Azumaya Algebras and Canonical Components

Let $M$ be a compact 3-manifold and $\Gamma =\pi _1(M)$. Work by Thurston and Culler–Shalen established the ${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$ character variety $X(\Gamma )$ as fundamental tool in the study of the geometry and topology of $M$. This is particularly the case when $M$ is the exterior of a hyperbolic knot $K$ in $S^3$. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of $X(\Gamma )$, as well as distinguished points on the canonical component, when $\Gamma $ is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.