About: Main conjecture of Iwasawa theory

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In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes byMazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture.There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on.

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rdfs:label	岩澤理論の主予想 (ja) Main conjecture of Iwasawa theory (en) Huvudförmodan inom Iwasawateori (sv)
rdfs:comment	In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes byMazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture.There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on. (en) 数学では、岩澤理論の主予想 (main conjecture of Iwasawa theory) は、p-進L-函数と円分体のイデアル類群との間の深い関係であり、でを満たす素数に対して証明され、すべての素数に対しては Mazur and Wiles により証明された。とが両方ともこの主予想より容易に導ける結果である。この主予想にはいくつかの一般化があり、総実体や CM体や、楕円曲線などへ一般化される。 (ja) Inom matematiken är huvudförmodan inom Iwasawateori en djup relation mellan och av , bevisad av ) för primtal som satisfierar och för alla primtal av Mazur och Wiles. och Gras förmodan är båda enkla konsekvenser av huvudförmodan. Det finns flera generaliseringar av huvudförmodan, exempelvis till , och elliptiska kurvor. (sv)
name	Main conjecture of Iwasawa theory (en)
dcterms:subject	Conjectures Cyclotomic fields Theorems in algebraic number theory
Wikipage page ID	22091332 (xsd:integer)
Wikipage revision ID	1068757632 (xsd:integer)
Link from a Wikipage to another Wikipage	Benedict Gross Totally real number field Totally real field Barry Mazur Annals of Mathematics Inventiones Mathematicae Iwasawa algebra Iwasawa theory Jacobian variety Conjectures Mathematics Elliptic curve Frobenius endomorphism Modular form Eric Urban Andrew Wiles Cyclotomic fields Ideal class group Kummer–Vandiver conjecture Wei Zhang (mathematician) Weil conjectures Algebraic number theory Cyclotomic field Finite field Christopher Skinner Graduate Texts in Mathematics Gras conjecture Theorems in algebraic number theory Karl Rubin Ken Ribet Kenkichi Iwasawa Herbrand–Ribet theorem Modular elliptic curve Teichmüller character Don Zagier Manjul Bhargava Rational number Selmer group Euler system Victor Kolyvagin P-adic L-function Thaine's method Springer-Verlag P-adic L function Hasse–Weil L-function Tate–Shafarevich conjecture Generalized Bernoulli number Birch–Swinnerton-Dyer conjecture CM field
Link from a Wikipage to an external page	https://mattbaker.blog/2014/03/10/the-bsd-conjecture-is-true-for-most-elliptic-curves/ http://link.springer.com/10.1007/s00222-013-0448-1 https://books.google.com/books%3Fisbn=0-387-94762-0 https://books.google.com/books%3Fisbn=0-387-96671-4
sameAs	Main conjecture of Iwasawa theory Main conjecture of Iwasawa theory Main conjecture of Iwasawa theory Main conjecture of Iwasawa theory Main conjecture of Iwasawa theory Main conjecture of Iwasawa theory
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field	Iwasawa theory Algebraic number theory
has abstract	In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes byMazur and Wiles. The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture.There are several generalizations of the main conjecture, to totally real fields, CM fields, elliptic curves, and so on. (en) 数学では、岩澤理論の主予想 (main conjecture of Iwasawa theory) は、p-進L-函数と円分体のイデアル類群との間の深い関係であり、でを満たす素数に対して証明され、すべての素数に対しては Mazur and Wiles により証明された。とが両方ともこの主予想より容易に導ける結果である。この主予想にはいくつかの一般化があり、総実体や CM体や、楕円曲線などへ一般化される。 (ja)

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