### Abstract

Let p be an odd prime satisfying Vandiver's conjecture. We consider two objects, the Galois group X of the maximal unramified abelian pro-p extension of the compositum of all Z _{p} -extensions of Q(μ _{p} ) and the Galois group G of the maximal unramified pro-p extension of Q μ_{p}. We give a lower bound for the height of the annihilator of X as an Iwasawa module. Under some mild assumptions on Bernoulli numbers, we provide a necessary and sufficient condition for G to be abelian. The bound and the condition in the two results are given in terms of special values of a cup product pairing on cyclotomic p-units. We obtain in particular that, for p < 1,000, Greenberg's conjecture that X is pseudo-null holds and G is in fact abelian.

Original language | English (US) |
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Pages (from-to) | 297-308 |

Number of pages | 12 |

Journal | Mathematische Annalen |

Volume | 342 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2008 |

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### ASJC Scopus subject areas

- Mathematics(all)