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Mathematische Annalen Integral points on character varieties

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. Before stating our results, we describe the setting in which we will work. Let M be a finite volume hyperbolic 3-manifold with a single cusp, and let X(M) denote the character variety associated to representations π1(M) → SL(2, C) (see §2.1 for definitions). By an algebraic curve we will mean an irreducible algebraic curve unless otherwise stated. The work of Thurston [28] shows that the component of the character variety containing the character of the faithful discrete representation of π1(M) is an algebraic curve. Throughout this paper, this algebraic curve component will be denoted X0. Work begun in [13] (and