Torsion in the Braid Monodromy of Elliptic Fibrations

Given an elliptic fibration $\pi : M \to S^2$ with singular locus $\Delta \subseteq S^2$, let $\operatorname{Br}(\pi) < \operatorname{Mod}(S^2,\Delta)$ be the subgroup of the spherical braid group consisting of those braids that lift to a fiber-preserving diffeomorphism of $M$. We classify the order $n = |\Delta|$ elements of $\operatorname{Br}(\pi)$ up to conjugacy in $\operatorname{Br}(\pi)$. To do so, we relate these conjugacy classes to special points on the $\operatorname{SL}_2$-character variety for $(S^2,\Delta)$ that correspond naturally to the exceptional elliptic curves $\mathbb{C}/\mathbb{Z}[\omega]$ and $\mathbb{C}/\mathbb{Z}[i]$ with their associated norms on $\mathbb{Z}[\omega]$ and $\mathbb{Z}[i]$. We also show that there are no elements of order $n-1$ or $n-2$ in $\operatorname{Br}(\pi)$, as there are in $\operatorname{Mod}(S^2,\Delta)$.