In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Iwasawa theory is the study of objects of arithmetic interest over infinite
towers
A tower is a tall Nonbuilding structure, structure, taller than it is wide, often by a significant factor. Towers are distinguished from guyed mast, masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting ...
of
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
s. It began as a
Galois module theory of
ideal class group
In mathematics, the ideal class group (or class group) of an algebraic number field K is the quotient group J_K/P_K where J_K is the group of fractional ideals of the ring of integers of K, and P_K is its subgroup of principal ideals. The ...
s, initiated by (), as part of the theory of
cyclotomic field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
s. In the early 1970s,
Barry Mazur 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
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
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 pol ...
isomorphic to the additive group of
p-adic integer
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infini ...
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,
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 any Noetherian local ring with unique maxi ...
, 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
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
.
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-functions that were defined in the 1960s by
Kubota 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 function ...
s, and use
interpolation
In the mathematics, 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 ...
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. By anal ...
s. 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'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 claimed a proof of a ''main conjecture'' for
GL(2). In 2010, they posted a preprint .
See also
*
Ferrero–Washington theorem
*
Tate module of a number field
References
Sources
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Citations
Further reading
*
* Masato Kurihara, Kenichi Bannai, Tadashi Ochiai, Takeshi Tsuji (EDs.): ''Development of Iwasawa Theory: The Centennial of K. Iwasawa's Birth'', Mathematical Soc of Japan, (Advanced Studies in Pure Mathematics, V.86), ISBN 978-4-86497092-1 (2020).
* Tadashi Ochiai: ''Iwasawa Theory and Its Perspective, Vol.1'', Amer. Math. Soc., (Mathematical Surveys and Monographs V.272), ISBN 978-1-4704-5672-6 (2023).
* Tadashi Ochiai: ''Iwasawa Theory and Its Perspective, Vol.2'', Amer. Math. Soc., (Mathematical Surveys and Monographs V.280), ISBN 978-1-4704-5673-3 (2024).
* Tadashi Ochiai: ''Iwasawa Theory and Its Perspective, Vol.3'', Amer. Math. Soc., (Mathematical Surveys and Monographs V.291), ISBN 978-1-4704-7732-5 (2025).
External links
*
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Field (mathematics)
Cyclotomic fields
Class field theory