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

Behrens, Mark¹ ; Ormsby, Kyle²

¹Department of Mathematics, University of Notre Dame, 287 Hurley Hall, Notre Dame, IN 46556, United States
²Department of Mathematics, Reed College, 3203 SE Woodstock Blvd, Portland, OR 97202, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 5, pp. 2459-2534

Résumé

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).

Détail

DOI : 10.2140/agt.2016.16.2459

Classification : 55Q45, 55Q51
Keywords: topological modular forms, $v_n$–periodic homotopy, elliptic curves

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