Vol. 7, No. 6, 2013

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Weak approximation for cubic hypersurfaces of large dimension

Mike Swarbrick Jones

Vol. 7 (2013), No. 6, 1353–1363
Abstract

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.

Keywords
cubic hypersurfaces, weak approximation, local-global principles, fibration method, circle method, many variables
Mathematical Subject Classification 2010
Primary: 11G35
Secondary: 11D25, 11D72, 11P55, 14G25
Milestones
Received: 25 October 2011
Revised: 24 July 2012
Accepted: 7 September 2012
Published: 19 September 2013
Authors
Mike Swarbrick Jones
School of Mathematics
University of Bristol
University Walk
Bristol BS8 1TW
United Kingdom