Density of rational points on diagonal quartic surfaces

Logan, Adam; McKinnon, David; van Luijk, Ronald

doi:10.2140/ant.2010.4.1

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Density of rational points on diagonal quartic surfaces

Adam Logan, David McKinnon and Ronald van Luijk

Vol. 4 (2010), No. 1, 1–20

DOI: 10.2140/ant.2010.4.1

Abstract

Let $a, b, c, d$ be nonzero rational numbers whose product is a square, and let $V$ be the diagonal quartic surface in $ℙ^{3}$ defined by $a x^{4} + b y^{4} + c z^{4} + d w^{4} = 0$ . We prove that if $V$ contains a rational point that does not lie on any of the $48$ lines on $V$ or on any of the coordinate planes, then the set of rational points on $V$ is dense in both the Zariski topology and the real analytic topology.

Keywords

rational points, K3 surfaces, elliptic surfaces, quartic surfaces

Mathematical Subject Classification 2000

Primary: 11D25

Secondary: 14J28, 14G05

Milestones

Received: 27 December 2008

Revised: 29 July 2009

Accepted: 9 November 2009

Published: 14 January 2010

Authors

Adam Logan
Centre de recherches mathématiques Université de Montréal Case postale 6128, Succursale Centre-ville Montréal, QC H3C 3J7 Canada

David McKinnon
Pure Mathematics Department University of Waterloo Waterloo, ON N2L 3G1 Canada

Ronald van Luijk
Universiteit Leiden Mathematisch Instituut Postbus 9512 2300 RA Leiden Netherlands
http://www.math.leidenuniv.nl/~rvl

Vol. 4, No. 1, 2010