On the class numbers of the fields of the [formula omitted]-torsion points of certain elliptic curves over [formula omitted

Abstract

Let E be an elliptic curve over Q with prime conductor p. For each positive integer n we put Kn:=Q(E[pn]). The aim of this paper is to estimate the order of the p-Sylow group of the ideal class group of Kn. We give a lower bound in terms of the Mordell–Weil rank of E(Q). As an application of our result, we give an example such that p2n divides the class number of the field Kn in the case of p=5077 for each positive integer n.