A modular description of the -local sphere at the prime 3

Using degree N isogenies of elliptic curves, we produce a spectrum Q ( N ) . This spectrum is built out of spectra related to tmf. At p = 3 we show that the K ( 2 ) -local sphere is built out of Q ( 2 ) and its K ( 2 ) -local Spanier–Whitehead dual. This gives a conceptual reinterpretation a resolution of Goerss, Henn, Mahowald, and Rezk.