Iterated extensions and relative Lubin-Tate groups

Let K be a finite extension of Qp with residue field Fq and let P(T) = Td+ ad-1Td-1+ ⋯ + a1T where d is a power of q and ai∈ mK for all i. Let u0 be a uniformizer of OK and let {un}n⩾0 be a sequence of elements of Q¯ p such that P(un+1) = un for all n⩾ 0. Let K∞ be the field generated over K by all the un. If K∞/ 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. © 2016, Fondation Carl-Herz and Springer International Publishing Switzerland.