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arXiv · 1211.0076

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

Mark Behrens·Kyle Ormsby
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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.

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Mark Behrens, Kyle Ormsby. 2012-11-01. On the homotopy of Q(3) and Q(5) at the prime 2. https://doi.org/10.2140/agt.2016.16.2459

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