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

Abstract

In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_{\infty }$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K_n^E$ of $\mathbb{Q}$ 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 $X_{\infty }$. 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 $X_{\infty }$.

45 pages