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单词 MordellWeilTheorem
释义

Mordell-Weil theorem


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

Theorem 1 (Mordell-Weil).

E(K) is a finitely generatedMathworldPlanetmath abeliangroupMathworldPlanetmath.

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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