The \(E(1)\)-local Picard graded homotopy groups of the sphere spectrum at the prime two (Q2203642)

Working at the prime two, and with the first Johnson-Wilson theory \(E(1)\), one can form the category of \(E(1)\)-local spectra. Sitting inside this category is the Picard group of isomorphism classes of invertible \(E(1)\)-local spectra. The \(E(1)\)-local Picard graded homotopy groups of an \(E(1)\)-local spectrum \(X\) contain the classical \(E(1)\)-local homotopy groups of \(X\). Building on [\textit{H. R. Miller}, J. Pure Appl. Algebra 20, 287--312 (1981; Zbl 0459.55012)], in [Am. J. Math. 106, 351--414 (1984; Zbl 0586.55003)], \textit{D. C. Ravenel} determined the structure of the classical homotopy groups of the \(E(1)\)-local homotopy groups of the sphere spectrum. In this paper, the author sets out to gain insight into the \(E(1)\)-local Picard graded homotopy groups of the sphere spectrum. As the main result, the author gives a description of the \(E(1)\)-local Picard graded homotopy groups of the sphere spectrum (as an algebra over the \(2\)-local integers) as a polynomial algebra modulo an ideal, which the author explicitly describes.