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
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
Wed Apr 24 15:09:57 EST 2013