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

TL;DR

This paper analyzes the structure of the $p$-adic elliptic spectrum's cofiber, revealing its decomposition into multiple copies of $K(2)$-theory lifts, providing insights into elliptic cohomology at primes greater than 3.

Contribution

It presents a detailed decomposition of the $p$-adic elliptic spectrum's cofiber, connecting it to $K(2)$-theory and elliptic curve loci, extending previous analyses.

Findings

Decomposition of the spectrum's cofiber into multiple $K(2)$)-theory lifts.

Identification of the spectrum's structure away from ordinary elliptic curves.

Connection between elliptic spectrum and height two cohomology theories.

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.