Isogenies of formal group laws and power operations in the cohomology theories \(E_ n\) (Q1902031)

Let \(\Phi\) be the Honda formal group over \(\mathbb{F}_p\) of height \(n\). Let \(E_n\) be the Lubin-Tate ring of lifts of \(\Phi\), \(E_n= \mathbb{Z}_p [[u_1, \dots, u_{n-1}]]\). A formal group \(F\) which is a universal lift of \(\Phi\) is classified by a homomorphism \(t: MU^* \to E_n\). The Landweber Exact Functor Theorem gives a 2-periodic cohomology theory (also called \(E_n (-).)\) whose degree-zero part is \(X \mapsto E_n \otimes_tM U^{2*} (X)\) and whose coefficient ring is \(E_n\). Adjoin the roots of the \(p\)-series of \(F\) to form the ring extension \(D_k\) of \(E_n\). The author uses complex cobordism operations to produce power operations for \(E_n\)-theory. These take the form of unstable transformations of ring-valued functors: \(\Psi^H: E_n(X) \to D_k \otimes_{E_n} E_n(X)\) and are parameterized by the subgroups of the formal group. The author gives formulas related to the operations, studies conditions which ensure that the range of operations is restricted to \(E_n (X)\), and relates his operations to \(E_n\) power operations constructed by Hopkins and Miller. The aim of this project is to have the power operations lead to insight into the \(E_n\)-theories.