Arithmetic of Certain $\ell $-Extensions Ramified at Three Places

Let $\ell $ be a regular odd prime number, $k$ the $\ell $th cyclotomic field, $k_\infty $ the cyclotomic $\mathbb Z_\ell $-extension of $k$, $K$ a cyclic extension of $k$ of degree $\ell $, and $K_\infty =K\cdot k_\infty $. Under the assumption that there are exactly three places not over $\ell $ that ramify in the extension $K_\infty /k_\infty $ and $K$ satisfies some additional conditions, we study the structure of the Iwasawa module $T_\ell (K_\infty )$ of $K_\infty $ as a Galois module. In particular, we prove that $T_\ell (K_\infty )$ is a cyclic $G(K_\infty /k_\infty )$-module and the Galois group $\Gamma =G(K_\infty /K)$ acts on $T_\ell (K_\infty )$ as $\sqrt {\varkappa }$, where $\varkappa \colon \Gamma \to \mathbb Z_\ell ^\times $ is the cyclotomic character.