Siegel's theorem on integral points

Remarks

Equivalently, an affine curve of genus at least $1$ over $K$, embedded in affine space, has only finitely many points with coordinates in $\mathcal{O}_K$. For genus $0$ the same conclusion holds when $f$ has at least three distinct poles over $\overline{K}$. The hypothesis $g \ge 1$ is necessary: the Pell equation $x^2 - 2y^2 = 1$ defines a curve of genus $0$ with infinitely many integral points, here $f = x$ has only two poles. Example: for each integer $k \ne 0$, the Mordell equation $y^2 = x^3 + k$ defines an elliptic curve, so it has only finitely many solutions with $x, y \in \Z$. The proof combines the weak Mordell–Weil theorem for the Jacobian with a Diophantine approximation theorem of Thue–Siegel type, now usually Roth's theorem; it is not effective in general. Siegel proved the theorem in 1929.

Going deeper

For a student

The theorem applies to elliptic curves, where rational points may be infinite in number by Mordell's theorem but integral points are finite. Baker's method in transcendence theory later gave effective bounds in many cases, such as Mordell's equation and Thue equations. For genus at least $2$, Faltings's theorem gives the stronger statement that even rational points are finite.