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The $SL_2(\mathbb{C})$ character variety $X(M)$ has emerged as an importanttool in studying the topology of hyperbolic 3-manifolds. Chinburg-Reid-Stoverconstructed arithmetic invariants stemming from a canonical Azumaya algebraover the normalization of an irreducible component of $X(M)$ containing a liftof the holonomy representation of $M$. We provide an explicit topologicalcriterion for extending the canonical Azumaya algebra over an ideal point,potentially leading to finer arithmetic invariants than those ofChinburg-Reid-Stover. This topological criterion involves Culler-Shalen theoryand, in some cases, JSJ decompositions of toroidal Dehn fillings of knotcomplements in the three-sphere. Inspired by the work of Paoluzzi-Porti andTillmann, we provide examples of several cases where these refined invariantsexist. Along the way, we show that certain families of Seifert surfaces inhyperbolic knot complements can be associated to ideal points of charactervarieties.

Punctured JSJ tori and tautological extensions of Azumaya algebras