In
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
-extensions - infinite extensions of a
number field 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 ...
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
-extensions in early papers.
) 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
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.
Example
Let
be a prime number and let
be the field generated over
by the
th roots of unity. Iwasawa considered the following tower of number fields:
:
where
is the field generated by adjoining to
the ''p''
''n''+1-st roots of unity and
:
The fact that
implies, by infinite Galois theory, that
In order to get an interesting Galois module, Iwasawa took the ideal class group of
, and let
be its ''p''-torsion part. There are
norm maps
whenever
, and this gives us the data of an
inverse system. If we set
:
then it is not hard to see from the inverse limit construction that
is a module over
In fact,
is a
module over the
Iwasawa algebra . 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
The motivation here is that the ''p''-torsion in the ideal class group of
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
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