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.
- Ray Class Groups
- Creation of Class Fields
AbelianExtension(D, U): DivFunElt, GrpAb → FldFunAbMaximalAbelianSubfield(K): FldFunG → FldFunAbHilbertClassField(K, p): FldFun, PlcFunElt → FldFunAbFunctionField(A): FldFunAb → FldFunMaximalOrderFinite(A): FldFunAb → RngFunOrdMaximalOrderInfinite(A): FldFunAb → RngFunOrd- Example: Creating Class Fields of a Function Field
- Example: Degree 4 Class Fields with Many Places
- Properties of Class Fields
Conductor(m): DivFunElt → DivFunEltConductor(m, U): DivFunElt, GrpAb → DivFunEltConductor(A): FldFunAb → DivFunEltDiscriminantDivisor(m, U): DivFunElt, GrpAb → DivFunEltDiscriminantDivisor(A): FldFunAb → DivFunEltDegreeOfExactConstantField(m): DivFunElt → RngIntEltDegreeOfExactConstantField(m, U): DivFunElt, GrpAb → RngIntEltDegreeOfExactConstantField(A): FldFunAb → RngIntEltGenus(m, U): DivFunElt, GrpAb → RngIntEltGenus(A): FldFunAb → RngIntEltDecompositionType(m, U, p): DivFunElt, GrpAb, PlcFunElt → [<f,e>]DecompositionType(A, p): FldFunAb, PlcFunElt → [<f,e>]NumberOfPlacesOfDegreeOne(m, U): DivFunElt, GrpAb → RngIntEltNumberOfPlacesOfDegreeOne(A): FldFunAb → RngIntEltDegree(A): FldFunAb → RngIntEltBaseField(A): FldFunAb → FldFunGA eq B: FldFunAb, FldFunAb → BoolEltA subset B: FldFunAb, FldFunAb → BoolEltA meet B: FldFunAb, FldFunAb → FldFunAbA * B: FldFunAb, FldFunAb → FldFunAb
- The Ring of Witt Vectors of Finite Length
WittRing(F, n): Fld, RngIntElt → RngWittW ! a: RngWitt, . → RngWittEltBaseRing(W): RngWitt → FldBaseField(W): RngWitt → FldLength(W): RngWitt → RngIntEltEltseq(a): RngWittElt → [FldElt]One(W): RngWitt → RngWittEltZero(W): RngWitt → RngWittEltW . 1: RngWitt, RngIntElt → RngWittEltx in W: ., RngWitt → BoolElta eq b: RngWittElt, RngWittElt → BoolElta eq b: RngWitt, RngWitt → BoolElta - b: RngWittElt, RngIntElt → RngWittElta - b: RngIntElt, RngWittElt → RngWittElta - b: RngWittElt, RngWittElt → RngWittElt- a: RngWittElt → RngWittElta + b: RngWittElt, RngIntElt → RngWittElta + b: RngIntElt, RngWittElt → RngWittElta + b: RngWittElt, RngWittElt → RngWittElta * b: RngWittElt, RngIntElt → RngWittElta * b: RngIntElt, RngWittElt → RngWittElta * b: RngWittElt, RngWittElt → RngWittElta ^ n: RngWittElt, RngIntElt → RngWittEltFrobeniusMap(W): RngWitt → MapFrobeniusImage(e): RngWittElt → RngWittEltVerschiebungMap(W): RngWitt → MapVerschiebungImage(e): RngWittElt → RngWittEltRandom(W): RngWitt → RngWittEltRandom(W, n): RngWitt, RngIntElt → RngWittEltTeichmuellerSystem(R): Any → [RngLocElt]TeichmuellerSystem(R): Rng → [RngLocElt]LocalRing(W): RngWitt → RngLoc, MapArtinSchreierMap(W): RngWitt → MapArtinSchreierImage(e): RngWittElt → RngWittEltFunctionField(e): RngWittElt → FldFun, MapMaximalOrderFinite(u): RngWittElt → RngFunOrdMaximalOrderInfinite(u): RngWittElt → RngFunOrdSMaximalOrder(u, S): RngWittElt, [PlcFunElt] → RngFunOrdMaximalOrders(u): RngWittElt → RngFunOrd, RngFunOrd
- 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
A + B: RngIntElt, RngUPolTwstElt → RngUPolTwstEltA + B: RngUPolTwstElt, RngIntElt → RngUPolTwstEltA + B: RngUPolTwstElt, RngUPolTwstElt → RngUPolTwstEltA - B: RngIntElt, RngUPolTwstElt → RngUPolTwstEltA - B: RngUPolTwstElt, RngIntElt → RngUPolTwstEltA - B: RngUPolTwstElt, RngUPolTwstElt → RngUPolTwstElt- A: RngUPolTwstElt → RngUPolTwstEltA * B: RngUPolTwstElt, RngUPolTwstElt → RngUPolTwstEltA ^ n: RngUPolTwstElt, RngIntElt → RngUPolTwstEltA eq B: RngUPolTwstElt, RngUPolTwstElt → BoolEltIsZero(A): RngUPolTwstElt → BoolEltLeadingCoefficient(F): RngUPolTwstElt → RngEltConstantCoefficient(F): RngUPolTwstElt → RngEltDegree(F): RngUPolTwstElt → RngIntEltQuotrem(F, G): RngUPolTwstElt, RngUPolTwstElt → RngUPolTwstElt, RngUPolTwstEltGCD(F, G): RngUPolTwstElt, RngUPolTwstElt → RngUPolTwstEltBaseRing(F): RngUPolTwstElt → RngPolynomial(G): RngUPolTwstElt → RngUPolEltSpecialEvaluate(F, x): RngUPolTwstElt, RngElt → RngEltSpecialEvaluate(F, x): RngUPolElt, Any → RngEltEltseq(F): RngUPolTwstElt → [RngElt]
- Analytic Theory
CarlitzModule(R, x): RngUPolTwst, RngUPolElt → RngUPolTwstEltCarlitzModule(R, x): RngUPolTwst, FldFunRatUElt → RngUPolTwstElt- Example: Abelian Extensions from the Carlitz Module
AnalyticDrinfeldModule(F, p): FldFun, PlcFunElt → RngUPolTwstEltExtend(D, x, p): RngUPolTwstElt, RngElt, PlcFunElt → RngUPolTwstElt- Example: Drinfeld Module on an Elliptic Curve
Exp(x, p): RngElt, PlcFunElt → RngUPolTwstEltAnalyticModule(x, p): RngElt, PlcFunElt → RngEltCanNormalize(F): RngUPolTwstElt → BoolElt, RngUPolTwstElt, RngEltCanSignNormalize(F): RngUPolTwstElt → BoolElt, RngUPolTwstElt, RngEltAlgebraicToAnalytic(F, p): RngUPolTwstElt, PlcFunElt → RngUPolTwstElt
- Related Functions
StrongApproximation(m, S): DivFunElt, [<PlcFunElt, FldFunElt>] → FldFunEltStrongApproximation(S, Z, V): [PlcFunElt], [FldFunGElt], [RngIntElt] → FldFunEltChineseRemainderTheorem(S, Z, V): [PlcFunElt], [FldFunGElt], [RngIntElt] → FldFunEltCRT(S, Z, V): [PlcFunElt], [FldFunGElt], [RngIntElt] → FldFunElt- Example: Strong Approximation
NonSpecialDivisor(m): DivFunElt → DivFunElt, RngIntEltNormGroup(F): FldFun → DivFunElt, GrpAbSign(a, p): FldFunElt, PlcFunElt → RngEltChangeModel(F, p): FldFun, PlcFunElt → FldFunArtinSchreierReduction(u, P): FldFunGElt, PlcFunElt → RngIntElt, FldFunElt
- Enumeration of Places
- Bibliography