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

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 $1$).