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This article is cited in 3 scientific papers (total in 3 papers)
Arithmetic of certain $\ell$-extensions ramified at three places. II
L. V. Kuz'min National Research Centre "Kurchatov Institute", Moscow
Abstract:
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.
Received: 09.06.2020
Citation:
L. V. Kuz'min, “Arithmetic of certain $\ell$-extensions ramified at three places. II”, Izv. RAN. Ser. Mat., 85:5 (2021), 132–151; Izv. Math., 85:5 (2021), 953–971
Linking options:
https://www.mathnet.ru/eng/im9070https://doi.org/10.1070/IM9070 https://www.mathnet.ru/eng/im/v85/i5/p132
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Abstract page: | 211 | Russian version PDF: | 26 | English version PDF: | 14 | Russian version HTML: | 74 | References: | 26 | First page: | 6 |
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