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 μp. 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.
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