Towards the homotopy of the K(2)-local Moore spectrum at p = 2

Let V(0) be the mod 2 Moore spectrum and let C be the supersingular elliptic curve over F4 defined by the Weierstrass equation y2+y=x3. Let FC be its formal group law and EC be the spectrum classifying the deformations of FC. The group of automorphisms of FC, which we denote by SC, acts on EC. Further, SC admits a surjective homomorphism to Z2 whose kernel we denote by SC1. The cohomology of SC1 with coefficients in (EC)⁎V(0) is the E2-term of a spectral sequence converging to the homotopy groups of EChSC1∧V(0), a spectrum closely related to LK(2)V(0). In this paper, we use the algebraic duality resolution spectral sequence to compute an associated graded for H⁎(SC1;(EC)⁎V(0)). These computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2.