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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

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 ΣTMF (a result originally due to Hopkins), while Pic(Tmf) = 24. In particular, we show that there exists an invertible Tmf–module which is not equivalent to a suspension of Tmf.

Picard groups and spectra, descent, topological modular forms
Mathematical Subject Classification 2010
Primary: 14C22, 55N34, 55P43, 55S35, 55T99
Secondary: 55P47
Received: 22 October 2014
Revised: 8 October 2015
Accepted: 19 November 2015
Published: 21 December 2016
Proposed: Mark Behrens
Seconded: Haynes Miller, Jesper Grodal
Akhil Mathew
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138
United States
Vesna Stojanoska
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W Green Steet
Urbana, IL 61801
United States