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Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

Published online by Cambridge University Press:  08 December 2014

ARNE SMEETS
ARNE SMEETS*
Affiliation:
Departement Wiskunde, KU Leuven, Leuven, Belgium, and Département de Mathématiques, Université Paris-Sud 11, Orsay, France. e-mail: [email protected]
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Abstract

Let k be a number field and T a k-torus. Consider a family of torsors under T, i.e. a morphism f : X → ℙ1k from a projective, smooth k-variety X to ℙ1k, the generic fibre Xη → η of which is a smooth compactification of a principal homogeneous space under T ⊗k η. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation for X, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional if k = Q and all non-split fibres of f are defined over Q. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

Résumé

Soit k un corps de nombres et soit T un k-tore. Considérons une fibration en torseurs sous T, c'est-à-dire un morphisme f : X → ℙ1k d'une k-variété projective et lisse X vers ℙ1k tel que sa fibre générique Xη → η soit une compactification lisse d'un espace principal homogène sous T ⊗k η. On étudie dans ce texte l'obstruction de Brauer-Manin au principe de Hasse et à l'approximation faible pour X, sous l'hypothèse de Schinzel. On généralise les résultats récents de Wei [21]. Nos résultats sont inconditionnels si k = Q et les fibres non-scindées de f sont définies sur Q. On établit également un analogue inconditionnel de notre résultat principal pour les zéro-cycles de degré 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 1 , January 2015 , pp. 131 - 145
Copyright
Copyright © Cambridge Philosophical Society 2014 

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