# 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).

$E(K)$

###### 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, . 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 MordellWeilTheorem 2013-03-22 12:16:35 2013-03-22 12:16:35 alozano (2414) alozano (2414) 10 alozano (2414) Theorem msc 14H52 WeakMordellWeilTheorem MazursTheoremOnTorsionOfEllipticCurves EllipticCurve RankOfAnEllipticCurve ArithmeticOfEllipticCurves