Torsion in the Braid Monodromy of Elliptic Fibrations

Abstract

Given an elliptic fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, let $\operatorname{Br}(π) < \operatorname{Mod}(S^2,Δ)$ 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 $n = |Δ|$ elements of $\operatorname{Br}(π)$ up to conjugacy in $\operatorname{Br}(π)$. To do so, we relate these conjugacy classes to special points on the $\operatorname{SL}_2$-character variety for $(S^2,Δ)$ that correspond naturally to the exceptional elliptic curves $\mathbb{C}/\mathbb{Z}[ω]$ and $\mathbb{C}/\mathbb{Z}[i]$ with their associated norms on $\mathbb{Z}[ω]$ and $\mathbb{Z}[i]$. We also show that there are no elements of order $n-1$ or $n-2$ in $\operatorname{Br}(π)$, as there are in $\operatorname{Mod}(S^2,Δ)$.