Martin's Blog

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 K be a number field and S a finite set of places of K. Let X be an affine K-curve of genus 0 such that there are at least 3 \bar{K}-points in the projective closure of X which are not in X. Then X has finitely many S-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 x^2 - Dy^2 = 1 for D a non-square positive integer has 2 points at infinity and infinitely many integer points.

2 comments Tags maths, number-theory Read more...

Shafarevich and Siegel's theorems

Posted by Martin Orr on Friday, 07 October 2011 at 09:00

In this post I will prove the Shafarevich conjecture for elliptic curves (also called Shafarevich's theorem). The proof is by reducing it to the finiteness of the number of solutions of a certain Diophantine equation, and then applying Siegel's theorem on integral points on curves.

Shafarevich's Theorem. Let K be a number field and S a finite set of places of K. Then there are only finitely many isomorphism classes of elliptic curves over K with good reduction outside S.

Siegel's Theorem. Let K be a number field and S a finite set of places of K. An absolutely irreducible affine curve C over K of genus at least 1 has only finitely many S-integral points.

Since the reduction of Shafarevich's theorem to Siegel's theorem is short, and Siegel's theorem is of independent interest, most of the post will be about Siegel's theorem.

2 comments Tags alg-geom, faltings, maths, number-theory Read more...

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