Picard sheaves, local Brauer groups, and topological modular forms

Benjamin Antieau, Lennart Meier*, Vesna Stojanoska

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real (Formula presented.) -theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of (Formula presented.) is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of (Formula presented.) and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2-torsion group.

Original languageEnglish (US)
Article numbere12333
JournalJournal of Topology
Volume17
Issue number2
DOIs
StatePublished - Jun 2024

ASJC Scopus subject areas

  • Geometry and Topology

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