On Galois groups of unramified pro-p extensions
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 Zp-extensions of the pth cyclotomic field and the Galois group G of the unramified pro-p extension of the cyclotomic field of all p-power roots of unity. 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 the special values of a cup product pairing on cyclotomic p-units. We obtain, in particular, that for p less than 1000, Greenberg's conjecture on the pseudo-nullity of X holds and G is in fact abelian.