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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

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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
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:
DateEvent
15 October 2017Published
16 November 2017Available
11 August 2017Accepted
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:
Project/Grant IDRIOXX Funder NameFunder ID
EP/K034383/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
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