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

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 $E$.