#### Volume 16, issue 5 (2016)

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

### Mark Behrens and Kyle M Ormsby

Algebraic & Geometric Topology 16 (2016) 2459–2534
##### 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 ${S}_{K\left(2\right)}$.

##### Keywords
topological modular forms, $v_n$–periodic homotopy, elliptic curves
##### Mathematical Subject Classification 2010
Primary: 55Q45, 55Q51