Torsion in the Braid Monodromy of Elliptic Fibrations

Abstract

Given an elliptic fibration $\pi :M\to S^2$ with singular locus $\Delta \subseteq S^2$, let $\mathrm{Br}(\pi )<\mathrm{Mod}(S^2,\Delta )$ 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 $\mathrm{Br}(\pi )$ up to conjugacy in $\mathrm{Br}(\pi )$. To do so, we relate these conjugacy classes to special points on the ${\mathrm{SL}}_2$-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 $\mathrm{Br}(\pi )$, as there are in $\mathrm{Mod}(S^2,\Delta )$.

15 pages, 3 figures