12. The Iwasawa Main Conjecture
We now move from arithmetic to algebra. To state the Iwasawa Main Conjecture we
use the structure theory of \Lam-modules. We first summarise that theory, then
introduce the \Lam-modules coming from the Galois theory of abelian extensions.
These modules carry an action of the Galois group
\GG = \Gal(F_\infty/\Q) \cong \Zpx, and so become \Lam(\GG)-modules. The
Main Conjecture describes the characteristic ideal of one of these Galois
modules in terms of the Kubota–Leopoldt p-adic L-function \zeta_p.
Throughout, p is an odd prime, L/\Qp is a finite extension with ring of
integers \cO_L and maximal ideal \mathfrak{p}, and
\Lam = \Lam(\Zp) = \varprojlim_n \cO_L[\Zp/p^n\Zp] \cong \cO_L[[T]] is the
Iwasawa algebra of \Zp over \cO_L. We write F_n = \Q(\mu_{p^n}) and
F_n^+ for its maximal totally real subfield, F_\infty = \cup_n F_n,
F_\infty^+ = \cup_n F_n^+, and \GG^+ = \Gal(F_\infty^+/\Q).