Coleman power series for a CM formal group ($f_u(t_n)=u_n$)

For a height-one CM elliptic curve formal group and compatible ${\mathfrak{p}}^n$-division parameters $t_n$, an integral Coleman series interpolates a norm-compatible sequence of local units $u_n$. Its norm relation is multiplication of its translates by the $\mathfrak{p}$-torsion, with coefficient Frobenius where required by the unramified base. Coates–Wiles homomorphisms and the Frobenius-corrected logarithm turn this local series into critical values and measures.

Assume $K_{\mathfrak{p}}={\mathbb{Q}}_p$, $p>2$, $f(t)\in {\mathbb{Z}}_p[[t]{]}^{\times }$, and $[\pi ]t\equiv t^p(\mathrm{mod}p)$. Then the displayed ratio lies in $1+p{\mathbb{Z}}_p[[t]]$, so its logarithm divided by $p$ is integral. A formal-group isomorphism over the completed unramified coefficient ring transfers it to an integral Amice transform. A norm relation makes the corresponding p-adic measure supported on units. Its $k$th moment has the Euler correction $1-{\pi }^k/p$.

Let $R={\mathbb{Z}}_p[[T]]$, $p$ odd, and $\varphi (T)=(1+T{)}^p-1$. An element $g\in R$ is the Frobenius-corrected logarithm of a unit $f$ with $f(0)\in 1+p{\mathbb{Z}}_p$ if and only if $g^{\prime }(0)=0$. Necessity follows by differentiating at zero, using ${\varphi }^{\prime }(0)=p$. For sufficiency put $H=\exp (pg)\in 1+pR$ and solve $f^p=H\varphi (f)$. Choose $f_0=\exp (p{g}_0/(p-1))$. The linear equation reduces to $H_1=0$, so $f_1$ is free. In degree $n\ge 2$ the unknown $f_n$ has coefficient $p{f}_0^{p-1}(1-p^{n-1})$. The remaining terms are divisible by $p$ because reduction of any integral truncated $f$ satisfies $f^p\equiv \varphi (f)(\mathrm{mod}p)$. Division gives a unique integral $f_n$. Two lifts differ by $(1+T{)}^c$, $c\in {\mathbb{Z}}_p$, as follows from the logarithmic equation in the coordinate $z=\log (1+T)$. If also $Sg=0$, the Coleman trace operator gives $\log (Nf/f)=pSg=0$. As $Nf/f\in 1+pR$, injectivity of the p-adic logarithm implies $Nf=f$.

The $k$th logarithmic derivative in the additive formal logarithm coordinate defines the Coates–Wiles homomorphism. Products of local units become sums under this map. For elliptic units, differentiating the underlying complex elliptic function expresses the result in terms of critical Hecke L-function values divided by a CM period.