Open main menu

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 Kenkichi Iwasawa (1959) (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.

FormulationEdit

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.

ExampleEdit

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.

Connections with p-adic analysisEdit

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   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.

GeneralizationsEdit

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.

In 2002, Chris Skinner and Eric Urban claimed a proof of a main conjecture for GL(2). In 2010, they posted a preprint (Skinner & Urban 2010).

See alsoEdit

ReferencesEdit

Further readingEdit

  • 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 978-0-12-210255-4, Zbl 0674.12004

External linksEdit