The 5-local homotopy of $eo_4$

TL;DR

This paper computes the 5-local cohomology and homotopy groups of a spectrum related to elliptic cohomology, providing new insights into the structure of $eo_4$ and its connection to higher real K-theory.

Contribution

It introduces the 5-local analogue of the Weierstrass Hopf algebroid and computes its Adams-Novikov differentials, advancing understanding of $eo_4$ homotopy.

Findings

Computed the 5-local cohomology of the $eo_4$-related algebroid

Determined the Adams-Novikov differentials for $eo_4$

Linked the homotopy of $eo_4$ to higher real K-theory spectrum $EO_4$

Abstract

We compute the 5-local cohomology of a 5-local analogue of the Weierstrass Hopf algebroid used to compute $tmf$ homology. We compute the Adams-Novikov differentials in the cohomology, giving the homotopy, V(0)-homology, and V(1)-homology of the putative spectrum $eo_4$. We also link this computation to the homotopy of the higher real $K$-theory spectrum $EO_4$.