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