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A Mordell-Weil theorem for cubic hypersurfaces of high dimension
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Papanikolopoulos, Stefanos and Siksek, Samir (2017) A Mordell-Weil theorem for cubic hypersurfaces of high dimension. Algebra & Number Theory, 11 (8). pp. 1953-1965. doi:10.2140/ant.2017.11.1953 ISSN 1937-0652.
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Official URL: http://doi.org/10.2140/ant.2017.11.1953
Abstract
Let X/Q be a smooth cubic hypersurface of dimension n ≥ 1.
It is well-known that new rational points may be obtained from old ones by secant and tangent constructions. In view of the Mordell–Weil theorem for n = 1, Manin (1968) asked if there exists a finite set S from which all other rational points can be thus obtained. We give an affirmative answer for n ≥ 48, showing in fact that we can take the generating set S to consist of just one point. Our proof makes use of a weak approximation theorem due to Skinner, a theorem of Browning, Dietmann and Heath-Brown on the existence of rational points on the intersection of a quadric and cubic in large dimension, and some elementary ideas from differential geometry, algebraic geometry and numerical analysis.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Hypersurfaces, Geometry, Differential, Geometry, Analytic | ||||||||
| Journal or Publication Title: | Algebra & Number Theory | ||||||||
| Publisher: | Mathematical Sciences Publishers | ||||||||
| ISSN: | 1937-0652 | ||||||||
| Official Date: | 15 October 2017 | ||||||||
| Dates: |
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| Volume: | 11 | ||||||||
| Number: | 8 | ||||||||
| Page Range: | pp. 1953-1965 | ||||||||
| DOI: | 10.2140/ant.2017.11.1953 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 3 October 2017 | ||||||||
| Date of first compliant Open Access: | 16 May 2018 | ||||||||
| RIOXX Funder/Project Grant: |
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