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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. [ABSTRACT FROM AUTHOR]

The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by p-adic L-functions under the relevant CM main conjectures. [ABSTRACT FROM AUTHOR]

We study commutative rings R for which all projective schemes over R have finite morphisms to a projective space of R. We then use our results to define a new kind of capacity for adelic subsets of projective flat schemes X over global fields. This capacity can be used to generalize the converse part of the Fekete-Szegő Theorem. [ABSTRACT FROM AUTHOR]