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Le complémentaire des puissances $n$-ièmes dans un corps de nombres est un ensemble diophantien

Part of: Arithmetic problems. Diophantine geometry Cycles and subschemes Arithmetic algebraic geometry Number theory: Connections with logic

Published online by Cambridge University Press:  22 June 2015

Jean-Louis Colliot-Thélène  and
Jan Van Geel
Jean-Louis Colliot-Thélène
Affiliation:
CNRS & Université Paris-Sud, Mathématiques, Bâtiment 425, F-91405 Orsay Cedex, France email jlct@math.u-psud.fr
Jan Van Geel
Affiliation:
Universiteit Gent, Vakgroep Wiskunde, Krijgslaan 281, S22, B-9000 Gent, Belgium email jvg@cage.ugent.be
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Abstract

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For $n=2$ the statement in the title is a theorem of B. Poonen (2009). He uses a one-parameter family of varieties together with a theorem of Coray, Sansuc and one of the authors (1980), on the Brauer–Manin obstruction for rational points on these varieties. For $n=p$, $p$ any prime number, A. Várilly-Alvarado and B. Viray (2012) considered analogous families of varieties. Replacing this family by its $(2p+1)$th symmetric power, we prove the statement in the title using a theorem on the Brauer–Manin obstruction for rational points on such symmetric powers. The latter theorem is based on work of one of the authors with Swinnerton-Dyer (1994) and with Skorobogatov and Swinnerton-Dyer (1998), work generalising results of Salberger (1988).

Samenvatting

Voor $n=2$ is de bewering in de titel een stelling van B. Poonen (2009). Hij gebruikt een één-parameter familie van variëteiten, en een stelling van Coray, Sansuc en één van de auteurs (1980), over de Brauer–Manin obstructie voor de rationale punten van deze variëteiten. Voor $n=p$, $p$ een willekeurig priemgetal, beschouwden A. Várilly-Alvarado en B. Viray (2012) een analoge familie van variëteiten. We bewijzen de bewering in de titel door deze familie te vervangen door de $(2p+1)$-de symmetrische macht ervan en door een stelling over de Brauer–Manin obstructie voor de rationale punten van zulke symmetrische machten toe te passen. Deze stelling steunt op werk van één van de auteurs met Swinnerton-Dyer (1994) en met Skorobogatov en Swinnerton-Dyer (1998). Dat werk veralgemeent resultaten van Salberger (1988).

Keywords

zero-cyclesBrauer–Manin obstructiondiophantine definitions

MSC classification

Primary: 14G25: Global ground fields
Secondary: 11G35: Varieties over global fields 14C25: Algebraic cycles 11U99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 10 , October 2015 , pp. 1965 - 1980
Copyright
© The Authors 2015 

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