The study of elliptic curves over the rationals and number fields are motivated by their relevance to major theoretical questions such as the conjecture of Birch and Swinnerton-Dyer which is one of the millennium problems. A practical application of computational techniques is to find the solutions of certain classes of diophantine equation.
Much of arithmetic geometry grew out of the study of Diophantine equations, with the basic problem being to describe the set of integral or rational solutions to a system of polynomial equations. Since the mid twentieth century, Diophantine equations have been studied more and more from the point of view of the geometry of the curve or variety they define, as this approach has yielded more powerful results than elementary arguments. Elliptic curves in Magma can be defined over ℤ, ℚ, a number field, a function field, a p-adic field or a finite field. The tools for creating elliptic curves and for computing basic information about them are largely common for each of the above rings. In this chapter we concentrate on curves over ℚ and number fields.
Below is a list of some information that can be obtained from an elliptic curve over ℚ:-
Below is a list of some information that can be obtained from an elliptic curve over a number field K:-
A routine is provided for finding elliptic curves with given conductor, or with good reduction outside a given set of primes. The aim is not to provably find all such curves; in most cases, this would be a very difficult task using current algorithms. Rather the aim is to efficiently search for these curves, using a variety of techniques, taking advantage of all available information, and taking advantage of all the tools available in Magma which can be applied to the problem.
The routine is very much more effective when some traces of Frobenius are known for the desired curve. The principal application of this tool is for finding elliptic curves that match known modular forms.
The Mordell–Weil theorem states that for an elliptic curve E defined over a global field F (such as 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 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 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 Sha(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 ℚ for n = 2, 4, 8, 3, 9, 5, 6, 12 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 ℚ. All these techniques (and some others) are combined in the main functions to determine ranks and generators.
The method of descent furnishes a “meta-algorithm” which, if standard conjectures hold, must succeed for all curves. It would be hard to encode this entire method in a finite program. Nevertheless, Magma goes much further than any other software, both for curves over ℚ and also over general number fields; it contains by far the most powerful array of tools available anywhere for obtaining rank bounds and for finding generators.
The first and simplest form of descent is 2-descent (implementations by G. Bailey, N. Bruin, M. Watkins, S. Donnelly). Magma contains the only implementation of full 2-descent over general number fields; that is, it has routines to express the 2-descendents as hyperelliptic curves and to find models of these that are nice enough to be well-suited for searching for points. The next simplest form is 4-descent (M. Watkins, N. Bruin). The remaining techniques are implemented only for curves over ℚ: 3-descent (M. Stoll, S. Donnelly, T. A. Fisher); 8-descent (Stamminger, T. A. Fisher); 6- and 12- descent by combining the earlier descents (T. A. Fisher); 9-descent (B. Creutz). Excluding 9-descent, these implementations all produce nice models of covering curves which are useful for searching for generators; the most powerful of them for this purpose is 12-descent, which is often as effective as the complementary Heegner point method (M. Watkins) where both are applicable.
A feature is the algorithm developed by S. Donnelly for finding integral points on an elliptic curve over either ℚ or a totally real number field. This assumes that generators for at least part of the Mordell–Weil group can be found.
A Heegner point package developed by M. Watkins (Magma) is designed to compute the generator of a rank 1 elliptic curve over ℚ. S. Donnelly has added additional functions that make it possible to compute the Heegner point directly over the Hilbert class field, rather than its trace to ℚ. In addition, this machinery allows a real approximation of the Heegner point to be computed on a 2-cover or 4-cover of the elliptic curve, greatly expediting the process in some cases.
Magma is able to compute the p-adic height of a point on an elliptic curve over ℚ for a good ordinary prime p ≥ 5. This is achieved using Kedlaya's algorithm to compute the Monsky–Washnitzer cohomology, and then applying an algorithm of Mazur, Stein, and Tate, as improved by Harvey. This has been used by various people, including J-S. Mueller, in computations with p-adic variants of the BSD conjecture, and can also be used in conjunction with Iwasawa theory to bound the p-part of the Tate–Shafarevich group.
Four types of maps between elliptic curves may be constructed: isogenies, isomorphisms, translations, and rational maps. A large number of constructions for isogenies are provided together with tools for working with them. Given two elliptic curves defined over the same field, a function is provided which determines whether the curves are isomorphic and if so returns the isomorphism. The endomorphism ring and automorphism group of an elliptic curve can be computed.