On Greenberg's generalized conjecture for families of number fields

For a number field k and an odd prime p, let k̃ be the compositum of all the Zₚ-extensions of k, Λ ̃ the associated Iwasawa algebra, and X(k̃) the Galois group over k̃ of the maximal abelian unramified pro-p-extension of k̃. Greenberg's generalized conjecture (GGC for short) asserts that the Λ̃-module X(k̃) is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special Z²ₚ-extension of k, and these conditions are partially or entirely fullfilled by certain families of number fields.