On the homotopy of Q(3) and Q(5) at the prime 2

TL;DR

This paper investigates the spectra Q(3) and Q(5) at the prime 2, developing new tools to analyze their homotopy groups and their relation to the K(2)-local sphere, with implications for understanding modular forms and elliptic curves.

Contribution

It introduces Hopf algebroid tools for Q(l) spectra and computes their homotopy groups, connecting modular approximations to the K(2)-local sphere at prime 2.

Findings

Computed homotopy groups of Q(3) and Q(5).

Determined the image of the divided beta-family in spectral sequences.

Explored the role of Q(3) and Q(5) as approximations to the K(2)-local sphere.

Abstract

We study modular approximations Q(l), l = 3,5, of the K(2)-local sphere at the prime 2 that arise from l-power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(l) and record Hill, Hopkins, and Ravenel's computation of the homotopy groups of TMF_0(5). Using these tools and formulas of Mahowald and Rezk for Q(3) we determine the image of Shimomura's 2-primary divided beta-family in the Adams-Novikov spectral sequences for Q(3) and Q(5). Finally, we use low-dimensional computations of the homotopy of Q(3) and Q(5) to explore the role of these spectra as approximations to the K(2)-local sphere.