Iwasawa spent the next two years at Institute for Advanced Study in Princeton, and in Spring of 1952 was offered a job at the Massachusetts Institute of Technology, where he worked until 1967.

Before that he worked on Lie groups and Lie algebras, introducing the general Iwasawa decomposition.

* Iwasawa decomposition ( KAN ) a mathematical process dealing with Lie groups

Also, starting with any compact real form of a semisimple Lie algebra g its complexification as a real Lie algebra of twice the dimension splits into g and a certain solvable Lie algebra ( the Iwasawa decomposition ), and this provides a canonical bicrossproduct quantum group associated to g. For su ( 2 ) one obtains a quantum group deformation of the Euclidean group E ( 3 ) of motions in 3 dimensions.

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

Combining this with the p-parity theorem by and the announced proof of the main conjecture of Iwasawa theory for GL ( 2 ) by, they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by, satisfy the Birch and Swinnerton-Dyer conjecture.

For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.

However, this same year Iwasawa became sick with pleurisy, and was unable to return to his position at the university until April 1947.

In the last ten years he has focused on the study of various aspects of non-commutative Iwasawa theory, for instance, the study of the arithmetic of elliptic curves in nonabelian infinite extensions.

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields.

Tate's thesis ( 1950 ) on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it ; independently and a little earlier, Kenkichi Iwasawa obtained a similar theory.

Iwasawa considered the following tower of number fields:

Kenkichi Iwasawa ( Iwasawa Kenkichi, September 11, 1917 – October 26, 1998 ) was a Japanese mathematician who is known for his influence on algebraic number theory.

* Local class field theory / Kenkichi Iwasawa ( 1986 ) ISBN 0-19-504030-9

* Mathematics: Approximation theory — Arakelov theory — Asymptotic theory — Bifurcation theory — Catastrophe theory — Category theory — Chaos theory — Choquet theory — Coding theory — Combinatorial game theory — Computability theory — Computational complexity theory — Deformation theory — Dimension theory — Ergodic theory — Field theory — Galois theory — Game theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory — Intersection theory — Invariant theory — Iwasawa theory — K-theory — KK-theory — Knot theory — L-theory — Lie theory — Littlewood – Paley theory — Matrix theory — Measure theory — Model theory — Morse theory — Nevanlinna theory — Number theory — Obstruction theory — Operator theory — PCF theory — Perturbation theory — Potential theory — Probability theory — Ramsey theory — Rational choice theory — Representation theory — Ring theory — Set theory — Shape theory — Small cancellation theory — Spectral theory — Stability theory — Stable theory — Sturm – Liouville theory — Twistor theory

It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields.

Iwasawa studied classical Galois modules over by asking questions about the structure of modules over.

In order to get an interesting Galois module here, Iwasawa took the ideal class group of, and let be its p-torsion part.

Ralph Greenberg has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motives in the context of Iwasawa theory.

* The Iwasawa decomposition G = KAN of a semisimple group G as the product of compact, abelian, and nilpotent subgroups generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix ( a consequence of Gram – Schmidt orthogonalization ).

In fact, is a module over the Iwasawa algebra ( i. e. the completed group ring of over ).

In mathematics, the Iwasawa decomposition KAN of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix ( a consequence of Gram-Schmidt orthogonalization ).

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.

* Lectures on p-adic L-functions / by Kenkichi Iwasawa ( 1972 )

More recently ( early 90s ), Ralph Greenberg has proposed an Iwasawa theory for motives.

These elements — named Beilinson elements after Alexander Beilinson who introduced them in — were used by Kazuya Kato in to prove one divisibility in Barry Mazur's main conjecture of Iwasawa theory for elliptic curves.

Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a ( disconnected ) maximal compact subgroup provided the center of G is finite.

Iwasawa is perhaps best known for introducing what is now called Iwasawa theory, which developed from researches on cyclotomic fields from the later parts of the 1950s.

* In Japan, according to M. Iwasawa at the National Institute of Population and Social Security Research, less than 3 % of females between 25-29 are currently cohabiting, but more than 1 in 5 have had some experience of an unmarried partnership, including cohabitation.

* Iwasawa, Kenkichi: On some types of topological groups.

Ribet's methods were pushed further by Barry Mazur and Andrew Wiles in order to prove the Main Conjecture of Iwasawa theory ,< ref > a corollary of which is a strengthening of the Herbrand-Ribet theorem: the power of p dividing B < sub > p − n </ sub > is exactly the power of p dividing the order of G < sub > n </ sub >.

A more recent Iwasawa study has shown that there has been a recent emergence of non-marital cohabitation.

Via the theory of zeta integrals initiated by Kenkichi Iwasawa and by John Tate in Tate's thesis it is related to the study of the zeta function of global fields.

The latter was explicitly introduced in papers of Kenkichi Iwasawa and John Tate.

* Iwasawa Yoshihiko 岩沢愿彥.

In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties.

* Greenberg, Ralph, Iwasawa Theory-Past & Present, Advanced Studies in Pure Math.

* Borel, A .; Chowla, S .; Herz, C. S .; Iwasawa, K .; Serre, J .- P. Seminar on complex multiplication.

This idea is much used in Iwasawa theory.

It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

Then the Iwasawa decomposition of is

and the Iwasawa decomposition of G is

In 1950, Iwasawa was invited to Cambridge, Massachusetts to give a lecture at the International Congress of Mathematicians on his method to study Dedekind zeta functions using integration over ideles and duality of adeles ; this method was also independently obtained by John Tate and it is sometimes called Tate's thesis or the Iwasawa-Tate theory.