Iwasawa theory

In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields.

Related Topics:
Number theory - Galois module - Ideal class group - Kenkichi Iwasawa - Cyclotomic field

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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 pro-finite groups). The group Γ itself is the inverse limit of the additive groups of the Z/pn.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.

Related Topics:
Algebraic number theory - Galois group - P-adic integer - Pro-finite group - Inverse limit - Prime number - Pontryagin duality - Roots of unity - Complex number

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A first and important example is in terms of the field K = Q(ζ) with ζ a primitive p-th root of unity. If Kn is the field generated by a primitive pn+1-th root of unity, then the tower of fields Kn (inside C) has a union L. Then the Galois group of L over K is isomorphic with Γ, because the Galois group of Kn over K is Z/pn.Z.

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In order to get an interesting Galois module here, Iwasawa took the ideal class group of Kn, and let In be its p-torsion part. There are norm mappings

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:Im → In

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when m > n, and so an inverse system. Letting I be the inverse limit, we can say that Γ acts on I: and ask for a description.

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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.

Related Topics:
Kummer - Fermat's last theorem

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In fact I is a module over the group ring Zp. 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.

Related Topics:
Module - Group ring - Regular

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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.

Related Topics:
P-adic L-function - Bernoulli number - Interpolation - Dirichlet L-function - Regular prime

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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 in generality, for totally real number fields, by Barry Mazur and Andrew Wiles.

Related Topics:
Totally real number field - Barry Mazur - Andrew Wiles

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