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ELLIPTIC CURVES OVER FUNCTION FIELDS

This section involves elliptic curves with coefficients in a function field k(C) where C is a regular projective curve over some field k (usually a number field or a finite field). The commands are largely parallel to those for elliptic curves over the rationals; one can compute local information (Tate's algorithm and so forth), a minimal model, the L-function, the 2-Selmer group, and the Mordell--Weil group. This goes in order of decreasing generality: Local information is available for curves over univariate function fields over any exact base field, while at the other extreme Mordell--Weil groups are available only for curves over rational function fields over finite fields for which the associated surface is a rational surface. The generality of many of the commands will be expanded in future releases.  
Acknowledgements
 
An Overview of Relevant Theory
 
Local Computations
 
Elliptic Curves of Given Conductor
 
Heights
 
The Torsion Subgroup
 
The Mordell--Weil Group
 
Two Descent
 
The L-function and Counting Points
 
Action of Frobenius
 
Extended Examples
 
Bibliography







DETAILS

 
An Overview of Relevant Theory

 
Local Computations
      BadPlaces(E) : CrvEll -> [ PlcFunElt ]
      Conductor(E) : CrvEll -> DivFunElt
      LocalInformation(E, Pl) : CrvEll[FldFun], PlcFunElt -> Tup, CrvEll
      LocalInformation(E) : CrvEll -> [ < Tup > ]
      KodairaSymbols(E) : CrvEll -> [ <SymKod, RngIntElt> ]
      NumberOfComponents(K) : SymKod -> RngIntElt
      MinimalModel(E) : CrvEll[FldFunG] -> CrvEll, MapIsoSch
      MinimalDegreeModel(E) : CrvEll[FldFunRat] -> CrvEll, Map, Map
      IsConstantCurve(E) : CrvEll[FldFunRat] -> BoolElt, CrvEll
      TraceOfFrobenius(E, p) : CrvEll[FldFunRat], RngElt -> BoolElt, CrvEll

 
Elliptic Curves of Given Conductor
      EllipticCurveSearch(N, Effort) : [], RngIntElt -> SeqEnum

 
Heights
      NaiveHeight(P) : PtEll -> FldPrElt
      Height(P) : PtEll -> FldRatElt
      LocalHeight(P, Pl) : PtEll, PlcFunElt -> FldPrElt
      HeightPairing(P, Q) : PtEll[FldFunG], PtEll[FldFunG] -> FldRatElt
      HeightPairingMatrix(S) : SeqEnum[PtEll[FldFunG]] -> AlgMatElt
      HeightPairingLattice(S) : [PtEll[FldFunG]] -> AlgMatElt, Map
      Basis(S) : [ PtEll ] -> [ PtEll ], ModMatAlgElt
      Basis(S, r, disc) : SeqEnum -> SeqEnum
      IsLinearlyDependent(points) : [PtEll] -> BoolElt, ModTupRngElt

 
The Torsion Subgroup
      TorsionSubgroup(E) : CrvEll[FldFunG] -> GrpAb, Map
      TorsionBound(E, n) : CrvEll[FldFunG], RngIntElt -> RngIntElt
      GeometricTorsionBound(E) : CrvEll[FldFunG] -> RngIntElt

 
The Mordell--Weil Group
      RankBounds(E) : CrvEll[FldFunG] -> RngIntElt, RngIntElt
      MordellWeilGroup(E : parameters) : CrvEll[FldFunRat] -> GrpAb, Map
      MordellWeilLattice(E) : CrvEll[FldFunRat] -> Lat, Map
      GeometricMordellWeilLattice(E) : CrvEll[FldFunRat] -> Lat, Map
      Generators(E) : CrvEll[FldFunRat] -> SeqEnum
      Example CrvEllFldFun_rank2 (H123E1)

 
Two Descent
      TwoSelmerGroup(E) : CrvEll[FldFunG] -> GrpAb, MapSch
      TwoDescent(E) : CrvEll[FldFunG] -> SeqEnum[CrvHyp], List[MapSch]
      QuarticMinimize(f) : RngMPolElt[FldFunRat] -> RngMPolElt[FldFunRat]
      Points(C : parameters) : CrvHyp -> [Pt]
      PointsQI(C, H) : Crv, RngIntElt -> [Pt]
      TwoIsogenySelmerGroups(E) : CrvEll[FldFunG] -> GrpAb, GrpAb, MapSch, MapSch

 
The L-function and Counting Points
      LFunction(E) : CrvEll[FldFunRat] -> RngUPolElt
      LFunction(E, e) : CrvEll[FldFunRat], RngIntElt -> RngUPolElt
      AnalyticRank(E) : CrvEll[FldFunG] -> RngIntElt
      AnalyticInformation(E) : CrvEll[FldFunG] -> Tup
      Example CrvEllFldFun_sha3 (H123E2)
      Example CrvEllFldFun_rank2-continued (H123E3)
      NumberOfPointsOnSurface(E, e) : CrvEll, RngIntElt -> RngIntElt
      NumbersOfPointsOnSurface(E, e) : CrvEll, RngIntElt -> [ RngIntElt ], [ RngIntElt ]
      BettiNumber(E, i) : CrvEll, RngIntElt -> RngIntElt
      CharacteristicPolynomialFromTraces(traces) : [ Fld ] -> RngUPolElt
      CharacteristicPolynomialFromTraces(traces, d, q, i) : [ Fld ], RngIntElt, RngIntElt, RngIntElt -> RngUPolElt, RngUPolElt

 
Action of Frobenius
      Frobenius(P, q) : PtEll[FldFunRat], RngIntElt -> PtEll
      FrobeniusActionOnPoints(S, q : parameters) : [ PtEll ], RngIntElt -> AlgMatElt
      FrobeniusActionOnReducibleFiber(L) : < Tup > -> AlgMatElt
      FrobeniusActionOnTrivialLattice(E) : CrvEll -> AlgMatElt

 
Extended Examples
      Example CrvEllFldFun_ellfunfld1 (H123E4)
      Example CrvEllFldFun_Reductionmodp (H123E5)
      Example CrvEllFldFun_LFunctionbyhand (H123E6)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012