Iwasawa theory

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

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 $\textrm{Gal}(F_n/F)\cong \mathbb{Z}/p^n\mathbb{Z}$. Iwasawa studied classical Galois modules over F_n 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.

Example

Let p be a prime number and let K = Q(μ_p) be the field generated over Q by the pth roots of unity. Iwasawa considered the following tower of number fields:

$K = K_{0} \subset K_{1} \subset \cdots \subset K_{\infty},$

where K_n is the field generated by adjoining to K the pⁿ⁺¹st 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 K_n, and let I_n 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 Iwasawa algebra $\Lambda=\mathbb{Z}_p[[\Gamma]]$ (i.e. the completed group ring of Γ over $\mathbb{Z}_p$). 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.

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 proved by Mazur & Wiles (1984) for Q, and for all totally real number fields by Wiles (1990). These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (so-called Herbrand-Ribet theorem).

Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systems, described in Lang (1990) and Washington (1997), and later proved other generalizations of the main conjecture for imaginary quadratic fileds.

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.

In 2002, Chris Skinner and Eric Urban claimed a proof of a main conjecture for GL(2). In 2010, they posted a preprint (Skinner & Urban 2010).

Notes

References

• Greenberg, Ralph, Iwasawa Theory - Past & Present, Advanced Studies in Pure Math. 30 (2001), 335-385.
• Coates, J. and Sujatha, R., Cyclotomic Fields and Zeta Values, Springer-Verlag, 2006
• Kato, Kazuya (2007), "Iwasawa theory and generalizations", in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis et al., International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich, pp. 335–357, doi:10.4171/022-1/14, ISBN 978-3-03719-022-7, MR2334196, http://www.icm2006.org/proceedings/Vol_I/18.pdf 
• Lang, Serge (1990), Cyclotomic fields I and II, Graduate Texts in Mathematics, 121 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96671-7, http://books.google.com/books?isbn=0-387-96671-4 
• Mazur, Barry; Wiles, Andrew (1984), "Class fields of abelian extensions of Q", Inventiones Mathematicae 76 (2): 179–330, doi:10.1007/BF01388599, ISSN 0020-9910, MR742853 
• Rubin, Karl (1991), "The ``main conjectures of Iwasawa theory for imaginary quadratic fields", Inventiones Mathematicae 103 (1): 25–68, doi:10.1007/BF01239508, ISSN 0020-9910 
• Skinner, Chris; Urban, Éric (2010), The Iwasawa main conjectures for GL₂, p. 219, http://www.math.columbia.edu/%7Eurban/eurp/MC.pdf 
• Washington, Lawrence C. (1997), Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94762-4, http://books.google.com/books?isbn=0-387-94762-0 
• Andrew Wiles (1990), "The Iwasawa Conjecture for Totally Real Fields", Annals of Mathematics (Annals of Mathematics) 131 (3): 493–540, doi:10.2307/1971468, JSTOR 1971468. 

External links

• Hazewinkel, Michiel, ed. (2001), "Iwasawa theory", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/i/i130090.htm