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.
- An Overview of Relevant Theory
- Local Computations
BadPlaces(E): CrvEll → [ PlcFunElt ]Conductor(E): CrvEll → DivFunEltLocalInformation(E, Pl): CrvEll[FldFun], PlcFunElt → Tup, CrvEllLocalInformation(E, f): CrvEll[FldFunRat], FldFunRatUElt → Tup, CrvEllLocalInformation(E): CrvEll → [ < Tup > ]KodairaSymbols(E): CrvEll → [ <SymKod, RngIntElt> ]NumberOfComponents(K): SymKod → RngIntEltMinimalModel(E): CrvEll[FldFunG] → CrvEll, MapIsoSchMinimalDegreeModel(E): CrvEll[FldFunRat] → CrvEll, Map, MapIsConstantCurve(E): CrvEll[FldFunRat] → BoolElt, CrvEllTraceOfFrobenius(E, p): CrvEll[FldFunRat], RngElt → BoolElt, CrvEll
- Elliptic Curves of Given Conductor
- Heights
NaiveHeight(P): PtEll → FldPrEltHeight(P): PtEll → FldRatEltLocalHeight(P, Pl): PtEll, PlcFunElt → FldPrEltHeightPairing(P, Q): PtEll[FldFunG], PtEll[FldFunG] → FldRatEltHeightPairingMatrix(S): SeqEnum[PtEll[FldFunG]] → AlgMatEltHeightPairingLattice(S): [PtEll[FldFunG]] → AlgMatElt, MapBasis(S): [ PtEll ] → [ PtEll ], ModMatAlgEltBasis(S, r, disc): SeqEnum, RngIntElt, RngIntElt → SeqEnumIsLinearlyDependent(points): [PtEll] → BoolElt, ModTupRngEltIsLinearlyIndependent(points): [PtEll] → BoolElt, ModTupRngEltIndependentGenerators(points): [PtEll] → [PtEll]
- The Torsion Subgroup
- The Mordell–Weil Group
RankBounds(E): CrvEll[FldFunG] → RngIntElt, RngIntEltRankBound(E): CrvEll[FldFunG] → RngIntEltMordellWeilGroup(E : parameters): CrvEll[FldFunRat] → GrpAb, MapMordellWeilLattice(E): CrvEll[FldFunRat] → Lat, MapGeometricMordellWeilLattice(E): CrvEll[FldFunRat] → Lat, MapGenerators(E): CrvEll[FldFunRat] → SeqEnumExample: rank2
- Two Descent
TwoSelmerGroup(E): CrvEll[FldFunG] → GrpAb, MapSchTwoDescent(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] → RngUPolEltLFunction(E, S): CrvEll[FldFunRat], <SeqEnum> → RngUPolEltLFunction(E, e): CrvEll[FldFunRat], RngIntElt → RngUPolEltAnalyticRank(E): CrvEll[FldFunG] → RngIntEltAnalyticInformation(E): CrvEll[FldFunG] → TupAnalyticInformation(E, L): CrvEll[FldFunG], RngUPolElt → TupAnalyticInformation(E, e): CrvEll[FldFunG], RngIntElt → TupExample: sha3Example: Rank2 ContinuedNumberOfPointsOnSurface(E, e): CrvEll, RngIntElt → RngIntEltNumbersOfPointsOnSurface(E, e): CrvEll, RngIntElt → [ RngIntElt ], [ RngIntElt ]BettiNumber(E, i): CrvEll, RngIntElt → RngIntEltCharacteristicPolynomialFromTraces(traces): [ Fld ] → RngUPolEltCharacteristicPolynomialFromTraces(traces, d, q, i): [ Fld ], RngIntElt, RngIntElt, RngIntElt → RngUPolElt, RngUPolElt
- Action of Frobenius
- Extended Examples