Iwasawa's constant $\mu$ vanishes in cyclotomic $\Z_p$-extensions of CM fields

TL;DR

This paper proves that the Iwasawa $d$ constant vanishes in the cyclotomic d-extension of CM fields, establishing finiteness of the p-rank of the inverse limit of class groups.

Contribution

It demonstrates the vanishing of Iwasawa's d constant d in cyclotomic d-extensions of CM fields, a significant result in Iwasawa theory.

Findings

The p-rank of the inverse limit of class groups is finite.

Iwasawa's d constant d vanishes in this setting.

Provides new evidence supporting conjectures in Iwasawa theory.

Abstract

Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show that the $p$-rank of $\rg{A}$ is finite, which is equivalent to the vanishing of Iwasawa's constant $\mu$ for $\rg{A}$. (Currently withdrawn)