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 s.

A first and important example is in terms of the field K = Q(ζ) with ζ a primitive p -th root of unity. If K n is the field generated by a primitive pn+1 -th root of unity, then the tower of fields K n (inside C) 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/ pn .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 field 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 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 (mathematics) over the group ring Z p [Γ]. This is a well-behaved ring (Regular local ring 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.