Finite submodules and Iwasawa \(\mu\)-invariants
type: Item
instance of: scholarly article
title: Finite submodules and Iwasawa \(\mu\)-invariants
zbMATH Open document ID: 1014.11065
author: Michael J. Drinen
published in: Journal of Number Theory
publication date: 2002-06-11T00:00:00Z
review text: Let \(K\) be a number field and let \(K_{\infty}/K\) be a \(\mathbb Z_p\)-extension. Let \(G_K=\)Gal\((\overline{K}/K)\) and \(A\) be a \(G_K\)-module that is cofinitely generated and cofree as a \(\mathbb Z_p\)-module. Under certain ordinariness conditions, \textit{R. Greenberg} [in Algebraic number theory -- in honor of K. Iwasawa, Adv. Stud. Pure Math. 17, 97-137 (1989; Zbl 0739.11045)] showed how to define a Selmer group \(S_A(K_{\infty})\), which is cofinitely generated as a \(\mathbb Z_pT\)-module. The \(\mu\)-invariant \(\mu(A)\) of \(S_A(K_{\infty})\) is positive if and only if the \(\mathbb Z_p\)-cotorsion group is infinite. The present paper studies the effect that finite submodules of \(A\) have on this \(\mu\)-invariant for the cyclotomic \(\mathbb Z_p\)-extension of \(K\). If \(\alpha\) is a finite submodule of \(A\), the author defines an integer \(\delta(\alpha)\) that is a lower bound for \(\mu(A)\). Let \(m(A)\) be the maximum value of \(\delta(\alpha)\) for all such \(\alpha\subset A\). It was shown by \textit{B. Perrin-Riou} [in Algebraic number theory -- in honor of K. Iwasawa, Adv. Stud. Pure Math. 17, 347-358 (1989; Zbl 0747.11056)] how \(\mu(A)\) behaves under isogenies. The present author shows that \(m(A)\) behaves similarly; namely, under certain hypotheses, if there is an isogeny from \(A\) to \(B\) with kernel \(\alpha\), then \(\mu(A)-\mu(B)=m(A)-m(B)=\delta(\alpha)\). Moreover, \(A\) is isogenous to some \(B\) with \(m(A)=0\). The hope is that \(\mu(A)=m(A)\) when \(A\) is defined over \(\mathbb Q\), which would mean that positive \(\mu\)-invariants could be explained completely by finite submodules of \(A\). The author discusses the case when \(A\) is not defined over \(\mathbb Q\), which is more complicated than when \(A\) is defined over \(\mathbb Q\) since there can be finite submodules of the representation induced to \(\mathbb Q\) that are not induced from finite submodules defined over the number field. He uses this fact to g…
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1753715
zbMATH Keywords: Selmer groups
zbMATH Keywords: Iwasawa theory
reviewed by: Lawrence C. Washington
MaRDI profile type: Publication
full work available at URL: jnth.2001.2717
OpenAlex ID: W1979947251
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DOI: 10.1006/JNTH.2001.2717
description: scientific article; zbMATH DE number 1753715