Elliptic curves related to cyclic cubic extensions

TL;DR

This paper investigates a family of elliptic curves derived from cubic surfaces linked to cyclic cubic extensions, revealing their 3-isogenies, connections to class groups, and bounds on Selmer groups.

Contribution

It establishes a relationship between the Selmer groups of these elliptic curves and unit/class groups of the cyclic cubic extension, and identifies the Shafarevich-Tate group with a class group.

Findings

Each elliptic curve admits a 3-isogeny over the base field.

The dual Selmer group is bounded by unit/class groups of the extension.

The Shafarevich-Tate group coincides with a class group in certain cases.

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 $\text{\cyr X} (\hat{\mathscr{X}_H}/\rat)[\hat{\phi}]$ coincides with a class group of $K$ as a special case.