The Gross-Kuz'min Connjecture for CM fields

2012/09/14 by Mihăilescu, Preda
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1209.3172

Abstract

Let A' = \varprojlim_n A'_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_∞/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.

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