Heights

NaiveHeight(P): PtEll -> FldPrElt

The naive \(x\)-coordinate height of a point \(P\) on an elliptic curve over a function field \(K\); in other words, the degree of the point \((x(P):1)\) on the projective line.

Height(P): PtEll -> FldRatElt

The Néron–Tate height \(\hat{h}(P)\) of the given point \(P\) on an elliptic curve defined over a function field.

LocalHeight(P, Pl): PtEll, PlcFunElt -> FldPrElt

Given a point \(P\) on an elliptic curve defined over a function field \(F\) and a place \(Pl\) of the function field \(F\), returns the local height \(\lambda_{Pl}(P)\) at \(Pl\) of \(P\).

HeightPairing(P, Q): PtEll[FldFunG], PtEll[FldFunG] -> FldRatElt

Returns the height pairing of the points \(P\) and \(Q\), defined as \(\left<P,Q\right> = (\hat{h}(P+Q) - \hat{h}(P) - \hat{h}(Q))/2\) (where as usual \(\hat{h}\) denotes the Néron–Tate height).

HeightPairingMatrix(S): SeqEnum[PtEll[FldFunG]] -> AlgMatElt

Given a sequence \(S\) of points \(P_i\) on an elliptic curve defined over a function field, this function returns the matrix \((\left< P_i, P_j \right>)\), where \(\left< , \right>\) is the height pairing.

HeightPairingLattice(S): [PtEll[FldFunG]] -> AlgMatElt, Map

The height pairing lattice of a sequence of independent points on an elliptic curve defined over a function field.

Basis(S): [ PtEll ] -> [ PtEll ], ModMatAlgElt

Given a sequence \(S\) of points on an elliptic curve, returns a sequence of points that form a basis for the free part of the subgroup generated by the points in \(S\). The second returned value is a Gram matrix for this basis with respect to the Néron–Tate pairing.

Basis(S, r, disc): SeqEnum, RngIntElt, RngIntElt -> SeqEnum

Given a sequence \(S\) of points on an elliptic curve, returns a sequence of independent points in the free part of the subgroup generated by \(S\) such that these points generate a lattice of rank \(r\) and discriminant \(disc\). The answer is returned as soon as such a lattice has been found, ignoring any additional points in the given sequence.

IsLinearlyDependent(points): [PtEll] -> BoolElt, ModTupRngElt

IsLinearlyIndependent(points): [PtEll] -> BoolElt, ModTupRngElt

IndependentGenerators(points): [PtEll] -> [PtEll]

These functions are available for elliptic curves over function fields, and behave the same way as for elliptic curves over the rationals.