Vol. 11, No. 8, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 10, 2213–2445
Issue 9, 1967–2212
Issue 8, 1739–1965
Issue 7, 1489–1738
Issue 6, 1243–1488
Issue 5, 1009–1241
Issue 4, 767–1007
Issue 3, 505–765
Issue 2, 253–503
Issue 1, 1–252

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
A Mordell–Weil theorem for cubic hypersurfaces of high dimension

Stefanos Papanikolopoulos and Samir Siksek

Vol. 11 (2017), No. 8, 1953–1965
Abstract

Let X 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.

Keywords
cubic hypersurfaces, rational points, Mordell–Weil problem
Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 11G35
Milestones
Received: 10 October 2016
Revised: 13 July 2017
Accepted: 11 August 2017
Published: 15 October 2017
Authors
Stefanos Papanikolopoulos
Mathematics Institute
University of Warwick
Coventry
United Kingdom
Samir Siksek
Mathematics Institute
University of Warwick
Coventry
United Kingdom