 
Summary: Digital Object Identifier (DOI) 10.1007/s002080020380y
Math. Ann. 325, 299321 (2003) MathematischeAnnalen
Integral points on character varieties
D.D. Long · A.W. Reid
Received: 7 July 1999 / Revised version: 7 July 2002 /
Published online: 2 December 2002 © SpringerVerlag 2002
1 Introduction
Let k be a number field and denote its ring of integers by Rk. If V is a com
plex algebraic variety defined over a number field k, and L a number field, by an
Lintegral point we mean a point P on V all of whose coordinates are in RL. A
good deal of attention has been devoted over the years to understanding Lintegral
points, most of which are rooted in the celebrated theorem of Siegel [26]:
Theorem 1.1. Let k and L be number fields, and X be an algebraic curve defined
overk withprojectivecompletionX.IfthegenusofX isatleastoneorifX contains
at least three points at infinity, then the collection of Lintegral points on X is finite.
The focus of this paper will be to show that one can prove even stronger results
using topological methods for certain naturally occurring varieties. For example
Siegel's theorem leaves open the question of whether, given a constant D, there
are finitely many RL points as we allow L to run over all number fields with
[L : Q] < D; we have a result in this direction for our varieties.
