The Picard group of topological modular forms via descent theory

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of E_∞–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(TMF) is cyclic of order 576, generated by the suspension †TMF (a result originally due to Hopkins), while Pic(Tmf)= Z⊕Z/24. In particular, we show that there exists an invertible Tmf–module which is not equivalent to a suspension of Tmf.