A consequence of Greenberg's generalized conjecture on Iwasawa invariants of ZpZp-extensions

For a prime number p and a number field k  , let k˜ be the compositum of all ZpZp-extensions of k  . Greenberg's Generalized Conjecture (GGC) claims the pseudo-nullity of the unramified Iwasawa module X(k˜) of k˜. It is known that, when k   is an imaginary quadratic field, GGC has a consequence on the Iwasawa invariants associated to ZpZp-extensions of k. In this paper, we partially generalize it to arbitrary number fields k.