## Siegel's theorem for curves of genus 0

Posted by Martin Orr on Friday, 28 October 2011 at 12:35

Last time we proved Siegel's theorem on the finiteness of integer points on affine curves of genus at least 1. The theorem applies also to curves of genus 0 with at least 3 points at infinity. I shall give a simple proof that deduces this from the higher genus case, then another proof using Baker's theorem from transcendental number theory which gives an effective bound on the heights of the points.

Theorem.Let`be a number field and`

`a finite set of places of`

`. Let`

`be an affine`

`-curve of genus 0 such that there are at least 3`

`-points in the projective closure of`

`which are not in`

`. Then`

`has finitely many`

`-integer points.`

The condition that there should be at least 3 points at infinity is necessary:
the affine line is a genus 0 curve with 1 point at infinity and infinitely many integer points,
and the curve ` for `

` a non-square positive integer has 2 points at infinity and infinitely many integer points.`