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

Abstract

Let $p$ be odd and $G^{\prime }=\mathbb{Z}/p⋊\mathbb{Z}/(p-1{)}^2$ the maximal finite subgroup of the Morava stabilizer group at height $p-1$. Inverse Galois theory produces from $G^{\prime }$ alone a curve $X$, the unique curve of minimal genus with $\mathrm{Aut}(X)\simeq G^{\prime }$; its ramification, its field of definition and its equation are consequences of the group, not choices. We prove a $G^{\prime }$-equivariant equivalence between the deformations of $X$ and Lubin--Tate space, so that the Lubin--Tate action of $G^{\prime }$ is the action of $\mathrm{Aut}(X)$ on deformations of the curve. The proof is a coordinate-free Kodaira--Spencer argument reducing to a single character count. The action becomes explicit: $G^{\prime }$ acts through ${\mathbb{F}}_p⋊{\mathbb{F}}_p^{\times }$ shifting and scaling $p+1$ points on ${\mathbb{P}}^1$. From this we compute $H^{\ast }(G^{\prime },{\pi }_{\ast }{E}_{p-1})$ and its Tate cohomology. One identity, ${\pi }^{p-1}=-p$, runs through every section.

v2: 71 pages. Coordinate-free Kodaira-Spencer proof of the main equivalence replaces the coordinate proof, ungraded and graded theorems combined; Complete proof of uniqueness of minimal genus curve with a faithful G'-action replaces previous incomplete proof; Introduction reorganized - physical motivation added and comparison of prior and current work expanded