### 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 language | English (US) |
---|---|

Pages (from-to) | 253-260 |

Number of pages | 8 |

Journal | Publicationes Mathematicae |

Volume | 93 |

Issue number | 1-2 |

DOIs | |

State | Published - Jan 1 2018 |

### Fingerprint

### Keywords

- Hilbert symbol
- Quartic surface

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Publicationes Mathematicae*,

*93*(1-2), 253-260. https://doi.org/10.5486/PMD.2018.8252

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

Research output: Contribution to journal › Article

*Publicationes Mathematicae*, vol. 93, no. 1-2, pp. 253-260. https://doi.org/10.5486/PMD.2018.8252

}

TY - JOUR

T1 - An interesting quartic surface, everywhere locally solvable, with cubic point but no global point

AU - Bremner, Andrew

AU - Xuan, Tho Nguyen

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Hilbert symbol

KW - Quartic surface

UR - http://www.scopus.com/inward/record.url?scp=85054585268&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054585268&partnerID=8YFLogxK

U2 - 10.5486/PMD.2018.8252

DO - 10.5486/PMD.2018.8252

M3 - Article

AN - SCOPUS:85054585268

VL - 93

SP - 253

EP - 260

JO - Publicationes Mathematicae

JF - Publicationes Mathematicae

SN - 0033-3883

IS - 1-2

ER -