An interesting quartic surface, everywhere locally solvable, with cubic point but no global point

Andrew Bremner, Tho Nguyen Xuan

Research output: Contribution to journalArticle

Abstract

There seem few examples in the literature of quartic surfaces defined over ℚ that are everywhere locally solvable, yet which have no global point. It is a delicate question as to whether such surfaces can possess points defined over an odd-degree number field, and to our knowledge no previous example is known. We give here an example of such a diagonal quartic surface which contains a point defined over a cubic extension field (and it follows that there exist number fields of every odd degree greater than 1 in which the surface has points). This surface is one member of a more general family of surfaces, each of which is also everywhere locally solvable but with no rational point.

Original languageEnglish (US)
Pages (from-to)253-260
Number of pages8
JournalPublicationes Mathematicae
Volume93
Issue number1-2
DOIs
StatePublished - Jan 1 2018

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Quartic
Number field
Odd
Field extension
Rational Points

Keywords

  • Hilbert symbol
  • Quartic surface

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

An interesting quartic surface, everywhere locally solvable, with cubic point but no global point. / Bremner, Andrew; Xuan, Tho Nguyen.

In: Publicationes Mathematicae, Vol. 93, No. 1-2, 01.01.2018, p. 253-260.

Research output: Contribution to journalArticle

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