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 Qp generated by all p-power roots of unity. Letformula herewhere μpn+1 denote the pn+1th roots of 1. Let [pscr ]n be the maximal ideal of the ring of integers of Φn and let Un be the units congruent to 1 modulo [pscr ]n.Let ζn be a fixed primitive pn+1th root of unity such that ζpn = ζn − 1, ∀n [ges ] 1. Put πn = ζn − 1. Thus πn is a local parameter for Φn. Letformula hereKummer already exploited the obvious fact that every u0 ∈ U0 can be written in the formformula herewhere f0(T) is some power series in Zp[[T]]. Here of course, the power series f0(T) is not uniquely determined. Letformula herethe inverse limit being with respect to norm maps. Coates and Wiles (see [3]) discovered that any unit u = (un) ∈ U∞ has a unique power series fu(T) in Zp[[T]] with fu(πn) = un. 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 Zp. Almost immediately, Coleman [4] gave a totally different proof of the existence of the fu(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].