---
tag: 67A4D4FAYJ
kind: statement
role: theorem
name: Iwasawa main conjecture
msc: [11R23, 11R42]
uses: [Y2SPPSQ5D4, NN77VYSGXR, WQ46BD3ARY, V6TJAFZ4MD, 9D2Z543EJQ, 7XB5VJKDPG, B4VK33PCXZ, 0KC00K7KY4]
about: [Y2SPPSQ5D4]
sources:
  - url: https://en.wikipedia.org/wiki/Main_conjecture_of_Iwasawa_theory
same_as:
  wikidata: Q6736209
  wikipedia: Main conjecture of Iwasawa theory
---

## Statement

Let $p$ be an odd prime.
Let $K_n = \Q(\zeta_{p^{n+1}})$ for $n \ge 0$ be the [[cyclotomic fields|NN77VYSGXR]] of $p$-power roots of unity, with union $K_\infty$.
Let $A_n$ be the $p$-part of the [[ideal class group|WQ46BD3ARY]] of $K_n$, and let $X = \varprojlim_n A_n$, the inverse limit under the norm maps.
Let $\Lambda = \Z_p[\![T]\!]$ be the Iwasawa algebra over the [[p-adic integers|V6TJAFZ4MD]], acting on $X$ through $\Gal(K_\infty/K_0)$ as in [[Iwasawa theory|Y2SPPSQ5D4]].
Let $\Delta = \Gal(K_0/\Q) \cong (\Z/p\Z)^\times$, and let $\omega \colon \Delta \to \Z_p^\times$ be the Teichmüller character.
For an integer $i$, let $X^{(i)}$ be the part of $X$ on which $\Delta$ acts by $\omega^i$.
Let $i$ be odd with $i \not\equiv 1 \pmod{p-1}$.
Then the $\Lambda$-module $X^{(i)}$ is finitely generated and torsion.
Its characteristic ideal is generated by the element of $\Lambda$ that interpolates the Kubota–Leopoldt $p$-adic $L$-function $L_p(s, \omega^{1-i})$.
Here the variable $T$ has its standard normalisation.

## Remarks

The $p$-adic $L$-function $L_p(s, \omega^{1-i})$ interpolates values at negative integers of [[Dirichlet L-functions|9D2Z543EJQ]].
These are attached to powers of $\omega$ viewed as [[Dirichlet characters|7XB5VJKDPG]], and the values are given by generalised [[Bernoulli numbers|0KC00K7KY4]].
So the statement links an algebraic object, built from class groups, to an analytic one, built from $L$-values.
Mazur and Wiles proved it using the geometry of modular curves, and Wiles later proved the analogue for totally real fields.
Rubin gave another proof using Kolyvagin's Euler systems of cyclotomic units.

Example of its content: by the structure theory of $\Lambda$-modules, $X^{(i)}$ is finite exactly when its characteristic ideal is all of $\Lambda$.
So $X^{(i)}$ is finite exactly when the interpolating power series has a constant term that is a $p$-adic unit.
When $p$ is a [[regular prime|B4VK33PCXZ]], $p$ does not divide the class number of $K_0$, and then all the $A_n$ are trivial.
Correspondingly, all the relevant power series are units of $\Lambda$.

## Explanations

### L1 human

The theorem says that two very different ways of measuring the same arithmetic agree.
One way counts ideal classes in the tower of fields generated by $p$-power roots of unity.
The other way uses $p$-adic versions of the special values of $L$-functions.
The agreement is exact, not only up to some error.

### L2 human

The result is a refinement of Kummer's criterion and of the Herbrand–Ribet theorem, which relate the $p$-divisibility of Bernoulli numbers to the class group of $\Q(\zeta_p)$.
Here the tower of all $p$-power cyclotomic fields is treated at once, with modules over $\Lambda$ in place of finite groups.
It is a model for many later main conjectures, for example for elliptic curves, where the class group is replaced by a Selmer group.