# 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).

$E(K)$ is a finitely generated^{} abelian
group^{}.

###### 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, 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 theorem^{} |
---|---|

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 |