Density of rational points on diagonal quartic surfaces

2008/12/27 by Adam Logan, Logan, Adam, David McKinnon +3
Mathematics · #11Dxx #11Gxx #14Gxx #14Jxx #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.AG #math.NT #msc:11Dxx #msc:11Gxx #msc:14Gxx #msc:14Jxx

paper · pdf · doi:10.48550/arxiv.0812.4779

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openalex publication_date 2008/12/27 · arxiv created 2009/02/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP³ 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.

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