On Greenberg's generalized conjecture for families of number fields

Abstract

For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{\Lambda }$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the maximal abelian unramified pro-$p$-extension of $\tilde{k}$. Greenberg's generalized conjecture (GGC for short) asserts that the $\tilde{\Lambda}$-module $X(\tilde{k})$ 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 ${\mathbb Z}^2_p$-extension of $k$, and these conditions are partially or entirely fullfilled by certain families of number fields.