Elliptic curves related to cyclic cubic extensions

Abstract

The aim of this paper is to study certain family of elliptic curves $\{{\mathcal{X}}_H{\}}_H$ defined over a number field $F$ arising from hyperplane sections of some cubic surface $\mathcal{X}/F$ associated to a cyclic cubic extension $K/F$. We show that each ${\mathcal{X}}_H$ admits a 3-isogeny $\varphi$ over $F$ and the dual Selmer group $S^{(\hat{\varphi })}(\widehat{{\mathcal{X}}_H}/F)$ is bounded by a kind of unit/class groups attached to $K/F$. This is proven via certain rational function on the elliptic curve ${\mathcal{X}}_H$ 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.

29 pages