Mordell–Weil theorem
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, called the Mordell-Weil 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.
History
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(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 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 left unanswered a number of questions:
- 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.
- For an curve 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 Raynaud, this is false unless it is the elliptic curve case.
See also
References
- ↑ Weil, André (1928). L'arithmétique sur les courbes algébriques (PhD). Almqvist & Wiksells Boktryckeri AB, Uppsala.
- Weil, André (1929). "L'arithmétique sur les courbes algébriques". Acta Mathematica 52 (1). pp. 281–315. doi:10.1007/BF02592688. MR 1555278.
- Mordell, Louis Joel (1922). "On the rational solutions of the indeterminate equations of the third and fourth degrees". Proc Cam. Phil. Soc. 21. pp. 179–192.
- Joseph H., Silverman (1986). The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106. Springer-Verlag. doi:10.1007/978-0-387-09494-6. ISBN 0-387-96203-4. MR 2514094.