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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(l) and record Hill, Hopkins, and Ravenels 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 Shimomuras 2-primary divided beta-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 role of these spectra as approximations to the K(2)-local sphere.
In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These homotopy
We calculate the rational homotopy and the K(1)-local homotopy of the K(2)-local sphere at the prime 3 and level 2. We use this to verify the chromatic splitting conjecture in this case.
We make a conjecture about all the relations in the $E_2$ page of the May spectral sequence and prove it in a subalgebra which covers a large range of dimensions. We conjecture that the $E_2$ page is nilpotent free and also prove it in this subalgebr
We calculate the homotopy type of the Brown-Comenetz dual $I_2$ of the K(2)-local sphere at the prime 3 and show that there is a twisting by a non-trivial element $P$ in the exotic part of the Picard group. We give a complete characterization of $P$
We give a calculation of Picard groups of K(2)-local invertible spectra and of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup of invertible spectra with the same Morava module as a sphere.