#### 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
##### 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