Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



On the failure of pseudo-nullity of Iwasawa modules

Authors: Yoshitaka Hachimori and Romyar T. Sharifi
Journal: J. Algebraic Geom. 14 (2005), 567-591
Published electronically: March 24, 2005
MathSciNet review: 2129011
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Abstract: Consider the family of CM-fields which are pro-$p$ $p$-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic $\mathbf {Z}_p$-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-$p$ extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given $p$-adic Lie groups. The proof uses Kida’s formula for the growth of $\lambda$-invariants in cyclotomic ${\mathbf Z}_p$-extensions of CM-fields. In fact, we give a new proof of Kida’s formula which includes a slight weakening of the usual $\mu = 0$ assumption. This proof uses certain exact sequences involving Iwasawa modules in procyclic extensions. These sequences are derived in an appendix by the second author.

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Yoshitaka Hachimori
Affiliation: CICMA, Department of Mathematics and Statistics, Concordia University, Montréal, Québec H3G 1M8, Canada

Romyar T. Sharifi
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada

Received by editor(s): June 27, 2004
Published electronically: March 24, 2005
Additional Notes: The first author was partially supported by Gakushuin University and the 21st Century COE Program at the Graduate School of Mathematical Sciences of the University of Tokyo. The second author was supported by the Max Planck Institute for Mathematics.