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main conjecture of Iwasawa theory

{{short description|Theorem in algebraic number theory relating p-adic L-functions and ideal class groups}}

{{Infobox mathematical statement

| name = Main conjecture of Iwasawa theory

| image =

| caption =

| field = Algebraic number theory
Iwasawa theory

| conjectured by = Kenkichi Iwasawa

| conjecture date = 1969

| first proof by = Barry Mazur
Andrew Wiles

| first proof date = 1984

}}

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 by

{{harvs|txt|last1=Mazur|last2=Wiles|year=1984}}. 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,{{harvnb|Wiles|1990}}, {{harvnb|Kakde|2013}} CM fields, elliptic curves, and so on.

Motivation

{{harvtxt|Iwasawa|1969a}} was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian variety. In this analogy,

  • The action of the Frobenius corresponds to the action of the group Γ.
  • The Jacobian of a curve corresponds to a module X over Γ defined in terms of ideal class groups.
  • The zeta function of a curve over a finite field corresponds to a p-adic L-function.
  • Weil's theorem relating the eigenvalues of Frobenius to the zeros of the zeta function of the curve corresponds to Iwasawa's main conjecture relating the action of the Iwasawa algebra on X to zeros of the p-adic zeta function.

History

The main conjecture of Iwasawa theory was formulated 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 fields by {{harvtxt|Wiles|1990}}. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the Herbrand–Ribet theorem).

Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in {{harvtxt|Lang|1990}} and {{harvtxt|Washington|1997}}, and later proved other generalizations of the main conjecture for imaginary quadratic fields.{{sfn|Manin|Panchishkin|2007|p=246}}

In 2014, Christopher Skinner and Eric Urban proved several cases of the main conjectures for a large class of modular forms.{{sfn|Skinner|Urban|2014|pp=1–277}} As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil L-function L(E, s) of E at s = 1 implies that the p-adic Selmer group of E is infinite. Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has infinitely many rational points if and only if L(E, 1) = 0, a (weak) form of the Birch–Swinnerton-Dyer conjecture. These results were used by Manjul Bhargava, Skinner, and Wei Zhang to prove that a positive proportion of elliptic curves satisfy the Birch–Swinnerton-Dyer conjecture.{{sfn|Bhargava|Skinner|Zhang|2014}}{{sfn|Baker|2014}}

Statement

  • p is a prime number.
  • Fn is the field Q(ζ) where ζ is a root of unity of order pn+1.
  • Γ is the largest subgroup of the absolute Galois group of F∞ isomorphic to the p-adic integers.
  • γ is a topological generator of Γ.
  • Ln is the p-Hilbert class field of Fn.
  • Hn is the Galois group Gal(Ln/Fn), isomorphic to the subgroup of elements of the ideal class group of Fn whose order is a power of p.
  • H∞ is the inverse limit of the Galois groups Hn.
  • V is the vector space H∞⊗ZpQp.
  • ω is the Teichmüller character.
  • Vi is the ωi eigenspace of V.
  • hp(ωi,T) is the characteristic polynomial of γ acting on the vector space Vi.
  • Lp is the p-adic L function with Lp(ωi,1–k) = –Bk(ωi–k)/k, where B is a generalized Bernoulli number.
  • u is the unique p-adic number satisfying γ(ζ) = ζu for all p-power roots of unity ζ.
  • Gp is the power series with Gp(ωi,us–1) = Lp(ωi,s).

The main conjecture of Iwasawa theory proved by Mazur and Wiles states that if i is an odd integer not congruent to 1 mod p–1 then the ideals of \mathbf Z_pT generated by hp(ωi,T) and Gp(ω1–i,T) are equal.

Notes

{{Reflist|22em}}

Sources

{{refbegin}}

  • {{citation| title = The BSD conjecture is true for most elliptic curves

| last = Baker | first = Matt

| website = Matt Baker's Math Blog

| url = https://mattbaker.blog/2014/03/10/the-bsd-conjecture-is-true-for-most-elliptic-curves/

| date = 2014-03-10 | access-date = 2019-02-24

}}

  • {{cite arXiv| title = A majority of elliptic curves over $\mathbb Q$ satisfy the Birch and Swinnerton-Dyer conjecture

| last1 = Bhargava | first1 = Manjul

| last2 = Skinner | first2 = Christopher

| last3 = Zhang | first3 = Wei

| class = math.NT

| date = 2014-07-07

| eprint = 1407.1826

| mode = cs2

}}

  • {{citation| title = Cyclotomic Fields and Zeta Values

| last1 = Coates | first1 = John

| last2 = Sujatha | first2 = R.

| author1-link = John Coates (mathematician)

| author2-link = Sujatha Ramdorai

| year = 2006

| publisher = Springer-Verlag

| series = Springer Monographs in Mathematics

| isbn = 978-3-540-33068-4 | zbl = 1100.11002

}}

  • {{Citation| title = On some modules in the theory of cyclotomic fields

| last = Iwasawa | first = Kenkichi | year = 1964

| journal = Journal of the Mathematical Society of Japan

| volume = 16 | pages = 42–82

| doi = 10.2969/jmsj/01610042 | issn = 0025-5645 | mr = 0215811

| doi-access = free

}}

  • {{Citation| chapter = Analogies between number fields and function fields

| last = Iwasawa | first = Kenkichi | year = 1969a

| title = Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966)

| publisher = Belfer Graduate School of Science, Yeshiva Univ., New York

| pages = 203–208

| mr = 0255510

}}

  • {{Citation| title = On p-adic L-functions

| last = Iwasawa | first = Kenkichi | year = 1969b

| journal = Annals of Mathematics

| volume = 89 | issue = 1 | pages = 198–205

| series = Second Series

| doi = 10.2307/1970817 | issn = 0003-486X | jstor = 1970817 | mr = 0269627

}}

  • {{citation

| last = Kakde | first = Mahesh | author-link = Mahesh Kakde

| arxiv = 1008.0142

| doi = 10.1007/s00222-012-0436-x

| issue = 3

| journal = Inventiones Mathematicae

| mr = 3091976

| pages = 539–626

| title = The main conjecture of Iwasawa theory for totally real fields

| volume = 193

| year = 2013| bibcode = 2013InMat.193..539K }}

  • {{Citation| title = Cyclotomic fields I and II | edition = Combined 2nd

| last = Lang | first = Serge | year = 1990

| author-link = Serge Lang

| others = With an appendix by Karl Rubin

| publisher = Springer-Verlag | location = Berlin, New York

| volume = 121 | series = Graduate Texts in Mathematics

| url = https://books.google.com/books?isbn=0-387-96671-4

| isbn = 978-0-387-96671-7 | zbl = 0704.11038

}}

  • {{citation| title = Introduction to Modern Number Theory | edition = Second

| last1 = Manin | first1 = Yu I.

| last2 = Panchishkin | first2 = A. A.

| author1-link = Yuri I. Manin

| year = 2007

| volume = 49 | series = Encyclopaedia of Mathematical Sciences

| isbn = 978-3-540-20364-3 | issn = 0938-0396 | zbl = 1079.11002

}}

  • {{Citation| title = Class fields of abelian extensions of Q

| last1 = Mazur | first1 = Barry

| last2 = Wiles | first2 = Andrew

| author1-link = Barry Mazur

| author2-link = Andrew Wiles

| journal = Inventiones Mathematicae

| year = 1984 | volume = 76 | issue = 2 | pages = 179–330

| doi = 10.1007/BF01388599 | bibcode = 1984InMat..76..179M | issn = 0020-9910 | mr = 742853

| s2cid = 122576427 }}

  • {{citation

| last1 = Skinner | first1 = Christopher

| last2 = Urban | first2 = Eric

| citeseerx = 10.1.1.363.2008

| doi = 10.1007/s00222-013-0448-1

| issue = 1

| journal = Inventiones Mathematicae

| mr = 3148103

| pages = 1–277

| s2cid = 120848645

| title = The Iwasawa main conjectures for {{math|GL2}}

| volume = 195

| year = 2014| bibcode = 2014InMat.195....1S

}}

  • {{Citation| title = Introduction to cyclotomic fields | edition = 2nd

| last = Washington | first = Lawrence C. | year = 1997

| author-link = Lawrence C. Washington

| publisher = Springer-Verlag | location = Berlin, New York

| volume = 83 | series = Graduate Texts in Mathematics

| url = https://books.google.com/books?isbn=0-387-94762-0

| isbn = 978-0-387-94762-4

}}

  • {{Citation| title = The Iwasawa conjecture for totally real fields

| last = Wiles | first = Andrew | year = 1990

| author-link = Andrew Wiles

| journal = Annals of Mathematics

| volume = 131 | issue = 3 | pages = 493–540

| series = Second Series

| doi = 10.2307/1971468 | issn = 0003-486X | jstor = 1971468 | mr = 1053488

}}

{{refend}}

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Category:Conjectures

Category:Cyclotomic fields

Category:Theorems in algebraic number theory