# Galois coinvariants of the unramified Iwasawa modules of multiple $\mathbb{Z}_p$-extensions

## TL;DR

This paper investigates the structure and properties of unramified Iwasawa modules over multiple zp-extensions of CM-fields, providing formulas for Galois coinvariants and criteria for cyclicity in special cases.

## Contribution

It offers new descriptions of Galois coinvariants and conditions for cyclicity of Iwasawa modules in multiple zp-extensions, extending prior understanding of their structure.

## Key findings

- Order of Galois coinvariants expressed via characteristic power series
- Necessary and sufficient conditions for cyclicity of Iwasawa modules
- Results depend on splitting behavior of primes and Iwasawa invariants

## Abstract

For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second one is that if $K$ is an imaginary quadratic field and $p$ does not split in $K$, we give a necessary and sufficient condition for which $X_{\widetilde K}$ is $\mathbb{Z}_p[[{\rm Gal}(\widetilde K/K)]]$-cyclic under several assumptions on the Iwasawa $\lambda$-invariant and the ideal class group of $K$, where $\widetilde K$ is the $\mathbb{Z}_p^2$-extension of $K$.