Vol. 4, No. 1, 2010

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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

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.

rational points, K3 surfaces, elliptic surfaces, quartic surfaces
Mathematical Subject Classification 2000
Primary: 11D25
Secondary: 14J28, 14G05
Received: 27 December 2008
Revised: 29 July 2009
Accepted: 9 November 2009
Published: 14 January 2010
Adam Logan
Centre de recherches mathématiques
Université de Montréal
Case postale 6128, Succursale Centre-ville
Montréal, QC H3C 3J7
David McKinnon
Pure Mathematics Department
University of Waterloo
Waterloo, ON N2L 3G1
Ronald van Luijk
Universiteit Leiden
Mathematisch Instituut
Postbus 9512
2300 RA Leiden