In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group. The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posed by Poincaré around 1908; it was proved by Louis Mordell in 1922.

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 with in establishing a result on the quotient group

E(Q)/2E(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 in his doctoral dissertation 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). The theorem left unanswered a number of questions:

• Calculation of the rank (still a demanding computational problem, and not always effective, as far as we know).
• Meaning of the rank: see Birch and Swinnerton-Dyer conjecture.
• For a curve C in its Jacobian variety as A, can the intersection of C with A(K) be infinite? (Not unless C = A, according to Mordell's conjecture, proved by Faltings.)
• In the same context, can C contain infinitely many torsion points of A? (No, according to the Manin-Mumford conjecture proved by Raynaud, other than in the elliptic curve case.)

See also: arithmetic of abelian varieties.