On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple $\mathbb{Z}_p$-Extensions Over Imaginary Quadratic Fields

TL;DR

This paper investigates the cyclicity of unramified Iwasawa modules over maximal multiple al extensions of imaginary quadratic fields, providing computational methods and examples without relying on Greenberg's conjecture.

Contribution

The authors develop new methods for computing and analyzing the cyclicity of Iwasawa modules in this setting, removing the need for certain conjectural assumptions.

Findings

Established criteria for cyclicity of Iwasawa modules

Provided explicit computational techniques and numerical examples

Demonstrated results without assuming Greenberg's generalized conjecture

Abstract

For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.