On Hopkins' Picard group $\mathrm{Pic}_2$ at the prime $3$

Abstract

In this paper we calculate the algebraic Hopkins Picard group ${Pic}_2^{alg}$ at the prime $p=3$, which is a subgroup of the group of isomorphism classes of invertible $K\left(2\right)$–local spectra, ie of Hopkins’ Picard group ${Pic}_2$. We use the resolution of the $K\left(2\right)$–local sphere introduced by Goerss, Henn, Mahowald and Rezk in [Ann. of Math (2) 162 (2005) 777-822] and the methods from Henn, Karamanov and Mahowald [to appear in Math. Zeit. arXiv:0811.0235] and Karamanov [PhD thesis, Universite Louis Pasteur (2006)].