Models of Genus One Curves

• Introduction
• Creation of Genus One Models
  • GenusOneModel(seq): [ RngElt ] → ModelG1
  • GenusOneModel(n, seq): RngIntElt, [ RngElt ] → ModelG1
  • GenusOneModel(R, n, seq): RngIntElt, [ RngElt ] → ModelG1
  • GenusOneModel(n, str): RngIntElt, MonStgElt → ModelG1
  • GenusOneModel(C): Crv → ModelG1
  • GenusOneModel(f): RngMPolElt → ModelG1
  • GenusOneModel(f): RngUPolElt → ModelG1
  • GenusOneModel(seq): [ RngMPolElt ] → ModelG1
  • GenusOneModel(n, E): RngIntElt, CrvEll → ModelG1, Crv, MapSch, MapSch
  • GenusOneModel(mat): Mtrx → ModelG1
  • GenusOneModel(mats): [ AlgMatElt ] → ModelG1
  • IsGenusOneModel(f): RngUPolElt → BoolElt, ModelG1
  • IsGenusOneModel(f): RngMPolElt → BoolElt, ModelG1
  • IsGenusOneModel(seq): [ RngMPolElt ] → BoolElt, ModelG1
  • IsGenusOneModel(mat): Mtrx → BoolElt, ModelG1
  • GenericModel(n): RngIntElt → ModelG1
  • RandomGenusOneModel(n): RngIntElt → ModelG1
  • RandomModel(n): RngIntElt → ModelG1
  • ChangeRing(model, R): ModelG1, Rng → ModelG1
  • CompleteTheSquare(model): ModelG1 → ModelG1
  • CubicFromPoint(E, P): CrvEll, PtEll → RngMPolElt, MapSch, Pt
  • HesseModel(n, seq): RngIntElt, [ RngElt ] → ModelG1
  • DiagonalModel(n, seq): RngIntElt, [ RngElt ] → ModelG1
  • Example: Generic Genus One Model of Degree 5
• Attributes of Genus One Models
  • Degree(model): ModelG1 → RngIntElt
  • DefiningEquations(model): ModelG1 → [ RngMPolElt ]
  • Equations(model): ModelG1 → [ RngMPolElt ]
  • Matrix(model): ModelG1 → Mtrx
  • Curve(model): ModelG1 → Crv
  • HyperellipticCurve(model): ModelG1 → CrvHyp
  • QuadricIntersection(model): ModelG1 → Crv
  • Matrices(model): ModelG1 → [ AlgMatElt ]
  • BaseRing(model): ModelG1 → Rng
  • PolynomialRing(model): ModelG1 → RngMPol
  • ModelToSequence(model): ModelG1 → [ RngElt ]
  • Eltseq(model): ModelG1 → [ RngElt ]
  • ModelToString(model): ModelG1 → MonStgElt
• Transformations between Genus One Models
  • IsTransformation(n, g): RngIntElt, Tup → BoolElt, TransG1
  • Tuple(g): TransG1 → Tup
  • ChangeRing(g, R): TransG1, Rng → TransG1
  • IdentityTransformation(n, R): RngIntElt, Rng → TransG1
  • RandomTransformation(n : parameters): RngIntElt → TransG1
  • g * model: TransG1, ModelG1 → ModelG1
  • ApplyTransformation(g, model): TransG1, ModelG1 → ModelG1
  • g1 * g2: TransG1, TransG1 → TransG1
  • ComposeTransformations(g1, g2): TransG1, TransG1 → TransG1
  • MultiplyTransformations(g1, g2): TransG1, TransG1 → TransG1
  • Inverse(g): TransG1 → TransG1
  • InverseTransformation(g): TransG1 → TransG1
  • InverseTransformation(n, g): RngIntElt, TransG1 → TransG1
  • ScalingFactor(g): TransG1 → RngElt
  • ScalingFactor(n, g): RngIntElt, TransG1 → RngElt
  • Determinant(g): TransG1 → RngElt
• Equivalence of Genus One Models
  • IsEquivalent(model1, model2): ModelG1, ModelG1 → BoolElt, TransG1
  • IsEquivalent(cubic1, cubic2): RngMPolElt, RngMPolElt → BoolElt, TransG1
• Minimisation and Reduction
  • Minimise(model : parameters): ModelG1 → ModelG1, TransG1
  • Minimise(f): RngMPolElt → RngMPolElt, TransG1
  • pMinimise(f, p): RngMPolElt, RngIntElt → RngMPolElt, AlgMatElt
  • Reduce(model): ModelG1 → ModelG1, TransG1
  • Reduce(f): RngMPolElt → RngMPolElt, TransG1
  • ReduceQuadrics(seq): [RngMPolElt] → [RngMPolElt], AlgMatElt, AlgMatElt
• Local Solubility
  • IsLocallySoluble(model, p): ModelG1, RngIntElt → BoolElt, Any
  • IsLocallySoluble(model): ModelG1 → BoolElt
• Genus One Models as Coverings
  • Jacobian(model): ModelG1 → CrvEll
  • Jacobian(C): Crv → CrvEll
  • nCovering(model : parameters): ModelG1 → Crv, CrvEll, MapSch
  • AddCubics(cubic1, cubic2 : parameters): RngMPolElt, RngMPolElt → RngMPolElt
  • AddCubics(model1, model2 : parameters): ModelG1, ModelG1 → ModelG1
  • model1 + model2: ModelG1, ModelG1 → ModelG1
  • DoubleGenusOneModel(model): ModelG1 → ModelG1
  • FourToTwoCovering(model : parameters): ModelG1 → Crv, Crv, MapSch
  • FourToTwoCovering(C : parameters): Crv → Crv, Crv, MapSch
• Families of Elliptic Curves with Prescribed \(n\)-Torsion
  • RubinSilverbergPolynomials(n, J : parameters): RngIntElt, RngElt → RngElt, RngElt
• Invariants for Genus One Models
  • aInvariants(model): ModelG1 → [ RngElt ]
  • bInvariants(model): ModelG1 → [ RngElt ]
  • cInvariants(model): ModelG1 → [ RngElt ]
  • Invariants(model): ModelG1 → RngElt, RngElt, RngElt
  • Discriminant(model): ModelG1 → RngElt
  • SL4Invariants(model): ModelG1 → [ RngElt ]
• Covariants and Contravariants for Genus One Models
  • Hessian(model): ModelG1 → ModelG1
  • CoveringCovariants(model): ModelG1 → [ RngMPolElt ]
  • Contravariants(model): ModelG1 → ModelG1, ModelG1
  • HesseCovariants(model, r): ModelG1, RngIntElt → ModelG1, ModelG1
  • HessePolynomials(n, r, invariants : parameters): RngIntElt, RngIntElt, [RngElt] → RngElt, RngElt, RngElt
• Examples
  • Example: A Cubic Violating the Hasse Principle
• Bibliography