Wilson spaces, snaith constructions, and elliptic orientations

We construct a canonical family of even periodic \(\mathbb{E}_{\infty}\)-ring spectra, with exactly one member of the family for every prime \(p\) and chromatic height \(n\). At height 1 our construction is due to Snaith, who built complex \(K\)-theory from \(\mathbb{CP}^{\infty}\). At height 2 we replace \(\mathbb{CP}^{\infty}\) with a \(p\)-local retract of \(\mathrm{BU} \langle 6 \rangle \), producing a new theory that orients elliptic, but not generic, height 2 Morava \(E\)-theories.

In general our construction exhibits a kind of redshift, whereby \(\mathrm{BP}\langle n-1 \rangle \) is used to produce a height \(n\) theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the \(K(n)\)-localization of our height \(n\) ring to work of Peterson and Westerland building \(E_{n}^{hS\mathbb{G}^{\pm}}\) from \(\mathrm{K}(\mathbb{Z},n+1)\).