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{{Orphan|date=February 2012}}
In [[number theory]], '''Iwasawa theory''' is a [[Galois module]] theory of [[ideal class group]]s, started by [[Kenkichi Iwasawa]], in the 1950s, as part of the theory of [[cyclotomic field]]s. In the early 1970s, [[Barry Mazur]] thought about generalizations of Iwasawa theory to Abelian Varieties. Later, in the early 90s, [[Ralph Greenberg]] has suggested an Iwasawa theory for [[motive (algebraic geometry)|motives]].

==Formulation==

The first thing Iwasawa noticed was that there are towers of fields in [[algebraic number theory]], having [[Galois group]] isomorphic with the additive group of [[p-adic integer]]s. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature [[pro-finite group]]s). The group <math> \Gamma</math> is the [[inverse limit]] of the additive groups <math> \mathbf Z/p^n \mathbf  Z </math>, where ''p'' is the fixed [[prime number]] and <math> n = 1,2, \dots </math>. We can express this by [[Pontryagin duality]] in another way: Γ is dual to the discrete group of all <math> p</math>-power [[roots of unity]] in the [[complex number]]s.

==Example==

Let <math>\zeta</math> be a primitive <math> p</math>-th root of unity and look at the following tower of number fields:

:<math> K = \mathbf{Q} (\zeta) \subset K_{1} \subset K_{2} \subset \cdots \subset \mathbf{C}, </math>

where <math> K_{n} </math> is the field generated by a primitive <math> p^{n+1}</math>-th root of unity. This tower of fields has a union <math> L</math>. Then the Galois group of <math> L </math> over <math> K </math> is isomorphic with <math> \Gamma</math>; because the Galois group of <math> K_n</math> over <math> K</math> is <math> \mathbf Z/p^n \mathbf Z </math>. In order to get an interesting Galois module here, Iwasawa took the ideal class group of <math> K_{n} </math>, and let <math> I_n </math> be its <math> p</math>-torsion part. There are [[field norm|norm]] mappings <math> I_m \rightarrow I_n </math> when <math> m > n </math>, and so an inverse system. Letting <math> I</math> be the [[inverse limit]], we can say that <math> \Gamma </math> acts on <math> I </math>, and it is good to have a description of this action.

The motivation here was undoubtedly that the <math> p</math>-torsion in the ideal class group of <math> K</math>  had already been identified by [[Kummer]] as the main obstacle to the direct proof of [[Fermat's last theorem]]. What Iwasawa did that was new, was to go 'off to infinity' in a new direction. In fact, <math> I</math>  is a [[module (mathematics)|module]] over the [[group ring]] <math> \mathbf Z_p [[\Gamma]] </math>. This is a well-behaved ring ([[Regular local ring|regular]] and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.

==History==

From this beginning, in the 1950s, a good-sized theory has been built up. A basic connection was noticed between the module theory, and the [[p-adic L-function]]s that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the [[Bernoulli number]]s, and use [[interpolation]] to define p-adic analogues of the [[Dirichlet L-function]]s. It became clear that the theory might be able to move ahead finally from Kummer's century-old results on [[regular prime]]s.

The '''main conjecture of Iwasawa theory''' was formulated as an assertion that two ways of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was eventually proved by [[Barry Mazur]] and [[Andrew Wiles]] for '''Q''', and for all [[totally real number field]]s by Andrew Wiles. These proofs were modeled upon [[Ken Ribet]]'s proof of the converse to Herbrand's theorem (so-called [[Herbrand-Ribet theorem]]).

More recently, also modeled upon Ribet's method, Chris Skinner and Eric Urban have announced a proof of a ''main conjecture'' for GL(2). An easier proof of the Mazur-Wiles theorem can be found by using [[Euler system]]s as developed by [[Kolyvagin]] (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been found by [[Karl Rubin]], amongst others.

== References ==

* Greenberg, Ralph, ''Iwasawa Theory - Past & Present'', Advanced Studies in Pure Math. 30 (2001), 335-385. Available at [http://www.math.washington.edu/~greenber/iwhi.ps].
* Coates, J. and Sujatha, R., ''Cyclotomic Fields and Zeta Values'', Springer-Verlag, 2006
* Lang, S., ''Cyclotomic Fields'', Springer-Verlag, 1978
* Washington, L., ''Introduction to Cyclotomic Fields, 2nd edition'', Springer-Verlag, 1997
* {{cite journal | author = [[Barry Mazur]] and [[Andrew Wiles]]| year = 1984 |  title = ''Class Fields of Abelian Extensions of Q'' | journal = Inventiones Mathematicae | volume = 76 | issue = 2 | pages = 179-330 }}
* {{cite journal | author = [[Andrew Wiles]]| year = 1990 |  title = ''The Iwasawa Conjecture for Totally Real Fields'' | url = https://archive.org/details/sim_annals-of-mathematics_1990-05_131_3/page/493| journal = Annals of Mathematics | volume = 131 | issue = 3 | pages = 493-540 }}
* {{cite journal | author = [[Chris Skinner]] and [[Eric Urban]]| year = 2002 |  title = ''Sur les deformations p-adiques des formes de Saito-Kurokawa'' | journal = C. R. Math. Acad. Sci. Paris | volume =335 | issue = 7 | pages = 581-586 }}

[[Category:Number theory]]