Compared with Kummer’s criterion and class number formula, Iwasawa theory is finer in the point that it describes not only the class number, i.e. the order of the ideal class group, but also the action of the Galois group on the ideal class group. In fact, one could even say that the aim of Iwasawa theory is to describe Galois actions on arithmetic objects in terms of zeta values. (Kato 06)

Via the 3-manifold/number field analogy of arithmetic topology, Iwasawa theory can be seen as the analog of Alexander-Fox theory (see sec. 7 of Morishita).

References

Ralph Greenberg (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 (ps file).

Ralph Greenberg, Topics in Iwasawa Theory, (online book in process of being written).

Kazuya Kato, Iwasawa theory and generalizations, (ICM 2006 talk).

Masanori Morishita, Analogies between Knots and Primes, 3-Manifolds and Number Rings, (arxiv)

Last revised on August 28, 2014 at 05:24:10.
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