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, "Ma ...
, 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 specific ...
of
number fields. It began as a
Galois module theory of
ideal class groups, initiated by (), as part of the theory of
cyclotomic fields. 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 ...
considered generalizations of Iwasawa theory to
abelian varieties. More recently (early 1990s),
Ralph Greenberg 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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
.
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
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
over the
Iwasawa algebra . This is a
2-dimensional,
regular local ring, 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 as the main obstruction to the direct proof of
Fermat's Last Theorem.
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 n ...
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, p ...
and Leopoldt. The latter begin from the
Bernoulli numbers, 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-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on
regular primes.
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
Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fe ...
's proof of the converse to Herbrand's theorem (the so-called
Herbrand–Ribet theorem).
Karl Rubin 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
Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory.
Career
Urban received his PhD in mathematics from Paris-Sud University in 1994 under the super ...
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 fields.
History
introduced the μ-invariant o ...
*
Tate module of a number field
In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', ...
References
Sources
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Citations
Further reading
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External links
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Field (mathematics)
Cyclotomic fields
Class field theory