On Hopkins' Picard groups for the prime 3 and chromatic level 2

TL;DR

This paper calculates the Picard groups of K(2)-local and E(2)-local invertible spectra at prime 3, focusing on those with Morava modules equivalent to spheres, advancing understanding of chromatic homotopy theory.

Contribution

It provides the first detailed calculation of the subgroup of invertible spectra sharing the Morava module with a sphere at prime 3 and chromatic level 2.

Findings

Determined the structure of Picard groups at prime 3 and chromatic level 2.

Identified the subgroup of invertible spectra with sphere-like Morava modules.

Enhanced understanding of invertible spectra in chromatic homotopy theory.

Abstract

We give a calculation of Picard groups of K(2)-local invertible spectra and of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup of invertible spectra with the same Morava module as a sphere.