Picard sheaves, local Brauer groups, and topological modular forms

We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real $K$-theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of $\mathrm{TMF}$ 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 $\mathrm{TMF}$ 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.