Algebraic & Geometric Topology

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

Mark Behrens and Kyle Ormsby

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We study modular approximations Q(), = 3,5, of the K(2)–local sphere at the prime 2 that arise from –power degree isogenies of elliptic curves. We develop Hopf algebroid level tools for working with Q(5) and record Hill, Hopkins and Ravenel’s computation of the homotopy groups of TMF0(5). Using these tools and formulas of Mahowald and Rezk for Q(3), we determine the image of Shimomura’s 2–primary divided β–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 rôle of these spectra as approximations to SK(2).

Article information

Algebr. Geom. Topol., Volume 16, Number 5 (2016), 2459-2534.

Received: 31 October 2012
Revised: 22 January 2016
Accepted: 31 January 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q45: Stable homotopy of spheres 55Q51: $v_n$-periodicity

topological modular forms $v_n$–periodic homotopy elliptic curves


Behrens, Mark; Ormsby, Kyle. On the homotopy of $Q(3)$ and $Q(5)$ at the prime $2$. Algebr. Geom. Topol. 16 (2016), no. 5, 2459--2534. doi:10.2140/agt.2016.16.2459.

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