$\mathbb{Z}_p$-extensions of CM-fields

Question

I am trying to prove some consequences of Iwasawa's Theorem for CM-fields. There is a sequence of CM-fields $$K=K_0\subseteq K_1 \subseteq \dots \subseteq K_\infty$$ so that $K_\infty/K$ is a $\mathbb{Z}_p$-extension for a prime $p$. So I think that $[K_{n+1}:K_n]$ must be some $p$-power for all $n\geq0$. Since each $K_n$ is a CM-field, there exist subfields $K_n^+$ which are totally real, so that $[K_n:K_n^+]=2$.
What can I say about the maximal unramified abelian extension $K_\infty/K$? How can I prove that $K_\infty$ also is a CM-field? (I need this result to talk about $K_\infty^+/K^+$.)

Answer 1

You can define a CM-field number $K$ as being a totally imaginary quadratic extension of a totally real field $k$. Then it can be shown that complex conjugation on $\mathbf C$ induces an automorphism on $K$ which is independent of the embedding of $K$ into $\mathbf C$ (see e.g. Washington, chap. 4, just before thm. 4.10). By abuse of language, this is called « the » complex conjugation on $K$, and its fixed field $k$ is denoted $K^+$ for obvious reason. Now consider a $\mathbf Z_p$-extension $K_\infty/K$, i.e. an infinite Galois extension whose Galois group is isomorphic to the additive group $\mathbf Z_p$ = inverse limit of $(\mathbf Z /p^n \mathbf Z$ , +). By definition and Galois theory, $K_\infty$ is the direct limit (actually, the union here) of the fields $K_n$ fixed by the subgroups of $\mathbf Z_p$ of index $p^n$ . Note that $K_n/K$ is the unique subextension of degree $p^n$, and $K_{n+1}/K_n$ has degree $p$. Moreover, if all the $K_n$ ‘s are CM, so is obviously the union $K_\infty$.

You ask what can be said of the maximal abelian unramified extension of $K_\infty$. Almost by defintion, it’s the direct limit of the maximal abelian unramified extensions of the $K_n$ ‘s, so by class field theory, its Galois group $X$ over $K_\infty$ is isomorphic to the inverse limit of the ideal class groups $C_n$ of the $K_n$ ‘s. The interesting point would be to describe the action of $\Gamma = Gal(K_\infty/K)$ on $X$. When decomposing $X$ into its (pro)-$l$- Sylow subgroups $X(l)$, a few things can be said if $l \neq p$ (see e.g. Washington, chap. ). But the heart of Iwasawa’s « business » is about the structure of the noetherian compact torsion $\mathbf Z_p[[\Gamma]]$- module $X(p)$. A general algebraic structure theorem is known for such modules (Wash., thm. 13.12), which generalizes in some way the classical description of finite abelian $p$-groups. By descent (i.e. by taking co-invariants under $\Gamma$), one obtains the celebrated Iwasawa formulae giving the asymptotic orders of the $p$-Sylow subgroups of the $C_n$ ‘s ( Wash. , thm. 13. 13).

What we have seen is just the algebraic part of the theory. The most interesting information actually concerns the precise relationship of the module $X(p)$ with the $p$-adic L-functions attached to totally real number fields. This is the so-called Main Conjecture, the « simplest » version of which is the Mazur-Wiles theorem : let $K$ be CM, abelian over $\mathbf Q$, of degree prime to $p$ ; let $K_\infty$ be the cyclotomic $\mathbf Z_p$-extension of $K$ (not just any $\mathbf Z_p$-extension); for any non trivial even character of Gal$(K/\mathbf Q)$, say $\chi$, let $X (\omega \chi^-1)$ (where $\omega$ is the Teichmûller character) be the $\omega \chi^-1$ part of $X(p)$, and let $f_\chi$ (T) be « the » « characteristic power series » associated to $X (\omega \chi^-1)$ via the previous structure theorem ; then $f_\chi ((1+p)s – 1) = L_p (\chi, s)$ for all $s$ in $\mathbf Z_p$ (see e.g. Wash., chapter 16). Note that this descibes only the « minus part » (inverted by complex conjugation) of $X(p)$, the « plus part » (fixed by complex conjugation) being still unknown. For example, if $K$ is the $p$-th cyclotomic field, the famous Vandiver conjecture is equivalent to the vanishing of the plus part of $X(p)$.