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

Behrens, Mark; Ormsby, Kyle

doi:10.2140/agt.2016.16.2459

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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

DOI: 10.2140/agt.2016.16.2459

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 ${TMF}_{0} (5)$ . Using these tools and formulas of Mahowald and Rezk for $Q (3)$ , we determine the image of Shimomura’s $2$ –primary divided $β$ –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 $S_{K (2)}$ .

Keywords

topological modular forms, $v_n$–periodic homotopy, elliptic curves

Mathematical Subject Classification 2010

Primary: 55Q45, 55Q51

References

Publication

Received: 31 October 2012

Revised: 22 January 2016

Accepted: 31 January 2016

Published: 7 November 2016

Authors

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

Volume 16, issue 5 (2016)