# Siegel’s theorem

The following theorem is a very deep application of Roth’s theorem. Let $K$ be a number field^{} and let $S$ be a finite set^{} of places of $K$. Let ${R}_{S}$ be the http://planetmath.org/node/RingOfSIntegersring of $S$-integers in $K$. Let $C/K$ be a smooth projective curve of genus $g$ defined over $K$ and let $f$ be a non-constant function in the function field^{} of $C/K$, i.e. $f\in K(C)$.

###### Theorem (Siegel’s Theorem).

Assume that $C\mathrm{/}K$ has genus $g\mathrm{\ge}\mathrm{1}$. Then the set $\mathrm{\{}P\mathrm{\in}C\mathit{}\mathrm{(}K\mathrm{)}\mathrm{:}f\mathit{}\mathrm{(}P\mathrm{)}\mathrm{\in}{R}_{S}\mathrm{\}}$ is finite.

In particular, when $f$ is the coordinate functions $x(P)$ and $y(P)$, Siegel’s theorem implies that a curve of genus $\ge 1$ has only finitely many integral points. For example, this shows that an elliptic curve^{} defined over $\mathbb{Q}$ can only have finitely many points defined over $\mathbb{Z}$.

Title | Siegel’s theorem |
---|---|

Canonical name | SiegelsTheorem |

Date of creation | 2013-03-22 15:57:24 |

Last modified on | 2013-03-22 15:57:24 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11G05 |

Related topic | FaltingsTheorem |