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Computer • algebra
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Introduction to Riemann Surfaces
Example
RieSrf_rie-srf-verbose (H124E1)
Creation Functions
Riemann Surfaces over Number Fields
RiemannSurface(f) : RngMPolElt -> RieSrf
RiemannSurface(f,sigma) : RngMPolElt, PlcNumElt -> RieSrf
Example
RieSrf_riesrf-ex-1 (H124E2)
Example
RieSrf_riesrf-ex-2 (H124E3)
Superelliptic Riemann Surfaces
RiemannSurface(p,m) : RngUPolElt, RngIntElt -> RieSrf
RiemannSurface(L,m) : SeqEnum[FldComElt], RngIntElt -> RieSrf
Example
RieSrf_riesrf-ex-2 (H124E4)
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 (H124E5)
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 (H124E6)
Basis for Period Matrix
HolomorphicDifferentials(X) : RieSrf -> Tup
Example
RieSrf_riesrf-ex-1 (H124E7)
Example
RieSrf_riesrf-ex-1 (H124E8)
HomologyBasis(L) : SeqEnum[GrpPermElt] -> SeqEnum[SeqEnum[RngIntElt]], Mtrx, Mtrx
HomologyBasis(X) : RieSrf -> SeqEnum[SeqEnum[RngIntElt]], Mtrx, Mtrx
Example
RieSrf_homology-basis1 (H124E9)
Example
RieSrf_homology-basis2 (H124E10)
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 (H124E11)
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 (H124E12)
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 (H124E13)
Example
RieSrf_abel-jacobi-gen-1 (H124E14)
Example
RieSrf_abel-jacobi-gen-2 (H124E15)
Period Matrix Functions
Example
RieSrf_iso-small-pm-1 (H124E16)
Bibliography
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V2.28, 13 July 2023