On the class numbers of the fields of the p ⁿ -torsion points of elliptic curves over Q

Let E be an elliptic curve over Q which has multiplicative reduction at a fixed prime p. Assume E has multiplicative reduction or potentially good reduction at any prime not equal to p. For each positive integer n we put K _n := Q(E[p ⁿ ]). The aim of this paper is to extend the authors’ previous results in [13] 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 given in terms of the Mordell–Weil rank of E(Q) and the ramification related to E.