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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
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
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
Linking options:
https://www.mathnet.ru/eng/tm4038https://doi.org/10.4213/tm4038 https://www.mathnet.ru/eng/tm/v307/p78
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Abstract page: | 203 | Full-text PDF : | 37 | References: | 20 | First page: | 7 |
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