Mordell–Weil Groups and Descent Methods#

The remainder of the chapter describes the functions and methods for determining the Mordell–Weil group (the group of rational points) for a curve defined over \({\mathbb{Q}}\) or a number field.

The Mordell–Weil theorem states that for an elliptic curve \(E\) defined over a global field \(F\) (such as \({\mathbb{Q}}\) or a number field), the set \(E(F)\) of points on \(E\) with \(F\)-rational coordinates forms a finitely generated abelian group. This is called the Mordell–Weil group, and the \({\mathbb{Z}}\)-rank of the free part is called the Mordell–Weil rank of \(E\).

It is an open problem to give an explicit algorithm which can determine the Mordell–Weil group of any elliptic curve over \({\mathbb{Q}}\) (or any number field). The general approach is to use the method of \(n\)-descent, for one or more suitable integers \(n\). There are two main problems with this. Firstly, \(n\)-descent cannot determine the rank of \(E\) if the Tate–Shafarevich group \(\rm III\hskip-10.2pt{\vrule width9.5pt depth0pt height0.3pt}\hskip 2pt(E)\) has elements of order \(n\). It is an open conjecture that, for fixed \(E\), this occurs for only finitely many values of \(n\). The second problem is more practical: one cannot implement \(n\)-descent for general \(n\).

Magma has implementations of descent over \({\mathbb{Q}}\) for \(n = 2, 4, 8, 3, 9, 5\) as well as for isogenies of various degrees. It is the only software that has a complete implementation of \(2\)-descent over number fields (in the sense that \(2\)-coverings are reduced and therefore can be used to search for points). In addition, the Cassels-Tate pairing on the \(2\)-Selmer group is implemented over all global fields, and also on the \(4\)-Selmer group over \({\mathbb{Q}}\). All these techniques (and some others) are combined in the main functions to determine ranks and generators. Each of the main functions returns a result together with the status of the result, i.e. whether the rank has been proven, or the group returned is known to be the full Mordell–Weil group.

Prior to version 2.21, the commands MordellWeilGroup, Rank, Generators, etc, did not use all available techniques. Furthermore they did not return the status of their results, but instead a warning was printed in the event of an unproven result; no such warnings are printed now.

Magma provides several ways to ask for information about ranks and Mordell–Weil groups.

(i): The main commands are Rank, RankBounds, MordellWeilGroup, Generators. These apply the available techniques in a suitable order.
(ii): At a slightly lower level, the routine MordellWeilShaInformation follows the same procedure, and also prints a summary of what is known about the rank, generators and sha after applying each technique. The information about sha is also returned at the end.
(iii): The techniques may be called individually. This is often necessary to get the best results in hard examples – the automated procedure used by the higher level routines cannot always make the best choices. All the descent techniques are described later in this chapter; the other main techniques are AnalyticRank and HeegnerPoint.

Torsion#

TorsionSubgroup(E): CrvEll -> GrpAb, Map#

TorsionSubgroup(H): SetPtEll -> GrpAb, Map#

Given an elliptic curve \(E\) defined over \({\mathbb{Q}}\) or a number field, this returns an abelian group \(A\) isomorphic to the torsion subgroup of the Mordell–Weil group, and a map from this abstract group \(A\) to the elliptic curve providing the isomorphism.

For a curve over \({\mathbb{Q}}\), by a theorem of Mazur, \(A\) is either \(C_k\) (for \(k\) in \(\{1..10\}\) or \(12\)) or \(C_2 \times C_{2k}\) (for k in \(\{1..4\}\)). When there are two generators, they are chosen in such a way that the first generator has order \(2\).

The algorithm (for curves over number fields) is based on the TorsionBound described in this section. (For curves over \({\mathbb{Q}}\) a similar method is used.)

TwoTorsionSubgroup(E): CrvEll -> GrpAb, Map#: This returns the group of \(2\)-torsion points on the elliptic curve \(E\) (returned in the same way as the TorsionSubgroup). Note that this is a very cheap computation.

TorsionBound(E, n): CrvEll, RngIntElt -> RngIntElt#: Given an elliptic curve \(E\) defined over a number field, returns a bound on the size of the torsion subgroup of \(E\). This is done by considering at least n primes of good reduction (with early exit in some cases, such as when the torsion is shown to be only two-torsion).

pPowerTorsion(E, p): CrvEll, RngIntElt -> GrpAb, Map#

Bound: RngIntElt                    Default: 

Given an elliptic curve \(E\) defined over \({\mathbb{Q}}\) or a number field, this returns the \(p\)-power torsion subgroup of \(E\) over its base field (returned in the same way as the TorsionSubgroup). A bound on the size of the \(p\)-power torsion subgroup may be given, as the parameter Bound, to cut short the computation in hard cases.

Mordell–Weil Group and Rank#

RankBounds(H: parameters): SetPtEll -> RngIntElt, RngIntElt#

RankBounds(E: parameters): CrvEll -> RngIntElt, RngIntElt#

MordellWeilRankBounds(H: parameters): SetPtEll -> RngIntElt, RngIntElt#

MordellWeilRankBounds(E: parameters): CrvEll -> RngIntElt, RngIntElt#

Effort: RngIntElt                    Default: 1

Given an elliptic curve \(E\) defined over \({\mathbb{Q}}\) or a number field, this returns lower and upper bounds on the rank of the Mordell–Weil group of \(E\).

The parameter Effort must be an integer at least \(1\) (the default). Its usage is explained under MordellWeilShaInformation.

Rank(H: parameters): SetPtEll -> RngIntElt, BoolElt#

Rank(E: parameters): CrvEll -> RngIntElt, BoolElt#

RankBound(E): CrvEll -> RngIntElt, BoolElt#

MordellWeilRank(H: parameters): SetPtEll -> RngIntElt, BoolElt#

MordellWeilRank(E: parameters): CrvEll -> RngIntElt, BoolElt#

Effort: RngIntElt                    Default: 1

Given an elliptic curve \(E\) defined over \({\mathbb{Q}}\) or a number field, this returns a lower bound \(r\) for the Mordell–Weil group of \(E\). The second value returned is a boolean, which is true iff it is known that the rank is exactly \(r\).

The parameter Effort must be an integer at least \(1\) (the default). Its usage is explained under MordellWeilShaInformation.

MordellWeilGroup(H: parameters): SetPtEll -> GrpAb, Map, BoolElt, BoolElt#

MordellWeilGroup(E: parameters): CrvEll -> GrpAb, Map, BoolElt, BoolElt#

AbelianGroup(H: parameters): SetPtEll -> GrpAb, Map, BoolElt, BoolElt#

AbelianGroup(E: parameters): CrvEll -> GrpAb, Map, BoolElt, BoolElt#

Effort     : RngIntElt                    Default: 1
HeightBound: RngIntElt                    Default: 15

The Mordell–Weil group of an elliptic curve \(E\) defined over \({\mathbb{Q}}\). The first two values returned are 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.

Additionally the function returns two booleans. The first boolean is true iff the rank of the group returned is known to be the rank of the curve. The second boolean is true iff the group returned is known to be the full Mordell–Weil group.

The parameter Effort must be an integer at least \(1\) (the default). Its usage is explained under MordellWeilShaInformation.

The parameter HeightBound limits the search for points directly on \(E\) to the given naive height. (Note: this affects only saturation of the group, not determination of the rank.)

Generators(H): SetPtEll -> [ PtEll ]#
Generators(E): CrvEll -> [ PtEll ]#: This function returns the generators of the MordellWeilGroup of the elliptic curve \(E\) (listed in the same order).

NumberOfGenerators(H): SetPtEll -> RngIntElt#
NumberOfGenerators(E): CrvEll -> RngIntElt#
Ngens(H): SetPtEll -> RngIntElt#
Ngens(E): CrvEll -> RngIntElt#: This is equivalent to NumberOfGenerators(MordellWeilGroup(E)).

Saturation(points, n): [ PtEll ], RngIntElt -> [ PtEll ]#

TorsionFree: BoolElt                          Default: false
OmitPrimes : [ RngIntElt ]                    Default: []
Check      : BoolElt                          Default: true

Given a sequence of points on an elliptic curve \(E\) over the rationals or a number field, and an integer \(n\), this function returns a sequence of points generating a subgroup of \(E({\mathbb{Q}})\) which contains the given points and which is \(p\)-saturated for all primes \(p\) up to \(n\). (A subgroup \(S\) is \(p\)-saturated in a group \(G\) if there is no intermediate subgroup \(H\) for which the index \([H:S]\) is finite and divisible by \(p\).)

If OmitPrimes is set to be a sequence of primes, then the group is not checked to be \(p\)-saturated for those primes. If TorsionFree is set to true, torsion points are omitted from the result (that is, the result contains independent generators modulo torsion). If Check is set to false, the input sequence of points are assumed to be independent modulo torsion.

Saturation(points): [ PtEll ] -> [ PtEll ]#

TorsionFree: BoolElt                    Default: false
Check      : BoolElt                    Default: true

Given a sequence of points on an elliptic curve \(E\) over the rationals or a number field, returns generators of the saturation of the given points in the Mordell–Weil group of \(E\).

Example: Mordell Weil (ex-0fa71f)#

> E := EllipticCurve([73, 0]);
> E;
Elliptic Curve defined by y^2 = x^3 + 73*x over Rational Field
> Factorization(Integers() ! Discriminant(E));
[ <2, 6>, <73, 3> ]
> BadPrimes(E);
[ 2, 73 ]
> LocalInformation(E);
[ <2, 6, 6, 1, II, true>, <73, 3, 2, 2, III, true> ]
> G, m := MordellWeilGroup(E);
> G;
Abelian Group isomorphic to Z/2 + Z + Z
Defined on 3 generators
Relations:
    2*G.1 = 0
> Generators(E);
[ (0 : 0 : 1), (36 : -222 : 1), (4/9 : 154/27 : 1) ]
> 2*m(G.1);
(0 : 1 : 0)

Mordell–Weil Groups and Descent Methods#

Torsion#

Mordell–Weil Group and Rank#

Two-Descent#

Two Descent Using Isogenies#

Invariants#

Selmer Groups#

The Cassels-Tate Pairing#

Four-Descent#

Eight-Descent#

Three-Descent and Five-Descent#

Six and Twelve Descent#

Nine-Descent#

Higher 2-power Isogeny Descents#

\(p\)-Isogeny Descent#