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Failures of weak approximation in families

Part of:Arithmetic problems. Diophantine geometryArithmetic algebraic geometry(Co)homology theoryMultiplicative number theoryFamilies, fibrationsHomological methods (field theory)

Published online by Cambridge University Press:  26 April 2016

M. J. Bright,
T. D. Browning and
D. Loughran
M. J. Bright
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands email m.j.bright@math.leidenuniv.nl
T. D. Browning
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK email t.d.browning@bristol.ac.uk
D. Loughran
Affiliation:
Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Welfengarten 1, 30167 Hannover, Germany email loughran@math.uni-hannover.de
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Abstract

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Given a family of varieties $X\rightarrow \mathbb{P}^{n}$ over a number field, we determine conditions under which there is a Brauer–Manin obstruction to weak approximation for 100% of the fibres which are everywhere locally soluble.

Keywords

fibrationsrational pointsweak approximationBrauer–Manin obstructionsieves

MSC classification

Primary:14G05: Rational points
Secondary:11G35: Varieties over global fields11N36: Applications of sieve methods12G05: Galois cohomology14D10: Arithmetic ground fields (finite, local, global)14F22: Brauer groups of schemes

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 7 , July 2016, pp. 1435 - 1475
Copyright
© The Authors 2016 

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