Title: Azumaya algebras and canonical components,
Abstract: Let $M$ be a compact 3-manifold and $\Gamma=\pi_1(M)$. The work of Thurston
and Culler--Shalen established the $\mathrm{SL}_2(\mathbb{C})$ character
variety $X(\Gamma)$ as fundamental tool in the study of the geometry and
topology of $M$. This is particularly so in 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.