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
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.
[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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