Iwasawa theory

Example

Let ${\displaystyle p}$ be a prime number and let ${\displaystyle K=\mathbb{Q}({\mu }_p)}$ be the field generated over ${\displaystyle \mathbb{Q}}$ by the ${\displaystyle p}$th roots of unity. Iwasawa considered the following tower of number fields:

${\displaystyle K=K_0\subset K_1\subset \cdots \subset K_{\mathrm{\infty }},}$

where ${\displaystyle K_n}$ is the field generated by adjoining to ${\displaystyle K}$ the pⁿ⁺¹-st roots of unity and

${\displaystyle K_{\mathrm{\infty }}=\bigcup K_n.}$

The fact that ${\displaystyle \mathrm{Gal}(K_n/K)\simeq \mathbb{Z}/p^n\mathbb{Z}}$ implies, by infinite Galois theory, that ${\displaystyle \mathrm{Gal}(K_{\mathrm{\infty }}/K)\simeq \underset{n}{\underleftarrow{\lim }}\mathbb{Z}/p^n\mathbb{Z}={\mathbb{Z}}_p.}$ In order to get an interesting Galois module, Iwasawa took the ideal class group of ${\displaystyle K_n}$, and let ${\displaystyle I_n}$ be its p-torsion part. There are norm maps ${\displaystyle I_m\to I_n}$ whenever ${\displaystyle m>n}$, and this gives us the data of an inverse system. If we set

${\displaystyle I=\underleftarrow{\lim }{I}_n,}$

then it is not hard to see from the inverse limit construction that ${\displaystyle I}$ is a module over ${\displaystyle {\mathbb{Z}}_p.}$ In fact, ${\displaystyle I}$ is a module over the Iwasawa algebra ${\displaystyle \mathrm{\Lambda }={\mathbb{Z}}_p[[\mathrm{\Gamma }]]}$. This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this description it is possible to recover information about the p-part of the class group of ${\displaystyle K.}$

The motivation here is that the p-torsion in the ideal class group of ${\displaystyle K}$ had already been identified by Kummer as the main obstruction to the direct proof of Fermat's Last Theorem.

Connections with p-adic analysis

From this beginning in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on regular primes.

Iwasawa formulated the main conjecture of Iwasawa theory as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by Mazur & Wiles (1984) for ${\displaystyle \mathbb{Q}}$ and for all totally real number fields by Wiles (1990). These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the so-called Herbrand–Ribet theorem).

Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systems, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary quadratic fields.

Generalizations

The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a main conjecture linking the tower to a p-adic L-function.

In 2002, Christopher Skinner and Eric Urban claimed a proof of a main conjecture for GL(2). In 2010, they posted a preprint (Skinner & Urban 2010).