### 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 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 λ-invariants in cyclotomic 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 μ = 0 assumption. This proof uses certain exact sequences involving Iwasawa modules in pro-cyclic extensions. These sequences are derived in an appendix by the second author.

Original language | English (US) |
---|---|

Pages (from-to) | 567-591 |

Number of pages | 25 |

Journal | Journal of Algebraic Geometry |

Volume | 14 |

Issue number | 3 |

State | Published - Jul 2005 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Algebraic Geometry*,

*14*(3), 567-591.

**On the failure of pseudo-nullity of Iwasawa modules.** / Hachimori, Yoshitaka; Sharifi, Romyar T.

Research output: Contribution to journal › Article

*Journal of Algebraic Geometry*, vol. 14, no. 3, pp. 567-591.

}

TY - JOUR

T1 - On the failure of pseudo-nullity of Iwasawa modules

AU - Hachimori, Yoshitaka

AU - Sharifi, Romyar T

PY - 2005/7

Y1 - 2005/7

N2 - 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 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 λ-invariants in cyclotomic Zp-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weakening of the usual μ = 0 assumption. This proof uses certain exact sequences involving Iwasawa modules in pro-cyclic extensions. These sequences are derived in an appendix by the second author.

AB - 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 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 λ-invariants in cyclotomic Zp-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weakening of the usual μ = 0 assumption. This proof uses certain exact sequences involving Iwasawa modules in pro-cyclic extensions. These sequences are derived in an appendix by the second author.

UR - http://www.scopus.com/inward/record.url?scp=20444452542&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=20444452542&partnerID=8YFLogxK

M3 - Article

VL - 14

SP - 567

EP - 591

JO - Journal of Algebraic Geometry

JF - Journal of Algebraic Geometry

SN - 1056-3911

IS - 3

ER -