The Picard group of topological modular forms via descent theory

TL;DR

This paper uses descent theory to compute the Picard groups of topological modular forms spectra, revealing their structure and identifying invertible modules beyond suspensions.

Contribution

It develops descent spectral sequence techniques for Picard spectra and explicitly calculates Picard groups of TMF and Tmf spectra, including new invertible modules.

Findings

Picard group of TMF is cyclic of order 576

Picard group of Tmf is Z plus Z/24

Existence of invertible Tmf-modules not equivalent to suspensions

Abstract

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\mathbf{E}_{\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$. We find that $\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspension $\Sigma TMF $ (a result originally due to Hopkins), while $\mathrm{Pic}(Tmf) = \mathbb{Z}\oplus \mathbb{Z}/24$. In particular, we show that there exists an invertible $Tmf$-module which is not equivalent to a suspension of $Tmf$.