Arithmetic of Certain ℓ-Extensions Ramified at Three Places

Abstract

Let l be a regular odd prime number, k the lth cyclotomic field, k∞ the cyclotomic ℤl-extension of k, K a cyclic extension of k of degree l, and = K · k∞. Under the assumption that there are exactly three places not over l that ramify in the extension K∞/k∞ and K satisfies some additional conditions, we study the structure of the Iwasawa module Tl(K∞) of K∞ as a Galois module. In particular, we prove that Tl(K∞) is a cyclic G(K∞/k∞)-module and the Galois group Γ = G(K∞/K) acts on Tl(K∞) as $$\sqrt \chi $$, where $$\chi :\Gamma \to \mathbb{Z}_\ell^ \times $$ is the cyclotomic character.