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09 February 2025
 
  » arxiv » 0711.0083

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Elliptic curves related to cyclic cubic extension
Rintaro Kozuma ;
Date 1 Nov 2007
AbstractThe 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
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