Picard groups and duality for Real Morava $E$-theories

TL;DR

This paper proves that at prime 2, the Picard group of invertible modules over the homotopy fixed points of Lubin--Tate spectra is cyclic and establishes their self-duality, extending known results for height 1.

Contribution

It demonstrates the cyclicity of the Picard group and determines the self-duality shift for $E_n^{hC_2}$ at prime 2, generalizing previous height 1 findings.

Findings

Picard group of $E_n^{hC_2}$ is cyclic

$E_n^{hC_2}$ is Gross--Hopkins self-dual

Results extend known height 1 cases

Abstract

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$.