Power Operations on $K(n-1)$-Localized Morava $E$-theory at Height $n$

TL;DR

This work analyzes power operations on the $K(n-1)$-local Morava $E_n$-theory $F=L_{K(n-1)}E_n$, establishing a modular interpretation of the total power operation $\psi^p_F$ via augmented deformations of the associated formal group and its degree $p^m$ subgroups. It proves freeness and rank results for $F^*B\Sigma_k$ and its quotients, derives a dual description of the Dyer–Lashof algebra for $K(n-1)$-local $E$-algebras, and shows that these structures are governed by augmented deformation theory. In the height $n=2$ case, it yields an explicit formula for $\psi^p_F$ by relating $\psi^p_E$ through a height-1 reduction, with connections to elliptic curves, modular forms, and $p$-divisible groups via a norm parameter for $\Gamma_0(p)$ and the Atkin–Lehner involution. The work also develops augmented deformation spectra and demonstrates that their underlying spectra are independent of the chosen formal group, providing a robust framework for transchromatic power operations with potential implications for TMF and modular-forms-based phenomena.

Abstract

We calculate the $K(n-1)$-localized $E_n$ theory for symmetric groups, and deduce a modular interpretation of the total power operation $ψ^p_F$ on $F=L_{K(n-1)}E_n$ in terms of augmented deformations of formal groups and their subgroups. We compute the Dyer-Lashof algebra structure over $K(n-1)$-local $E_n$-algebra. Then we specify our calculation to the $n=2$ case. We calculate an explicit formula for $ψ^p_F$ using the formula of $ψ^p_E$, and explain connections between these computations and elliptic curves, modular forms and $p$-divisible groups.

Figures (1)

• Figure 1: Spectra on $p$ completed stack $\overline{\mathcal{M}}_{ell}$

Theorems & Definitions (45)

• Theorem A: Theorem \ref{['t2.12']}
• Theorem B: Proposition \ref{['p2.13']}
• Theorem C: Theorem \ref{['thm3.5']}
• Remark 1.1
• Remark 1.2
• Theorem 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof