Elliptic curves related to cyclic cubic extensions
| dc.creator | Kozuma, Rintaro | |
| dc.date | 2007-11-01 | |
| dc.date.accessioned | 2026-07-07T08:39:52Z | |
| dc.date.available | 2026-07-07T08:39:52Z | |
| dc.description | The aim of this paper is to study certain family of elliptic curves $\{\mathscr{X}_H\}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathscr{X}/F$ associated to a cyclic cubic extension $K/F$. We show that each $\mathscr{X}_H$ admits a 3-isogeny $ϕ$ over $F$ and the dual Selmer group $S^{(\hatϕ)}(\hat{\mathscr{X}_H}/F)$ is bounded by a kind of unit/class groups attached to $K/F$. This is proven via certain rational function on the elliptic curve $\mathscr{X}_H$ with nice property. We also prove that the Shafarevich-Tate group $\text{\cyr X} (\hat{\mathscr{X}_H}/\rat)[\hatϕ]$ coincides with a class group of $K$ as a special case. | |
| dc.description | 29 pages | |
| dc.identifier | https://arxiv.org/abs/0711.0083 | |
| dc.identifier | http://arxiv.org/abs/0711.0083 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/141221 | |
| dc.subject | Number Theory | |
| dc.subject | 11G05; 11R16; 11G07 | |
| dc.title | Elliptic curves related to cyclic cubic extensions | |
| dc.type | text |