Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2

TL;DR

This paper constructs a specific E_infinity-ring spectrum related to topological modular forms and Brown-Peterson spectra at prime 2, revealing new connections in chromatic homotopy theory.

Contribution

It demonstrates a natural E_infinity-ring map from topological modular forms to a truncated Brown-Peterson spectrum, extending the understanding of their algebraic and topological relationships.

Findings

Established an E_infinity-ring map between spectra

Connected topological modular forms with Brown-Peterson spectra

Organized Morava's K-theory forms into a sheaf of spectra

Abstract

Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E_infinity-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E_infinity-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation fits into a diagram of E_infinity-ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix we document how to organize Morava's forms of K-theory into a sheaf of E_infinity-ring spectra.