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

TL;DR

This paper explores the relationship between relative p-class groups, p-Selmer groups, and root numbers for elliptic curves with specific j-invariants, revealing new insights into their structure and applications to rank and number theory problems.

Contribution

It establishes a link between relative p-class groups, root numbers, and p-Selmer groups for certain elliptic curves, providing a method to construct large rank p-class groups.

Findings

Relative p-class groups determine p-Selmer group dimensions.

Construction of families with large p-class group rank.

Connection between congruent number problem and relative p-class groups.

Abstract

Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. 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.