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) |
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Pages (from-to) | 253-260 |
Number of pages | 8 |
Journal | Publicationes Mathematicae |
Volume | 93 |
Issue number | 1-2 |
DOIs | |
State | Published - Jan 1 2018 |
Keywords
- Hilbert symbol
- Quartic surface
ASJC Scopus subject areas
- Mathematics(all)