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

TL;DR

This paper explores the relationship between class groups and cyclotomic Iwasawa modules for elliptic curves over , providing asymptotic formulas for class group quotients using Iwasawa invariants.

Contribution

It establishes a connection between ideal class group quotients and Iwasawa modules of elliptic curves, extending Iwasawa's class number formula to this context.

Findings

Asymptotic behavior of class group quotients described

Relation between ideal class groups and Iwasawa modules established

Asymptotic formulas derived under certain conditions

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 $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ 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 $A^E_n$ by using Iwasawa's invariants of $X_\infty$.