On Greenberg's generalized conjecture

Abstract

For a number field $F$ and an odd prime number $p,$ let $\tilde{F}$ be the compositum of all $\mathbb{Z}_p$-extensions of $F$ and $\tilde{\Lambda}$ the associated Iwasawa algebra. Let $G_{S}(\tilde{F})$ be the Galois group over $\tilde{F}$ of the maximal extension which is unramified outside $p$-adic and infinite places. In this paper we study the $\tilde{\Lambda}$-module $\mathfrak{X}_{S}^{(-i)}(\tilde{F}):=H_1(G_S(\tilde{F}), \mathbb{Z}_p(-i))$ and its relationship with $X(\tilde{F}(\mu_p))(i-1)^\Delta,$ the $\Delta:=\mathrm{Gal}(\tilde{F}(\mu_{p})/\tilde{F})$-invariant of the Galois group over $\tilde{F}(\mu_{p})$ of the maximal abelian unramified pro-$p$-extension of $\tilde{F}(\mu_{p}).$ More precisely, we show that under a decomposition condition, the pseudo-nullity of the $\tilde{\Lambda}$-module $X(\tilde{F}(\mu_p))(i-1)^\Delta$ is implied by the existence of a $\mathbb{Z}_{p}^d$-extension $L$ with $\mathfrak{X}_{S}^{(-i)}(L):=H_1(G_S(L), \mathbb{Z}_p(-i))$ being without torsion over the Iwasawa algebra associated to $L,$ and which contains a $\mathbb{Z}_{p}$-extension $F_{\infty}$ satisfying $H^2(G_{S}(F_{\infty}),\mathbb{Q}_{p}/\mathbb{Z}_p(i))=0.$ As a consequence we obtain a sufficient condition for the validity of Greenberg's generalized conjecture when the integer $i\equiv 1 \mod{[F(\mu_p):F]}.$ This existence is fulfilled for $(p, i)$-regular fields.