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 Iwasawa (1959), 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 1990s), 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 $\Gamma$ isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of $\Gamma$ 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 $\Lambda ={\mathbb {Z}}_{p}[[\Gamma ]]$ . This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this description 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.

Iwasawa formulated the main conjecture of Iwasawa theory 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 (the 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 fields.

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).

References

Coates, J.; Sujatha, R. (2006), Cyclotomic Fields and Zeta Values, Springer Monographs in Mathematics, Springer-Verlag, ISBN 3-540-33068-2, Zbl 1100.11002
Greenberg, Ralph (2001), "Iwasawa theory---past and present", in Miyake, Katsuya, Class field theory---its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math., 30, Tokyo: Math. Soc. Japan, pp. 335–385, ISBN 978-4-931469-11-2, MR 1846466, Zbl 0998.11054
Iwasawa, Kenkichi (1959), "On Γ-extensions of algebraic number fields", Bulletin of the American Mathematical Society, 65 (4): 183–226, doi:10.1090/S0002-9904-1959-10317-7, ISSN 0002-9904, MR 0124316, Zbl 0089.02402
Kato, Kazuya (2007), "Iwasawa theory and generalizations", in Sanz-Solé, Marta; Soria, Javier; Varona, Juan Luis; et al., International Congress of Mathematicians. Vol. I (PDF), Eur. Math. Soc., Zürich, pp. 335–357, doi:10.4171/022-1/14, ISBN 978-3-03719-022-7, MR 2334196
Lang, Serge (1990), Cyclotomic fields I and II, Graduate Texts in Mathematics, 121, With an appendix by Karl Rubin (Combined 2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96671-7, Zbl 0704.11038
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, MR 742853, Zbl 0545.12005
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323 (Second ed.), Berlin: Springer-Verlag, ISBN 978-3-540-37888-4, MR 2392026, Zbl 1136.11001
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, Zbl 0737.11030
Skinner, Chris; Urban, Éric (2010), The Iwasawa main conjectures for GL₂ (PDF), p. 219
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
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, Zbl 0719.11071.

External links

Hazewinkel, Michiel, ed. (2001), "Iwasawa theory", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

L-functions in number theory

Analytic examples	Riemann zeta function Dirichlet L-functions L-functions of Hecke characters Automorphic L-functions Selberg class

Algebraic examples	Dedekind zeta functions Artin L-functions Hasse–Weil L-functions Motivic L-functions

Theorems	Analytic class number formula Riemann–von Mangoldt formula Weil conjectures

Analytic conjectures	Riemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjecture

Algebraic conjectures	Birch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecture

p-adic L-functions	Main conjecture of Iwasawa theory Selmer group Euler system

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