Mordell-Weil's theorem

The Mordell-Weil Theorem is a mathematical set of the field of algebraic geometry . It says that for an Abelian variety over a number field the Abelian group of -rational points is finitely generated. ${\ displaystyle A}$ ${\ displaystyle K}$ ${\ displaystyle A (K)}$ ${\ displaystyle K}$

The special case, which is an elliptic curve and the field of rational numbers, is called Mordell's theorem after Louis Mordell , who proved it in 1922. In 1901 Henri Poincaré asked what values the rank of can assume. ${\ displaystyle A}$ ${\ displaystyle E (K)}$ ${\ displaystyle K = \ mathbb {Q}}$ ${\ displaystyle E (\ mathbb {Q})}$

The generalization was proven by André Weil in his doctoral thesis published in 1928.

statement

Let be a number field, i.e. a finite field extension of and an Abelian variety, i.e. an algebraic variety , which at the same time bears the structure of an Abelian group and some other additional properties. An example of this are elliptic curves . Then the group of points of which are defined by is finitely generated. ${\ displaystyle K}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle A}$ ${\ displaystyle A (K)}$ ${\ displaystyle A}$ ${\ displaystyle K}$

From the theorem it follows that the Mordell-Weil group, where the torsion group is a finite Abelian group (the group of torsion points) and r is the rank of the Mordell-Weil group (with generators ). ${\ displaystyle A (K) = A_ {tors} \ oplus P_ {1} \ mathbb {Z} \ oplus \ cdot \ cdot \ cdot \ oplus P_ {r} \ mathbb {Z}}$ ${\ displaystyle A_ {tors}}$ ${\ displaystyle P_ {1}, \ cdot \ cdot \ cdot, P_ {r} \ in A (K)}$

Proof idea for elliptic curves

To show the theorem for elliptic curves, one first proves the so-called weak theorem of Mordell-Weil. This means that for every whole number the group is finite. The Mordell-Weil theorem can be obtained from this with the help of height functions and a descent argument . ${\ displaystyle m \ geq 2}$ ${\ displaystyle E (K) / mE (K)}$

Further questions

According to Mordell-Weil's theorem, the group of rational points of an elliptic curve has finite rank; Birch and Swinnerton-Dyer's conjecture gives a method for determining this.
More generally, one can also ask about the number of rational points in an algebraic curve . According to a now proven guess by Mordell , this is finally for curves with gender 2 or higher (that is, applies to their rank ). ${\ displaystyle r = 0}$

literature

André Weil: L'arithmétique sur les courbes algébriques. Acta Math 52, 1929, pp. 281-315.
Louis Mordell: On the rational solutions of the indeterminate equation of the 3rd and 4th degrees. Proc. Cambridge Philosophical Society, Vol. 21, 1922, pp. 179-192.
Joseph Silverman : The arithmetic of elliptic curves. Graduate Texts in Mathematics. Springer-Verlag, 1986, ISBN 0-387-96203-4 .
Jean-Pierre Serre : Lectures on the Mordell-Weil theorem. Vieweg, 1997, ISBN 978-3-528-28968-3 .

Web links

Eric W. Weisstein : Mordell-Weil Theorem . In: MathWorld (English).