Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

Fabio Caldarola[a]

Invariants and coinvariants of class groups in \(\mathbb{Z}_p\)-extensions and Greenberg's Conjecture

Pages: 181-192
Received: 4 January 2016
Accepted in revised form: 23 February 2016
Mathematics Subject Classification (2010): 11R23, 11R29.
Keywords: Iwasawa modules, \(\mathbb{Z}_p\)-extensions, class groups, capitulation of ideals, lower central series.
Author address:
[a] : University of Calabria, Department of Mathematics and Computer Science, Cubo 31/B, ponte Pietro Bucci, Arcavacata di Rende, 87036, Italy

Abstract: Let \(K/k\) be a \(\mathbb{Z}_p\)-extension of a number field \(k\), \(k_n\) its \(n\)-th layer and \(A_n\) the \(p\)-class group of \(k_n\). In this paper we give two criteria, both based on the group of invariants \(B_n\) of \(A_n\), which imply the finiteness of the Iwasawa module \(X(K/k)\) and we discuss some of their consequences. The first criterion deals with stabilization and capitulation of the \(B_n\), while the second one uses the nilpotency of the Galois group \({\rm Gal}(L(K)/k)\), where \(L(K)\) is the maximal unramified abelian pro-\(p\)-extension of \(K\).


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