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This paper starts with an exposition of descent-theoretic techniques in thestudy of Picard groups of $\mathbf{E}_{\infty}$-ring spectra, which naturallylead to the study of Picard spectra. We then develop tools for the efficientand explicit determination of differentials in the associated descent spectralsequences for the Picard spectra thus obtained. As a major application, wecalculate the Picard groups of the periodic spectrum of topological modularforms $TMF$ 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 aninvertible $Tmf$-module which is not equivalent to a suspension of $Tmf$.

The Picard group of topological modular forms via descent theory