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On Galois groups of unramified pro-p extensions

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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 \({\mathfrak{G}}\) 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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  1. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada

    Romyar T. Sharifi

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  1. Romyar T. Sharifi
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Correspondence to Romyar T. Sharifi.

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Sharifi, R.T. On Galois groups of unramified pro-p extensions. Math. Ann. 342, 297–308 (2008). https://doi.org/10.1007/s00208-008-0236-1

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  • DOI: https://doi.org/10.1007/s00208-008-0236-1

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