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In number theory, 'Iwasawa theory' is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. 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.
Iwasawa's starting observation was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature profinite groups). The group Γ is the inverse limit of the additive groups
:'Z'/''p''''n'''Z',
where ''p'' is the fixed prime number and ''n'' = 1, 2,... . We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all ''p''-power roots of unity in the complex numbers (the Prüfer ''p''-group).
Let ζ be a primitive ''p''-th root of unity and consider the following tower of number fields:
:
where ''K''n is the field generated by a primitive ''p''n+1-th root of unity. This tower of fields has a union ''L''. Then the Galois group of ''L'' over ''K'' is isomorphic with Γ because the Galois group of ''K''n over ''K'' is 'Z'/''p''n'Z'. In order to get an interesting Galois module here, Iwasawa took the ideal class group of ''K''n, and let ''I''n be its ''p''-torsion part. There are norm mappings ''I''m → ''I''n when ''m'' > ''n'', and so an inverse system. Letting ''I'' be the inverse limit, we can say that Γ acts on ''I'', and it is desirable to have a description of this action.
The motivation here was undoubtedly that the ''p''-torsion in the ideal class group of ''K'' had already been identified by Kummer as the main obstruction to the direct proof of Fermat's last theorem. Iwasawa's originality was to go 'off to infinity' in a novel direction. In fact ''I'' is a module over the group ring 'Z'''p'' [Γ]. This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.
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 numbers, 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 eventually proved by Barry Mazur and Andrew Wiles for 'Q', and for all totally real number fields by Andrew Wiles. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).
A more elementary proof of the Mazur-Wiles theorem can be obtained by using Euler systems as developed by Kolyvagin (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been obtained by Karl Rubin, amongst others.
More recently, in 2002, also modeled upon Ribet's method, Chris Skinner and Eric Urban have claimed they had a proof of a ''main conjecture'' for GL(2). However, so far, they have not substantiated their claim by releasing any preprint or article.
★ Greenberg, Ralph, ''Iwasawa Theory - Past & Present'', Advanced Studies in Pure Math. 30 (2001), 335-385. Available at [1].
★ Coates, J. and Sujatha, R., ''Cyclotomic Fields and Zeta Values'', Springer-Verlag, 2006
★ Lang, S., ''Cyclotomic Fields'', Springer-Verlag, 1978
★ Washington, L., ''Introduction to Cyclotomic Fields, 2nd edition'', Springer-Verlag, 1997
★ ''Class Fields of Abelian Extensions of Q'', Barry Mazur and Andrew Wiles, , , Inventiones Mathematicae, 1984
★ ''The Iwasawa Conjecture for Totally Real Fields'', Andrew Wiles, , , Annals of Mathematics, 1990
| Contents |
| Formulation |
| Example |
| History |
| References |
Formulation
Iwasawa's starting observation was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers. That group, usually written Γ in the theory and with multiplicative notation, can be found as a subgroup of Galois groups of infinite field extensions (which are by their nature profinite groups). The group Γ is the inverse limit of the additive groups
:'Z'/''p''''n'''Z',
where ''p'' is the fixed prime number and ''n'' = 1, 2,... . We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all ''p''-power roots of unity in the complex numbers (the Prüfer ''p''-group).
Example
Let ζ be a primitive ''p''-th root of unity and consider the following tower of number fields:
:
where ''K''n is the field generated by a primitive ''p''n+1-th root of unity. This tower of fields has a union ''L''. Then the Galois group of ''L'' over ''K'' is isomorphic with Γ because the Galois group of ''K''n over ''K'' is 'Z'/''p''n'Z'. In order to get an interesting Galois module here, Iwasawa took the ideal class group of ''K''n, and let ''I''n be its ''p''-torsion part. There are norm mappings ''I''m → ''I''n when ''m'' > ''n'', and so an inverse system. Letting ''I'' be the inverse limit, we can say that Γ acts on ''I'', and it is desirable to have a description of this action.
The motivation here was undoubtedly that the ''p''-torsion in the ideal class group of ''K'' had already been identified by Kummer as the main obstruction to the direct proof of Fermat's last theorem. Iwasawa's originality was to go 'off to infinity' in a novel direction. In fact ''I'' is a module over the group ring 'Z'''p'' [Γ]. This is a well-behaved ring (regular and two-dimensional), meaning that it is quite possible to classify modules over it, in a way that is not too coarse.
History
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 numbers, 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 eventually proved by Barry Mazur and Andrew Wiles for 'Q', and for all totally real number fields by Andrew Wiles. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).
A more elementary proof of the Mazur-Wiles theorem can be obtained by using Euler systems as developed by Kolyvagin (see Washington's book). Other generalizations of the main conjecture proved using the Euler system method have been obtained by Karl Rubin, amongst others.
More recently, in 2002, also modeled upon Ribet's method, Chris Skinner and Eric Urban have claimed they had a proof of a ''main conjecture'' for GL(2). However, so far, they have not substantiated their claim by releasing any preprint or article.
References
★ Greenberg, Ralph, ''Iwasawa Theory - Past & Present'', Advanced Studies in Pure Math. 30 (2001), 335-385. Available at [1].
★ Coates, J. and Sujatha, R., ''Cyclotomic Fields and Zeta Values'', Springer-Verlag, 2006
★ Lang, S., ''Cyclotomic Fields'', Springer-Verlag, 1978
★ Washington, L., ''Introduction to Cyclotomic Fields, 2nd edition'', Springer-Verlag, 1997
★ ''Class Fields of Abelian Extensions of Q'', Barry Mazur and Andrew Wiles, , , Inventiones Mathematicae, 1984
★ ''The Iwasawa Conjecture for Totally Real Fields'', Andrew Wiles, , , Annals of Mathematics, 1990
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