Iterated extensions and relative Lubin-Tate groups
Tóm tắt
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.
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