$3$-Selmer group, ideal class groups and cube sum problem

TL;DR

This paper establishes bounds on the 3-Selmer group ranks of Mordell curves over cyclotomic fields, linking them to ideal class groups, and applies these results to the cube sum problem and explicit curve families.

Contribution

It provides new bounds on the 3-Selmer groups of Mordell curves over _3, connecting them to ideal class groups and solving cases of the cube sum problem.

Findings

Bounds on the _3-Selmer group ranks in terms of ideal class groups.

Identification of families of Mordell curves with trivial or rank 1 _3-Selmer groups.

Application to positive proportion of curves with specific Selmer group structures.

Abstract

Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \in \mathbb Z$. These curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_a$ over $\mathbb Q(\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\mathbb Q(\zeta_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\rm Sel}^3(E_{a}/\mathbb Q)=0$ (respectively ${\rm Sel}^3(E_{a}/\mathbb Q)$ has $\mathbb F_3$-rank $1$).