#### Volume 20, issue 6 (2016)

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The Picard group of topological modular forms via descent theory

### Akhil Mathew and Vesna Stojanoska

Geometry & Topology 20 (2016) 3133–3217
##### Abstract

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of ${E}_{\infty }$–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 nonperiodic and nonconnective $Tmf$. We find that $Pic\left(TMF\right)$ is cyclic of order $576$, generated by the suspension $\Sigma \phantom{\rule{0.3em}{0ex}}TMF$ (a result originally due to Hopkins), while $Pic\left(Tmf\right)=ℤ\oplus ℤ∕24$. In particular, we show that there exists an invertible $Tmf$–module which is not equivalent to a suspension of $Tmf$.

##### Keywords
Picard groups and spectra, descent, topological modular forms
##### Mathematical Subject Classification 2010
Primary: 14C22, 55N34, 55P43, 55S35, 55T99
Secondary: 55P47