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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Volume 307, Pages 78–99
DOI: https://doi.org/10.4213/tm4038
(Mi tm4038)
 

This article is cited in 4 scientific papers (total in 4 papers)

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

L. V. Kuz'min

National Research Center “Kurchatov Institute,” pl. Akademika Kurchatova 1, Moscow, 123182 Russia
Full-text PDF (334 kB) Citations (4)
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Abstract: 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.
Received: May 8, 2019
Revised: June 23, 2019
Accepted: June 30, 2019
English version:
Proceedings of the Steklov Institute of Mathematics, 2019, Volume 307, Pages 65–84
DOI: https://doi.org/10.1134/S008154381906004X
Bibliographic databases:
Document Type: Article
UDC: 511.62
Language: Russian
Citation: L. V. Kuz'min, “Arithmetic of Certain $\ell $-Extensions Ramified at Three Places”, Algebra, number theory, and algebraic geometry, Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 307, Steklov Mathematical Institute of RAS, Moscow, 2019, 78–99; Proc. Steklov Inst. Math., 307 (2019), 65–84
Citation in format AMSBIB
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\by L.~V.~Kuz'min
\paper Arithmetic of Certain $\ell $-Extensions Ramified at Three Places
\inbook Algebra, number theory, and algebraic geometry
\bookinfo Collected papers. Dedicated to the memory of Academician Igor Rostislavovich Shafarevich
\serial Trudy Mat. Inst. Steklova
\yr 2019
\vol 307
\pages 78--99
\publ Steklov Mathematical Institute of RAS
\publaddr Moscow
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\crossref{https://doi.org/10.4213/tm4038}
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\jour Proc. Steklov Inst. Math.
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\vol 307
\pages 65--84
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    This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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