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

Abstract

We study modular approximations $Q\left(\ell \right)$, $\ell =3,5$, of the $K\left(2\right)$–local sphere at the prime $2$ that arise from $\ell$–power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with $Q\left(5\right)$ and record Hill, Hopkins and Ravenel’s computation of the homotopy groups of ${TMF}_0\left(5\right)$. Using these tools and formulas of Mahowald and Rezk for $Q\left(3\right)$, we determine the image of Shimomura’s $2$–primary divided $\beta$–family in the Adams–Novikov spectral sequences for $Q\left(3\right)$ and $Q\left(5\right)$. Finally, we use low-dimensional computations of the homotopy of $Q\left(3\right)$ and $Q\left(5\right)$ to explore the rôle of these spectra as approximations to $S_{K\left(2\right)}$.