Hyperelliptic Curves

• Introduction
• Creation Functions
  • Creation of a Hyperelliptic Curve
    • HyperellipticCurve(f, h): RngUPolElt, RngUPolElt → CrvHyp
    • HyperellipticCurve(f, h): RngElt, RngUPolElt → CrvHyp
    • HyperellipticCurve(f, h): RngUPolElt, RngElt → CrvHyp
    • HyperellipticCurve(f): RngUPolElt → CrvHyp
    • HyperellipticCurve([f, h]): [ RngUPolElt ] → CrvHyp
    • HyperellipticCurve(P, f, h): Prj, RngUPolElt, RngUPolElt → CrvHyp
    • HyperellipticCurveOfGenus(g, f, h): RngIntElt, RngUPolElt, RngUPolElt → CrvHyp
    • HyperellipticCurveOfGenus(g, f, h): RngIntElt, RngElt, RngUPolElt → CrvHyp
    • HyperellipticCurveOfGenus(g, f, h): RngIntElt, RngUPolElt, RngElt → CrvHyp
    • HyperellipticCurveOfGenus(g, f): RngIntElt, RngUPolElt → CrvHyp
    • HyperellipticCurveOfGenus(g, [f, h]): RngIntElt, [RngUPolElt] → CrvHyp
    • HyperellipticCurve(E): CrvEll → CrvHyp, Map
  • Creation Predicates
    • IsHyperellipticCurve([f, h]): [ RngUPolElt ] → BoolElt, CrvHyp
    • IsHyperellipticCurveOfGenus(g, [f, h]): RngIntElt, [RngUPolElt] → BoolElt, CrvHyp
    • Example: Creating a Hyperelliptic Curve
  • Changing the Base Ring
    • BaseChange(C, K): Sch, Fld → Sch
    • BaseExtend(C, K): Sch, Fld → Sch
    • BaseChange(C, j): Sch, Map → Sch
    • BaseExtend(C, j): Sch, Map → Sch
    • BaseChange(C, n): Sch, RngIntElt → Sch
    • BaseExtend(C, n): Sch, RngIntElt → Sch
    • ChangeRing(C, K): Sch, Rng → Sch
    • Example: Reducing a Curve Modulo 101
  • Models
    • SimplifiedModel(C): CrvHyp → CrvHyp, MapIsoSch
    • HasOddDegreeModel(C): CrvHyp → BoolElt, CrvHyp, MapIsoSch
    • IntegralModel(C): CrvHyp → CrvHyp, MapIsoSch
    • MinimalWeierstrassModel(C): CrvHyp → CrvHyp, MapIsoSch
    • pIntegralModel(C, p): CrvHyp, RngIntElt → CrvHyp, MapIsoSch
    • pIntegralModel(C, p): CrvHyp, FldRatElt → CrvHyp, MapIsoSch
    • pIntegralModel(C, p): CrvHyp, RngUPolElt → CrvHyp, MapIsoSch
    • pIntegralModel(C, p): CrvHyp, FldFunRatUElt → CrvHyp, MapIsoSch
    • pIntegralModel(C, p): CrvHyp, Infty → CrvHyp, MapIsoSch
    • pNormalModel(C, p): CrvHyp, RngIntElt → CrvHyp, MapIsoSch
    • pNormalModel(C, p): CrvHyp, FldRatElt → CrvHyp, MapIsoSch
    • pNormalModel(C, p): CrvHyp, RngUPolElt → CrvHyp, MapIsoSch
    • pNormalModel(C, p): CrvHyp, FldFunRatUElt → CrvHyp, MapIsoSch
    • pNormalModel(C, p): CrvHyp, Infty → CrvHyp, MapIsoSch
    • pMinimalWeierstrassModel(C, p): CrvHyp, RngIntElt → CrvHyp, MapIsoSch
    • pMinimalWeierstrassModel(C, p): CrvHyp, FldRatElt → CrvHyp, MapIsoSch
    • pMinimalWeierstrassModel(C, p): CrvHyp, RngUPolElt → CrvHyp, MapIsoSch
    • pMinimalWeierstrassModel(C, p): CrvHyp, FldFunRatUElt → CrvHyp, MapIsoSch
    • pMinimalWeierstrassModel(C, p): CrvHyp, Infty → CrvHyp, MapIsoSch
    • ReducedModel(C): CrvHyp → CrvHyp, MapIsoSch
    • ReducedMinimalWeierstrassModel(C): CrvHyp → CrvHyp, MapIsoSch
    • SetVerbose("CrvHypReduce", v): MonStgElt, RngIntElt
  • Minimization and Reduction of Binary Forms
    • SetVerbose("Minimize", v): MonStgElt, RngIntElt
    • MinimizeAtP(f, p): RngMPolElt, RngIntElt → RngMPolElt, AlgMatElt, RngIntElt
    • MinRedBinaryForm(f): RngMPolElt → RngMPolElt, AlgMatElt, RngIntElt
    • MinRedBinaryForm(f): RngUPolElt → RngUPolElt, AlgMatElt, RngIntElt
    • Example: Minimizing and Reducing a Binary Form
  • Predicates on Models
    • IsSimplifiedModel(C): CrvHyp → BoolElt
    • IsIntegral(C): CrvHyp → BoolElt
    • IspIntegral(C, p): CrvHyp, RngIntElt → BoolElt
    • IspIntegral(C, p): CrvHyp, RngUPolElt → BoolElt
    • IspIntegral(C, p): CrvHyp, Infty → BoolElt
    • IspNormal(C, p): CrvHyp, RngIntElt → BoolElt
    • IspNormal(C, p): CrvHyp, RngUPolElt → BoolElt
    • IspNormal(C, p): CrvHyp, Infty → BoolElt
    • IspMinimal(C, p): CrvHyp, RngIntElt → BoolElt, BoolElt
    • IspMinimal(C, p): CrvHyp, RngUPolElt → BoolElt, BoolElt
    • IspMinimal(C, p): CrvHyp, Infty → BoolElt, BoolElt
  • Type Change Predicates
    • IsEllipticCurve(C): CrvHyp → BoolElt, CrvEll, MapIsoSch, MapIsoSch
• Operations on Curves
  • Quadratic Twists
    • QuadraticTwist(C, d): CrvHyp, RngElt → CrvHyp
    • QuadraticTwist(C): CrvHyp → CrvHyp
    • QuadraticTwists(C): CrvHyp → SeqEnum
    • IsQuadraticTwist(C, D): CrvHyp, CrvHyp → BoolElt, RngElt
    • Example: Quadratic Twists over a Finite Field
    • Example: Quadratic Twist over the Rationals
  • Elementary Invariants
    • HyperellipticPolynomials(C): CrvHyp → RngUPolElt, RngUPolElt
    • Degree(C): CrvHyp → RngIntElt
    • Degree(C): SetPtHyp → RngIntElt
    • Discriminant(C): CrvHyp → RngElt
    • Genus(C): CrvHyp → RngIntElt
    • Conductor(C): CrvHyp → RngIntElt
    • Conductor(C, p): CrvHyp[FldRat], RngIntElt → RngIntElt
    • Conductor(C, P): CrvHyp[FldNum], RngOrdIdl → RngIntElt
    • Example: Conductor over the Rationals
    • ConductorExponent(C): CrvHyp[FldPad] → RngIntElt
    • Conductor(C): CrvHyp[FldPad] → FldPadElt
    • Example: Conductor over a p-Adic Field
    • EulerFactor(C, p): CrvHyp[FldRat], RngIntElt → RngUPolElt
    • EulerFactor(C, P): CrvHyp[FldNum], RngOrdIdl → RngUPolElt
    • EulerFactor(C): CrvHyp[FldPad] → RngUPolElt
    • EulerFactor(C, p): CrvHyp[FldNum], RngIntElt → RngUPolElt
    • Example: Euler Factors and Conductors at 3
  • Base Ring
    • BaseField(C): Sch → Fld
    • BaseRing(C): Sch → Fld
    • CoefficientRing(C): Sch → Fld
• Function Field
  • Function Field and Polynomial Ring
    • FunctionField(C): Sch → FldFunG
    • DefiningPolynomial(C): Sch → RngMPolElt
    • EvaluatePolynomial(C, a, b, c): CrvHyp, RngElt, RngElt, RngElt → RngElt
    • EvaluatePolynomial(C, [a, b, c]): CrvHyp, [RngElt] → RngElt
• Points
  • Creation of Points
    • C ! [x, y]: CrvHyp, [RngElt] → PtHyp
    • C ! [x, y, z]: CrvHyp, [RngElt] → PtHyp
    • elt< PS | x, y >: SetPtHyp, RngElt, RngElt, RngElt → PtHyp
    • elt< PS | x, y, z >: SetPtHyp, RngElt, RngElt, RngElt → PtHyp
    • C ! P: CrvHyp, PtHyp → PtHyp
    • Points(C, x): CrvHyp, RngElt → SetIndx
    • RationalPoints(C, x): CrvHyp, RngElt → SetIndx
    • Points(C, x): CrvHyp, Infty → SetIndx
    • RationalPoints(C, x): CrvHyp, Infty → SetIndx
    • PointsAtInfinity(C): CrvHyp → SetIndx
    • IsPoint(C, S): CrvHyp, SeqEnum → BoolElt, PtHyp
    • Example: Points at Infinity
  • Random Points
    • Random(C): CrvHyp → PtHyp
  • Predicates on Points
    • P eq Q: PtHyp, PtHyp → BoolElt
    • P ne Q: PtHyp, PtHyp → BoolElt
  • Access Operations
    • P[i]: PtHyp, RngIntElt → RngElt
    • Eltseq(P): PtHyp → SeqEnum
    • ElementToSequence(P): PtHyp → SeqEnum
  • Arithmetic of Points
    • - P: PtHyp → PtHyp
    • Involution(P): PtHyp → PtHyp
  • Enumeration and Counting Points
    • NumberOfPointsAtInfinity(C): CrvHyp → RngIntElt
    • PointsAtInfinity(C): CrvHyp → SetIndx
    • # C: CrvHyp → RngIntElt
    • Points(C): CrvHyp → SetIndx
    • RationalPoints(C): CrvHyp → SetIndx
    • RationalPoints(C): SetPtHyp → SetIndx
    • PointsGenus2(C): CrvHyp → SetIndx, BoolElt, RngIntElt
    • RationalPointsGenus2(C): CrvHyp → SetIndx, BoolElt, RngIntElt
    • PointsKnown(C): CrvHyp → BoolElt
    • Example: Points on a Curve of Diophantus
  • Frobenius
    • Frobenius(P, F): PtHyp, FldFin → PtHyp
    • FrobeniusMatrix(C, p): CrvHyp, RngIntElt → Mtrx
• Jacobians
  • Creation of a Jacobian
    • Jacobian(C): CrvHyp → JacHyp
  • Access Operations
    • Curve(J): JacHyp → CrvHyp
    • Dimension(J): JacHyp → RngIntElt
  • Base Ring
    • BaseField(J): JacHyp → Fld
    • BaseRing(J): JacHyp → Rng
    • CoefficientRing(J): JacHyp → Rng
  • Changing the Base Ring
    • BaseChange(J, F): JacHyp, Rng → JacHyp
    • BaseExtend(J, F): JacHyp, Rng → JacHyp
    • BaseChange(J, j): JacHyp, Map → JacHyp
    • BaseExtend(J, j): JacHyp, Map → JacHyp
    • BaseChange(J, n): JacHyp, RngIntElt → JacHyp
    • BaseExtend(J, n): JacHyp, RngIntElt → JacHyp
• Richelot Isogenies
  • RichelotIsogenousSurfaces(J): JacHyp → List, List
  • RichelotIsogenousSurfaces(C): CrvHyp → List, List
  • RichelotIsogenousSurface(J, kernel): JacHyp, RngUPolElt[RngUPolRes] → .
  • RichelotIsogenousSurface(C, kernel): CrvHyp, RngUPolElt[RngUPolRes] → .
  • Example: Three Richelot Isogenies
  • DoubleRichelotIsogenies(J): JacHyp → SeqEnum
  • DoubleRichelotIsogenies(C): CrvHyp → SeqEnum
  • TwoPowerIsogenies(J): JacHyp → SeqEnum, SeqEnum, SeqEnum
  • Example: Double Richelot Isogenies
• Points on the Jacobian
  • Creation of Points
    • J ! 0: JacHyp, RngIntElt → JacHypPt
    • Id(J): JacHyp → JacHypPt
    • Identity(J): JacHyp → JacHypPt
    • J ! [a, b]: JacHyp, [ RngUPolElt ] → JacHypPt
    • elt< J | a, b >: JacHyp, RngUPolElt, RngUPolElt → JacHypPt
    • elt< J | [a, b] >: JacHyp, [ RngUPolElt ] → JacHypPt
    • elt< J | a, b, d >: JacHyp, RngUPolElt, RngUPolElt, RngIntElt → JacHypPt
    • elt< J | [a, b], d >: JacHyp, [ RngUPolElt ], RngIntElt → JacHypPt
    • P - Q: PtHyp, PtHyp → JacHypPt
    • J ! [P, Q]: JacHyp, [PtHyp] → JacHypPt
    • elt< J | P, Q >: JacHyp, PtHyp, PtHyp → JacHypPt
    • J ! [S, T]: JacHyp, [SeqEnum] → JacHypPt
    • elt< J | S, T >: JacHyp, [PtHyp], [PtHyp] → JacHypPt
    • JacobianPoint(J, D): JacHyp, DivCrvElt → JacHypPt
    • J ! P: JacHyp, JacHypPt → JacHypPt
    • Points(J, a, d): JacHyp, RngUPolElt, RngIntElt → SetIndx
    • RationalPoints(J, a, d): JacHyp, RngUPolElt, RngIntElt → SetIndx
    • Example: Points on a Curve and Its Jacobian
    • Example: Nontrivial 2-Torsion Point
    • Example: Jacobian Point Not from the Curve
  • Random Points
    • Random(J): JacHyp → JacHypPt
  • Booleans and Predicates for Points
    • P eq Q: JacHypPt, JacHypPt → BoolElt
    • P ne Q: JacHypPt, JacHypPt → BoolElt
    • IsZero(P): JacHypPt → BoolElt
    • IsIdentity(P): JacHypPt → BoolElt
  • Access Operations
    • P[i]: JacHypPt, RngIntElt → RngElt
    • Eltseq(P): PtHyp → SeqEnum, RngIntElt
    • ElementToSequence(P): PtHyp → SeqEnum, RngIntElt
  • Arithmetic of Points
    • - P: JacHypPt → JacHypPt
    • P + Q: JacHypPt, JacHypPt → JacHypPt
    • P +:= Q: JacHypPt, JacHypPt
    • P - Q: JacHypPt, JacHypPt → JacHypPt
    • P -:= Q: JacHypPt, JacHypPt
    • n * P: RngIntElt, JacHypPt → JacHypPt
    • P * n: JacHypPt, RngIntElt → JacHypPt
    • P *:= n: JacHypPt, RngIntElt
  • Order of Points on the Jacobian
    • Order(P): JacHypPt → RngIntElt
    • Order(P, l, u): JacHypPt, RngIntElt, RngIntElt → RngIntElt
    • Order(P, l, u, n, m): JacHypPt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt
    • HasOrder(P, n): JacHypPt, RngIntElt → BoolElt
  • Frobenius
    • Frobenius(P, k): JacHypPt, FldFin → JacHypPt
  • Weil Pairing
    • WeilPairing(P, Q, m): JacHypPt, JacHypPt, RngIntElt → RngElt
    • Example: MOV Reduction on a Jacobian
• Rational Points and Group Structure over Finite Fields
  • Enumeration of Points
    • Points(J): JacHyp → SetIndx
    • RationalPoints(J): JacHyp → SetIndx
  • Counting Points on the Jacobian
    • SetVerbose("JacHypCnt", v): MonStgElt, RngIntElt
    • # J: JacHyp → RngIntElt
    • Order(J): JacHyp → RngIntElt
    • Example: Direct Point Counting
    • Example: Kedlaya's Algorithm
    • Example: Vercauteren's Algorithm
    • Example: Mestre's Algorithm
    • Example: Comparing Shanks and Pollard
    • Example: Schoof and Halving in Genus 2
    • FactoredOrder(J): JacHyp → [ <RngIntElt, RngIntElt> ]
    • EulerFactor(J): JacHyp → RngUPolElt
    • EulerFactorModChar(J): JacHyp → RngUPolElt
    • EulerFactor(J, K): JacHyp, FldFin → RngUPolElt
  • Deformation Point Counting
    • JacobianOrdersByDeformation(Q, Y): RngMPolElt, SeqEnum → SeqEnum
    • EulerFactorsByDeformation(Q, Y): RngMPolElt, SeqEnum → SeqEnum
    • ZetaFunctionsByDeformation(Q, Y): RngMPolElt, SeqEnum → SeqEnum
    • Example: Deformation Point Counting for a Family
  • Abelian Group Structure
    • Sylow(J, p): JacHyp, RngIntElt → GrpAb, Map, Eseq
    • AbelianGroup(J): JacHyp → GrpAb, Map
    • HasAdditionAlgorithm(J): JacHyp → Bool
• Jacobians over Number Fields or \({\mathbb{Q}}\)
  • Searching For Points
    • Points(J): JacHyp → SetIndx
    • RationalPoints(J): JacHyp → SetIndx
  • Torsion
    • TwoTorsionSubgroup(J): JacHyp → GrpAb, Map
    • TorsionBound(J, n): JacHyp, RngIntElt → RngIntElt
    • TorsionSubgroup(J): JacHyp → GrpAb, Map
    • Example: Torsion on a Jacobian
  • Heights and Regulator
    • NaiveHeight(P): JacHypPt → FldPrElt
    • NaiveHeight(P): SrfKumPt → FldPrElt
    • Height(P: parameters): JacHypPt → FldReElt
    • Height(P: Precision): SrfKumPt → FldReElt
    • CanonicalHeight(P: parameters): JacHypPt → FldReElt
    • CanonicalHeight(P: Precision): SrfKumPt → FldReElt
    • HeightConstant(J: parameters): JacHyp → FldPrElt, FldPrElt
    • HeightPairing(P, Q: parameters): JacHypPt, JacHypPt → FldReElt
    • HeightPairingMatrix(S: Precision): [JacHypPt] → AlgMat
    • Regulator(S: Precision): [JacHypPt] → FldReElt
    • ReducedBasis(S: Precision): [JacHypPt] → SeqEnum, AlgMatElt, SeqEnum
    • ReducedBasis(S: Precision): {@JacHypPt@} → SeqEnum, AlgMatElt, SeqEnum
    • ReducedBasis(S: Precision): {JacHypPt} → SeqEnum, AlgMatElt, SeqEnum
    • Example: Heights on a Genus 2 Jacobian
    • Example: Reduced Basis via Heights
  • Saturation
    • IsDivisibleBy(P, n): JacHypPt, RngIntElt → BoolElt, JacHypPt
    • Saturation(bas, p): [ JacHypPt ], RngIntElt → [ JacHypPt ]
  • The \(2\)-Selmer Group
    • BadPrimes(C): CrvHyp → SeqEnum
    • BadPrimes(J): JacHyp → SeqEnum
    • HasSquareSha(J): JacHyp → BoolElt
    • IsEven(J): JacHyp → BoolElt
    • IsDeficient(C, p): CrvHyp, RngIntElt → BoolElt
    • HasIndexOne(C, p): CrvHyp, RngIntElt → BoolElt
    • HasIndexOne(C, p): CrvHyp, RngOrdIdl → BoolElt
    • HasIndexOneEverywhereLocally(C): CrvHyp → BoolElt
    • TwoSelmerGroup(J): JacHyp → GrpAb, Map, Any, Any
    • RankBound(J): JacHyp → RngIntElt
    • RankBounds(J): JacHyp → RngIntElt, RngIntElt
    • Example: 2-Selmer Group of a Jacobian
    • Example: Tate-Shafarevich Groups of Nonsquare Order
    • Example: Visibility of the Tate-Shafarevich Group
    • Example: Better Rank Bounds by Torsor Descent
    • Example: Rank Bound Without Local Index One
  • The Mordell–Weil Group
    • MordellWeilGroupGenus2(J): JacHyp → GrpAb, Map, BoolElt, BoolElt, RngIntElt
    • MordellWeilGroup(J): JacHyp → GrpAb, Map, BoolElt, BoolElt
    • Example: Mordell-Weil Group of a Genus 2 Jacobian
    • Example: Inverting the Mordell-Weil Map
• Two-Selmer Set of a Curve
  • TwoCoverDescent(C): CrvHyp → SetEnum, Map, [Map, SeqEnum]
  • Example: Two-Cover Descent on Genus 2 Curves
• Chabauty’s Method
  • Chabauty0(J): JacHyp → SetIndx
  • Chabauty(P : ptC): JacHypPt → SetIndx
  • Chabauty(P, p: Precision): JacHypPt, RngIntElt → SetIndx
  • Example: Both Chabauty Implementations
  • Example: Chabauty on a Rank 1 Jacobian
  • Example: Points in a Rank 1 Subgroup
  • Example: Chabauty for a Rank 0 Jacobian
• Cyclic Covers of \({\mathbb{P}}^1\)
  • Points
    • RationalPoints(f, q): RngUPolElt, RngIntElt → SetIndx
    • Points(f, q): RngUPolElt, RngIntElt → SetIndx
    • HasPoint(f, q, v): RngUPolElt, RngIntElt, RngIntElt → BoolElt, SeqEnum
    • HasPoint(f, q, v): RngUPolElt, RngIntElt, RngOrdIdl → BoolElt, SeqEnum
    • HasPointsEverywhereLocally(f, q): RngUPolElt, RngIntElt → BoolElt
  • Descent
    • qCoverDescent(f, q): RngUPolElt, RngIntElt → Set, Map
    • Example: Descent Proving No Rational Points
  • Monic Models
    • MonicModel(f, q): RngUPolElt, RngIntElt → RngUPolElt, SeqEnum
  • Descent on the Jacobian
    • PhiSelmerGroup(f, q): RngUPolElt, RngIntElt → GrpAb, Map
    • PicnDescent(f, q): RngUPolElt, RngIntElt → RngIntElt, GrpAb, Tup, RngIntElt, Map, GrpAb
    • RankBound(f, q): RngUPolElt, RngIntElt → RngIntElt
    • RankBounds(f, q): RngUPolElt, RngIntElt → RngIntElt, RngIntElt
    • Example: Rank and 3-Primary Sha by Descent
  • Partial Descent
    • qCoverPartialDescent(f, factors, q): RngUPolElt, [* RngUPolElt *], RngIntElt → Set, Map
    • Example: Partial Descent on a Cyclic Cover
• Kummer Surfaces
  • Creation of a Kummer Surface
    • KummerSurface(J): JacHyp → SrfKum
  • Structure Operations
    • DefiningPolynomial(K): SrfKum → RngMPolElt
  • Base Ring
    • BaseField(K): SrfKum → Fld
    • BaseRing(K): SrfKum → Rng
    • CoefficientRing(K): SrfKum → Rng
  • Changing the Base Ring
    • BaseChange(K, F): SrfKum, Rng → SrfKum
    • BaseExtend(K, F): SrfKum, Rng → SrfKum
    • BaseChange(K, j): SrfKum, Map → SrfKum
    • BaseExtend(K, j): SrfKum, Map → SrfKum
    • BaseChange(K, n): SrfKum, RngIntElt → SrfKum
    • BaseExtend(K, n): SrfKum, RngIntElt → SrfKum
• Points on the Kummer Surface
  • Creation of Points
    • K ! 0: SrfKum, RngIntElt → SrfKumPt
    • K ! [x1, x2, x3, x4]: SrfKum, [ RngElt ] → SrfKumPt
    • K ! P: SrfKum, SrfKumPt → SrfKumPt
    • K ! P: SrfKum, JacHypPt → SrfKumPt
    • IsPoint(K, S): SrfKum, [RngElt] → BoolElt, SrfKumPt
    • Points(K,[x1, x2, x3]): SrfKum, [RngElt] → SetIndx
  • Access Operations
    • P[i]: SrfKumPt, RngIntElt → RngElt
    • Eltseq(P): SrfKumPt → SeqEnum
    • ElementToSequence(P): PtHyp → SeqEnum
  • Predicates on Points
    • P eq Q: SrfKumPt, SrfKumPt → BoolElt
    • P ne Q: SrfKumPt, SrfKumPt → BoolElt
  • Arithmetic of Points
    • - P: SrfKumPt → SrfKumPt
    • n * P: RngIntElt, SrfKumPt → SrfKumPt
    • P * n: SrfKumPt, RngIntElt → SrfKumPt
    • Double(P): SrfKumPt → SrfKumPt
    • PseudoAdd(P1, P2, P3): SrfKumPt, SrfKumPt, SrfKumPt → SrfKumPt
    • PseudoAddMultiple(P1, P2, P3, n): SrfKumPt, SrfKumPt, SrfKumPt, RngIntElt → SrfKumPt
  • Rational Points on the Kummer Surface
    • RationalPoints(K, Q): SrfKum, [RngElt] → SetIndx
    • Example: Searching Points on a Kummer Surface
  • Pullback to the Jacobian
    • Points(J, P): JacHyp, SrfKumPt → SetIndx
    • RationalPoints(J, P): JacHyp, SrfKumPt → SetIndx
• Analytic Jacobians of Hyperelliptic Curves
  • Creation and Access Functions
    • AnalyticJacobian(f): RngUPolElt → AnHcJac
    • HyperellipticPolynomial(A): AnHcJac → RngUPolElt
    • HomologyBasis(A): AnHcJac → SeqEnum, SeqEnum, Mtrx
    • Dimension(A): AnHcJac → RngIntElt
    • Genus(A): AnHcJac → RngIntElt
    • BaseField(A): JacHyp → Fld
    • BaseRing(A): JacHyp → Rng
    • CoefficientRing(A): JacHyp → Rng
  • Period Matrices
    • SmallPeriodMatrix(A): AnHcJac → AlgMatElt
    • BigPeriodMatrix(A): AnHcJac → AlgMatElt
    • PeriodMapping(A, n): ModSym, RngIntElt → Map
    • Periods(M, n): ModSym, RngIntElt → SeqEnum
  • Maps between Jacobians
    • ToAnalyticJacobian(x, y, A): FldComElt, FldComElt, AnHcJac → Mtrx
    • FromAnalyticJacobian(z, A): Mtrx, AnHcJac → SeqEnum
    • Example: Addition on the Analytic Jacobian
    • Isomorphisms, Isogenies, and Endomorphism Rings of Analytic Jacobians
      • To2DUpperHalfSpaceFundamentalDomain(z): Mtrx → Mtrx, Mtrx
      • AnalyticHomomorphisms(t1, t2): Mtrx, Mtrx → SeqEnum
      • IsIsomorphicSmallPeriodMatrices(t1, t2): Mtrx, Mtrx → Bool, Mtrx
      • IsIsomorphicBigPeriodMatrices(P1, P2): Mtrx, Mtrx → Bool, Mtrx, Mtrx
      • IsIsomorphic(A1, A2): AnHcJac, AnHcJac → Bool, Mtrx, Mtrx
      • IsIsogenousPeriodMatrices(P1, P2): Mtrx, Mtrx → Bool, Mtrx
      • IsIsogenous(A1, A2): AnHcJac, AnHcJac → Bool, Mtrx, Mtrx
      • EndomorphismRing(P): Mtrx → AlgMat
      • EndomorphismRing(A): AnHcJac → AlgMat, SeqEnum
      • Example: Finding a Rational Isogeny
      • ToAnalyticJacobianMumford(pt, AJ): JacHypPt, AnHcJac → Mtrx
      • ToAnalyticJacobianMumford(pt, AJ, conj): JacHypPt, AnHcJac, RngIntElt → Mtrx
      • FromAnalyticJacobianProjective(z, A): Mtrx[FldCom], AnHcJac → SeqEnum
  • From Period Matrix to Curve
    • RosenhainInvariants(t): Mtrx → Set
    • Example: Finding a Genus 2 CM Curve
  • Voronoi Cells
    • Delaunay(sites): SeqEnum → SeqEnum
    • Voronoi(sites): SeqEnum → SeqEnum, SeqEnum, SeqEnum
• Invariants
  • Igusa Invariants
    • ClebschInvariants(C): CrvHyp → SeqEnum
    • ClebschInvariants(f): RngUPolElt → SeqEnum
    • IgusaClebschInvariants(C: parameters): CrvHyp → SeqEnum
    • IgusaClebschInvariants(f, h): RngUPolElt, RngUPolElt → SeqEnum
    • IgusaClebschInvariants(f: parameters): RngUPolElt → SeqEnum
    • IgusaInvariants(C: parameters): CrvHyp → SeqEnum, SeqEnum
    • JInvariants(C: parameters): CrvHyp → SeqEnum
    • IgusaInvariants(f, h: parameters): RngUPolElt, RngUPolElt → SeqEnum, SeqEnum
    • JInvariants(f, h: parameters): RngUPolElt, RngUPolElt → SeqEnum, SeqEnum
    • IgusaInvariants(f: parameters): RngUPolElt → SeqEnum
    • IgusaInvariants(f: parameters): RngMPolElt → SeqEnum
    • JInvariants(f: parameters): RngUPolElt → SeqEnum
    • IgusaAlgebraicRelations(JI): SeqEnum → SeqEnum
    • IgusaInvariantsEqual(JI1, JI2): SeqEnum, SeqEnum → BoolElt
    • DiscriminantFromIgusaInvariants(JI): SeqEnum → Any
    • ScaledIgusaInvariants(f, h): RngUPolElt, RngUPolElt → SeqEnum
    • ScaledIgusaInvariants(f): RngUPolElt → SeqEnum
    • AbsoluteInvariants(C): CrvHyp → SeqEnum
    • ClebschToIgusaClebsch(Q): SeqEnum → SeqEnum
    • IgusaClebschToIgusa(S): SeqEnum → SeqEnum
    • G2Invariants(C): CrvHyp → SeqEnum
    • G2ToIgusaInvariants(GI): SeqEnum → SeqEnum
    • IgusaToG2Invariants(JI): SeqEnum → SeqEnum
  • Shioda Invariants
    • ShiodaInvariants(C): CrvHyp → SeqEnum, SeqEnum
    • ShiodaInvariants(f): RngUPolElt → SeqEnum, SeqEnum
    • ShiodaInvariants(f): RngMPolElt → SeqEnum, SeqEnum
    • ShiodaInvariants(fh): SeqEnum → SeqEnum, SeqEnum
    • ShiodaInvariantsEqual(V1, V2): SeqEnum, SeqEnum → BoolElt
    • DiscriminantFromShiodaInvariants(JI): SeqEnum → RngElt
    • ShiodaAlgebraicInvariants(FJI): SeqEnum → SeqEnum
    • Example: Shioda Invariants of a Genus 3 Curve
    • MaedaInvariants(C): CrvHyp → SeqEnum
    • MaedaInvariants(f): RngUPolElt → SeqEnum
  • Creation from Invariants
    • HyperellipticCurveFromIgusaClebsch(S): SeqEnum → CrvHyp
    • HyperellipticCurveFromIgusaInvariants(S): SeqEnum → CrvHyp, GrpPerm
    • HyperellipticCurveFromG2Invariants(S): SeqEnum → CrvHyp, GrpPerm
    • HyperellipticCurveFromShiodaInvariants(JI): SeqEnum → CrvHyp, GrpPerm
    • HyperellipticPolynomialFromShiodaInvariants(JI): SeqEnum → SeqEnum, GrpPerm
    • Example: Curve from Igusa-Clebsch Invariants
• Isomorphisms and Transformations
  • Creation of Isomorphisms
    • Aut(C): CrvHyp → PowAutSch
    • Iso(C1, C2): CrvHyp, CrvHyp → PowIsoSch
    • Transformation(C, t): CrvHyp, [RngElt] → CrvHyp, MapIsoSch
    • Transformation(C, u): CrvHyp, RngUPolElt → CrvHyp, MapIsoSch
    • Transformation(C, e): CrvHyp, RngElt → CrvHyp, MapIsoSch
    • Transformation(C, e, u): CrvHyp, RngElt, RngUPolElt → CrvHyp, MapIsoSch
    • Transformation(C, t, e, u): CrvHyp, [RngElt], RngElt, RngUPolElt → CrvHyp, MapIsoSch
    • Example: Transforming a Hyperelliptic Curve
  • Invariants of Isomorphisms
    • Parent(f): MapIsoSch → PowIsoSch
    • Domain(f): MapIsoSch → CrvHyp
    • Codomain(f): MapIsoSch → CrvHyp
  • Automorphism Group and Isomorphism Testing
    • IsIsomorphic(C1, C2): CrvHyp, CrvHyp → BoolElt, MapIsoSch
    • AutomorphismGroup(C): CrvHyp → GrpPerm, Map, Map
    • Example: Automorphisms of a Genus One Curve
    • IsIsomorphicHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → BoolElt, List
    • IsIsomorphicHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → BoolElt, List
    • Example: Isomorphisms Between Two Curves
    • IsomorphismsOfHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → List
    • IsomorphismsOfHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → List
    • AutomorphismsOfHyperellipticCurve(X): CrvHyp → List
    • AutomorphismsOfHyperellipticCurve(f): RngUPolElt → List
    • AutomorphismGroupOfHyperellipticCurve(X, Autos): CrvHyp, List → GrpPerm, Map
    • AutomorphismGroupOfHyperellipticCurve(f, Autos): RngUPolElt, List → GrpPerm, Map
    • AutomorphismGroupOfHyperellipticCurve(X): CrvHyp → GrpPerm, Map
    • AutomorphismGroupOfHyperellipticCurve(f): RngUPolElt → GrpPerm, Map
    • Example: Automorphisms over a Finite Field
    • GeometricAutomorphismGroup(C): CrvHyp → GrpPerm
    • GeometricAutomorphismGroupFromIgusaInvariants(GI): SeqEnum → SeqEnum, SeqEnum
    • GeometricAutomorphismGroupFromShiodaInvariants(JI): SeqEnum → GrpPerm
    • Example: Geometric Automorphism Groups
    • GeometricAutomorphismGroupGenus2Classification(F): FldFin → SeqEnum, SeqEnum
    • GeometricAutomorphismGroupGenus3Classification(F): FldFin → SeqEnum, SeqEnum
    • Example: Classifying Genus 2 Automorphism Groups
  • Twisting Hyperelliptic Curves
    • TwistsOfHyperellipticPolynomials(f: parameters): RngUPolElt → SeqEnum[RngUPolElt], GrpPerm
    • TwistsOfHyperellipticPolynomials(fh): SeqEnum[RngUPolElt] → SeqEnum, GrpPerm
    • TwistedHyperellipticPolynomialsFromShiodaInvariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum, GrpPerm
    • Twists(C: parameters): CrvHyp → SeqEnum[CrvHyp], GrpPerm
    • TwistsFromIgusaInvariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum[CrvHyp], GrpPerm
    • TwistsFromG2Invariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum[CrvHyp], GrpPerm
    • TwistsFromShiodaInvariants(JI: parameters): SeqEnum[FldFinElt] → SeqEnum[CrvHyp], GrpPerm
    • Twists(C, Autos: parameters): Crv, SeqEnum → SeqEnum[Crv], GrpPerm
    • HyperellipticPolynomialsFromShiodaInvariants(JI): SeqEnum → SeqEnum, GrpPerm
    • Example: Twists over a Finite Field
    • Example: Twists of a Supersingular Curve
• Reduced Automorphism Group and Reduced Isomorphism Testing
  • IsGL2Equivalent(f, g, n): RngUPolElt, RngUPolElt, RngIntElt → BoolElt, SeqEnum
  • IsGL2EquivalentExtended(f1, f2, deg): RngUPolElt, RngUPolElt, RngIntElt → BoolElt, List
  • IsReducedIsomorphicHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → BoolElt, List
  • IsReducedIsomorphicHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → BoolElt, List
  • ReducedIsomorphismsOfHyperellipticCurves(X1, X2): CrvHyp, CrvHyp → List
  • ReducedIsomorphismsOfHyperellipticCurves(f1, f2): RngUPolElt, RngUPolElt → List
  • ReducedAutomorphismsOfHyperellipticCurve(X): CrvHyp → List
  • ReducedAutomorphismsOfHyperellipticCurve(f): RngUPolElt → List
  • ReducedAutomorphismGroupOfHyperellipticCurve(X, Autos): CrvHyp, List → GrpPerm, Map
  • ReducedAutomorphismGroupOfHyperellipticCurve(f, Autos): RngUPolElt, List → GrpPerm, Map
  • ReducedAutomorphismGroupOfHyperellipticCurve(X): CrvHyp → GrpPerm, Map
  • ReducedAutomorphismGroupOfHyperellipticCurve(f): RngUPolElt → GrpPerm, Map
• Bibliography