Iwasawa theory

In number theory, Iwasawa theory is a Galois module theory of ideal class groups, started by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur thought about generalizations of Iwasawa theory to Abelian Varieties. Later, in the early 90s, Ralph Greenberg has suggested an Iwasawa theory for motives.

Formulation

The first thing Iwasawa noticed 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 [math]\displaystyle{ \Gamma }[/math] is the inverse limit of the additive groups [math]\displaystyle{ \mathbf Z/p^n \mathbf Z }[/math], where p is the fixed prime number and [math]\displaystyle{ n = 1,2, \dots }[/math]. We can express this by Pontryagin duality in another way: Γ is dual to the discrete group of all [math]\displaystyle{ p }[/math]-power roots of unity in the complex numbers.

Example

Let [math]\displaystyle{ \zeta }[/math] be a primitive [math]\displaystyle{ p }[/math]-th root of unity and look at the following tower of number fields:

[math]\displaystyle{ K = \mathbf{Q} (\zeta) \subset K_{1} \subset K_{2} \subset \cdots \subset \mathbf{C}, }[/math]

where [math]\displaystyle{ K_{n} }[/math] is the field generated by a primitive [math]\displaystyle{ p^{n+1} }[/math]-th root of unity. This tower of fields has a union [math]\displaystyle{ L }[/math]. Then the Galois group of [math]\displaystyle{ L }[/math] over [math]\displaystyle{ K }[/math] is isomorphic with [math]\displaystyle{ \Gamma }[/math]; because the Galois group of [math]\displaystyle{ K_n }[/math] over [math]\displaystyle{ K }[/math] is [math]\displaystyle{ \mathbf Z/p^n \mathbf Z }[/math]. In order to get an interesting Galois module here, Iwasawa took the ideal class group of [math]\displaystyle{ K_{n} }[/math], and let [math]\displaystyle{ I_n }[/math] be its [math]\displaystyle{ p }[/math]-torsion part. There are norm mappings [math]\displaystyle{ I_m \rightarrow I_n }[/math] when [math]\displaystyle{ m \gt n }[/math], and so an inverse system. Letting [math]\displaystyle{ I }[/math] be the inverse limit, we can say that [math]\displaystyle{ \Gamma }[/math] acts on [math]\displaystyle{ I }[/math], and it is good to have a description of this action.

The motivation here was undoubtedly that the [math]\displaystyle{ p }[/math]-torsion in the ideal class group of [math]\displaystyle{ K }[/math] had already been identified by Kummer as the main obstacle to the direct proof of Fermat's last theorem. What Iwasawa did that was new, was to go 'off to infinity' in a new direction. In fact, [math]\displaystyle{ I }[/math] is a module over the group ring [math]\displaystyle{ \mathbf Z_p [[\Gamma]] }[/math]. 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 good-sized theory has been built up. A basic 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 might be able to move ahead finally from Kummer's century-old results on regular primes.

The main conjecture of Iwasawa theory was formulated as an assertion that two ways 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).

More recently, also modeled upon Ribet's method, Chris Skinner and Eric Urban have announced a proof of a main conjecture for GL(2). An easier proof of the Mazur-Wiles theorem can be found 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 found by Karl Rubin, amongst others.

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
Barry Mazur and Andrew Wiles (1984). "Class Fields of Abelian Extensions of Q". Inventiones Mathematicae. 76 (2): 179–330. Bibcode:1984InMat..76..179M. doi:10.1007/BF01388599. S2CID 122576427.
Andrew Wiles (1990). "The Iwasawa Conjecture for Totally Real Fields". Annals of Mathematics. 131 (3): 493–540. doi:10.2307/1971468. JSTOR 1971468.
Chris Skinner and Eric Urban (2002). "Sur les deformations p-adiques des formes de Saito-Kurokawa". C. R. Math. Acad. Sci. Paris. 335 (7): 581–586. doi:10.1016/S1631-073X(02)02540-2.