The Gross-Kuz'min Connjecture for CM fields

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Let A' = ₙ A'ₙ be the projective limit of the p-parts of the ideal class groups of the p integers in the Zₚ-cyclotomic extension K∞/K of a CM number field K. We prove in this paper that the T-part (A')-(T) = { 1 } for CM extensions K/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/Q. Federer and Gross had shown in 1981 that (A')-(T) = { 1 } is equivalent to the non-vanishing of the p-adic regulator of the p-units of K.