Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 859-872.

The present paper is a sequel to the previous paper [4] (by Satoshi Fujii and the author). Let k be an algebraic number field, p a prime number, and k c /k the cyclotomic p -extension. For a finite set S of prime numbers which does not contain p, the Iwasawa module X S (k c ) (with respect to the maximal pro-p abelian extension unramified outside S) has been studied in several papers. We will give some non-trivial examples such that X S (k c ) has no non-trivial finite submodules even when k is totally real. We also give a similar example for the case of the p 2 -extension of an imaginary quadratic field. Moreover, weak analogs of Greenberg’s conjecture for X S (k c ) are also discussed in the appendix.

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 p -extension cyclotomique. Pour un ensemble fini S de nombres premiers qui ne contient pas p, le module d’Iwasawa (par rapport à la pro-p 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 p 2 -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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Accepted:
Published online:
DOI: 10.5802/jtnb.1053
Classification: 11R23
Keywords: Iwasawa modules, non-existence of non-trivial pseudo-null submodules
Itoh, Tsuyoshi 1

1 Division of Mathematics, Education Center Faculty of Social Systems Science Chiba Institute of Technology 2-1-1 Shibazono, Narashino Chiba, 275-0023, Japan
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Itoh, Tsuyoshi. Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 859-872. doi : 10.5802/jtnb.1053. http://www.numdam.org/articles/10.5802/jtnb.1053/

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