Arithmetic of the canonical component of the Knot $7_4$

TL;DR

This paper investigates the arithmetic properties of Dehn surgery points on the canonical component of the $ ext{SL}_2( extbf{C})$-character variety of the knot $7_4$, revealing infinite ramified places and infinite order in the Mordell-Weil group.

Contribution

It establishes two new arithmetic properties of Dehn surgery points on the canonical component, supporting conjectures and linking to elliptic curve theory.

Findings

Residue characteristics of ramified places form an infinite set.

Dehn surgery points have infinite order in the Mordell-Weil group.

Provides evidence for a conjecture of Chinburg, Reid, and Stover.

Abstract

We prove two arithmetic properties of Dehn surgery points on the canonical component of the $\mathrm{SL}_2(\mathbf{C})$-character variety of the knot $7_4$. The first is that the residue characteristics of the ramified places of the Dehn surgery points form an infinite set, providing evidence for a conjecture of Chinburg, Reid, and Stover. The second is that the Dehn surgery points have infinite order in the Mordell-Weil group of the elliptic curve obtained by a simple birational transformation of the canonical component into Weierstrass form.