Mordell–Weil theorem explained

Mordell–Weil theorem
Field:	Number theory
Conjectured By:	Henri Poincaré
Conjecture Date:	1901
First Proof By:	André Weil
First Proof Date:	1929
Generalizations:	Faltings's theorem Bombieri–Lang conjecture Mordell–Lang conjecture

A over a number field K , the group A(K) of K-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case with A an elliptic curve E and K the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.

History

E( Q )/ 2 E( Q ) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E( Q ) to be finitely generated; and it shows that the rank is finite. This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point on E.

Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation^[1] published in 1928. More abstract methods were required, to carry out a proof with the same basic structure. The second half of the proof needs some type of height function, in terms of which to bound the 'size' of points of

A(K) . Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set of homogeneous coordinates. For an abelian variety, there is no a priori preferred representation, though, as a projective variety.

Both halves of the proof have been improved significantly by subsequent technical advances: in Galois cohomology as applied to descent, and in the study of the best height functions (which are quadratic forms).

Further results

The theorem leaves a number of questions still unanswered:

• Calculation of the rank. This is still a demanding computational problem, and does not always have effective solutions.
• Meaning of the rank: see Birch and Swinnerton-Dyer conjecture.
• Possible torsion subgroups: Barry Mazur proved in 1978 that the Mordell–Weil group can have only finitely many torsion subgroups. This is the elliptic curve case of the torsion conjecture.

C in its Jacobian variety as A , can the intersection of C with A(K) be infinite? Because of Faltings's theorem, this is false unless C = A .

• In the same context, can

C contain infinitely many torsion points of A ? Because of the Manin–Mumford conjecture, proved by Michel Raynaud, this is false unless it is the elliptic curve case.

See also

• Arithmetic geometry
• Mordell–Weil group

Further reading

• Weil . André . André Weil. L'arithmétique sur les courbes algébriques. Acta Mathematica. 52. 1929. 281–315. 1. 10.1007/BF02592688. 1555278.
• Mordell . Louis Joel . Louis J. Mordell. On the rational solutions of the indeterminate equations of the third and fourth degrees. Mathematical Proceedings of the Cambridge Philosophical Society. 21. 1922. 179–192.
• Book: Silverman, Joseph H. . Joseph H. Silverman

. Joseph H. Silverman . 1986 . The Arithmetic of Elliptic Curves . . 106 . Springer-Verlag . 0-387-96203-4 . 2514094 . 10.1007/978-0-387-09494-6 .

Notes and References

1. Weil . André . André Weil. L'arithmétique sur les courbes algébriques. PhD. 1928. Almqvist & Wiksells Boktryckeri AB, Uppsala.