Main conjecture of Iwasawa theory (Mazur–Wiles)

Let $p$ be a prime, $F_n = \mathbb{Q}(\mu_{p^{n+1}})$, $H_\infty = \varprojlim A_n$ the inverse limit of the $p$-parts of the class groups of $F_n$ under the norm maps, $V = H_\infty \otimes_{\mathbb{Z}_p} \mathbb{Q}_p$, and $V_i$ the $\omega^i$-eigenspace of $V$, where $\omega$ is the Teichmüller character. Let $\gamma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ and $u \in \mathbb{Z}_p^\times$ satisfy $\gamma(\zeta) = \zeta^u$ for every $p$-power root of unity $\zeta$, and suppose $u$ topologically generates $1 + p\mathbb{Z}_p$ (so $\gamma$ restricts to a topological generator of $\Gamma = \mathrm{Gal}(F_\infty/F_0)$). Let

be the characteristic polynomial of $\gamma$ on $V_i$ in the Iwasawa variable $T = \gamma - 1$.

Theorem (Mazur–Wiles). If $i$ is an odd integer with $i \not\equiv 1 \pmod{p-1}$, then there is a power series $G_p(\omega^{1-i}, T) \in \mathbb{Z}_p[[T]]$ with

Moreover $h_p(\omega^i, T)$ has coefficients in $\mathbb{Z}_p$, and

as ideals of $\mathbb{Z}_p[[T]]$.

This identifies the characteristic polynomial of the Iwasawa module with the $p$-adic $L$-function. The Herbrand–Ribet theorem and the Gras conjecture are consequences.

Formalization Note The power series $G_p$ is stated existentially, and a power series with this interpolation property is unique. $h_p \in \mathbb{Q}_p[T]$ is compared with $G_p$ through its (unique) preimage in $\mathbb{Z}_p[T]$. Because $h_p$ is defined to be $0$ if $V_i$ were infinite-dimensional, the statement also asserts that $V_i$ is finite-dimensional. The source describes $h_p$ as "the characteristic polynomial of $\gamma$ acting on $V_i$"; for the stated ideal equality to hold, the polynomial must be read in the variable $T = \gamma - 1$ of $\mathbb{Z}_p[[T]] \cong \mathbb{Z}_p[[\Gamma]]$, i.e. as $\det(T - (\gamma - 1))$. Taken literally, $\det(T - \gamma)$ has a unit constant term and would generate the unit ideal.