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.

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

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

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

This idea is much used in Iwasawa theory.

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 >.

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

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.

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.

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

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.

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.

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.

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

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.

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 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.

* 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

From 1937 to 1940 Iwasawa studied as an undergraduate at Tokyo University, after which he entered graduate school at Tokyo University and became an assistant in the Department of Mathematics.

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

Algebraic K-groups are used in conjectures on special values of L-functions and the formulation of an non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.

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

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 considered the following tower of number fields:

* 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.

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

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

* Iwasawa Yoshihiko 岩沢愿彥.

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.

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

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

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

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

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.

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

He thought of the jungles below him, and of the wild, strange, untracked beauty there and he promised himself that someday he would return, on foot perhaps, to hunt in this last corner of the world where man is sometimes himself the hunted, and animals the lords.

A third, one of at least equal and perhaps even greater importance, is now being traversed: American immersion and involvement in world affairs.

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

It is perhaps difficult to conceive, but imagine that tonight on London bridge the Teddy boys of the East End will gather to sing Marlowe, Herrick, Shakespeare, and perhaps some lyrics of their own.

The key word in my plays is ' perhaps ' ''.

As Lipton puts it: `` The Eros is felt in the magic circle of marijuana with far greater force, as a unifying principle in human relationships, than at any other time except, perhaps, in the mutual metaphysical orgasms.

Years ago this was true, but with the replacement of wires or runners by radio and radar ( and perhaps television ), these restrictions have disappeared and now again too much is heard.

What I want to point out here is that all of them are ex-liberals, or modified liberals, with perhaps one exception.

There is another side of love, more nearly symbolized by the croak of the mating capercailzie, or better still perhaps by the mute antics of the slug.

However, it was not of innocence in general that I was speaking, but of perhaps the frailest and surely the least important side of it which is innocence in romantic love.

Of all the Whig tracts written in support of the Succession, The Crisis is perhaps the most significant.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

He tends to underestimate -- or perhaps to view charitably -- the brutality and the violence of the age, so that there is an idyllic quality in these pages which hazes over some of its sharp reality.

If only for this modest masterpiece of military history, Blenheim is likely to be read and reread long after newer interpretations have perhaps altered our picture of the Marlborough wars.

The most famous document that comes out of this dispute is perhaps Sir Philip Sidney's An Apologie For Poetrie, published in 1595.

His credulity is perhaps best illustrated in his introduction to The Emancipation Of Massachusetts, which purports to examine the trials of Moses and to draw a parallel between the leader of the Israelite exodus from Egypt and the leadership of the Puritan clergy in colonial New England.

the mere fact that he was selected, though as a substitute, to act as interlocutor or moderator for it, or perhaps we should say with Buck as ' father of the act ', is in itself a difficult phase of his development to grasp.

If it proclaims that the best is yet to be, it always arouses, at least in the young, either a suspicious question or perhaps the exclamation of the Negro youth who saw on a tombstone the inscription, `` I am not dead but sleeping ''.

It is perhaps too late now to talk of mandate because it is inconsistent with what is termed political realism.

The only response we can think of is the humble one that at least we aren't playing the marimba with our shoes in the United Nations, but perhaps the heavy domes in the house of delegates can improve on this feeble effort.

-- Is this, perhaps, one of the things that is wrong with our country??

Since appeals to morality, to humanity, and to sanity have had such small effect, perhaps our last recourse is the deterrent example.