Faltings’ theorem

Let K be a number field and let C/K be a non-singularPlanetmathPlanetmath curve defined over K and genus g. When the genus is 0, the curve is isomorphicPlanetmathPlanetmathPlanetmath to 1 (over an algebraic closureMathworldPlanetmath K¯) and therefore C(K) is either empty or equal to 1(K) (in particular C(K) is infiniteMathworldPlanetmathPlanetmath). If the genus of C is 1 and C(K) contains at least one point over K then C/K is an elliptic curveMathworldPlanetmath and the Mordell-Weil theoremMathworldPlanetmath shows that C(K) is a finitely generatedMathworldPlanetmath abelian groupMathworldPlanetmath (in particular, C(K) may be finite or infinite). However, if g2, 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/K be a non-singular curve defined over K of genus g2. Then C(K) 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