On mirror equalities and certain weak forms of Greenberg's conjecture
type: Item
instance of: scholarly article
title: On mirror equalities and certain weak forms of Greenberg's conjecture
zbMATH Open document ID: 1073.11067
author: Raphaël Badino
author: Thong Nguyen Quang Do
published in: Manuscripta Mathematica
publication date: 2005-04-15T00:00:00Z
review text: Let \(p\) be an odd rational prime, \(K\) a fixed number field and \(\lambda = \lambda _ p (K)\) the Iwasawa \(\lambda\) invariant associated to the cyclotomic \({\mathbb Z}_ p\)-extension \(K_ \infty/K\). When \(K\) is a CM-field containing a primitive \(p\)-th root of unity \(\zeta _ p\), we consider the Iwasawa \(\lambda\) invariant \(\lambda ^ +=\lambda _ p (K^ +)\) of the maximal totally real subfield \(K^ +\) of \(K\). Iwasawa theory yields \(\lambda ^ + \leq \lambda\). Then \(\lambda=\lambda ^ + + \lambda ^ -\) for some non-negative integer \(\lambda ^ -\). Class field theory and Kummer theory show the well known mirror inequality \(\lambda ^ + \leq \lambda ^ -\). Ralph Greenberg conjectured that \(\lambda ^ + = 0\). In particular Greenberg's conjecture predicts that if \(\lambda ^ ->0\) then \(\lambda ^ + <\lambda ^ -\). This justifies to study a weak form of Greenberg's conjecture, namely: \(\lambda ^ + = \lambda ^ -\) iff \(\lambda ^ + =0 = \lambda ^ -\). This weak form of Greenberg's conjecture is considered in this paper. The authors study only number fields \(K\) such that both Leopoldt conjecture at \(p\) and the nullity of the Iwasawa \(\mu\) invariant hold. First, using some Iwasawa modules, conditions equivalent to the condition \(\lambda ^ +=\lambda ^ -\) are found and in particular to the conditions \(\lambda ^ + =0\) and \(\lambda ^ + =\lambda ^ - =0\). The authors apply these conditions to the cases when \(K^ +\) admits only one \(p\)-place which is totally ramified in \(K_ \infty ^ +\) and when \(p\) splits in \(K ^ +\). Along the paper it is studied the same conjecture for the invariants \(\lambda (\chi)\) and \(\lambda(\chi^ \ast)\) where \(\chi\) is a \(p\)-adic character of \(K\), \(\chi^ \ast = \omega \chi ^ {-1}\) is the mirror character, \(\omega\) being the Teichmüller character and \(K\) is a totally real number field.
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R80
zbMATH DE Number: 2156129
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: cyclotomic extensions
zbMATH Keywords: Iwasawa \(\lambda\) invariant
zbMATH Keywords: \(CM\) number field
zbMATH Keywords: totally real number field
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s00229-004-0531-4
OpenAlex ID: W1978176831
Wikidata QID: Q122987174
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DOI: 10.1007/S00229-004-0531-4
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description: scientific article; zbMATH DE number 2156129