Algebra & Number Theory

Weak approximation for cubic hypersurfaces of large dimension

Mike Swarbrick Jones

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We address the problem of weak approximation for general cubic hypersurfaces defined over number fields with arbitrary singular locus. In particular, weak approximation is established for the smooth locus of projective, geometrically integral, nonconical cubic hypersurfaces of dimension at least 17. The proof utilises the Hardy–Littlewood circle method and the fibration method.

Article information

Algebra Number Theory, Volume 7, Number 6 (2013), 1353-1363.

Received: 25 October 2011
Revised: 24 July 2012
Accepted: 7 September 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G35: Varieties over global fields [See also 14G25]
Secondary: 11D25: Cubic and quartic equations 11D72: Equations in many variables [See also 11P55] 11P55: Applications of the Hardy-Littlewood method [See also 11D85] 14G25: Global ground fields

cubic hypersurfaces weak approximation local-global principles fibration method circle method many variables


Swarbrick Jones, Mike. Weak approximation for cubic hypersurfaces of large dimension. Algebra Number Theory 7 (2013), no. 6, 1353--1363. doi:10.2140/ant.2013.7.1353.

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