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.
Iwasawa worked with so-called -extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form , so by Galois theory, a -extension is the same thing as a tower of fields such that . Iwasawa studied classical Galois modules over by asking questions about the structure of modules over .
More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.
Let p be a prime number and let K = Q(μp) be the field generated over Q by the pth roots of unity. Iwasawa considered the following tower of number fields:
where is the field generated by adjoining to the pn+1st roots of unity and . The fact that implies, by infinite Galois theory, that is isomorphic to . In order to get an interesting Galois module here, Iwasawa took the ideal class group of , and let be its p-torsion part. There are norm maps whenever , and this gives us the data of an inverse system. If we set , then it is not hard to see from the inverse limit construction that is a module over . In fact, is a module over the Iwasawa algebra . 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 .
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.
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.
- 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.
- de Shalit, Ehud (1987), Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions, Perspectives in Mathematics 3, Boston etc.: Academic Press, ISBN 0-12-210255-X, Zbl 0674.12004