Some remarks on pseudo-null submodules of tamely ramified Iwasawa modules

Fujii, Satoshi; Itoh, Tsuyoshi

doi:10.5802/jtnb.1038

Cited by 4 publications

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“…, and this is not trivial because S 1 = ∅ (see also [10]). Hence, the assertion follows from the fact (which is shown in [4]) that X S 1 (Q c ) contains a non-trivial finite submodule. (The same type result for imaginary quadratic fields is given in [4].…”

Section: Totally Real Fieldsmentioning

confidence: 83%

“…In [4], it was shown that if p is odd and X S (Q c ) = 0, then X S (Q c ) always contains a non-trivial finite submodule. On the other hand, when p = 2, Mizusawa's result [16,Theorem 7.3] implies the existence of the case that X S (Q c ) ∼ = Z 2 as a Z 2 -module.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Next, we will consider the case of the Z ⊕2 p -extension of an imaginary quadratic field k. Concerning this paragraph, see also [4] for the details. Let k/k be the unique Z ⊕2 p -extension.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…is a finitely generated torsion Λ k/k -submodule. In [4], some sufficient conditions such that X S ( k) has a non-trivial pseudo-null Λ k/k -submodule were given. However, there is a non-trivial example such that X S ( k) does not contain a non-trivial pseudo-null submodule.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…(see, e.g., [18, p. We will prove Theorems 1.1 and 1.2 by using Proposition 2.1 directly. (Note that a similar idea is already used in [4] to show the existence of a non-trivial finite submodule of X S (Q c ) when p is odd. )…”

Section: Preliminariesmentioning

confidence: 99%

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Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules

Itoh¹

2018

J. Théor. Nombres Bordeaux

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Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules Tome 30, n o 3 (2018), p. 859-872. © Société Arithmétique de Bordeaux, 2018, tous droits réservés. L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 30 (2018), 859-872 Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules par Tsuyoshi ITOH Résumé. Ce travail fait suite à l'article [4] de Satoshi Fujii et l'auteur. Soient k un corps de nombres, p un nombre premier, et k c /k la Z p-extension cyclotomique. Pour un ensemble fini S de nombres premiers qui ne contient pas p, le module d'Iwasawa (par rapport à la prop extension abélienne maximale non ramifiée en dehors de S) a été étudié dans plusieurs articles. Nous donnons des exemples non-triviaux où X S (k c) a un sous-module fini non-nul avec k totalement réel. Nous donnons également un exemple similaire dans le cas de la Z ⊕2 p-extension d'un corps quadratique imaginaire. De plus, nous discutons en appendice des analogues faibles de la conjecture de Greenberg pour X S (k c).

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Section: Totally Real Fieldsmentioning

confidence: 83%

Section: Introduction and Resultsmentioning

confidence: 99%

Section: Introduction and Resultsmentioning

confidence: 99%

Section: Introduction and Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules

Itoh¹

2018

J. Théor. Nombres Bordeaux

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On the Structure of the Galois Group of the Maximal Pro-$p$ Extension with Restricted Ramification over the Cyclotomic $\mathbb{Z}_p$-extension

Itoh¹

2020

Tokyo J. Math.

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On the structure of the Galois group of the maximal pro-$p$ extension with restricted ramification over the cyclotomic $\mathbb{Z}_p$-extension

Itoh¹

2018

Preprint

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Let k ∞ be the cyclotomic Z p -extension of an algebraic number field k. We denote by S a finite set of prime numbers which does not contain p, and S(k ∞ ) the set of primes of k ∞ lying above S. In the present paper, we will study the structure of the Galois group X S (k ∞ ) of the maximal pro-p extension unramified outside S(k ∞ ) over k ∞ . We mainly consider the question whether X S (k ∞ ) is a non-abelian free pro-p group or not. In the former part, we treat the case when k is an imaginary quadratic field and S = ∅ (here p is an odd prime number which does not split in k). In the latter part, we treat the case when k is a totally real field and S = ∅.

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Some remarks on pseudo-null submodules of tamely ramified Iwasawa modules

Cited by 4 publications

References 13 publications

Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules

Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules

On the Structure of the Galois Group of the Maximal Pro-$p$ Extension with Restricted Ramification over the Cyclotomic $\mathbb{Z}_p$-extension

On the structure of the Galois group of the maximal pro-$p$ extension with restricted ramification over the cyclotomic $\mathbb{Z}_p$-extension

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