Relative $p$-class groups and $p$-Selmer groups

Abstract

Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\tilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\tilde{E}$, a degree $k$ relative $p$-class group and the root number of $\tilde{E}$ determines the dimension of the $p$-Selmer group of $\tilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group.