Mordell–Weil theorem

The set of Mordell -Weil is a mathematical theorem in the field of algebraic geometry. He states that for an abelian variety over a number field, the group of - rational points is finitely generated. The special case that an elliptic curve and the body is the rational numbers is called the set of Mordell by Louis Mordell, who proved it in 1922. The generalization was proved by André Weil in his doctoral thesis published in 1928.

Statement

Be a number field, ie a finite field extension of an abelian variety and, therefore an algebraic variety, which also carries the structure of an abelian group and some other additional properties. An example of this are elliptic curves. Then the set of points is finitely generated of which are defined.

Idea of ​​proof for elliptic curves

To prove the theorem for elliptic curves, one first proves the so-called weak Mordell -Weil theorem of. This indicates that the group is for each integer finite. The set of Mordell -Weil obtained therefrom by means of height functions and a descent argument.

Any more questions

• By the theorem of Mordell - Weil group of rational points of an elliptic curve finite rank, the conjecture of Birch and Swinnerton - Dyer provides a method of how to determine this.
• It is also common for the number of rational points of an algebraic curve questions. After a now proven conjecture of Mordell this is finite for curves with genus 2 or higher.