Arithmetic of certain -extensions ramified at three places. II

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...

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Bibliographic Details
Published in:Izvestiya. Mathematics 2021-10, Vol.85 (5), p.953-971
Main Author: Kuz’min, L. V.
Format: Article
Language:English
Subjects:
extensions with restricted ramification
Integers
Iwasawa theory
Modules
Tate module
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Summary: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.
ISSN:1064-5632
1468-4810
DOI:10.1070/IM9070