The Gross - Kuz'min conjecture for CM fields

TL;DR

This paper proves a key conjecture by Kuz'min for CM fields, showing the vanishing of a specific part of the Iwasawa module, linking it to the non-vanishing of the p-adic regulator of p-units.

Contribution

It establishes the vanishing of the minus part of the T-module in the Iwasawa theory of CM fields, confirming Kuz'min's conjecture and extending previous results beyond abelian extensions.

Findings

Proves $(A')^-(T) = 0$ for CM fields.

Links the vanishing to the non-vanishing of the p-adic regulator.

Extends Greenberg's results to non-abelian CM fields.

Abstract

Let $A' = \varprojlim_n$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the $\Z_p$-cyclotomic extension $\K_{\infty}/\K$ of a CM number field $\K$. We prove in this paper that the $T$ part $(A')^-(T) = 0$. This fact has been explicitly conjecture by Kuz'min in 1972 and was proved by Greenberg in 1973, for abelian extensions $\K/\Q$. Federer and Gross had shown in 1981 that $(A')^-(T) = 0$ is equivalent to the non-vanishing of the $p$-adic regulator of the $p$-units of $\K$.