Riemann Surfaces

• Introduction to Riemann Surfaces
  • Example: Verbose Output for Riemann Surfaces
• Creation Functions
  • Riemann Surfaces over Number Fields
    • RiemannSurface(f): RngMPolElt → RieSrf
    • RiemannSurface(f,sigma): RngMPolElt, PlcNumElt → RieSrf
    • Example: Riemann Surface over the Rationals
    • Example: Riemann Surface over a Number Field
  • Superelliptic Riemann Surfaces
    • RiemannSurface(p,m): RngUPolElt, RngIntElt → RieSrf
    • RiemannSurface(L,m): SeqEnum[FldComElt], RngIntElt → RieSrf
    • Example: Superelliptic Riemann Surface of Genus 12
• 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: Invariants of a Riemann Surface
  • Fundamental Group
    • DiscriminantPoints(f): RngMPolElt → SeqEnum[FldComElt]
    • DiscriminantPoints(f, sigma): RngMPolElt, PlcNumElt → SeqEnum[FldComElt]
    • DiscriminantPoints(X): RieSrf → 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: Fundamental Group of a Riemann Surface
  • Basis for Period Matrix
    • HolomorphicDifferentials(X): RieSrf → Tup
    • Example: Holomorphic Differentials of a Plane Curve
    • Example: Holomorphic Differentials of a Superelliptic Surface
    • HomologyBasis(L): SeqEnum[GrpPermElt] → SeqEnum[SeqEnum[RngIntElt]], Mtrx, Mtrx
    • HomologyBasis(X): RieSrf → SeqEnum[SeqEnum[RngIntElt]], Mtrx, Mtrx
    • Example: Homology Basis from Monodromy
    • Example: Homology Basis for the Big Period Matrix
• Points on Riemann Surfaces
  • Points
    • IsCoercible(X, S): RieSrf, Any → BoolElt, .
    • Point(X, S): RieSrf, SeqEnum → RieSrfPt
    • Point(X, S): RieSrf, Tup → RieSrfPt
    • Example: Creating a Point on a Riemann Surface
  • 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: Point Access Functions on a Riemann Surface
• 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(DX): DivRieSrfElt → Mtrx
  • AbelJacobi(DF, X): DivFunElt, RieSrf → Mtrx
  • AbelJacobi(P, Q): RieSrfPt, RieSrfPt → Mtrx
  • AbelJacobi(D, P): DivRieSrfElt, RieSrfPt → Mtrx
  • Example: Abel-Jacobi Map on a Superelliptic Curve
  • Example: Abel-Jacobi Map of a Principal Divisor
  • Example: Abel-Jacobi Map of Function Field Divisors
• Period Matrix Functions
  • Example: Period Matrices of a Hyperelliptic Curve
• Bibliography