Mordell-Weil theorem
Let be a number field and let be an elliptic curve
over. By we denote the set of points in with coordinatesin .
Theorem 1 (Mordell-Weil).
is a finitely generated abeliangroup
.
Proof.
The proof of this theorem is fairly involved. Themain two ingredients are the so called weak Mordell-Weil theorem (http://planetmath.org/WeakMordellWeilTheorem), the concept of height function for abelian groups andthe “descent (http://planetmath.org/HeightFunction)” theorem.
See [2], Chapter VIII, page189.∎
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 inthe Arithmetic of Elliptic Curves. Springer-Verlag, New York,1994.
- 4 Goro Shimura, Introduction to theArithmetic Theory of Automorphic Functions. Princeton UniversityPress, Princeton, New Jersey, 1971.