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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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