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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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

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).

topological modular forms, $v_n$–periodic homotopy, elliptic curves
Mathematical Subject Classification 2010
Primary: 55Q45, 55Q51
Received: 31 October 2012
Revised: 22 January 2016
Accepted: 31 January 2016
Published: 7 November 2016
Mark Behrens
Department of Mathematics
University of Notre Dame
287 Hurley Hall
Notre Dame, IN 46556
United States
Kyle M Ormsby
Department of Mathematics
Reed College
3203 SE Woodstock Blvd
Portland, OR 97202
United States