On the Hasse principle for Witt groups
type: Item
instance of: scholarly article
title: On the Hasse principle for Witt groups
zbMATH Open document ID: 0822.11030
DOI: 10.1007/BF01444494
author: R. Sujatha
published in: Mathematische Annalen
publication date: 1995-10-23T00:00:00Z
full work available at URL: https://eudml.org/doc/165330
review text: Let \(k\) be a global field of characteristic different from 2. For a field \(F\), let \(W(F)\) denote the Witt group of quadratic forms over \(F\). The set of places of \(k\) is denoted by \(P(k)\). The classical Hasse principle for the Witt group states that the natural map \(W(k)\to \prod_{\nu\in P(k)} W(k_ \nu)\) is injective, where \(k_ \nu\) denotes the completion of \(k\) at \(\nu\). This can be used to prove that the map \(W(K)\to \prod_{\nu\in P(k)} W(K_ \nu)\) is injective for the rational function field \(K= k(T)\) with \(K_ \nu= k_ \nu (T)\). In this paper, we prove that if \(K\) is the function field of a smooth projective curve \(X/k\) such that \(X(k)\neq \emptyset\), and \(K_ \nu= k_ \nu (X)\), then the kernel of the natural map \(W(K)\to \prod_{\nu\in P(k)} W(K_ \nu)\) is finite. In fact we show that this obstruction to the Hasse principle is a subgroup of the 2-torsion group of the Tate- Shafarevich group of \(J\) where \(J\) is the Jacobian of \(\overline {X}= X\times_ k \overline{k}\). We also give examples to show that this obstruction is non-trivial.
reviewed by: R. Sujatha
Mathematics Subject Classification ID: 11E70
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 778942
zbMATH Keywords: rational function field
zbMATH Keywords: Jacobian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 778942