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 ∞ 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 ∞ . 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 ∞ .