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

Abstract

We calculate the $K(n-1)$-localized $E_n$ theory for symmetric groups, and deduce a modular interpretation of the total power operation ${\psi }_F^p$ 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 ${\psi }_F^p$ using the formula of ${\psi }_E^p$, and explain connections between these computations and elliptic curves, modular forms and $p$-divisible groups.