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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
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(KS) : SymKod -> RngIntElt
MinimalModel(E) : CrvEll[FldFunG] -> CrvEll, MapIsoSch
MinimalDegreeModel(E) : CrvEll[FldFunRat] -> CrvEll, Map, Map
IsConstantCurve(E) : CrvEll[FldFunRat] -> BoolElt, CrvEll
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(seq,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 (H105E1)
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 (H105E2)
Example CrvEllFldFun_rank2-continued (H105E3)
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 (H105E4)
Example CrvEllFldFun_Reductionmodp (H105E5)
Bibliography
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