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Concept# Iwasawa theory

Summary

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 (岩澤 健吉), 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. (These were called -extensions in early papers.) 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 be a prime number and let be the field generated over by the th roots of unity. Iwasawa considered the following tower of number fields:
where is the field generated by adjoining to the pn+1-st roots of unity and
The fact that implies, by infinite Galois theory, that In order to get an interesting Galois module, 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
The motivation here is that the p-torsion in the ideal class group of had already been identified by Kummer as the main obstruction to the direct proof of Fermat's Last Theorem.
From this beginning in the 1950s, a substantial theory has been built up.

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