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Approximation faible pour les 0-cycles sur un produit de variétés rationnellement connexes

Published online by Cambridge University Press:  20 March 2017

YONGQI LIANG
YONGQI LIANG*
Affiliation:
Bâtiment Sophie Germain, Université Paris Diderot - Paris 7, Institut de Mathématiques de Jussieu - Paris Rive Gauche, 75013 Paris, France. e-mail: [email protected]
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Abstract

Consider weak approximation for 0-cycles on a smooth proper variety defined over a number field, it is conjectured to be controlled by the Brauer group of the variety. Let X be a Châtelet surface or a smooth compactification of a homogeneous space of a connected linear algebraic group with connected stabilizer. Let Y be a rationally connected variety. We prove that weak approximation for 0-cycles on the product X × Y is controlled by its Brauer group if it is the case for Y after every finite extension of the base field. We do not suppose the existence of 0-cycles of degree 1 neither on X nor on Y.

Résumé

Considérons l'approximation faible de 0-cycles sur une variété propre lisse définie sur un corps de nombres, elle est conjecturée d'étre contrôlée par le groupe de Brauer de la variété. Soit X une surface de Châtelet ou une compactification lisse d'un espace homogéne d'un groupe algébrique linéaire connexe à stabilisateur connexe. Soit Y une variété rationnellement connexe. Nous montrons que l'approximation faible de 0-cycles sur le produit X × Y est contrôlée par son groupe de Brauer si c'est le cas pour Y après toute extension finie du corps de base. Nous ne supposons l'existence de 0-cycles de degré 1 ni sur X ni sur Y.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 485 - 491
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

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