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| In [[number theory]], '''Iwasawa theory''' is the study of objects of arithmetic interest over infinite [[Tower of fields|towers]] of [[number field]]s. It began as a [[Galois module]] theory of [[ideal class group]]s, initiated by {{harvs|txt|authorlink=Kenkichi Iwasawa|last=Iwasawa|year=1959}}, as part of the theory of [[cyclotomic field]]s. In the early 1970s, [[Barry Mazur]] considered generalizations of Iwasawa theory to [[abelian variety|abelian varieties]]. More recently (early 90s), [[Ralph Greenberg]] has proposed an Iwasawa theory for [[motive (algebraic geometry)|motives]].
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| ==Formulation==
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| Iwasawa worked with so-called <math>\mathbb{Z}_p</math>-extensions: infinite extensions of a [[number field]] <math> F </math> with [[Galois group]] <math> \Gamma </math> isomorphic to the additive group of [[p-adic integer]]s for some prime ''p''. Every closed subgroup of <math> \Gamma </math> is of the form <math> \Gamma^{p^n} </math>, so by Galois theory, a <math> \mathbb{Z}_p </math>-extension <math> F_\infty/F </math> is the same thing as a tower of fields <math> F = F_0 \subset F_1 \subset F_2 \subset \ldots \subset F_\infty </math> such that <math>\textrm{Gal}(F_n/F)\cong \mathbb{Z}/p^n\mathbb{Z}</math>. Iwasawa studied classical Galois modules over <math> F_n </math> by asking questions about the structure of modules over <math>F_\infty</math>.
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| More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a [[p-adic Lie group]].
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| ==Example== | |
| Let ''p'' be a prime number and let ''K'' = '''Q'''(μ<sub>''p''</sub>) be the field generated over '''Q''' by the ''p''th roots of unity. Iwasawa considered the following tower of number fields:
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| :<math> K = K_{0} \subset K_{1} \subset \cdots \subset K_{\infty}, </math>
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| where <math>K_n</math> is the field generated by adjoining to <math>K</math> the ''p<sup>n''+1</sup>st roots of unity and <math> K_\infty = \bigcup K_n </math>. The fact that <math>\textrm{Gal}(K_n/K)\simeq \mathbb{Z}/p^n\mathbb{Z}</math> implies, by infinite Galois theory, that <math>\textrm{Gal}(K_{\infty}/K)</math> is isomorphic to <math> \mathbb{Z}_p </math>. In order to get an interesting Galois module here, Iwasawa took the ideal class group of <math>K_n</math>, and let <math>I_n</math> be its ''p''-torsion part. There are [[field norm|norm]] maps <math>I_m\rightarrow I_n</math> whenever <math>m>n</math>, and this gives us the data of an [[inverse limit|inverse system]]. If we set <math>I = \varprojlim I_n</math>, then it is not hard to see from the inverse limit construction that <math> I </math> is a module over <math> \mathbb{Z}_p</math>. In fact, <math>I</math> is a [[module (mathematics)|module]] over the [[Iwasawa algebra]] <math>\Lambda=\mathbb{Z}_p[[\Gamma]]</math>. This is a [[Krull dimension|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 <math> K</math>.
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| The motivation here is that the ''p''-torsion in the ideal class group of <math>K</math> had already been identified by [[Ernst Kummer|Kummer]] as the main obstruction to the direct proof of [[Fermat's last theorem]].
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| ==Connections with p-adic analysis==
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| 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-function]]s that were defined in the 1960s by [[Tomio Kubota|Kubota]] and Leopoldt. The latter begin from the [[Bernoulli number]]s, and use [[interpolation]] to define p-adic analogues of the [[Dirichlet L-function]]s. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on [[regular prime]]s.
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| 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 {{harvtxt|Mazur|Wiles|1984}} for '''Q''', and for all [[totally real number field]]s by {{harvtxt|Wiles|1990}}. These proofs were modeled upon [[Ken Ribet]]'s proof of the converse to Herbrand's theorem (so-called [[Herbrand-Ribet theorem]]).
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| [[Karl Rubin]] found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's [[Euler system]]s, described in {{harvtxt|Lang|1990}} and {{harvtxt|Washington|1997}}, and later proved other generalizations of the main conjecture for imaginary quadratic fields.
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| ==Generalizations==
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| 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.
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| In 2002, Chris Skinner and Eric Urban claimed a proof of a ''main conjecture'' for [[General linear group|GL]](2). In 2010, they posted a preprint {{harv|Skinner|Urban|2010}}.
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| ==See also==
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| *[[Ferrero–Washington theorem]]
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| *[[Tate module of a number field]]
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| == References ==
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| * {{citation | first1=J. | last1=Coates | authorlink1=John Coates (mathematician) | first2=R. | last2=Sujatha | authorlink2=Sujatha Ramdorai | title=Cyclotomic Fields and Zeta Values | series=Springer Monographs in Mathematics | publisher=[[Springer-Verlag]] | year=2006 | isbn=3-540-33068-2 | zbl=1100.11002 }}
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| *{{Citation | last1=Greenberg | first1=Ralph | author1-link=Ralph Greenberg | editor1-last=Miyake | editor1-first=Katsuya | title=Class field theory---its centenary and prospect (Tokyo, 1998) | url=http://www.math.washington.edu/~greenber/iwhi.ps | publisher=Math. Soc. Japan | location=Tokyo | series=Adv. Stud. Pure Math. | isbn=978-4-931469-11-2 | mr=1846466 | year=2001 | volume=30 | chapter=Iwasawa theory---past and present | pages=335–385 | zbl=0998.11054 }}
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| *{{Citation | last1=Iwasawa | first1=Kenkichi | authorlink=Kenkichi Iwasawa | title=On Γ-extensions of algebraic number fields | doi=10.1090/S0002-9904-1959-10317-7 | mr=0124316 | year=1959 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=65 | issue=4 | pages=183–226| zbl=0089.02402 | issn=0002-9904 }}
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| *{{Citation | last1=Kato | first1=Kazuya | author1-link=Kazuya Kato | editor1-last=Sanz-Solé | editor1-first=Marta | editor2-last=Soria | editor2-first=Javier | editor3-last=Varona | editor3-first=Juan Luis | editor4-last=Verdera | editor4-first=Joan | title=International Congress of Mathematicians. Vol. I | url=http://www.icm2006.org/proceedings/Vol_I/18.pdf | publisher=Eur. Math. Soc., Zürich | isbn=978-3-03719-022-7 | doi=10.4171/022-1/14 | mr=2334196 | year=2007 | chapter=Iwasawa theory and generalizations | pages=335–357}}
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| * {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Cyclotomic fields I and II | url=http://books.google.com/books?isbn=0-387-96671-4 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=Combined 2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-96671-7 | year=1990 | volume=121 | zbl=0704.11038 | others=With an appendix by [[Karl Rubin]] }}
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| *{{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | last2=Wiles | first2=Andrew | author2-link=Andrew Wiles | title=Class fields of abelian extensions of '''Q''' | doi=10.1007/BF01388599 | mr=742853 | year=1984 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=76 | issue=2 | pages=179–330 | zbl=0545.12005 }}
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| *{{Citation
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| | last1=Neukirch | first1=Jürgen | author-link=Jürgen Neukirch
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| | last2=Schmidt | first2=Alexander | last3=Wingberg | first3=Kay
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| | title=Cohomology of Number Fields | chapter=
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| | publisher=[[Springer-Verlag]] | location=Berlin
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| | series=''Grundlehren der Mathematischen Wissenschaften''
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| | volume=323 | year=2008 | page=
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| | isbn=978-3-540-37888-4 | id={{MathSciNet | id = 2392026 }}
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| | zbl= 1136.11001 | edition=Second
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| }}
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| *{{Citation | last1=Rubin | first1=Karl | title=The ‘main conjectures’ of Iwasawa theory for imaginary quadratic fields | doi=10.1007/BF01239508 | year=1991 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=103 | issue=1 | pages=25–68 | unused_data=DUPLICATE DATA: doi=10.1007/BF01239508 | zbl=0737.11030 }}
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| *{{citation| last=Skinner| first=Chris| last2=Urban| first2=Éric| title=The Iwasawa main conjectures for GL<sub>2</sub>| year=2010| url=http://www.math.columbia.edu/%7Eurban/eurp/MC.pdf| page=219}}
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| *{{Citation | last1=Washington | first1=Lawrence C. | title=Introduction to cyclotomic fields | url=http://books.google.com/books?isbn=0-387-94762-0 | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-94762-4 | year=1997 | volume=83}}
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| * {{Citation | author = [[Andrew Wiles]]| year = 1990 | title = The Iwasawa Conjecture for Totally Real Fields | journal = Annals of Mathematics | volume = 131 | issue = 3 | pages = 493–540 | doi = 10.2307/1971468 | publisher = Annals of Mathematics | ref = harv | postscript = . | jstor = 1971468 | zbl=0719.11071 }}
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| ==Further reading==
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| * {{citation | last=de Shalit | first=Ehud | title=Iwasawa theory of elliptic curves with complex multiplication. ''p''-adic ''L'' functions | series=Perspectives in Mathematics | volume=3 | location=Boston etc. | publisher=Academic Press | year=1987 | isbn=0-12-210255-X | zbl=0674.12004 }}
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| ==External links==
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| *{{Springer|title=Iwasawa theory|id=i/i130090}}
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| {{L-functions-footer}}
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| [[Category:Field theory]]
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| [[Category:Cyclotomic fields]]
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| [[Category:Class field theory]]
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