Mazur’s rational torsion result for pointless genus one curves: examples
| Autor: | Arjan Dwarshuis, Majken Roelfszema, Jaap Top |
|---|---|
| Přispěvatelé: | Algebra |
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Algebra and Number Theory 010102 general mathematics 0102 computer and information sciences Algebraic geometry Computer Science::Digital Libraries 01 natural sciences Elliptic curve Number theory 010201 computation theory & mathematics Rational point Genus (mathematics) Computer Science::Mathematical Software Torsion (algebra) 0101 mathematics Mathematics |
| Zdroj: | Research in Number Theory, 7(1):7. Springer Nature |
| ISSN: | 2363-9555 2522-0160 |
| DOI: | 10.1007/s40993-020-00231-z |
| Popis: | This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over$$\mathbb {Q}$$Qin a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over$$\mathbb {Q}$$Qcorresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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