Mordell-Weil theorem

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

Theorem 1 (Mordell-Weil).

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


  • 1 James Milne, Elliptic Curves, online course notes.
  • 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 theoremMathworldPlanetmath
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