On the Picard group graded homotopy groups of a finite type two $K(2)$-local spectrum at the prime three

. Consider Hopkins’ Picard group of the stable homotopy category of E (2)-local spectra at the prime three, consisting of homotopy classes of invertible spectra [1]. Then, it is isomorphic to the direct sum of an in(cid:12)nite cyclic group and two cyclic groups of order three. We consider the Smith-Toda spectrum V (1) and the co(cid:12)ber V 2 of the square (cid:11) 2 of the Adams map, which is a ring spectrum. In this paper, we introduce imaginary elements which make computation clearer, and determine the module structures of the Picard group graded homotopy groups (cid:25) ⋆ ( V (1)) and (cid:25) ⋆ ( V 2 ).