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

TL;DR

This paper computes the homotopy groups related to the $K(2)$-local Moore spectrum at prime 2 by analyzing the cohomology of automorphism groups of supersingular elliptic curves using algebraic duality spectral sequences.

Contribution

It introduces new computations of the cohomology of the automorphism group of a supersingular elliptic curve's formal group law at chromatic level 2, utilizing algebraic duality spectral sequences.

Findings

Computed an associated graded for the cohomology of S_C^1 with coefficients in (E_C)_*V(0)

Relied on elliptic curve geometry at chromatic level 2

Provided insights into the homotopy fixed points related to L_{K(2)}V(0)

Abstract

Let V(0) be the mod 2 Moore spectrum and let C be the supersingular elliptic curve over F_4 defined by the Weierstrass equation y^2+y=x^3. Let F_C be its formal group law and E_C be the spectrum classifying the deformations of F_C. The group of automorphisms of F_C, which we denote by S_C, acts on E_C. Further, S_C admits a surjective homomorphism to the 2-adic integers whose kernel we denote by S_C^1. The cohomology of S_C^1 with coefficients in (E_C)_*V(0) is the E_2-term of a spectral sequence converging to the homotopy groups of the homotopy fix points of E_C smash V(0) with respect to S_C^1, a spectrum closely related to L_{K(2)}V(0). In this paper, we use the algebraic duality resolution spectral sequence to compute an associated graded for H^*(S_C^1;(E_C)_*V(0)). These computations rely heavily on the geometry of elliptic curves made available to us at chromatic level 2.