On Greenberg's generalized conjecture

TL;DR

This paper investigates the structure of certain Iwasawa modules over number fields and establishes conditions under which Greenberg's generalized conjecture holds, especially for regular primes and specific integer congruences.

Contribution

It introduces new criteria linking the torsion properties of Iwasawa modules over larger extensions to Greenberg's conjecture, extending previous results in Iwasawa theory.

Findings

Pseudo-nullity of certain modules implies conjecture validity.

Existence of specific $bZ_p^d$-extensions ensures torsion-free modules.

Results apply to $(p,i)$-regular fields and particular congruences.

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.