On the class numbers of the fields of the pnpn-torsion points of certain elliptic curves over QQ

Let E   be an elliptic curve over QQ with prime conductor p. For each positive integer n   we put Kn:=Q(E[pn])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 KnKn. We give a lower bound in terms of the Mordell–Weil rank of E(Q)E(Q). As an application of our result, we give an example such that p2np2n divides the class number of the field KnKn in the case of p=5077p=5077 for each positive integer n.