Strictly Commutative Realizations of Diagrams Over the Steenrod Algebra and Topological Modular Forms at the Prime 2

Previous work constructed a generalized truncated Brown–Peterson spectrum of chromatic height 2 at the prime 2 as an
$\varepsilon_{\infty}$
-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
$\varepsilon_{\infty}$
-ring map from the spectrum of topological modular forms to this truncated Brown–Peterson spectrum, and that this orientation fits into a diagram of
$\varepsilon_{\infty}$
-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
$\varepsilon_{\infty}$
-ring spectra.