Azumaya algebras associated to knot groups

Ted Chinburg
University of Pennsylvania

This is a report on work with M. Stover and A. Reid on Azumaya algebras $A_\Gamma$ associated to knot groups $\Gamma$. Thurston defined a canonical affine curve $C$ inside the $\mathrm{SL}_2$ character variety of $\Gamma$. The Azumaya algebra $A_\Gamma$ is defined over an open dense subset of $C$. I will describe some sufficient conditions for $A_\Gamma$ to have a continuation over a projective closure $\overline{C}$ of $C$. This is possible if and only if the quaternion algebras which arise from specialization (e.g. from Dehn surgeries) are unramified outside a fixed finite set of primes. I will also discuss Brauer-Manin obstructions and elements of Tate-Shafarevitch groups arising from $A_\Gamma$.

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