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RATIONAL CURVES AND CONICS

 
Acknowledgements
 
Introduction
 
Rational Curves and Conics
      Rational Curve and Conic Creation
      Access Functions
      Rational Curve and Conic Examples
 
Conics
      Elementary Invariants
      Alternative Defining Polynomials
      Alternative Models
      Other Functions on Conics
 
Local-Global Correspondence
      Local Conditions for Conics
      Norm Residue Symbol
 
Rational Points on Conics
      Finding Points
      Point Reduction
 
Isomorphisms
      Isomorphisms with Standard Models
 
Automorphisms
      Automorphisms of Rational Curves
      Automorphisms of Conics
 
Bibliography







DETAILS

 
Introduction

 
Rational Curves and Conics

      Rational Curve and Conic Creation
            Conic(coeffs) : [RngElt] -> CrvCon
            Conic(M) : Mtrx -> CrvCon
            Conic(X, f) : Prj, RngMPolElt -> CrvCon
            IsConic(S) : Sch -> BoolElt, CrvCon
            RationalCurve(X, f) : Prj, RngMPolElt -> CrvRat
            IsRationalCurve(S) : Sch -> BoolElt, CrvRat
            Example CrvCon_ConicCreation (H119E1)

      Access Functions
            DefiningPolynomial(C) : Crv -> RngMPolElt
            DefiningIdeal(C) : Crv -> RngMPol

      Rational Curve and Conic Examples
            Example CrvCon_RationalCurveExample (H119E2)
            Example CrvCon_ConicCurve (H119E3)
            Example CrvCon_ConicAccess (H119E4)

 
Conics

      Elementary Invariants
            Discriminant(C) : CrvCon -> FldElt

      Alternative Defining Polynomials
            LegendrePolynomial(C) : CrvCon -> RngMPolElt, ModMatRngElt
            ReducedLegendrePolynomial(C) : CrvCon -> RngMPolElt, ModMatRngElt

      Alternative Models
            LegendreModel(C) : CrvCon -> CrvCon, MapIsoSch
            ReducedLegendreModel(C) : CrvCon -> CrvCon, MapIsoSch

      Other Functions on Conics
            MinimalModel(C) : CrvCon -> CrvCon, Map
            Example CrvCon_ConicMinimalModel (H119E5)

 
Local-Global Correspondence

      Local Conditions for Conics
            BadPrimes(C) : CrvCon -> SeqEnum

      Norm Residue Symbol
            NormResidueSymbol(a, b, p) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
            HilbertSymbol(a, b, p : parameters) : FldRatElt, FldRatElt, RngIntElt -> RngIntElt
            Example CrvCon_LocalGlobal (H119E6)

 
Rational Points on Conics

      Finding Points
            HasRationalPoint(C) : CrvCon -> BoolElt, Pt
            RationalPoint(C) : CrvCon -> Pt
            Random(C : parameters) : CrvCon -> Pt
            Points(C : parameters) : CrvCon -> SetIndx
            Example CrvCon_rational-point-enum (H119E7)

      Point Reduction
            IsReduced(p) : Pt -> BoolElt
            Reduction(p) : Pt -> Pt
            Example CrvCon_PointReduction (H119E8)
            Example CrvCon_PointFinding (H119E9)

 
Isomorphisms

      Isomorphisms with Standard Models
            Conic(C) : Crv -> MapSch
            Example CrvCon_ (H119E10)
            ParametrizationMatrix(C) : CrvCon -> ModMatRngElt
            Example CrvCon_ (H119E11)
            Parametrization(C) : CrvCon -> MapSch
            Example CrvCon_RationalParametrization (H119E12)

 
Automorphisms

      Automorphisms of Rational Curves
            Automorphism(C, S, T) : CrvRat, SetIndx, SetIndx -> MapIsoSch

      Automorphisms of Conics
            QuaternionAlgebra(C) : CrvCon-> AlgQuat
            Automorphism(C, a) : CrvCon, AlgQuatElt -> MapIsoSch
            Example CrvCon_ConicAutomorphisms (H119E13)

 
Bibliography

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