Extending Azumaya algebras associated to arithmetic 2-bridge links

Abstract

Let {\Gamma} be a finitely generated group and consider the set of all characters of representations of {\Gamma} into SL2(C). This set, denoted by X({\Gamma}), admits an algebraic structure and is called the character variety of {\Gamma}. When {\Gamma} is the fundamental group of a hyperbolic 3-manifold M, X({\Gamma}) turns out to be a powerful tool in the study of the geometry and topology of M. Chinburg-Reid-Stover have borrowed tools from algebraic and arithmetic geometry to understand algebraic and number-theoretic properties of the canonical curves of X({\Gamma}). In this paper, we will partly generalize their results to certain hyperbolic link complements, and prove that the associated canonical quaternion algebra will not extend to an Azumaya algebra over the canonical surfaces.