# 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.

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.

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.

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

Im → In

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.

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

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