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.

Iwasawa worked with so-called ${\displaystyle \mathbb {Z} _{p}}$-extensions: infinite extensions of a number field ${\displaystyle F}$ with Galois group ${\displaystyle \Gamma }$ isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of ${\displaystyle \Gamma }$ is of the form ${\displaystyle \Gamma ^{p^{n}}}$, so by Galois theory, a ${\displaystyle \mathbb {Z} _{p}}$-extension ${\displaystyle F_{\infty }/F}$ is the same thing as a tower of fields ${\displaystyle F=F_{0}\subset F_{1}\subset F_{2}\subset \ldots \subset F_{\infty }}$ such that ${\displaystyle {\textrm {Gal}}(F_{n}/F)\cong \mathbb {Z} /p^{n}\mathbb {Z} }$. Iwasawa studied classical Galois modules over ${\displaystyle F_{n}}$ by asking questions about the structure of modules over ${\displaystyle F_{\infty }}$.

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