## Algebraic & Geometric Topology

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

#### 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 $Q(5)$ 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

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

Dates
Revised: 22 January 2016
Accepted: 31 January 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841218

Digital Object Identifier
doi:10.2140/agt.2016.16.2459

Mathematical Reviews number (MathSciNet)
MR3572338

Zentralblatt MATH identifier
1366.55009

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

#### Citation

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. https://projecteuclid.org/euclid.agt/1510841218

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