Introduction to Riemann Surfaces
      Example RieSrf_rie-srf-verbose (H127E1)

 
Creation Functions

      Riemann Surfaces over Number Fields
            RiemannSurface(f) : RngMPolElt -> RieSrf
            RiemannSurface(f,sigma) : RngMPolElt, PlcNumElt -> RieSrf
            Example RieSrf_riesrf-ex-1 (H127E2)
            Example RieSrf_riesrf-ex-2 (H127E3)

      Superelliptic Riemann Surfaces
            RiemannSurface(p,m) : RngUPolElt, RngIntElt -> RieSrf
            RiemannSurface(L,m) : SeqEnum[FldComElt], RngIntElt -> RieSrf
            Example RieSrf_riesrf-ex-2 (H127E4)

 
Properties of Riemann Surfaces

      Basic Invariants
            BasePoint(X) : RieSrf -> RieSrfPt
            Genus(X) : RieSrf -> RngIntElt
            Degree(X) : RieSrf -> RngIntElt
            Precision(X) : RieSrf -> RngIntElt
            Embedding(X) : RieSrf -> PlcNumElt
            BigPeriodMatrix(X) : RieSrf -> Mtrx
            SmallPeriodMatrix(X) : RieSrf -> Mtrx
            FunctionField(X) : RieSrf -> FldFun
            Example RieSrf_invariants (H127E5)

      Fundamental Group
            DiscriminantPoints(f) : RngMPolElt -> SeqEnum[FldComElt]
            BranchPoints(X) : RieSrf -> Tup
            RamificationPoints(X) : RieSrf -> SeqEnum[RieSrfPt]
            SingularPoints(X) : RieSrf -> SeqEnum
            FundamentalGroup(P) : SeqEnum[FldComElt] -> FldComElt, SeqEnum[FldComElt], SeqEnum[CPath], SeqEnum[SeqEnum[RngIntElt]]
            FundamentalGroup(X) : RieSrf -> SeqEnum[CChain]
            MonodromyRepresentation(X): RieSrf -> SeqEnum
            Example RieSrf_riesrf-ex-1 (H127E6)

      Basis for Period Matrix
            HolomorphicDifferentials(X) : RieSrf -> Tup
            Example RieSrf_riesrf-ex-1 (H127E7)
            Example RieSrf_riesrf-ex-1 (H127E8)
            HomologyBasis(L) : SeqEnum[GrpPermElt] -> SeqEnum[SeqEnum[RngIntElt]], Mtrx, Mtrx
            HomologyBasis(X) : RieSrf -> SeqEnum[SeqEnum[RngIntElt]], Mtrx, Mtrx
            Example RieSrf_homology-basis1 (H127E9)
            Example RieSrf_homology-basis2 (H127E10)

 
Points on Riemann Surfaces

      Points
            IsCoercible(X, S) : RieSrf, Any -> BoolElt, .
            Point(X, S): RieSrf, SeqEnum -> RieSrfPt
            Point(X, S) : RieSrf, Tup -> RieSrfPt
            Example RieSrf_rie-points (H127E11)

      Access Functions
            RiemannSurface(P) : RieSrfPt -> RieSrf
            Representation(P) : RieSrfPt -> Tup
            Coordinates(P) : RieSrfPt -> SeqEnum[FldComElt]
            RamificationIndex(P) : RieSrfPt -> RngIntElt
            PointsOverDiscriminantPoint(X, k) : RieSrf, RngIntElt -> SeqEnum[RieSrfPt]
            RandomPoint(X) : RieSrf -> RieSrfPt
            Example RieSrf_create-pts-1 (H127E12)

 
Divisors on Riemann Surfaces
      Divisor(S,V) : SeqEnum[RieSrfPt], SeqEnum[RngIntElt] -> DivRieSrfElt
      ZeroDivisor(X) : RieSrfElt -> DivRieSrfElt
      RiemannSurface(D) : DivRieSrfElt -> RieSrf
      Support(D) : DivRieSrfElt -> SeqEnum[RieSrfPt], SeqEnum[RngIntElt]
      Degree(D) : DivRieSrfElt -> RngIntElt
      RandomDivisor(X,d) : RieSrf, RngIntElt -> RieSrfDivElt

 
Abel--Jacobi Map
      AbelJacobi(P) : RieSrfPt -> Mtrx
      AbelJacobi(P, Q) : RieSrfPt, RieSrfPt -> Mtrx
      AbelJacobi(D, P) : DivRieSrfElt, RieSrfPt -> Mtrx
      Example RieSrf_abel-jacobi-sup (H127E13)
      Example RieSrf_abel-jacobi-gen-1 (H127E14)
      Example RieSrf_abel-jacobi-gen-2 (H127E15)

 
Period Matrix Functions
      Example RieSrf_iso-small-pm-1 (H127E16)

 
Bibliography

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