Iwasawa descent and co-descent for units modulo circular units. With an appendix by J.-R. Belliard.
type: Item
instance of: scholarly article
title: Iwasawa descent and co-descent for units modulo circular units. With an appendix by J.-R. Belliard.
zbMATH Open document ID: 1129.11047
author: M. Lescop
author: Thong Nguyen Quang Do
published in: Pure and Applied Mathematics Quarterly
publication date: 2006-11-21T00:00:00Z
review text: Let \(F\) be a totally real abelian number field, \(p\) an odd prime number, \(F_\infty\) the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\) and \(U_n\), \(C_n\) the group of units, respectively circular units, of its \(n\)th layer \(F_n\) (the fixed field of \(\Gamma_n\)). Define \(B_\infty= \varinjlim \mathbb{Z}_p\otimes_{\mathbb{Z}}U_n/C_n\) and \(Y_\infty= \varprojlim \mathbb{Z}_p\otimes_{\mathbb{Z}} U_n/C_n\), with connecting maps induced by extension and the norm map, respectively. This paper studies descent and co-descent for \(B_\infty\) and \(Y_\infty\), i.e., the kernels and cokernels of \(\mathbb{Z}_p\otimes U_n/C_n@>{j_n}>> (B_\infty)^{\Gamma_n}\) and \((Y_\infty)_{\Gamma_n}@>{h_n}>> \mathbb{Z}_p\otimes U_n/C_n\) in terms of module theoretic invariants attached to \(F_\infty/F\). If \(U_n/C_n\) is replaced by the ideal class group of \(F_n\) one is back in the situation of the classical main conjecture of Iwasawa theory; however, now things are more difficult because there is no Galois theoretic interpretation of \(U_n/C_n\) in terms of class field theory. We summarize two theorems (in part). If \(\varprojlim \mathbb{Z}_p\otimes C_n\) is \(\Lambda\)-free, then \(j_n\) is injective and \(\text{coker\,}j_n\), \(\ker h_n\) and \(\text{coker\,}h_n\) stabilize with respect to extension maps. Moreover, if \(p\nmid[F:\mathbb{Q}]\), a characterwise discussion of \(\mathbb{Z}_p\otimes U_n/C_n\) is possible. The tools applied in the paper are, besides the Mazur-Wiles Theorem (proof of the main conjecture of Iwasawa theory), an exact sequence of Sinnott's involving the Artin map and the Gross kernel of everywhere local universal norms and, assuming the \(\Lambda\)-freeness of \(\varprojlim \mathbb{Z}_p\otimes C_n\), a thorough study of the cohomology of units and circular units. Related work is [\textit{J.-R. Belliard}, Submodules of units in Iwasawa theory, Number theory, Besançon (2002; Zbl 1161.11394); \textit{J.-R. Belliard} and \textit{N.-Q.-D. Thong}, Nagoya …
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5074490
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
full work available at URL: pamq.2006.v2.n2.a4
OpenAlex ID: W4299817633
DOI: 10.4310/PAMQ.2006.V2.N2.A4
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description: scientific article; zbMATH DE number 5074490