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 $i\equiv 1\mathrm{mod}[F({\mu }_p):F].$ This existence is fulfilled for $(p,i)$-regular fields.

A modified version, 24 pages