What $Ell$ sees that $K$ doesn't (when $p >3$)

Abstract

We use Andrew Baker's analysis of the cofiber of the endomorphism of the $p$-adic elliptic spectrum ($p>3$) defined by multiplication by the `Hasse invariant' $E_{p-1}$ to present its completion away from the locus of ordinary elliptic curves as a sum of roughly $p/12$ copies (indexed by supesingular elliptic curves) of $p$-adic lifts of the height two mod $p$ cohomology theory $K(2)$. See a recent paper of Zhu Yifei for a much deeper exploration of the topics considered in this note.