We study modular approximations , , of the –local sphere at the prime that arise from –power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with and record Hill, Hopkins and Ravenel’s computation of the homotopy groups of . Using these tools and formulas of Mahowald and Rezk for , we determine the image of Shimomura’s –primary divided –family in the Adams–Novikov spectral sequences for and . Finally, we use low-dimensional computations of the homotopy of and to explore the rôle of these spectra as approximations to .
"On the homotopy of $Q(3)$ and $Q(5)$ at the prime $2$." Algebr. Geom. Topol. 16 (5) 2459 - 2534, 2016. https://doi.org/10.2140/agt.2016.16.2459