On the Stark-Shintani conjecture and cyclotomic \({\mathbb{Z}}_ p\)-extensions of class fields over real quadratic fields. II
type: Item
instance of: scholarly article
title: On the Stark-Shintani conjecture and cyclotomic \({\mathbb{Z}}_ p\)-extensions of class fields over real quadratic fields. II
zbMATH Open document ID: 0549.12009
author: Jin Nakagawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: For some notations, see the previous review. In this paper the author considers the cyclotomic \({\mathbb{Z}}_ p\)-extension \(M_{\infty}\) of M for which the \(X_{{\mathfrak f}_ n}(c,M_ n)\) are units in \(M_ n\) for all n, and studies, under some hypothesis, the closure of these units in the local units in \(M_{\infty}\); he proves that the situation is analogous to the classical cyclotomic one [this uses generalizations of Iwasawa- Coates-Wiles computations, on formal series, which are in \textit{S. Lang}, Cyclotomic fields (1978; Zbl 0395.12005), some results of T. Arakawa on analytic class number formulas, and results of \textit{R. F. Coleman}, Invent. Math. 53, 91-116 (1979; Zbl 0429.12010)]. Remark on theorem 2: It seems that the classical ambiguous class number formula in \(M_ n/M\), \(F_ n/F\), \(n\geq 1\), gives directly the property: \(p| h^-_ 0\Leftrightarrow p| h^-_ n\quad for\quad all\quad n.\)
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 3875417
zbMATH Keywords: Stark-Shintani conjectures
zbMATH Keywords: partial zeta function
zbMATH Keywords: cyclotomic \({\mathbb{Z}}_ p\)-extension
zbMATH Keywords: units
zbMATH Keywords: local units
zbMATH Keywords: formal series
reviewed by: Georges Gras
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228809
OpenAlex ID: W2042997766
Wikidata QID: Q123017588
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DOI: 10.2748/TMJ/1178228809
description: scientific article; zbMATH DE number 3875417