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

Abstract

For a CM-field $K$ and an odd prime number $p$, let ${\tilde{K}}^{\prime }$ be a certain multiple ${\mathbb{Z}}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{{\tilde{K}}^{\prime }}$ of ${\tilde{K}}^{\prime }$ as a ${\mathbb{Z}}_p[[\mathrm{Gal}({\tilde{K}}^{\prime }/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{{\tilde{K}}^{\prime }}$ 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 $X_{\tilde{K}}$ is ${\mathbb{Z}}_p[[\mathrm{Gal}(\tilde{K}/K)]]$-cyclic under several assumptions on the Iwasawa $\lambda$-invariant and the ideal class group of $K$, where $\tilde{K}$ is the ${\mathbb{Z}}_p^2$-extension of $K$.

v2: 24pages. replace Lemma 3.1 and the proof, typos corrected