 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 Krational points of A is a finitelygenerated abelian group, called the MordellWeil 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 tangentchord 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 finitelygenerated; 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 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 coordinates 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 it is currently known).
 Meaning of the rank: see Birch and SwinnertonDyer 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 ManinMumford conjecture proved by Raynaud, other than in the elliptic curve case.)
See also
References
 A. Weil, L'arithmétique sur les courbes algébriques, Acta Math 52, (1929) p. 281315, reprinted in vol 1 of his collected papers ISBN 0387903305
 L.J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc Cam. Phil. Soc. 21, (1922) p. 179.
 J. H. Silverman, The arithmetic of elliptic curves, ISBN 0387962034 second edition
Categories: Diophantine geometry
 Elliptic curves
 Abelian varieties
 Theorems in algebra
 Theorems in number theory
Wikimedia Foundation. 2010.
Look at other dictionaries:
Satz von MordellWeil — Der Satz von Mordell Weil ist ein mathematischer Satz aus dem Gebiet der algebraischen Geometrie. Er besagt, dass für eine abelsche Varietät A über einem Zahlkörper K die Gruppe A(K) der K rationalen Punkte endlich erzeugt ist. Den Spezialfall,… … Deutsch Wikipedia
Mordell — Louis Mordell in Nizza, 1970 Louis Joel Mordell (* 28. Januar 1888 in Philadelphia, USA; † 12. März 1972 in Cambridge, England) war ein amerikanisch britischer Mathematiker, der vor allem in der Zahlentheorie, spezi … Deutsch Wikipedia
Louis Joel Mordell — Louis Mordell in Nizza, 1970 Louis Joel Mordell (* 28. Januar 1888 in Philadelphia, USA; † 12. März 1972 in Cambridge, England) war ein amerikanisch britischer Mathematiker, der vor allem in der Zahlentheorie, spezi … Deutsch Wikipedia
Faltings' theorem — In number theory, the Mordell conjecture is the conjecture made by Mordell (1922) that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing… … Wikipedia
Louis Mordell — in Nizza, 1970 Louis Joel Mordell (* 28. Januar 1888 in Philadelphia, USA; † 12. März 1972 in Cambridge, England) war ein amerikanisch britischer Mathematiker, der vor allem in der Zahlentheorie, speziell der … Deutsch Wikipedia
André Weil — Infobox Scientist name = André Weil image width = caption = birth date = birth date190656 birth place = Nantes death date = death date and age199886190656 death place = field = Mathematics work institutions = Lehigh University… … Wikipedia
Louis Mordell — Louis Joel Mordell est un mathématicien américano britannique, né le 28 janvier 1888 à Philadelphie et mort le 12 mars 1972 à Cambridge. Pionnier par ses recherches en théorie des nombres, il est un spécialiste reconnu des équations… … Wikipédia en Français
Louis J. Mordell — Louis Mordell Louis Mordell in Nizza, 1970. Born 28 January 1888( … Wikipedia
De Franchis theorem — In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism… … Wikipedia
Louis Mordell — Louis Joel Mordell (28 January 1888 12 March 1972) was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction. He came in 1906 to Cambridge to take… … Wikipedia