In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, 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 90s), Ralph Greenberg has proposed an Iwasawa theory for motives. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense... In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... Kenkichi Iwasawa (岩澤 健吉 Iwasawa Kenkichi, September 11, 1917 - October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... Barry Mazur (born December 19, 1937) is a professor of mathematics at Harvard University. ... In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ...

Formulation

Iwasawa's starting observation was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers. 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 groups). The group Γ is the inverse limit of the additive groups , where p is the fixed prime number and . We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all p-power roots of unity in the complex numbers. This article or section does not cite its references or sources. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... The title given to this article is incorrect due to technical limitations. ... In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

Example

Let ζ be a primitive p-th root of unity and consider the following tower of number fields:

where K_n is the field generated by a primitive pⁿ⁺¹-th root of unity. This tower of fields has a union L. Then the Galois group of L over K is isomorphic with Γ because the Galois group of K_n over K is Z/pⁿZ. In order to get an interesting Galois module here, Iwasawa took the ideal class group of K_n, and let I_n be its p-torsion part. There are norm mappings when m > n, and so an inverse system. Letting I be the inverse limit, we can say that Γ acts on I, and it is desirable to have a description of this action. In mathematics, the (field) norm is a mapping defined in field theory, to map elements of a larger field into a smaller one. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ...

The motivation here was undoubtedly that the p-torsion in the ideal class group of K had already been identified by Kummer as the main obstruction to the direct proof of Fermat's last theorem. Iwasawa's originality was to go 'off to infinity' in a novel direction. In fact I is a module over the group ring . This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse. Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension. ...

History

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. In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ... In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ... In mathematics, a Dirichlet L-series, named in honour of Johann Peter Gustav Lejeune Dirichlet, is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1. ... In mathematics, regular primes are a certain kind of prime numbers. ...

The main conjecture of Iwasawa theory was formulated 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 eventually proved by Barry Mazur and Andrew Wiles for Q, and for all totally real number fields by Andrew Wiles. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem). Barry Mazur (born December 19, 1937) is a professor of mathematics at Harvard University. ... Andrew Wiles should not be confused with André Weil, another famous mathematician who, like Wiles, did important work in the area of elliptic curves. ... In number theory , a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers. ... Kenneth Alan Ken Ribet is an American mathematician, currently a professor of mathematics at the University of California, Berkeley. ... The Herbrand-Ribet theorem is a strengthening of Kummers theorem to the effect that the prime p divides the class number of the cyclotomic field of pth roots of unity if and only if p divides the denominator of the nth Bernoulli number Bn for some n, 0 < n...

More recently, also modeled upon Ribet's method, Chris Skinner and Eric Urban have announced a proof of a main conjecture for GL(2). A more elementary proof of the Mazur-Wiles theorem can be obtained by using Euler systems as developed by Kolyvagin (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been obtained by Karl Rubin, amongst others. In mathematics, an Euler system is a technical device in the theory of Galois modules, first noticed as such in the work around 1990 by Victor Kolyvagin on Heegner points on modular elliptic curves. ... Victor Kolyvagin (Russian: ) is an American mathematician. ... Karl Rubin is currently (2006) the Thorp Professor of Mathematics at the University of California, Irvine. ...

References

• Greenberg, Ralph, Iwasawa Theory - Past & Present, Advanced Studies in Pure Math. 30 (2001), 335-385. Available at [1].
• 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
• Barry Mazur and Andrew Wiles (1984). "Class Fields of Abelian Extensions of Q". Inventiones Mathematicae 76 (2): 179-330. 
• Andrew Wiles (1990). "The Iwasawa Conjecture for Totally Real Fields". Annals of Mathematics 131 (3): 493-540. 
• Chris Skinner and Eric Urban (2002). "Sur les deformations p-adiques des formes de Saito-Kurokawa". C. R. Math. Acad. Sci. Paris 335 (7): 581-586.