A modular description of the K(2)K(2)-local sphere at the prime 3

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