A consequence of Greenberg's generalized conjecture on Iwasawa invariants of $\mathbb{Z}_p$-extensions

TL;DR

This paper explores the implications of Greenberg's Generalized Conjecture on Iwasawa invariants across various number fields, extending previous results known for imaginary quadratic fields to a broader class.

Contribution

It generalizes the known consequences of GGC from imaginary quadratic fields to arbitrary number fields, broadening the scope of Iwasawa theory implications.

Findings

Partial generalization of GGC consequences to all number fields

Insights into Iwasawa invariants for $bZ_p$-extensions

Extension of previous results beyond quadratic fields

Abstract

For a prime number $p$ and a number field $k$, let $\tilde{k}$ be the compositum of all $\mathbb{Z}_p$-extensions of $k$. Greenberg's Generalized Conjecture (GGC) claims the pseudo-nullity of the unramified Iwasawa module $X(\tilde{k})$ of $\tilde{k}$. It is known that, when $k$ is an imaginary quadratic field, GGC has a consequence on the Iwasawa invariants associated to $\mathbb{Z}_p$-extensions of $k$. In this paper, we partially generalize it to arbitrary number fields $k$.