Iwasawa theory
A theory of -extensions introduced by K. Iwasawa [a8]. Its motivation has been a strong analogy between number fields and curves over finite fields. One of the most fruitful results in this theory is the Iwasawa main conjecture, which has been proved for totally real number fields [a19]. The conjecture is considered as an analogue of Weil's result that the characteristic polynomial of the Frobenius automorphism acting on the Jacobian of a curve over a finite field is the essential part of the zeta-function of the curve. A lot of methods and ideas developed in the theory appeared to be widely applicable and have given rise to major advances, for example, results on the Birch–Swinnerton-Dyer conjecture [a3], [a6], [a16], [a22] and on Fermat's last theorem [a20] (cf. also Fermat last theorem). For details and generalizations of Iwasawa theory, see [a10], [a7], [a12], [a18].
-extension of a number field.
Let be a prime number and let
be a finite extension of the rational number field
. A
-extension of
is an extension
with
, where
is the additive group of
-adic integers. Then there is a sequence of fields
![]() |
where is a cyclic extension of
of degree
. Class field theory shows that there are at least
independent
-extensions of
(cf. below, the section Leopoldt conjecture). Every
has at least one
-extension, namely the cyclotomic
-extension
. It is obtained by letting
be an appropriate subfield of
, where
is the group of
th roots of unity.
Leopoldt conjecture.
Let be the group of units of
which are congruent to
modulo every prime ideal
of
lying above
. By Dirichlet's unit theorem,
, where
(resp.
) is the number of embeddings of
in
(resp.
). Let
be the group of local units of
congruent to
modulo
. There is an embedding
(
). Let
denote the topological closure of the image. It is Leopoldt's conjecture that the equality
![]() |
holds for every . A. Brumer [a1] proved the conjecture for Abelian extensions
(or an imaginary quadratic field). Put
. Then class field theory shows that there are
independent
-extensions of
.
Iwasawa module.
Let be the integer ring of a finite extension of
and
a uniformizer of
. Let
be a compact Abelian group isomorphic to
and
, where the inverse limit is taken with respect to
(
) for
. Fix a topological generator
of
. Let
be the ring of formal power series in an indeterminate
with coefficients in
.
is called a distinguished polynomial if
with
for
. The prime ideals of
are
,
,
,
, where
is distinguished and irreducible. The classification of compact
-modules in [a8] was simplified by J.-P. Serre, who pointed out that
is topologically isomorphic to
, hence each compact
-module
admits the unique structure of a compact
-module such that
for every
. Finitely-generated
-modules are called Iwasawa modules. They are classified as follows: for an Iwasawa module
, there is a
-homomorphism
![]() |
with and
finite
-modules, where
and
is distinguished and irreducible. For a torsion
-module
, i.e.,
, one defines
![]() |
![]() |
Iwasawa invariant.
Let be a
-extension. Let
denote the
-Sylow subgroup of the ideal class group of
. Let
be the order of
. Iwasawa [a8] proved that there exist integers
,
and
such that
![]() |
for all sufficiently large . The invariants
and
can be obtained from the Iwasawa module
, where the inverse limit is taken with respect to the relative norm mappings. Put
.
is a compact
-module in a natural way. One fixes a topological generator
of
. Then
is considered as a compact
-module (cf. the section on Iwasawa module above). Since
is finite,
is a finitely-generated torsion
-module. One has that
and
.
Iwasawa [a9] constructed infinitely many non-cyclotomic -extensions
with
. There are infinitely many
-extensions
with
. For
,
if and only if
is irregular (cf. also Irregular prime number). It is Iwasawa's conjecture that
for every
. B. Ferrero and L. Washington [a5] proved this conjecture for Abelian extensions
. W. Sinnott [a17] gave a new proof of this using the
-transform of a rational function.
It is Greenberg's conjecture that for every totally real
. For small
, it was proved that there are infinitely many real quadratic fields
with
[a14], [a15]. There exists a lot of numerical work verifying this conjecture, mainly for real quadratic fields.
It is Vandiver's conjecture that does not divide the class number of the maximal real subfield
of
for all
, which implies that
. This conjecture was verified for all
[a2].
Iwasawa main conjecture.
Let be an odd prime number and
a totally real number field. Fix an embedding of
into
. Let
be a
-adic valued Artin character for
of order prime to
. Let
be the extension of
attached to
. Assume that
is also totally real. Fix a topological generator
of
and let
be such that
for all
.
Let be the Teichmüller character
![]() |
and let be the classical
-function for
. Following T. Kubota and H.W. Leopoldt [a11], P. Deligne and K. Ribet [a4] proved the existence of a
-adic
-function
on
(
if
is trivial) satisfying the following interpolation property:
![]() |
for . There exists a unique power series
such that
(if
is trivial,
), where
is the ring generated over
by the values of
. By the
-adic Weierstrass preparation theorem (cf. also Weierstrass theorem), one can write
, where
,
is a distinguished polynomial,
is a uniformizer of
, and
is a unit power series. Let
be such that
(if
is trivial,
). One can similarly define
and a distinguished polynomial
for
.
Let , let
be the maximal unramified Abelian
-extension of
and
the maximal Abelian
-extension of
, which are both unramified outside the primes above
. By class field theory,
. Extend
to
. Then
acts on
by
. Put
and
. Let
,
![]() |
![]() |
Then one can regard and
as
-modules.
Following [a13], A. Wiles proved the following equality, i.e., the Iwasawa main conjecture for totally real fields:
![]() |
This equality is equivalent to
![]() |
The proof uses delicate techniques from modular forms, especially Hida's theory of modular forms, to construct unramified extensions.
Following Stickelberger's theorem, F. Thaine and V. Kolyvagin invented techniques for constructing relations in ideal class groups. These methods, which use Gauss sums (cyclotomic units or elliptic units) satisfying properties known as the Euler system, have given elementary proofs of the Iwasawa main conjecture for [a12], [a21].
References
[a1] | A. Brumer, "On the units of algebraic number fields" Mathematika , 14 (1967) pp. 121–124 |
[a2] | J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, M.A. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million" J. Symbolic Comput. , 31 (2001) pp. 89–96 |
[a3] | J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" Invent. Math. , 39 (1977) pp. 223–251 |
[a4] | P. Deligne, K. Ribet, "Values of abelian ![]() |
[a5] | B. Ferrero, L. Washington, "The Iwasawa invariant ![]() |
[a6] | R. Greenberg, "On the Birch and Swinnerton-Dyer conjecture" Invent. Math. , 72 (1983) pp. 241–265 |
[a7] | K. Iwasawa, "On ![]() |
[a8] | K. Iwasawa, "On ![]() |
[a9] | K. Iwasawa, "On the ![]() ![]() |
[a10] | J. Coates, R. Greenberg, B. Mazur, I. Satake, "Algebraic Number Theory—In Honor of K. Iwasawa" , Adv. Studies in Pure Math. , 17 , Acad. Press (1989) |
[a11] | T. Kubota, H.W. Leopoldt, "Eine ![]() ![]() ![]() |
[a12] | S. Lang, "Cyclotomic fields I—II" , Graduate Texts in Math. , 121 , Springer (1990) (with an appendix by K. Rubin) |
[a13] | B. Mazur, A. Wiles, "Class fields of abelian extensions of ![]() |
[a14] | J. Nakagawa, K. Horie, "Elliptic curves with no rational points" Proc. Amer. Math. Soc. , 104 (1988) pp. 20–24 |
[a15] | K. Ono, "Indivisibility of class numbers of real quadratic fields" Compositio Math. , 119 (1999) pp. 1–11 |
[a16] | K. Rubin, "Tate–Shafarevich groups and ![]() |
[a17] | W. Sinnott, "On the ![]() ![]() |
[a18] | L. Washington, "Introduction to cyclotomic fields" , Graduate Texts in Math. , 83 , Springer (1997) (Edition: Second) |
[a19] | A. Wiles, "The Iwasawa conjecture for totally real fields" Ann. of Math. , 131 (1990) pp. 493–540 |
[a20] | A. Wiles, "Modular elliptic curves and Fermat's last theorem" Ann. of Math. , 141 (1995) pp. 443–551 |
[a21] | K. Rubin, "The "main conjectures" of Iwasawa theory for imaginary quadratic fields" Invent. Math. , 103 (1991) pp. 25–68 |
[a22] | K. Rubin, "Euler systems and modular elliptic curves" , Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996) , London Math. Soc. Lecture Notes , 284 , Cambridge Univ. Press (1998) pp. 351–367 |
Iwasawa theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iwasawa_theory&oldid=17710