On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$

TL;DR

This paper investigates the class numbers of fields generated by $p^n$-torsion points of elliptic curves over $Q$, extending previous results and relating class group properties to the elliptic curve's rank and ramification.

Contribution

The paper generalizes earlier work on class numbers of fields from elliptic curves with multiplicative reduction, incorporating the Mordell-Weil rank and ramification into lower bounds.

Findings

Extended bounds on the $p$-Sylow subgroup of class groups

Connected class number growth to Mordell-Weil rank

Refined lower bounds involving ramification details

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results concerning with the order of the $p$-Sylow group of the ideal class group of $K_n$ to more general setting. We also modify the previous lower bound of the order and describe the new lower bound in terms of the Mordell-Weil rank of $E(\mathbb{Q})$ and the ramification related to $E$.