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number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, Iwasawa theory is the study of objects of arithmetic interest over infinite
towers A tower is a tall structure, taller than it is wide, often by a significant factor. Towers are distinguished from masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting structures. Towers are specifi ...
of number fields. It began as a Galois module theory of
ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a ...
s, initiated by (), as part of the theory of
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because o ...
s. In the early 1970s,
Barry Mazur Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...
considered generalizations of Iwasawa theory to
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
. More recently (early 1990s),
Ralph Greenberg Ralph Greenberg (born 1944) is an American mathematician who has made contributions to number theory, in particular Iwasawa theory. He was born in Chester, Pennsylvania and studied at the University of Pennsylvania, earning a B.A. in 1966, after ...
has proposed an Iwasawa theory for motives.


Formulation

Iwasawa worked with so-called \Z_p-extensions - infinite extensions of a number field F with
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
\Gamma isomorphic to the additive group of
p-adic integer In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extens ...
s for some prime ''p''. (These were called \Gamma-extensions in early papers.) Every closed subgroup of \Gamma is of the form \Gamma^, so by Galois theory, a \Z_p-extension F_\infty/F is the same thing as a tower of fields :F=F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_\infty such that \operatorname(F_n/F)\cong \Z/p^n\Z. Iwasawa studied classical Galois modules over F_n by asking questions about the structure of modules over F_\infty. More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.


Example

Let p be a prime number and let K=\Q(\mu_p) be the field generated over \Q by the pth roots of unity. Iwasawa considered the following tower of number fields: : K = K_ \subset K_ \subset \cdots \subset K_, where K_n is the field generated by adjoining to K the ''p''''n''+1-st roots of unity and :K_\infty = \bigcup K_n. The fact that \operatorname(K_n/K)\simeq \Z/p^n\Z implies, by infinite Galois theory, that \operatorname(K_/K) \simeq \varprojlim_n \Z/p^n\Z = \Z_p. In order to get an interesting Galois module, Iwasawa took the ideal class group of K_n, and let I_n be its ''p''-torsion part. There are norm maps I_m\to I_n whenever m>n, and this gives us the data of an inverse system. If we set :I = \varprojlim I_n, then it is not hard to see from the inverse limit construction that I is a module over \Z_p. In fact, I is a module over the Iwasawa algebra \Lambda=\Z_p \Gamma. This is a
2-dimensional In mathematics, a plane is a Euclidean (flat), two- dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
,
regular local ring 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 equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal id ...
, 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 K. The motivation here is that the ''p''-torsion in the ideal class group of K had already been identified by
Kummer Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist *Clare Kummer (1873—1958), American composer, lyricist and playwright *Clarence Kummer (1899–1930), American jockey * Christo ...
as the main obstruction to the direct proof of
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...
.


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-function In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime nu ...
s that were defined in the 1960s by
Kubota Kubota machine is a Japanese multinational corporation based in Osaka. It was established in 1890. The corporation produces many products including tractors and other agricultural machinery, construction equipment, engines, vending machines, pi ...
and Leopoldt. The latter begin from the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions ...
s, and use
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...
to define p-adic analogues of the
Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. ...
s. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on
regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli nu ...
s. 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 \Q 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 Karl Cooper Rubin (born January 27, 1956) is an American mathematician at University of California, Irvine as Edward O. Thorp, Thorp Professor of Mathematics. Between 1997 and 2006, he was a professor at Stanford, and before that worked at Ohio S ...
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 .


See also

*
Ferrero–Washington theorem In algebraic number theory, the Ferrero–Washington theorem, proved first by and later by , states that Iwasawa's μ-invariant vanishes for cyclotomic Z''p''-extensions of abelian algebraic number field In mathematics, an algebraic number fie ...
* Tate module of a number field


References

Sources * * * * * * * * * * * Citations


Further reading

*


External links

* {{Authority control Field (mathematics) Cyclotomic fields Class field theory