Number theory Algebraic number theory Theorem

Iwasawa main conjecture

Builds on Iwasawa theory Cyclotomic field Ideal class group p-adic integer Dirichlet L-function Dirichlet character Regular prime Bernoulli number

Statement

Let $p$ be an odd prime. Let $K_n = \Q(\zeta_{p^{n+1}})$ for $n \ge 0$ be the cyclotomic fields of $p$ -power roots of unity, with union $K_\infty$ . Let $A_n$ be the $p$ -part of the ideal class group 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, acting on $X$ through $\Gal(K_\infty/K_0)$ as in Iwasawa theory. 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. These are attached to powers of $\omega$ viewed as Dirichlet characters, and the values are given by generalised Bernoulli numbers. 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, $p$ 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 $\Lambda$ .

Going deeper

For a student

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.

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Tag	`67A4D4FAYJ`
URI	`m2://67A4D4FAYJ`
Kind	statement, theorem
Writing	W0 stub
MSC	11R23, 11R42
Status	This copy of the site has no build data. Run `m2 status 67A4D4FAYJ`.
About	Iwasawa theory
Uses	Iwasawa theory, Cyclotomic field, Ideal class group, p-adic integer, Dirichlet L-function, Dirichlet character, Regular prime, Bernoulli number
Same as	Wikidata Q6736209 Wikipedia Main conjecture of Iwasawa theory
Changed	2 November 2026

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