Iwasawa

Iwasawa is a Japanese surname that has been associated with several important ideas in mathematics, particularly in number theory and Lie group theory. The best known usage in mathematics comes from Kenkichi Iwasawa, a 20th‑century mathematician whose work led to two influential concepts that bear his name: Iwasawa theory and the Iwasawa decomposition.

Iwasawa theory is a framework in algebraic number theory that studies how arithmetic invariants grow in infinite towers of number fields, especially in p-adic settings. Central objects are p-adic L-functions and Iwasawa modules, which encode information about class groups and units in cyclotomic Z_p-extensions. The theory introduces invariants such as lambda, mu, and nu that measure growth rates, and it culminates in conjectures and theorems connecting algebraic data to analytic p-adic L-functions, including the main conjecture of Iwasawa theory.

The Iwasawa decomposition is a structural result in Lie group theory. For many real semisimple Lie groups G, one can factor G into a product G = KAN, where K is a maximal compact subgroup, A is an abelian (split) component, and N is a nilpotent subgroup. This decomposition is foundational in harmonic analysis and representation theory on Lie groups, enabling the study of functions and representations via simpler components.

In addition, the Iwasawa algebra refers to completed group algebras like Z_p[[G]] for profinite groups G, which play a key role in modern Iwasawa theory and related areas of arithmetic geometry.

Overall, Iwasawa denotes both a surname and a pair of foundational mathematical constructions, each named in honor of his contributions.