Mordell-Weil theorem

Let $K$ be a number field and let $E$ be an elliptic curve over $K$ . By $E(K)$ we denote the set of points in $E$ with coordinates in $K$ .

Theorem 1 (Mordell-Weil).

$E(K)$ is a finitely generated abelian group.

Proof.

The proof of this theorem is fairly involved. The main two ingredients are the so called weak Mordell-Weil theorem (http://planetmath.org/WeakMordellWeilTheorem), the concept of height function for abelian groups and the “descent (http://planetmath.org/HeightFunction)” theorem.
See [2], Chapter VIII, page 189. ∎

References

1 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
4 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

Title	Mordell-Weil theorem
Canonical name	MordellWeilTheorem
Date of creation	2013-03-22 12:16:35
Last modified on	2013-03-22 12:16:35
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	10
Author	alozano (2414)
Entry type	Theorem
Classification	msc 14H52
Related topic	WeakMordellWeilTheorem
Related topic	MazursTheoremOnTorsionOfEllipticCurves
Related topic	EllipticCurve
Related topic	RankOfAnEllipticCurve
Related topic	ArithmeticOfEllipticCurves