Picard sheaves, local Brauer groups, and topological modular forms

TL;DR

This paper establishes an isomorphism between the Brauer group of topological modular forms (TMF) and that of the derived moduli stack of elliptic curves, and computes the local Brauer group, revealing new connections in algebraic topology.

Contribution

It proves the isomorphism of Brauer groups for TMF and the moduli stack, and explicitly computes the local Brauer group, advancing understanding of algebraic structures in topology.

Findings

Brauer group of TMF is isomorphic to that of the derived moduli stack of elliptic curves

Computed the local Brauer group, identifying its structure up to 2-torsion

Established new links between algebraic topology and algebraic geometry

Abstract

We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some \'etale cover of the moduli stack, up to a finite 2-torsion group.