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Izvestiya: Mathematics, 2021, Volume 85, Issue 5, Pages 953–971
DOI: https://doi.org/10.1070/IM9070
(Mi im9070)
 

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
References:
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.
Funding agency Grant number
National Research Centre "Kurchatov Institute" 1807
This paper was written with the support of the NRC ‘Kurchatov Institute’ (order 14.08.2019 no. 1807).
Received: 09.06.2020
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2021, Volume 85, Issue 5, Pages 132–151
DOI: https://doi.org/10.4213/im9070
Bibliographic databases:
Document Type: Article
UDC: 511.62
MSC: 11R23, 11R18
Language: English
Original paper language: Russian
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
Citation in format AMSBIB
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\by L.~V.~Kuz'min
\paper Arithmetic of certain $\ell$-extensions ramified at three places.~II
\jour Izv. RAN. Ser. Mat.
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\vol 85
\issue 5
\pages 132--151
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\crossref{https://doi.org/10.4213/im9070}
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\jour Izv. Math.
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\vol 85
\issue 5
\pages 953--971
\crossref{https://doi.org/10.1070/IM9070}
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  • https://www.mathnet.ru/eng/im/v85/i5/p132
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    Citing articles in Google Scholar: Russian citations, English citations
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    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    English version PDF:14
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    References:26
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