Iwasawa theory
Encyclopedia
In number theory
, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module
theory of ideal class group
s, initiated by Kenkichi Iwasawa
, in the 1950s, as part of the theory of cyclotomic field
s. In the early 1970s, Barry Mazur
considered generalizations of Iwasawa theory to abelian varieties
. More recently (early 90s), Ralph Greenberg has proposed an Iwasawa theory for motives
.
-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 
. 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.
where
is the field generated by adjoining to 
the pn+1st roots of unity and 
. The fact that 
implies, by infinite Galois theory, that 
is isomorphic to 
. In order to get an interesting Galois module here, 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
(i.e. the completed group ring
of
over 
). This is a well-behaved ring (a 2-dimensional
, regular local ring
), and this makes it possible to classify modules over it in a way that is not too coarse. From this classification 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
.
s that were defined in the 1960s by Kubota
and Leopoldt. The latter begin from the Bernoulli number
s, and use interpolation
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.
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. This was proved by for Q, and for all totally real number field
s by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).
Karl Rubin
found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler system
s, described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fileds.
In 2002, Chris Skinner and Eric Urban claimed a proof of a main conjecture for GL
(2). In 2010, they posted a preprint .
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module
Galois module
In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...
theory of ideal class group
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
s, initiated by Kenkichi Iwasawa
Kenkichi Iwasawa
Kenkichi Iwasawa was a Japanese mathematician who is known for his influence on algebraic number theory.Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture...
, in the 1950s, as part of the theory of cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s. In the early 1970s, Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
considered generalizations of Iwasawa theory to abelian varieties
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and 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 functions...
. More recently (early 90s), Ralph Greenberg has proposed an Iwasawa theory for motives
Motive (algebraic geometry)
In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
.
Formulation
Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois groupGalois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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 . 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 p be a prime number and let K = Q(μp) be the field generated over Q by the pth roots of unity. Iwasawa considered the following tower of number fields:where
is the field generated by adjoining to the pn+1st roots of unity and . The fact that implies, by infinite Galois theory, that is isomorphic to . In order to get an interesting Galois module here, Iwasawa took the ideal class group of , and let be its p-torsion part. There are normField norm
In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...
maps
whenever , and this gives us the data of an inverse systemInverse limit
In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...
. If we set
, then it is not hard to see from the inverse limit construction that is a module over . In fact, is a moduleModule (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over the Iwasawa algebra
(i.e. the completed group ringGroup ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
of
over ). This is a well-behaved ring (a 2-dimensionalKrull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
, regular local ring
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 ideal m, and suppose a1, ..., an is a minimal set of...
), and this makes it possible to classify modules over it in a way that is not too coarse. From this classification 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 KummerErnst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
as the main obstruction to the direct proof of Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
.
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-functionP-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...
s that were defined in the 1960s by Kubota
Tomio Kubota
is a Japanese mathematician, who studies number theory. His contributions include works on p-adic L functions and real-analytic automorphic forms.His work on p-adic L-functions, later recognised as an aspect of Iwasawa theory, was jointly with Leopoldt....
and Leopoldt. The latter begin from the Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s, and use interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
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.
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. This was proved by for Q, and for all totally real number field
Totally real number field
In number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers...
s by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).
Karl Rubin
Karl Rubin
Karl Rubin is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. His research interest is in elliptic curves. He was the first mathematician to show that some elliptic curves over the rationals have finite Tate-Shafarevich groups...
found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler system
Euler system
In mathematics, an Euler system is collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of...
s, described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fileds.
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, Chris Skinner and Eric Urban claimed a proof of a main conjecture for GL
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
(2). In 2010, they posted a preprint .