On the failure of pseudo-nullity of Iwasawa modules

Authors:
Yoshitaka Hachimori and Romyar T. Sharifi

Journal:
J. Algebraic Geom. **14** (2005), 567-591

DOI:
https://doi.org/10.1090/S1056-3911-05-00396-6

Published electronically:
March 24, 2005

MathSciNet review:
2129011

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Abstract | References | Additional Information

Abstract: Consider the family of CM-fields which are pro- -adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic -extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro- extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given -adic Lie groups. The proof uses Kida's formula for the growth of -invariants in cyclotomic -extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weakening of the usual 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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Additional Information

**Yoshitaka Hachimori**

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

Email:
yhachi@mathstat.concordia.ca

**Romyar T. Sharifi**

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

Email:
sharifi@math.mcmaster.ca

DOI:
https://doi.org/10.1090/S1056-3911-05-00396-6

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.