Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives.
Formulation.
Iwasawa worked with so-called formula_1-extensions: infinite extensions of a number field formula_2 with Galois group formula_3 isomorphic to the additive group of p-adic integers for some prime "p". Every closed subgroup of formula_4 is of the form formula_5 so by Galois theory, a formula_1-extension formula_7 is the same thing as a tower of fields 
such that formula_9 Iwasawa studied classical Galois modules over formula_10 by asking questions about the structure of modules over formula_11
More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.
Example.
Let formula_12 be a prime number and let formula_13 be the field generated over formula_14 by the formula_12th roots of unity. Iwasawa considered the following tower of number fields:
where formula_17 is the field generated by adjoining to formula_18 the "p""n"+1-st roots of unity and 
The fact that formula_20 implies, by infinite Galois theory, that formula_21 In order to get an interesting Galois module, Iwasawa took the ideal class group of formula_17, and let formula_23 be its "p"-torsion part. There are norm maps formula_24 whenever formula_25, and this gives us the data of an inverse system. If we set 
then it is not hard to see from the inverse limit construction that formula_27 is a module over formula_28 In fact, formula_29 is a module over the Iwasawa algebra formula_30. 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 formula_31
The motivation here is that the "p"-torsion in the ideal class group of formula_18 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 for formula_14 and for all totally real number fields by . 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 and , 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 .