Crystals and derived local moduli for ordinary K3 surfaces

It is shown that the K3 spectra which refine the local rings of the moduli stack of ordinary p-primitively polarized K3 surfaces in characteristic p allow for an E∞ structure which is unique up to equivalence. This uses the E∞ obstruction theory of Goerss and Hopkins and the description of the deformation theory of such K3 surfaces in terms of their Hodge F-crystals due to Deligne and Illusie. Furthermore, all automorphisms of such K3 surfaces can be realized by E∞ maps which are unique up to homotopy, and this can be rigidified to an action if the automorphism group is tame.