A simple proof of a lemma of Coleman

Abstract

Let p be an odd prime. The results in this paper concern the units of the infinite extension of Q_p generated by all p-power roots of unity. Let

formula here

where μ_pⁿ⁺¹ denote the pⁿ⁺¹th roots of 1. Let [pscr ]_n be the maximal ideal of the ring of integers of Φ_n and let U_n be the units congruent to 1 modulo [pscr ]_n.

Let ζ_n be a fixed primitive pⁿ⁺¹th root of unity such that ζ^p_n = ζ_{n − 1}, ∀n [ges ] 1. Put π_n = ζ_n − 1. Thus π_n is a local parameter for Φ_n. Let

formula here

Kummer already exploited the obvious fact that every u₀ ∈ U₀ can be written in the form

formula here

where f₀(T) is some power series in Z_p[[T]]. Here of course, the power series f₀(T) is not uniquely determined. Let

formula here

the inverse limit being with respect to norm maps. Coates and Wiles (see [3]) discovered that any unit u = (u_n) ∈ U_∞ has a unique power series f_u(T) in Z_p[[T]] with f_u(π_n) = u_n. The uniqueness of such a power series is obvious by Weierstrass Preparation Theorem, but the existence is in no way obvious. They worked with the formal group of height one attached to an elliptic curve with complex multiplication at an ordinary prime, but their ideas apply to any Lubin–Tate group defined over Z_p. Almost immediately, Coleman [4] gave a totally different proof of the existence of the f_u(T), which holds for all Lubin–Tate groups. We refer to such a power series as a Coleman power series. In this paper we adopt the same approach as [3]. We first prove the following result which is stronger than the original one in [3].