# Faltings’ theorem

Let $K$ be a number field and let $C/K$ be a non-singular^{} curve defined over $K$ and genus $g$. When the genus is $0$, the curve is isomorphic^{} to ${\mathbb{P}}^{1}$ (over an algebraic closure^{} $\overline{K}$) and therefore $C(K)$ is either empty or equal to ${\mathbb{P}}^{1}(K)$ (in particular $C(K)$ is infinite^{}). If the genus of $C$ is $1$ and $C(K)$ contains at least one point over $K$ then $C/K$ is an elliptic curve^{} and the Mordell-Weil theorem^{} shows that $C(K)$ is a finitely generated^{} abelian group^{} (in particular, $C(K)$ may be finite or infinite). However, if $g\ge 2$, Mordell conjectured in $1922$ that $C(K)$ cannot be infinite. This was first proven by Faltings in $1983$.

###### Theorem (Faltings’ Theorem (Mordell’s conjecture)).

Let $K$ be a number field and let $C\mathrm{/}K$ be a non-singular curve defined over $K$ of genus $g\mathrm{\ge}\mathrm{2}$. Then $C\mathit{}\mathrm{(}K\mathrm{)}$ is finite.

The reader may also be interested in Siegel’s theorem.

Title | Faltings’ theorem |
---|---|

Canonical name | FaltingsTheorem |

Date of creation | 2013-03-22 15:57:21 |

Last modified on | 2013-03-22 15:57:21 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 14G05 |

Classification | msc 14H99 |

Synonym | Mordell’s conjecture |

Related topic | SiegelsTheorem |