Mathematische Annalen

, Volume 342, Issue 2, pp 297–308

On Galois groups of unramified pro-p extensions



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 Zp-extensions of Qp) and the Galois group \({\mathfrak{G}}\) of the maximal unramified pro-p extension of Q\({(\mu_{p^{\infty}})}\). 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 \({\mathfrak{G}}\) 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 \({\mathfrak{G}}\) is in fact abelian.


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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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