Hyperelliptic Curves
- Introduction
- Creation Functions
- Creation of a Hyperelliptic Curve
HyperellipticCurve(f, h): RngUPolElt, RngUPolElt → CrvHypHyperellipticCurve(f, h): RngElt, RngUPolElt → CrvHypHyperellipticCurve(f, h): RngUPolElt, RngElt → CrvHypHyperellipticCurve(f): RngUPolElt → CrvHypHyperellipticCurve([f, h]): [ RngUPolElt ] → CrvHypHyperellipticCurve(P, f, h): Prj, RngUPolElt, RngUPolElt → CrvHypHyperellipticCurveOfGenus(g, f, h): RngIntElt, RngUPolElt, RngUPolElt → CrvHypHyperellipticCurveOfGenus(g, f, h): RngIntElt, RngElt, RngUPolElt → CrvHypHyperellipticCurveOfGenus(g, f, h): RngIntElt, RngUPolElt, RngElt → CrvHypHyperellipticCurveOfGenus(g, f): RngIntElt, RngUPolElt → CrvHypHyperellipticCurveOfGenus(g, [f, h]): RngIntElt, [RngUPolElt] → CrvHypHyperellipticCurve(E): CrvEll → CrvHyp, Map
- Creation Predicates
- Changing the Base Ring
BaseChange(C, K): Sch, Fld → SchBaseExtend(C, K): Sch, Fld → SchBaseChange(C, j): Sch, Map → SchBaseExtend(C, j): Sch, Map → SchBaseChange(C, n): Sch, RngIntElt → SchBaseExtend(C, n): Sch, RngIntElt → SchChangeRing(C, K): Sch, Rng → SchExample: Base Extension
- Models
SimplifiedModel(C): CrvHyp → CrvHyp, MapIsoSchHasOddDegreeModel(C): CrvHyp → BoolElt, CrvHyp, MapIsoSchIntegralModel(C): CrvHyp → CrvHyp, MapIsoSchMinimalWeierstrassModel(C): CrvHyp → CrvHyp, MapIsoSchpIntegralModel(C, p): CrvHyp, RngIntElt → CrvHyp, MapIsoSchpIntegralModel(C, p): CrvHyp, FldRatElt → CrvHyp, MapIsoSchpIntegralModel(C, p): CrvHyp, RngUPolElt → CrvHyp, MapIsoSchpIntegralModel(C, p): CrvHyp, FldFunRatUElt → CrvHyp, MapIsoSchpIntegralModel(C, p): CrvHyp, Infty → CrvHyp, MapIsoSchpNormalModel(C, p): CrvHyp, RngIntElt → CrvHyp, MapIsoSchpNormalModel(C, p): CrvHyp, FldRatElt → CrvHyp, MapIsoSchpNormalModel(C, p): CrvHyp, RngUPolElt → CrvHyp, MapIsoSchpNormalModel(C, p): CrvHyp, FldFunRatUElt → CrvHyp, MapIsoSchpNormalModel(C, p): CrvHyp, Infty → CrvHyp, MapIsoSchpMinimalWeierstrassModel(C, p): CrvHyp, RngIntElt → CrvHyp, MapIsoSchpMinimalWeierstrassModel(C, p): CrvHyp, FldRatElt → CrvHyp, MapIsoSchpMinimalWeierstrassModel(C, p): CrvHyp, RngUPolElt → CrvHyp, MapIsoSchpMinimalWeierstrassModel(C, p): CrvHyp, FldFunRatUElt → CrvHyp, MapIsoSchpMinimalWeierstrassModel(C, p): CrvHyp, Infty → CrvHyp, MapIsoSchReducedModel(C): CrvHyp → CrvHyp, MapIsoSchReducedMinimalWeierstrassModel(C): CrvHyp → CrvHyp, MapIsoSchSetVerbose("CrvHypReduce", v): MonStgElt, RngIntElt
- Minimization and Reduction of Binary Forms
SetVerbose("Minimize", v): MonStgElt, RngIntEltMinimizeAtP(f, p): RngMPolElt, RngIntElt → RngMPolElt, AlgMatElt, RngIntEltMinRedBinaryForm(f): RngMPolElt → RngMPolElt, AlgMatElt, RngIntEltMinRedBinaryForm(f): RngUPolElt → RngUPolElt, AlgMatElt, RngIntEltExample: Bin Form Min Red
- Predicates on Models
IsSimplifiedModel(C): CrvHyp → BoolEltIsIntegral(C): CrvHyp → BoolEltIspIntegral(C, p): CrvHyp, RngIntElt → BoolEltIspIntegral(C, p): CrvHyp, RngUPolElt → BoolEltIspIntegral(C, p): CrvHyp, Infty → BoolEltIspNormal(C, p): CrvHyp, RngIntElt → BoolEltIspNormal(C, p): CrvHyp, RngUPolElt → BoolEltIspNormal(C, p): CrvHyp, Infty → BoolEltIspMinimal(C, p): CrvHyp, RngIntElt → BoolElt, BoolEltIspMinimal(C, p): CrvHyp, RngUPolElt → BoolElt, BoolEltIspMinimal(C, p): CrvHyp, Infty → BoolElt, BoolElt
- Type Change Predicates
- Operations on Curves
- Quadratic Twists
- Elementary Invariants
HyperellipticPolynomials(C): CrvHyp → RngUPolElt, RngUPolEltDegree(C): CrvHyp → RngIntEltDegree(C): SetPtHyp → RngIntEltDiscriminant(C): CrvHyp → RngEltGenus(C): CrvHyp → RngIntEltConductor(C): CrvHyp → RngIntEltConductor(C, p): CrvHyp[FldRat], RngIntElt → RngIntEltConductor(C, P): CrvHyp[FldNum], RngOrdIdl → RngIntEltExample: Crvhyp Conductor QConductorExponent(C): CrvHyp[FldPad] → RngIntEltConductor(C): CrvHyp[FldPad] → FldPadEltExample: Crvhyp Conductor PadicEulerFactor(C, p): CrvHyp[FldRat], RngIntElt → RngUPolEltEulerFactor(C, P): CrvHyp[FldNum], RngOrdIdl → RngUPolEltEulerFactor(C): CrvHyp[FldPad] → RngUPolEltEulerFactor(C, p): CrvHyp[FldNum], RngIntElt → RngUPolEltExample: Crvhyp Eulerfactor
- Base Ring
- Function Field
- Function Field and Polynomial Ring
FunctionField(C): Sch → FldFunGDefiningPolynomial(C): Sch → RngMPolEltEvaluatePolynomial(C, a, b, c): CrvHyp, RngElt, RngElt, RngElt → RngEltEvaluatePolynomial(C, [a, b, c]): CrvHyp, [RngElt] → RngElt
- Points
- Creation of Points
C ! [x, y]: CrvHyp, [RngElt] → PtHypC ! [x, y, z]: CrvHyp, [RngElt] → PtHypelt< PS | x, y >: SetPtHyp, RngElt, RngElt, RngElt → PtHypelt< PS | x, y, z >: SetPtHyp, RngElt, RngElt, RngElt → PtHypC ! P: CrvHyp, PtHyp → PtHypPoints(C, x): CrvHyp, RngElt → SetIndxRationalPoints(C, x): CrvHyp, RngElt → SetIndxPoints(C, x): CrvHyp, Infty → SetIndxRationalPoints(C, x): CrvHyp, Infty → SetIndxPointsAtInfinity(C): CrvHyp → SetIndxIsPoint(C, S): CrvHyp, SeqEnum → BoolElt, PtHypExample: Points At Infinity On Hypcurves
- Random Points
- Predicates on Points
- Access Operations
- Arithmetic of Points
- Enumeration and Counting Points
- Frobenius
- Jacobians
- Creation of a Jacobian
- Access Operations
- Base Ring
- Changing the Base Ring
BaseChange(J, F): JacHyp, Rng → JacHypBaseExtend(J, F): JacHyp, Rng → JacHypBaseChange(J, j): JacHyp, Map → JacHypBaseExtend(J, j): JacHyp, Map → JacHypBaseChange(J, n): JacHyp, RngIntElt → JacHypBaseExtend(J, n): JacHyp, RngIntElt → JacHyp
- Richelot Isogenies
- Points on the Jacobian
- Creation of Points
J ! 0: JacHyp, RngIntElt → JacHypPtId(J): JacHyp → JacHypPtIdentity(J): JacHyp → JacHypPtJ ! [a, b]: JacHyp, [ RngUPolElt ] → JacHypPtelt< J | a, b >: JacHyp, RngUPolElt, RngUPolElt → JacHypPtelt< J | [a, b] >: JacHyp, [ RngUPolElt ] → JacHypPtelt< J | a, b, d >: JacHyp, RngUPolElt, RngUPolElt, RngIntElt → JacHypPtelt< J | [a, b], d >: JacHyp, [ RngUPolElt ], RngIntElt → JacHypPtP - Q: PtHyp, PtHyp → JacHypPtJ ! [P, Q]: JacHyp, [PtHyp] → JacHypPtelt< J | P, Q >: JacHyp, PtHyp, PtHyp → JacHypPtJ ! [S, T]: JacHyp, [SeqEnum] → JacHypPtelt< J | S, T >: JacHyp, [PtHyp], [PtHyp] → JacHypPtJacobianPoint(J, D): JacHyp, DivCrvElt → JacHypPtJ ! P: JacHyp, JacHypPt → JacHypPtPoints(J, a, d): JacHyp, RngUPolElt, RngIntElt → SetIndxRationalPoints(J, a, d): JacHyp, RngUPolElt, RngIntElt → SetIndxExample: Point Creation JacobianExample: Point Creation Jacobian2Example: Point Creation Jacobian3
- Random Points
- Booleans and Predicates for Points
- Access Operations
- Arithmetic of Points
- Order of Points on the Jacobian
Order(P): JacHypPt → RngIntEltOrder(P, l, u): JacHypPt, RngIntElt, RngIntElt → RngIntEltOrder(P, l, u, n, m): JacHypPt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntEltHasOrder(P, n): JacHypPt, RngIntElt → BoolElt
- Frobenius
- Weil Pairing
- Rational Points and Group Structure over Finite Fields
- Jacobians over Number Fields or \({\mathbb{Q}}\)
- Searching For Points
- Torsion
- Heights and Regulator
- Saturation
- The \(2\)-Selmer Group
BadPrimes(C): CrvHyp → SeqEnumBadPrimes(J): JacHyp → SeqEnumHasSquareSha(J): JacHyp → BoolEltIsEven(J): JacHyp → BoolEltIsDeficient(C, p): CrvHyp, RngIntElt → BoolEltHasIndexOne(C, p): CrvHyp, RngIntElt → BoolEltHasIndexOne(C, p): CrvHyp, RngOrdIdl → BoolEltHasIndexOneEverywhereLocally(C): CrvHyp → BoolEltTwoSelmerGroup(J): JacHyp → GrpAb, Map, Any, AnyRankBound(J): JacHyp → RngIntEltRankBounds(J): JacHyp → RngIntElt, RngIntEltExample: 2 Selmer GroupExample: Nonsquare ShaExample: Sha VisibilityExample: Better Rank BoundsExample: Disregard The Warning
- The Mordell–Weil Group
- Two-Selmer Set of a Curve
- Chabauty’s Method
- Cyclic Covers of \({\mathbb{P}}^1\)
- Points
RationalPoints(f, q): RngUPolElt, RngIntElt → SetIndxPoints(f, q): RngUPolElt, RngIntElt → SetIndxHasPoint(f, q, v): RngUPolElt, RngIntElt, RngIntElt → BoolElt, SeqEnumHasPoint(f, q, v): RngUPolElt, RngIntElt, RngOrdIdl → BoolElt, SeqEnumHasPointsEverywhereLocally(f, q): RngUPolElt, RngIntElt → BoolElt
- Descent
- Monic Models
- Descent on the Jacobian
PhiSelmerGroup(f, q): RngUPolElt, RngIntElt → GrpAb, MapPicnDescent(f, q): RngUPolElt, RngIntElt → RngIntElt, GrpAb, Tup, RngIntElt, Map, GrpAbRankBound(f, q): RngUPolElt, RngIntElt → RngIntEltRankBounds(f, q): RngUPolElt, RngIntElt → RngIntElt, RngIntEltExample: qcoverdescent
- Partial Descent
- Kummer Surfaces
- Creation of a Kummer Surface
- Structure Operations
- Base Ring
- Changing the Base Ring
BaseChange(K, F): SrfKum, Rng → SrfKumBaseExtend(K, F): SrfKum, Rng → SrfKumBaseChange(K, j): SrfKum, Map → SrfKumBaseExtend(K, j): SrfKum, Map → SrfKumBaseChange(K, n): SrfKum, RngIntElt → SrfKumBaseExtend(K, n): SrfKum, RngIntElt → SrfKum
- Points on the Kummer Surface
- Creation of Points
K ! 0: SrfKum, RngIntElt → SrfKumPtK ! [x1, x2, x3, x4]: SrfKum, [ RngElt ] → SrfKumPtK ! P: SrfKum, SrfKumPt → SrfKumPtK ! P: SrfKum, JacHypPt → SrfKumPtIsPoint(K, S): SrfKum, [RngElt] → BoolElt, SrfKumPtPoints(K,[x1, x2, x3]): SrfKum, [RngElt] → SetIndx
- Access Operations
- Predicates on Points
- Arithmetic of Points
- P: SrfKumPt → SrfKumPtn * P: RngIntElt, SrfKumPt → SrfKumPtP * n: SrfKumPt, RngIntElt → SrfKumPtDouble(P): SrfKumPt → SrfKumPtPseudoAdd(P1, P2, P3): SrfKumPt, SrfKumPt, SrfKumPt → SrfKumPtPseudoAddMultiple(P1, P2, P3, n): SrfKumPt, SrfKumPt, SrfKumPt, RngIntElt → SrfKumPt
- Rational Points on the Kummer Surface
- Pullback to the Jacobian
- Analytic Jacobians of Hyperelliptic Curves
- Creation and Access Functions
- Period Matrices
- Maps between Jacobians
ToAnalyticJacobian(x, y, A): FldComElt, FldComElt, AnHcJac → MtrxFromAnalyticJacobian(z, A): Mtrx, AnHcJac → SeqEnumExample: Analytic Jacobian Addition- Isomorphisms, Isogenies and Endomorphism Rings of Analytic Jacobians
To2DUpperHalfSpaceFundamentalDomain(z): Mtrx → Mtrx, MtrxAnalyticHomomorphisms(t1, t2): Mtrx, Mtrx → SeqEnumIsIsomorphicSmallPeriodMatrices(t1, t2): Mtrx, Mtrx → Bool, MtrxIsIsomorphicBigPeriodMatrices(P1, P2): Mtrx, Mtrx → Bool, Mtrx, MtrxIsIsomorphic(A1, A2): AnHcJac, AnHcJac → Bool, Mtrx, MtrxIsIsogenousPeriodMatrices(P1, P2): Mtrx, Mtrx → Bool, MtrxIsIsogenous(A1, A2): AnHcJac, AnHcJac → Bool, Mtrx, MtrxEndomorphismRing(P): Mtrx → AlgMatEndomorphismRing(A): AnHcJac → AlgMat, SeqEnumExample: Find Rational IsogenyToAnalyticJacobianMumford(pt, AJ): JacHypPt, AnHcJac → MtrxToAnalyticJacobianMumford(pt, AJ, conj): JacHypPt, AnHcJac, RngIntElt → MtrxFromAnalyticJacobianProjective(z, A): Mtrx[FldCom], AnHcJac → SeqEnum
- From Period Matrix to Curve
- Voronoi Cells
- Invariants
- Igusa Invariants
ClebschInvariants(C): CrvHyp → SeqEnumClebschInvariants(f): RngUPolElt → SeqEnumIgusaClebschInvariants(C: parameters): CrvHyp → SeqEnumIgusaClebschInvariants(f, h): RngUPolElt, RngUPolElt → SeqEnumIgusaClebschInvariants(f: parameters): RngUPolElt → SeqEnumIgusaInvariants(C: parameters): CrvHyp → SeqEnum, SeqEnumJInvariants(C: parameters): CrvHyp → SeqEnumIgusaInvariants(f, h: parameters): RngUPolElt, RngUPolElt → SeqEnum, SeqEnumJInvariants(f, h: parameters): RngUPolElt, RngUPolElt → SeqEnum, SeqEnumIgusaInvariants(f: parameters): RngUPolElt → SeqEnumIgusaInvariants(f: parameters): RngMPolElt → SeqEnumJInvariants(f: parameters): RngUPolElt → SeqEnumIgusaAlgebraicRelations(JI): SeqEnum → SeqEnumIgusaInvariantsEqual(JI1, JI2): SeqEnum, SeqEnum → BoolEltDiscriminantFromIgusaInvariants(JI): SeqEnum → AnyScaledIgusaInvariants(f, h): RngUPolElt, RngUPolElt → SeqEnumScaledIgusaInvariants(f): RngUPolElt → SeqEnumAbsoluteInvariants(C): CrvHyp → SeqEnumClebschToIgusaClebsch(Q): SeqEnum → SeqEnumIgusaClebschToIgusa(S): SeqEnum → SeqEnumG2Invariants(C): CrvHyp → SeqEnumG2ToIgusaInvariants(GI): SeqEnum → SeqEnumIgusaToG2Invariants(JI): SeqEnum → SeqEnum
- Shioda Invariants
- Creation from Invariants
- Isomorphisms and Transformations
- Creation of Isomorphisms
Aut(C): CrvHyp → PowAutSchIso(C1, C2): CrvHyp, CrvHyp → PowIsoSchTransformation(C, t): CrvHyp, [RngElt] → CrvHyp, MapIsoSchTransformation(C, u): CrvHyp, RngUPolElt → CrvHyp, MapIsoSchTransformation(C, e): CrvHyp, RngElt → CrvHyp, MapIsoSchTransformation(C, e, u): CrvHyp, RngElt, RngUPolElt → CrvHyp, MapIsoSchTransformation(C, t, e, u): CrvHyp, [RngElt], RngElt, RngUPolElt → CrvHyp, MapIsoSchExample: Transformation
- Invariants of Isomorphisms
- Automorphism Group and Isomorphism Testing
IsIsomorphic(C1, C2): CrvHyp, CrvHyp → BoolElt, MapIsoSchAutomorphismGroup(C): CrvHyp → GrpPerm, Map, MapExample: Automorphism GroupIsIsomorphicHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → BoolElt, ListIsIsomorphicHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → BoolElt, ListExample: Is Isomorphic Hyperelliptic CurvesIsomorphismsOfHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → ListIsomorphismsOfHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → ListAutomorphismsOfHyperellipticCurve(X): CrvHyp → ListAutomorphismsOfHyperellipticCurve(f): RngUPolElt → ListAutomorphismGroupOfHyperellipticCurve(X, Autos): CrvHyp, List → GrpPerm, MapAutomorphismGroupOfHyperellipticCurve(f, Autos): RngUPolElt, List → GrpPerm, MapAutomorphismGroupOfHyperellipticCurve(X): CrvHyp → GrpPerm, MapAutomorphismGroupOfHyperellipticCurve(f): RngUPolElt → GrpPerm, MapExample: Automorphism Group Of HyperellipticCurveGeometricAutomorphismGroup(C): CrvHyp → GrpPermGeometricAutomorphismGroupFromIgusaInvariants(GI): SeqEnum → SeqEnum, SeqEnumGeometricAutomorphismGroupFromShiodaInvariants(JI): SeqEnum → GrpPermExample: Geometric Automorphism GroupGeometricAutomorphismGroupGenus2Classification(F): FldFin → SeqEnum, SeqEnumGeometricAutomorphismGroupGenus3Classification(F): FldFin → SeqEnum, SeqEnumExample: Aut Class
- Twisting Hyperelliptic Curves
TwistsOfHyperellipticPolynomials(f: parameters): RngUPolElt → SeqEnum[RngUPolElt], GrpPermTwistsOfHyperellipticPolynomials(fh): SeqEnum[RngUPolElt] → SeqEnum, GrpPermTwistedHyperellipticPolynomialsFromShiodaInvariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum, GrpPermTwists(C: parameters): CrvHyp → SeqEnum[CrvHyp], GrpPermTwistsFromIgusaInvariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum[CrvHyp], GrpPermTwistsFromG2Invariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum[CrvHyp], GrpPermTwistsFromShiodaInvariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum[CrvHyp], GrpPermTwists(C, Autos: parameters): Crv, SeqEnum → SeqEnum[Crv], GrpPermHyperellipticPolynomialsFromShiodaInvariants(JI): SeqEnum → SeqEnum, GrpPermExample: TwistsExample: Quadratic Twists
- Reduced Automorphism Group and Reduced Isomorphism Testing
IsGL2Equivalent(f, g, n): RngUPolElt, RngUPolElt, RngIntElt → BoolElt, SeqEnumIsGL2EquivalentExtended(f1, f2, deg): RngUPolElt, RngUPolElt, RngIntElt → BoolElt, ListIsReducedIsomorphicHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → BoolElt, ListIsReducedIsomorphicHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → BoolElt, ListReducedIsomorphismsOfHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → ListReducedIsomorphismsOfHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → ListReducedAutomorphismsOfHyperellipticCurve(X): CrvHyp → ListReducedAutomorphismsOfHyperellipticCurve(f): RngUPolElt → ListReducedAutomorphismGroupOfHyperellipticCurve(X, Autos): CrvHyp, List → GrpPerm, MapReducedAutomorphismGroupOfHyperellipticCurve(f, Autos): RngUPolElt, List → GrpPerm, MapReducedAutomorphismGroupOfHyperellipticCurve(X): CrvHyp → GrpPerm, MapReducedAutomorphismGroupOfHyperellipticCurve(f): RngUPolElt → GrpPerm, Map
