View source for Iwasawa theory

A theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300901.png" />-extensions introduced by K. Iwasawa [[#References|[a8]]]. Its motivation has been a strong analogy between number fields and curves over finite fields. One of the most fruitful results in this theory is the Iwasawa main conjecture, which has been proved for totally real number fields [[#References|[a19]]]. The conjecture is considered as an analogue of Weil's result that the characteristic polynomial of the Frobenius automorphism acting on the Jacobian of a curve over a finite field is the essential part of the zeta-function of the curve. A lot of methods and ideas developed in the theory appeared to be widely applicable and have given rise to major advances, for example, results on the [[Birch–Swinnerton-Dyer conjecture]] [[#References|[a3]]], [[#References|[a6]]], [[#References|[a16]]], [[#References|[a22]]] and on Fermat's last theorem [[#References|[a20]]] (cf. also [[Fermat last theorem|Fermat last theorem]]). For details and generalizations of Iwasawa theory, see [[#References|[a10]]], [[#References|[a7]]], [[#References|[a12]]], [[#References|[a18]]].

==<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300902.png" />-extension of a number field.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300903.png" /> be a prime number and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300904.png" /> be a finite extension of the rational number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300905.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300907.png" />-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300908.png" /> is an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i1300909.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009011.png" /> is the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009012.png" />-adic integers. Then there is a sequence of fields

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009014.png" /> is a cyclic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009015.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009016.png" />. [[Class field theory|Class field theory]] shows that there are at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009017.png" /> independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009018.png" />-extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009019.png" /> (cf. below, the section Leopoldt conjecture). Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009020.png" /> has at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009021.png" />-extension, namely the cyclotomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009022.png" />-extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009023.png" />. It is obtained by letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009024.png" /> be an appropriate subfield of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009025.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009026.png" /> is the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009027.png" />th roots of unity.

==Leopoldt conjecture.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009028.png" /> be the group of units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009029.png" /> which are congruent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009030.png" /> modulo every prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009032.png" /> lying above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009033.png" />. By Dirichlet's unit theorem, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009035.png" /> (resp. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009036.png" />) is the number of embeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009038.png" /> (resp. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009039.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009040.png" /> be the group of local units of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009041.png" /> congruent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009042.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009043.png" />. There is an embedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009045.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009046.png" /> denote the topological closure of the image. It is Leopoldt's conjecture that the equality

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holds for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009048.png" />. A. Brumer [[#References|[a1]]] proved the conjecture for Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009049.png" /> (or an imaginary quadratic field). Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009050.png" />. Then class field theory shows that there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009051.png" /> independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009052.png" />-extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009053.png" />.

==Iwasawa module.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009054.png" /> be the integer ring of a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009056.png" /> a uniformizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009057.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009058.png" /> be a compact Abelian group isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009060.png" />, where the inverse limit is taken with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009061.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009062.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009063.png" />. Fix a topological generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009065.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009066.png" /> be the ring of formal power series in an indeterminate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009067.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009068.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009069.png" /> is called a distinguished polynomial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009070.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009071.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009072.png" />. The prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009073.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009078.png" /> is distinguished and irreducible. The classification of compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009079.png" />-modules in [[#References|[a8]]] was simplified by J.-P. Serre, who pointed out that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009080.png" /> is topologically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009081.png" />, hence each compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009082.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009083.png" /> admits the unique structure of a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009084.png" />-module such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009085.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009086.png" />. Finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009087.png" />-modules are called Iwasawa modules. They are classified as follows: for an Iwasawa module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009088.png" />, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009089.png" />-homomorphism

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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009092.png" /> finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009093.png" />-modules, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009095.png" /> is distinguished and irreducible. For a torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009096.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009097.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i13009098.png" />, one defines

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==Iwasawa invariant.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090101.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090102.png" />-extension. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090103.png" /> denote the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090104.png" />-Sylow subgroup of the ideal class group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090105.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090106.png" /> be the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090107.png" />. Iwasawa [[#References|[a8]]] proved that there exist integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090110.png" /> such that

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for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090112.png" />. The invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090114.png" /> can be obtained from the Iwasawa module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090115.png" />, where the inverse limit is taken with respect to the relative norm mappings. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090116.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090117.png" /> is a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090118.png" />-module in a natural way. One fixes a topological generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090119.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090120.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090121.png" /> is considered as a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090122.png" />-module (cf. the section on Iwasawa module above). Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090123.png" /> is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090124.png" /> is a finitely-generated torsion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090125.png" />-module. One has that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090127.png" />.

Iwasawa [[#References|[a9]]] constructed infinitely many non-cyclotomic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090128.png" />-extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090129.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090130.png" />. There are infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090131.png" />-extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090132.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090133.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090134.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090135.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090136.png" /> is irregular (cf. also [[Irregular prime number|Irregular prime number]]). It is Iwasawa's conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090137.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090138.png" />. B. Ferrero and L. Washington [[#References|[a5]]] proved this conjecture for Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090139.png" />. W. Sinnott [[#References|[a17]]] gave a new proof of this using the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090140.png" />-transform of a rational function.

It is Greenberg's conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090141.png" /> for every totally real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090142.png" />. For small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090143.png" />, it was proved that there are infinitely many real quadratic fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090144.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090145.png" /> [[#References|[a14]]], [[#References|[a15]]]. There exists a lot of numerical work verifying this conjecture, mainly for real quadratic fields.

It is Vandiver's conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090146.png" /> does not divide the class number of the maximal real subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090147.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090148.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090149.png" />, which implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090150.png" />. This conjecture was verified for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090151.png" /> [[#References|[a2]]].

==Iwasawa main conjecture.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090152.png" /> be an odd prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090153.png" /> a totally real number field. Fix an embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090154.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090155.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090156.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090157.png" />-adic valued Artin character for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090158.png" /> of order prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090159.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090160.png" /> be the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090161.png" /> attached to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090162.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090163.png" /> is also totally real. Fix a topological generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090164.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090165.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090166.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090167.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090168.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090169.png" /> be the Teichmüller character

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090170.png" /></td> </tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090171.png" /> be the classical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090173.png" />-function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090174.png" />. Following T. Kubota and H.W. Leopoldt [[#References|[a11]]], P. Deligne and K. Ribet [[#References|[a4]]] proved the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090177.png" />-adic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090178.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090179.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090180.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090181.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090182.png" /> is trivial) satisfying the following interpolation property:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090183.png" /></td> </tr></table>

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090184.png" />. There exists a unique power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090185.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090186.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090187.png" /> is trivial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090188.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090189.png" /> is the ring generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090190.png" /> by the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090191.png" />. By the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090193.png" />-adic Weierstrass preparation theorem (cf. also [[Weierstrass theorem|Weierstrass theorem]]), one can write <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090194.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090195.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090196.png" /> is a distinguished polynomial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090197.png" /> is a uniformizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090198.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090199.png" /> is a unit power series. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090200.png" /> be such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090201.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090202.png" /> is trivial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090203.png" />). One can similarly define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090204.png" /> and a distinguished polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090205.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090206.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090207.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090208.png" /> be the maximal unramified Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090210.png" />-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090211.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090212.png" /> the maximal Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090214.png" />-extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090215.png" />, which are both unramified outside the primes above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090216.png" />. By class field theory, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090217.png" />. Extend <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090218.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090219.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090220.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090221.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090222.png" />. Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090223.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090224.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090225.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090226.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090227.png" /></td> </tr></table>

Then one can regard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090228.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090229.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090230.png" />-modules.

Following [[#References|[a13]]], A. Wiles proved the following equality, i.e., the Iwasawa main conjecture for totally real fields:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090231.png" /></td> </tr></table>

This equality is equivalent to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090232.png" /></td> </tr></table>

The proof uses delicate techniques from modular forms, especially Hida's theory of modular forms, to construct unramified extensions.

Following Stickelberger's theorem, F. Thaine and V. Kolyvagin invented techniques for constructing relations in ideal class groups. These methods, which use Gauss sums (cyclotomic units or elliptic units) satisfying properties known as the Euler system, have given elementary proofs of the Iwasawa main conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090233.png" /> [[#References|[a12]]], [[#References|[a21]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Brumer, "On the units of algebraic number fields" ''Mathematika'' , '''14''' (1967) pp. 121–124 {{MR|0220694}} {{ZBL|0171.01105}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, M.A. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million" ''J. Symbolic Comput.'' , '''31''' (2001) pp. 89–96 {{MR|1806208}} {{ZBL|1001.11061}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" ''Invent. Math.'' , '''39''' (1977) pp. 223–251 {{MR|0463176}} {{ZBL|0359.14009}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> P. Deligne, K. Ribet, "Values of abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090234.png" />-functions at negative integers over totally real fields" ''Invent. Math.'' , '''59''' (1980) pp. 227–286 {{MR|579702}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> B. Ferrero, L. Washington, "The Iwasawa invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090235.png" /> vanishes for abelian number fields" ''Ann. of Math.'' , '''109''' (1979) pp. 377–395 {{MR|528968}} {{ZBL|0443.12001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> R. Greenberg, "On the Birch and Swinnerton-Dyer conjecture" ''Invent. Math.'' , '''72''' (1983) pp. 241–265 {{MR|0700770}} {{ZBL|0546.14015}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> K. Iwasawa, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090236.png" />-extensions of algebraic number fields" ''Ann. of Math.'' , '''98''' (1973) pp. 246–326 {{MR|349627}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> K. Iwasawa, "On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090237.png" />-extensions of algebraic number fields" ''Bull. Amer. Math. Soc.'' , '''65''' (1959) pp. 183–226 {{MR|124316}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> K. Iwasawa, "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090238.png" />-invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090239.png" />-extensions" , ''Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki'' , Kinokuniya (1973) pp. 1–11 {{MR|357371}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> J. Coates, R. Greenberg, B. Mazur, I. Satake, "Algebraic Number Theory—In Honor of K. Iwasawa" , ''Adv. Studies in Pure Math.'' , '''17''' , Acad. Press (1989) {{MR|}} {{ZBL|0721.00006}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> T. Kubota, H.W. Leopoldt, "Eine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090240.png" />-adische Theorie der Zetawerte, I. Einführung der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090241.png" />-adischen Dirichletschen <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i130/i130090/i130090242.png" />-Funktionen" ''J. Reine Angew. Math.'' , '''214/215''' (1964) pp. 328–339 {{MR|0163900}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> S. Lang, "Cyclotomic fields I—II" , ''Graduate Texts in Math.'' , '''121''' , Springer (1990) (with an appendix by K. Rubin)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> B. Mazur, A. 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