The structure of the Iwasawa module associated with a Z_p^r-extension of a p-adic local field of characteristic 0

Abstract

Let K be a finite extension of Q_p. Let K_∞, _r be a Galois extension of K such that G_r: = Gal(K_∞,_r/K) ∃ Z_p^r for some integer r ≥ 1. Let K_∞, _r^ab, _p be the maximal abelian pro-p extension of K_∞, _r, M_r = Gal K_∞, _r^ab, _p/K_∞, _r, and Λ_r = lim Z_p[[ G _r/pⁿ G _r]]. When r = 1, Iwasawa has determined the Λ_r-module structure of M_r. In this article we determine the rank and depth of the Λr-module M_r for any integer r ≥ 1.