Moduli Stacks of $G$-Curves in Homotopy Theory at $h=p-1$

Abstract

We study the action on the deformation space of a formal group by the maximal finite subgroup $G$ of its automorphisms, at the first height where the group has nontrivial $p$-torsion for odd $p$. We show given this group $G$ there is a universal construction of a geometric model of the $G$-action via inverse Galois theory which generalizes the use of level structure to ramification data. We use configuration spaces to understand the model, and conclude that the Lubin-Tate action at $h=p-1$ is a subgroup of the symmetric group action on the configuration space of $p+1$ points on $\mathbb{P}^1$.