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2016 On the homotopy of $Q(3)$ and $Q(5)$ at the prime $2$
Mark Behrens, Kyle Ormsby
Algebr. Geom. Topol. 16(5): 2459-2534 (2016). DOI: 10.2140/agt.2016.16.2459


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).


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Mark Behrens. Kyle Ormsby. "On the homotopy of $Q(3)$ and $Q(5)$ at the prime $2$." Algebr. Geom. Topol. 16 (5) 2459 - 2534, 2016.


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

zbMATH: 1366.55009
MathSciNet: MR3572338
Digital Object Identifier: 10.2140/agt.2016.16.2459

Primary: 55Q45 , 55Q51

Keywords: $v_n$–periodic homotopy , Elliptic curves , topological modular forms

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 5 • 2016
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