Weak Greenberg's generalized conjecture for imaginary quadratic fields

TL;DR

This paper investigates a weak form of Greenberg's conjecture for imaginary quadratic fields where an odd prime splits, proving it under specific conditions related to Iwasawa invariants and characteristic ideals.

Contribution

It proves a weak form of Greenberg's generalized conjecture for certain imaginary quadratic fields under new assumptions involving Iwasawa invariants and characteristic ideal properties.

Findings

Proved the weak Greenberg conjecture under vanishing Iwasawa lambda-invariant.

Established the conjecture when the characteristic ideal has a square-free generator.

Extended understanding of Iwasawa modules in the context of imaginary quadratic fields.

Abstract

Let $p$ be an odd prime number and $k$ an imaginary quadratic field in which $p$ splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for $p$ and $k$, which states that the non-trivial Iwasawa module of the maximal multiple $\mathbb{Z}_p$-extension field over $k$ has a non-trivial pseudo-null submodule. We prove this conjecture for $p$ and $k$ under the assumption that the Iwasawa $\lambda$-invariant for a certain $\mathbb{Z}_p$-extension over a finite abelian extension of $k$ vanishes and that the characteristic ideal of the Iwasawa module associated to the cyclotomic $\mathbb{Z}_p$-extension over $k$ has a square-free generator.