Synthetic Spectra and the Picard Group of Derived Moduli Stacks of Formal Group Laws

This paper introduces a novel framework based on the notion of "synthetic spectra" to investigate the global geometric properties of derived moduli stacks in chromatic homotopy theory. Our primary object of study is the derived moduli stack of one-dimensional formal group laws, MderFG. By equipping this stack with a synthetic structure sheaf, which simultaneously encodes algebraic and homotopy-theoretic information, we construct a new descent spectral sequence. This spectral sequence converges to the group of homotopy classes of maps into a synthetic classifying space for line bundles and serves as a powerful computational tool. We analyze the structure of its E2-page and the behavior of its differentials to provide a systematic computation of the Picard group of MderFG, which classifies isomorphism classes of derived line bundles. Our main result gives a formula for Pic(MderFG) that decomposes it into classical algebro-geometric contributions and purely homotopy-theoretic obstructions involving the cohomology of the Morava stabilizer groups. We apply this framework to recover the classical computation for the moduli stack of elliptic curves (chromatic height 1) and provide new, explicit calculations at chromatic height 2, demonstrating the efficacy of our synthetic approach.