Norm coherence for descent of level structures on formal deformations

We give a formulation for descent of level structures on deformations of formal groups and study the compatibility between descent and a norm construction. Under this framework, we generalize Ando's construction of H-infinity complex orientations for Morava E-theories associated to the Honda formal groups over F-p. We show the existence and uniqueness of such an orientation for any Morava E-theory associated to a formal group over an algebraic extension of F-p and, in particular, orientations for a family of elliptic cohomology theories. These orientations correspond to coordinates on deformations of formal groups that are compatible with norm maps along descent. (C) 2020 Elsevier B.V. All rights reserved.