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

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.