH DE Number: 2075732
zbMATH Keywords: Witt ring
zbMATH Keywords: Witt equivalence
zbMATH Keywords: tame equivalence
zbMATH Keywords: algebraic function field
zbMATH Keywords: real closed field
zbMATH Keywords: Harrison map
MaRDI profile type: Publication
full work available at URL: s002090100336
OpenAlex ID: W2044159338
DOI: 10.1007/S002090100336
description: scientific article; zbMATH DE number 2075732