Arithmetic of certain $\ell$-extensions ramified at three places. II

Let $\ell$ be a regular odd prime, $k$ the $\ell$ th cyclotomic field and $K=k(\sqrt[\ell]{a})$, where $a$ is a positive integer. Under the assumption that there are exactly three places not over $\ell$ that ramify in $K_\infty/k_\infty$, we continue to study the structure of the Tate module (Iwasawa module) $T_\ell(K_\infty)$ as a Galois module. In the case $\ell=3$, we prove that for finite $T_\ell(K_\infty)$ we have $|T_\ell(K_\infty)|\,{=}\,\ell^r$ for some odd positive integer $r$. Under the same assumptions, if $\overline T_\ell(K_\infty)$ is the Galois group of the maximal unramified Abelian $\ell$-extension of $K_\infty$, then the kernel of the natural epimorphism $\overline T_\ell(K_\infty)\to T_\ell (K_\infty)$ is of order $9$. Some other results are obtained.