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Article overview
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Elliptic curves related to cyclic cubic extension | Rintaro Kozuma
; | Date: |
1 Nov 2007 | Abstract: | 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 $phi$
over $F$ and the dual Selmer group $S^{(hat{phi})}(hat{mathscr{X}_H}/F)$ 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 $ ext{cyr X}
(hat{mathscr{X}_H}/
at)[hat{phi}]$ coincides with a class group of $K$ as
a special case. | Source: | arXiv, 0711.0083 | Services: | Forum | Review | PDF | Favorites |
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