Mordell–Weil theorem

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 E(\mathbb{Q})/2E(\mathbb{Q}) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(\mathbb{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{{cite thesis 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. • For a curve C in its Jacobian variety as A, can the intersection of C with A(K) be infinite? Faltings' theorem implies 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==