**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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