Siegel’s theorem

The following theorem is a very deep application of Roth’s theorem. Let K be a number fieldMathworldPlanetmath and let S be a finite setMathworldPlanetmath of places of K. Let RS be the 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 fieldMathworldPlanetmath of C/K, i.e. fK(C).

Theorem (Siegel’s Theorem).

Assume that C/K has genus g1. Then the set {PC(K):f(P)RS} is finite.

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

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