The Gross-Kuz'min Connjecture for CM fields

Abstract

Let $A^{\prime }={\underleftarrow{\lim }}_n{A}_n^{\prime }$ be the projective limit of the $p$-parts of the ideal class groups of the $p$ integers in the ${\mathbb{Z}}_p$-cyclotomic extension $K_{\infty }/K$ of a CM number field $K$. We prove in this paper that the $T$-part $(A^{\prime }{)}^{-}(T)=\{1\}$ for CM extensions $K/\mathbb{Q}$. This fact has been conjectured for arbitrary fields $K$ by Kuz'min in 1972 and was proved by Greenberg in 1973, for abelian extensions $K/\mathbb{Q}$. Federer and Gross had shown in 1981 that $(A^{\prime }{)}^{-}(T)=\{1\}$ is equivalent to the non-vanishing of the $p$-adic regulator of the $p$-units of $K$.

Version 4.0, with improved exposition and minor corrections, thanks to seminar discussions and remarks of Jens Franke