Dehn surgeries on hyperbolic knots produce quaternion algebras over number fields. The ramification of these algebras is in some cases bounded over all hyperbolic Dehn surgeries on a given knot. In other cases the ramification is not bounded. This talk is about work with Matt Stover and Alan Reid which explains this phenomenon using invariants of knots in the Brauer group of Thurston's canonical curve associated to the knot. One consequence is to link the existence of non-abelian SU(2) representations of knot groups to elements of the Tate-Shafarevitch group of the Jacobian of Thurston's curve.