Geodesic
Arithmetic of certain $\ell$-extensions ramified at three places.~II
Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 953-971.

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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.
Keywords: Iwasawa theory, Tate module, extensions with restricted ramification. 
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     title = {Arithmetic of  certain $\ell$-extensions ramified at three {places.~II}},
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L. V. Kuz'min. Arithmetic of  certain $\ell$-extensions ramified at three places.~II. Izvestiya. Mathematics , Tome 85 (2021) no. 5, pp. 953-971. http://geodesic.mathdoc.fr/item/IM2_2021_85_5_a5/

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