Field

Number theory

Henri Poincaré

1901

André Weil

1929

Faltings's theorem
Bombieri–Lang conjecture
Mordell–Lang conjecture

History[edit]

The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for 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 . 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).

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

For a in its Jacobian variety as , can the intersection of with be infinite? Because of Faltings's theorem, this is false unless .

curve

In the same context, can contain infinitely many torsion points of ? Because of the , proved by Michel Raynaud, this is false unless it is the elliptic curve case.

Manin–Mumford conjecture

The theorem leaves a number of questions still unanswered:

Arithmetic geometry

Mordell–Weil group

(1929). "L'arithmétique sur les courbes algébriques". Acta Mathematica. Vol. 52, no. 1. pp. 281–315. doi:10.1007/BF02592688. MR 1555278.

Weil, André

Mordell, Louis Joel (1922). . Proc. Camb. Phil. Soc. Vol. 21. pp. 179–192.

"On the rational solutions of the indeterminate equations of the third and fourth degrees"

(1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 106. Springer-Verlag. doi:10.1007/978-0-387-09494-6. ISBN 0-387-96203-4. MR 2514094.

Silverman, Joseph H.