[Next][Prev] [Right] [Left] [Up] [Index] [Root]

CLASS FIELD THEORY FOR GLOBAL FUNCTION FIELDS

Global function fields admit a class field theory in the same way as number fields do (Chapter CLASS FIELD THEORY). From a computational point of view the main difference is the use of divisors rather than ideals and the availability in general of analytical methods; see Section Analytic Theory.

Class field theory deals with the abelian extensions of a given field. In the number field case, all abelian extensions can be parameterized using more general class groups, in the case of global function fields, the same will be achieved using the divisor class group and extensions of it.  
Acknowledgements
 
Ray Class Groups
 
Creation of Class Fields
 
Properties of Class Fields
 
The Ring of Witt Vectors of Finite Length
 
The Ring of Twisted Polynomials
      Creation of Twisted Polynomial Rings
      Operations with the Ring of Twisted Polynomials
      Creation of Twisted Polynomials
      Operations with Twisted Polynomials
 
Analytic Theory
 
Related Functions
 
Enumeration of Places
 
Bibliography







DETAILS

 
Ray Class Groups
      RayResidueRing(D) : DivFunElt -> GrpAb, Map
      RayClassGroup(D) : DivFunElt -> GrpAb, Map
      RayClassGroupDiscLog(y, D) : DivFunElt, DivFunElt -> GrpAbElt
      Example FldFunAb_classfield-structures (H43E1)

 
Creation of Class Fields
      AbelianExtension(D, U) : DivFunElt, GrpAb -> FldFunAb
      MaximalAbelianSubfield(K) : FldFunG -> FldFunAb
      HilbertClassField(K, p) : FldFun, PlcFunElt -> FldFunAb
      FunctionField(A) : FldFunAb -> FldFun
      MaximalOrderFinite(A) : FldFunAb -> RngFunOrd
      Example FldFunAb_classfields (H43E2)

 
Properties of Class Fields
      Conductor(m) : DivFunElt -> DivFunElt
      Conductor(m, U) : DivFunElt, GrpAb -> DivFunElt
      Conductor(A) : FldFunAb -> DivFunElt
      DiscriminantDivisor(m, U) : DivFunElt, GrpAb -> DivFunElt
      DiscriminantDivisor(A) : FldFunAb -> DivFunElt
      DegreeOfExactConstantField(m) : DivFunElt -> RngIntElt
      DegreeOfExactConstantField(m, U) : DivFunElt, GrpAb -> RngIntElt
      DegreeOfExactConstantField(A) : FldFunAb -> RngIntElt
      Genus(m, U) : DivFunElt, GrpAb -> RngIntElt
      Genus(A) : FldFunAb -> RngIntElt
      DecompositionType(m, U, p) : DivFunElt, GrpAb, PlcFunElt -> [<f,e>]
      DecompositionType(A, p) : FldFunAb, PlcFunElt -> [<f,e>]
      NumberOfPlacesOfDegreeOne(m, U) : DivFunElt, GrpAb -> RngIntElt
      NumberOfPlacesOfDegreeOne(A) : FldFunAb -> RngIntElt
      Degree(A) : FldFunAb -> RngIntElt
      BaseField(A) : FldFunAb -> FldFunG
      A eq B : FldFunAb, FldFunAb -> BoolElt
      A subset B : FldFunAb, FldFunAb -> BoolElt
      A meet B : FldFunAb, FldFunAb -> FldFunAb
      A * B : FldFunAb, FldFunAb -> FldFunAb

 
The Ring of Witt Vectors of Finite Length
      WittRing(F, n) : Fld, RngIntElt -> RngWitt
      W ! a : RngWitt, . -> RngWittElt
      BaseRing(W) : RngWitt -> Fld
      Length(W) : RngWitt -> RngIntElt
      Eltseq(a) : RngWittElt -> [FldElt]
      Unity(W) : RngWitt -> RngWittElt
      W . 1 : RngWitt, RngIntElt -> RngWittElt
      FrobeniusMap(W) : RngWitt -> Map
      FrobeniusImage(e) : RngWittElt -> RngWittElt
      VerschiebungMap(W) : RngWitt -> Map
      VerschiebungImage(e) : RngWittElt -> RngWittElt
      Random(W) : RngWitt -> RngWittElt
      Random(W, n) : RngWitt, RngIntElt -> RngWittElt
      TeichmuellerSystem(R) : Any -> [RngLocElt]
      LocalRing(W) : RngWitt -> RngLoc, Map
      ArtinSchreierMap(W) : RngWitt -> Map
      ArtinSchreierImage(e) : RngWittElt -> RngWittElt
      FunctionField(e) : RngWittElt -> FldFun, Map

 
The Ring of Twisted Polynomials

      Creation of Twisted Polynomial Rings
            TwistedPolynomials(R) : Rng -> RngUPolTwst

      Operations with the Ring of Twisted Polynomials
            Unity(R) : RngUPolTwst -> RngUPolTwstElt
            Zero(R) : RngUPolTwst -> RngUPolTwstElt
            R eq S : RngUPolTwst, RngUPolTwst -> BoolElt
            BaseRing(R) : RngUPolTwst -> Rng
            R . i : RngUPolTwst, RngIntElt -> RngUPolTwstElt

      Creation of Twisted Polynomials
            AdditivePolynomialFromRoots(x, P) : RngElt, PlcFunElt -> RngUPolTwstElt
            Random(F, n) : RngUPolTwst, RngIntElt -> RngUPolTwstElt
            Example FldFunAb_additive-polynomial (H43E3)

      Operations with Twisted Polynomials
            LeadingCoefficient(F) : RngUPolTwstElt -> RngElt
            ConstantCoefficient(F) : RngUPolTwstElt -> RngElt
            Degree(F) : RngUPolTwstElt -> RngIntElt
            Quotrem(F, G) : RngUPolTwstElt, RngUPolTwstElt -> RngUPolTwstElt, RngUPolTwstElt
            GCD(F, G) : RngUPolTwstElt, RngUPolTwstElt -> RngUPolTwstElt
            BaseRing(F) : RngUPolTwstElt -> Rng
            Polynomial(G) : RngUPolTwstElt -> RngUPolElt
            SpecialEvaluate(F, x) : RngUPolTwstElt, RngElt -> RngElt
            SpecialEvaluate(F, x) : RngUPolElt, Any -> RngElt
            Eltseq(F) : RngUPolTwstElt -> [RngElt]

 
Analytic Theory
      CarlitzModule(R, x) : RngUPolTwst, RngUPolElt -> RngUPolTwstElt
      Example FldFunAb_carlitz-module (H43E4)
      AnalyticDrinfeldModule(F, p) : FldFun, PlcFunElt -> RngUPolTwstElt
      Extend(D, x, p) : RngUPolTwstElt, RngElt, PlcFunElt -> RngUPolTwstElt
      Example FldFunAb_drinfeld (H43E5)
      Exp(x,p) : RngElt, PlcFunElt -> RngUPolTwstElt
      AnalyticModule(x, p) : RngElt, PlcFunElt -> RngElt
      CanNormalize(F) : RngUPolTwstElt -> BoolElt, RngUPolTwstElt, RngElt
      CanSignNormalize(F) : RngUPolTwstElt -> BoolElt, RngUPolTwstElt, RngElt
      AlgebraicToAnalytic(F, p) : RngUPolTwstElt, PlcFunElt -> RngUPolTwstElt

 
Related Functions
      StrongApproximation(m, S): DivFunElt, [<PlcFunElt, FldFunElt>] -> FldFunElt
      Example FldFunAb_strong-approximation (H43E6)
      NonSpecialDivisor(m): DivFunElt -> DivFunElt, RngIntElt
      NormGroup(F) : FldFun -> DivFunElt, GrpAb
      Sign(a, p) : FldFunElt, PlcFunElt -> RngElt
      ChangeModel(F, p) : FldFun, PlcFunElt -> FldFun

 
Enumeration of Places
      PlaceEnumInit(K) : FldFun -> PlcEnum
      PlaceEnumInit(P) : PlcFunElt -> PlcEnum
      PlaceEnumInit(K, Pos) : FldFun, [RngIntElt]) -> PlcEnum
      PlaceEnumCopy(R) : PlcEnum -> PlcEnum
      PlaceEnumPosition(R) : PlcEnum -> [RngIntElt]
      PlaceEnumNext(R) : PlcEnum -> PlcFunElt
      PlaceEnumCurrent(R) : PlcEnum -> PlcFunElt

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]
Version: V2.19 of Mon Dec 17 14:40:36 EST 2012