Inmathematics, theMordell–Weil theorem states that for anabelian variety${\displaystyle A}$ over anumber field${\displaystyle K}$, the group${\displaystyle A(K)}$ ofK-rational points of${\displaystyle A}$ is afinitely-generated abelian group, called theMordell–Weil group. The case with${\displaystyle A}$ anelliptic curve${\displaystyle E}$ and${\displaystyle K}$ the field ofrational numbers isMordell's theorem, answering a question apparently posed byHenri Poincaré around 1901; it was proven byLouis Mordell in 1922. It is a foundational theorem ofDiophantine geometry and thearithmetic of abelian varieties.

Thetangent-chord process (one form ofaddition theorem on acubic curve) had been known as far back as the seventeenth century. The process ofinfinite descent ofFermat was well known, but Mordell succeeded in establishing the finiteness of thequotient group${\displaystyle E(\mathbb{Q})/2E(\mathbb{Q})}$ which forms a major step in the proof. Certainly the finiteness of this group is anecessary condition for${\displaystyle E(\mathbb{Q})}$ to be finitely generated; and it shows that therank is finite. This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point onE.

Some years laterAndré 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 ofheight function, in terms of which to bound the 'size' of points of${\displaystyle A(K)}$. 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 ofhomogeneous coordinates. For an abelian variety, there is noa priori preferred representation, though, as aprojective variety.

Both halves of the proof have been improved significantly by subsequent technical advances: inGalois cohomology as applied to descent, and in the study of the best height functions (which arequadratic forms).

The theorem leaves a number of questions still unanswered:

• Calculation of the rank. This is still a demanding computational problem, and does not always haveeffective solutions.
• Meaning of the rank: seeBirch 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 thetorsion conjecture.
• For acurve${\displaystyle C}$ in itsJacobian variety as${\displaystyle A}$, can the intersection of${\displaystyle C}$ with${\displaystyle A(K)}$ be infinite? Because ofFaltings's theorem, this is false unless${\displaystyle C=A}$.
• In the same context, can${\displaystyle C}$ contain infinitely many torsion points of${\displaystyle A}$? Because of theManin–Mumford conjecture, proved by Michel Raynaud, this is false unless it is the elliptic curve case.

• Arithmetic geometry
• Mordell–Weil group

• Weil, André (1929). "L'arithmétique sur les courbes algébriques".Acta Mathematica.52 (1):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".Mathematical Proceedings of the Cambridge Philosophical Society.21:179–192.
• Silverman, Joseph H. (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.

• Diophantine geometry
• Elliptic curves
• Abelian varieties
• Theorems in algebraic number theory

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