The Picard group of topological modular forms via descent theory

Akhil Mathew, Vesna Stojanoska

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)3133-3217
Number of pages85
JournalGeometry and Topology
Issue number6
StatePublished - Dec 21 2016

ASJC Scopus subject areas

  • Geometry and Topology


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