Iwasawa theory

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as an 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.

Contents

[edit] Formulation

Iwasawa worked with so-called \mathbb{Z}_p-extensions: infinite extensions of a number field F with Galois group Γ isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of Γ is of the form \Gamma^{p^n} , so by Galois theory, a \mathbb{Z}_p -extension F_\infty/F is the same thing as a tower of fields F = F_0 \subset F_1 \subset F_2 \subset \ldots \subset F_\infty such that \textrm{Gal}(F_n/F)\simeq \mathbb{Z}/p^n. Iwasawa studied classical Galois modules over Fn by asking questions about the structure of modules over F_\infty.

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

[edit] Example

Let K be a number field and suppose that K contains the p-th roots of unity. Iwasawa considered the following tower of number fields:

K = K_{1} \subset K_{2} \subset \cdots \subset K_{\infty},

where Kn is the field generated by adjoining to K the pn-th roots of unity and K_\infty = \bigcup K_n . The fact that \textrm{Gal}(K_n/K)\simeq \mathbb{Z}/p^n\mathbb{Z} implies, by infinite Galois theory, that \textrm{Gal}(K_{\infty}/K) is isomorphic to \mathbb{Z}_p . 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 maps I_m\rightarrow I_n whenever m > n, and this gives us the data of an inverse system. If we set I = \varprojlim I_n, then it is not hard to see from the inverse limit construction that I is a module over \mathbb{Z}_p. In fact, I is a module over the completed group ring \mathbb{Z}_p[[\Gamma]]. This is a well-behaved ring (a 2-dimensional, regular local ring), and this makes it possible to classify modules over it in a way that is not too coarse. From this classification it is possible to recover information about the p-part of the class group of K.

The motivation here is 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.

[edit] Connections with p-adic analysis

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. Some use the more specific "horizontal Iwasawa theory" which focuses on the horizontal maps within its commutative diagrams.

[edit] Generalizations

The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a main conjecture linking the tower to a 'p;-adic L-function.

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.

[edit] 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. doi:10.1007/BF01388599. 
  • Andrew Wiles (1990). "The Iwasawa Conjecture for Totally Real Fields". Annals of Mathematics 131 (3): 493–540. doi:10.2307/1971468. 

[edit] External links

Hazewinkel, Michiel, ed. (2001), "Iwasawa theory", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

Retrieved from "http://en.wikipedia.org/wiki/Iwasawa_theory"