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 coordinatesin $K$ .

Theorem 1 (Mordell-Weil).

$E(K)$ 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.

单词	MordellWeilTheorem
释义	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 coordinatesin $K$ . Theorem 1 (Mordell-Weil). $E(K)$ 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.
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