Asymptotic behavior of class groups and cyclotomic Iwasawa theory of elliptic curves

In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module X-infinity of the Pontrjagin dual of the fine Selmer group of an elliptic curve E defined over Q. We consider the Galois extension field K-n(E) of Q generated by coordinates of all p(n)-torsion points of E, and introduce a quotient A(n)(E) of the p-Sylow subgroup of the ideal class group of K-n(E) cut out by the modulo p(n) Galois representation E[p(n)]. We describe the asymptotic behavior of A(n)(E) by using the Iwasawa module X-infinity. In particular, under certain conditions, we obtain an asymptotic formula as Iwasawa's class number formula on the order of A(n)(E) by using Iwasawa's invariants of X-infinity.