Constructions of diagonal quartic and sextic surfaces with infinitely many rational points

Andrew Bremner, Ajai Choudhry, Maciej Ulas

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

In this paper, we construct several infinite families of diagonal quartic surfaces ax4 + by4 + cz4 + dw4 = 0 (where a, b, c, d are non-zero integers) with infinitely many rational points and satisfying the condition abcd is not a square. In particular, we present an infinite family of diagonal quartic surfaces defined over ℚ with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type ax6 + by6 + cz6 + dwi = 0, i = 2, 3, or 6, with infinitely many rational points.

Original languageEnglish (US)
Pages (from-to)1675-1698
Number of pages24
JournalInternational Journal of Number Theory
Volume10
Issue number7
DOIs
StatePublished - Nov 16 2014

Keywords

  • Diagonal quartic surface
  • sextic surface

ASJC Scopus subject areas

  • Algebra and Number Theory

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