On Greenberg's generalized conjecture for families of number fields

TL;DR

This paper investigates Greenberg's generalized conjecture in Iwasawa theory, showing that for imaginary number fields, the conjecture can be deduced from specific pseudo-nullity conditions on certain extensions, with these conditions satisfied by some number field families.

Contribution

It establishes a link between Greenberg's generalized conjecture and pseudo-nullity conditions on ${f Z}_p^2$-extensions for imaginary fields, providing new theoretical insights.

Findings

GGC is implied by pseudo-nullity conditions on ${f Z}_p^2$-extensions.

Certain families of number fields satisfy these pseudo-nullity conditions.

The paper advances understanding of GGC in the context of imaginary 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.