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 |
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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 journal › Article
}
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 -