The machinery in this section and the next section is for curves defined over a rational function field k(t) whose field of constants k is finite.
The Mordell--Weil group can be computed (and generators found) for a curve y2 + a1xy + a3y=x3 + a2x2 + a4x + a6 where each ai is a polynomial in k[t] ⊂k(t) of degree at most i.
When this hypothesis is not satisfied, it may still be possible to bound the Mordell--Weil rank and find Mordell--Weil generators using the 2-descent routines described in the next section.
These functions return lower and upper bounds (or just an upper bound) on the rank of the Mordell--Weil group E(F) for an elliptic curve E defined over a function field F with finite constant field. The bound is obtained by applying all the available tools (those described in this section together with the AnalyticInformation obtained from the L-function).
Al: MonStgElt Default: "Geometric"
This function computes the Mordell--Weil group of an elliptic curve E that satisfies the hypotheses stated in the introduction immediately above. The function returns two values: an abelian group A and a map m from A to E. The map m provides an isomorphism between the abstract group A and the Mordell--Weil group.The algorithm used by default is the geometric method described above. However, when Al is set to "Descent" it instead uses the 2-descent tools described in the next section; if the curve admits 2-isogenies then it uses a separate implementation of descent by 2-isogenies (described in [Rob07]). These descent methods do not always determine the full Mordell--Weil group (in which case a warning is printed); their advantages are that they do not require the degrees of the coefficients to be bounded and in many cases are very efficient.
This function computes the Mordell--Weil lattice of an elliptic curve E that satisfies the hypotheses stated in the introduction immediately above. This is the free part of the Mordell--Weil group with an inner product given by the Néron--Tate height pairing. The function returns two values: the lattice L and a map m from L to E.
This function computes the geometric Mordell--Weil lattice of an elliptic curve E that satisfies the hypotheses stated in the introduction immediately above. This consists of the free part of the group of points on E that are defined over the function field with a possibly extended constant field, together with the Néron--Tate pairing. The function returns two values: a lattice L and a map m from L to E', where E' is a base change of E over the larger field.
Given an elliptic curve E over a rational function field F that satisfies the hypotheses stated in the introduction immediately above, this function returns a sequence of points in E(F) which are generators of the Mordell--Weil group.
We find that the curve y2 = x3 + (t4 + 2t2)x + t3 + 2t over F3(t) has rank 2 and has no 2-torsion in its Tate--Shafarevich group.
> F<t> := FunctionField(GF(3));
> E := EllipticCurve([ t^4 + 2*t^2, t^3 + 2*t ]);
> S2E := TwoSelmerGroup(E);
> S2E;
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*S2E.1 = 0
2*S2E.2 = 0
> MordellWeilGroup(E);
Abelian Group isomorphic to Z + Z
Defined on 2 generators (free)
Mapping from: Abelian Group isomorphic to Z + Z
Defined on 2 generators (free) to CrvEll: E given by a rule [no inverse]
Furthermore, we may compute the regulator of E as follows.
> Determinant(HeightPairingMatrix(Generators(E))); 12