# 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 $C/K$ has genus $g\geq 1$. Then the set $\{P\in C(K):f(P)\in R_{S}\}$ is finite.

In particular, when $f$ is the coordinate functions $x(P)$ and $y(P)$, Siegel’s theorem implies that a curve of genus $\geq 1$ 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 SiegelsTheorem 2013-03-22 15:57:24 2013-03-22 15:57:24 alozano (2414) alozano (2414) 6 alozano (2414) Theorem msc 11G05 FaltingsTheorem