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The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil. 

Abstract Let $$M$$ be a compact 3-manifold and $$\Gamma =\pi _1(M)$$. Work by Thurston and Culler–Shalen established the $${\operatorname{\textrm{SL}}}_2({\mathbb{C}})$$ character variety $$X(\Gamma )$$ as fundamental tool in the study of the geometry and topology of $$M$$. This is particularly 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.