Extending Azumaya algebras associated to arithmetic 2-bridge links

TL;DR

The paper extends the arithmetic-geometry framework for canonical quaternion algebras from knot complements to two-component arithmetic 2-bridge links. It constructs the canonical quaternion algebra $A_{k(C)}$ over the function field of canonical components of SL$_2$ character varieties and tests extension to Azumaya algebras on the canonical surfaces by analyzing tame symbols along curves in the reducible locus. For the three arithmetic links $5_{1}^{2}$, $6_{2}^{2}$, and $6_{3}^{2}$, it proves that $A_{k(S)}$ does not extend to an Azumaya algebra on the canonical surface $S$, using explicit geometric models: lines for the Whitehead case, an elliptic curve in one case, and a genus-3 hyperelliptic curve in the other, together with divisor-theoretic and Cantor-algorithm arguments. This work connects representation-theoretic invariants with Brauer-group obstructions on complex surfaces, shedding light on how geometric properties of canonical components influence arithmetic extensions and Dehn-surgery phenomena in hyperbolic link complements.

Abstract

Let Γ be a finitely generated group and consider the set of all characters of representations of Γ into SL2(C). This set, denoted by X(Γ), admits an algebraic structure and is called the character variety of Γ. When Γ is the fundamental group of a hyperbolic 3-manifold M, X(Γ) turns out to be a powerful tool in the study of the geometry and topology of M. Chinburg-Reid-Stover have borrowed tools from algebraic and arithmetic geometry to understand algebraic and number-theoretic properties of the canonical curves of X(Γ). In this paper, we will partly generalize their results to certain hyperbolic link complements, and prove that the associated canonical quaternion algebra will not extend to an Azumaya algebra over the canonical surfaces.

Theorems & Definitions (25)

• Theorem 1
• Definition 1
• Proposition 1
• Lemma 2
• proof
• Lemma 3
• Theorem 4
• Theorem 5
• Definition 2
• Definition 3