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 ring 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. $f\in K(C)$ .

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 $\Rats$ can only have finitely many points defined over $\Ints$ .