I am currently trying to learn Iwasawa theory and am following J. Coates and R. Sujatha's book 'Cyclotomic Fields and Zeta Values'. The setup is the following:
- Let $\mathcal{F}_n:=\mathbb{Q}(\mu_{p^{n+1}})$ where $\mu_{p^{n+1}}$ is the group of all $p^{n+1}$-th roots of unity in some fixed algebraic closure of $\mathbb{Q}$.
- Let $F_n:=\mathbb{Q}(\mu_{p^{n+1}})^+$ be the maximal real subfield of $\mathcal{F}_n$, which is defined to be the fixed field of $\mathcal{F}_n\subseteq\mathbb{C}$ under complex conjugation.
- Set $\mathcal{F}_\infty:=\mathbb{Q}(\mu_{p^\infty})$ where $\mu_{p^\infty}$ is the group of all $p$-th power roots of unity.
- $G:=Gal(F_\infty|\mathbb{Q})$ is abelian.
- $\Lambda(G):=\varprojlim\mathbb{Z}_p[G/H]$ where $H$ runs over all open subgroups of $G$, is the Iwasawa algebra of $G$.
- $\mathcal{M}_\infty:=$ maximal abelian $p$-extension of $\mathcal{F}_n$ which is unramified everywhere and $M_\infty$ is the same extension but for $F_\infty$.
- We define $X_\infty:=Gal(M_\infty|F_\infty)$ and know that it is a finitely generated $\Lambda(G)$-torsion module.
- $\mu$ is a pseudo-measure on $G$, defined to be an element of the ring of fractions of $\Lambda(G)$ such that $(g-1)\mu\in\Lambda(G)\,\forall g\in G$. Then, the integral $\int_G\nu d\mu$ is well-defined for any non-trivial continuous homomorphism $\nu:G\to \mathbb{Z}_p^\times$.
- By Theorem 1.4.2., $\exists!$ pseudo-measure $\zeta_p$ on $G$ such that \begin{equation*} \int_G\chi(g)^kd\zeta_p=(1-p^{k-1})\zeta(1-k) \end{equation*}for $\chi$ the cyclotomic character and $\zeta$ the Riemann-Zeta function.
- Define $I(G):=\ker(\epsilon)$ where $\epsilon:\Lambda(G)\to \mathbb{Z}_p$ is the augmentation map. With $\zeta_p$ a pseudo-measure, $I(G)\zeta_p\vartriangleleft \lambda(G)$ is an ideal in $\Lambda(G)$.
The main conjecture now tells me that the characteristic ideal of $X_\infty$, denoted by $ch_G(X_\infty)$, is in relation to $I(G)\zeta_p$: \begin{equation*} ch_G(X_\infty)=I(G)\zeta_p \end{equation*} I see that the LHS is arithmetic and the RHS is analytic, but I do not see what exactly the characteristic ideal tells me about the extension $F_\infty$, or rather $\mathcal{F}_\infty$.
If anyone knows anything, I'd be happy to hear your suggestions.