Hasse principle and weak approximation on certain hypersurfaces
type: Item
instance of: scholarly article
title: Hasse principle and weak approximation on certain hypersurfaces
zbMATH Open document ID: 0870.14014
author: David Harari
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1997-06-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73365
review text: The main object of the paper under review is an affine hypersurface \(V\) given by an equation of the form \( y^2-az^2=P(x_1,\dots ,x_{n-2}) \) where \(a\) is non-square and \(P\) is a polynomial of total degree at most \(4\). The ground field \(k\) is assumed to be a number field. The author is interested in the Hasse principle and weak approximation for a smooth projective model of \(V\) as above. To be more precise, the goal is to compute the Brauer-Manin obstruction which turned out to be the only one for this class of varieties [ cf. \textit{D. Harari}, Duke Math. J. 75, No. 1, 221-260 (1994; Zbl 0847.14001)]. The obtained results provide subclasses of varieties for which the Hasse principle and weak approximation hold as well as numerical counter-examples to both properties.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14F22
zbMATH DE Number: 941203
zbMATH Keywords: Hasse principle
zbMATH Keywords: weak approximation
zbMATH Keywords: Brauer group
zbMATH Keywords: affine hypersurfaces
zbMATH Keywords: Brauer-Manin obstruction
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
OpenAlex ID: W2050823167
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DOI: 10.5802/AFST.810
description: scientific article; zbMATH DE number 941203