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 ℤ_ℓ-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 T_ℓ(K_∞) of K_∞ as a Galois module. In particular, we prove that T_ℓ(K_∞) is a cyclic G(K_∞/k_∞)-module and the Galois group Γ = G(K_∞/K) acts on T_ℓ(K_∞) as \(\sqrt \chi \), where \(\chi :\Gamma \to \mathbb{Z}_\ell^ \times \) is the cyclotomic character.