  Recent work of Chinburg, Reid, and Stover has shown that certain arithmetic
and algebro-geometric properties of the character variety of a hyperbolic knot
complement in the 3-sphere $M=S^3\setminus K$ yields topological and number
theoretic information about Dehn fillings of M. Specifically, they show how the
study of a certain extension problem for quaternion Azumaya algebras is related
to topological invariants associated to these fillings.
  In this paper, we extend their work to the setting of hyperbolic once
punctured torus bundles. Along the way, we exhibit new phenomenon in the
relevant extension problem not visible in the case of a hyperbolic knot
complement, which is related to the more complicated non-abelian reducible
representation theory of hyperbolic once punctured torus bundles. We then apply
these results to a series of examples from the literature and list some
remaining questions from both works.
