Vol. 4, No. 1, 2010

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ISSN: 1944-7833 (e-only)
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Density of rational points on diagonal quartic surfaces

Adam Logan, David McKinnon and Ronald van Luijk

Vol. 4 (2010), No. 1, 1–20
Abstract

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in 3 defined by ax4 + by4 + cz4 + dw4 = 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