Iterated extensions and relative Lubin-Tate groups

Abstract

Let K be a finite extension of \({\mathbf {Q}}_{p}\) with residue field \({\mathbf {F}}_q\) and let \(P(T) = T^d + a_{d-1}T^{d-1} + \cdots +a_1 T\) where d is a power of q and \(a_i \in {\mathfrak {m}}_K\) for all i. Let \(u_0\) be a uniformizer of \({\mathcal {O}}_K\) and let \(\{u_n\}_{n \geqslant 0}\) be a sequence of elements of \({\overline{\mathbf {Q}}}_{p}\) such that \(P(u_{n+1}) = u_n\) for all \(n \geqslant 0\). Let \(K_\infty \) be the field generated over K by all the \(u_n\). If \(K_\infty /K\) is a Galois extension, then it is abelian, and our main result is that it is generated by the torsion points of a relative Lubin-Tate group (a generalization of the usual Lubin-Tate groups). The proof of this involves generalizing the construction of Coleman power series, constructing some p-adic periods in Fontaine’s rings, and using local class field theory.