We present a construction that manufactures _ orientations of Tate fixed-point objects together with useful formulas for these maps, and then give a number of applications. For example, we produce a formula for the Frobenius homomorphisms of Thom spectra such as as well as certain lifts of Frobenius. We prove a rigidity property of as a cyclotomic object. We construct a general obstruction theory for ₙ complex orientations and establish various non-existence results for p-typical ₙ orientations for low values of p and n. We end with some miscellaneous further applications.
Carmeli et al. (Wed,) studied this question.