  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 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.
