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For odd $p$ and a norm-compatible sequence of local units in ${\mathbb{Q}}_p({\mu }_{p^{n+1}})$, there is a unique $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ interpolating all its entries, and the Coleman norm operator fixes it. To construct it, choose integral unit polynomials $F_j$ interpolating entry $u_j$, and take a convergent subsequence of $N^j{F}_{2j}$ in the compact $(p,T)$-adic unit space. The congruence $NF\equiv F(\mathrm{mod}p)$ and Coleman norm contraction show that its evaluation at any fixed level tends to the desired entry. Uniqueness follows from the Weierstrass preparation theorem: a nonzero integral power series has only finitely many zeros in the open unit disk. The norm relation on evaluations then proves norm fixation. Principal units correspond to $f_u(0)\equiv 1(\mathrm{mod}p)$.

Every unit $\overline{f}\in {\mathbb{F}}_p[[T]{]}^{\times }$ has a unique Coleman norm operator-fixed lift to ${\mathbb{Z}}_p[[T]{]}^{\times }$. Choose any lift $f$ and use $Nf\equiv f(\mathrm{mod}p)$. The consecutive quotients of the iterates are $N^n(Nf/f)\in 1+p^{n+1}R$ by Coleman norm contraction, so they converge coefficientwise uniformly in the p-adic integers. Their limit is norm-fixed by continuity. Two such lifts have norm-fixed quotient in $1+pR$ and therefore coincide. Uniqueness also makes the lifting map multiplicative and independent of the initial lift.

Let $u=(u_n)\in U_{\infty }$. The Coleman interpolation theorem for cyclotomic local units gives a unique unit series $f_u\in {\mathbb{Z}}_p[[T]{]}^{\times }$ withSince the entries are principal units, $f_u(0)\equiv 1(\mathrm{mod}p)$. The existence statement can be seen from the norm operator as follows. Choose integral unit polynomials $F_j$ interpolating $u_j$, possible because ${\mathcal{O}}_{K_j}={\mathbb{Z}}_p[{\zeta }_j-1]$. The congruence $NF\equiv F(\mathrm{mod}p)$ and Coleman norm contraction imply that $N^j{F}_{2j}$ differs from $N^{2j-m}{F}_{2j}$ by a factor in $1+p^{j+1}R$ for each fixed $m\le j$. The second series evaluates at ${\zeta }_m-1$ to $u_m$ by repeated finite-level norms. Compactness in the $(p,T)$-adic topology gives a subsequential limit interpolating all the entries. A nonzero difference between two interpolating integral series would have infinitely many zeros in the open unit disk, contradicting the Weierstrass preparation theorem. Applying the norm identity at all levels then proves norm fixation.

Set $D=(1+T)d/dT$. For $k\ge 1$, define the cyclotomic higher logarithmic derivative byThe quotient $D{f}_u/f_u$ belongs to ${\mathbb{Z}}_p[[T]]$, and $D$ preserves that ring, so this indeed takes values in ${\mathbb{Z}}_p$. Equivalently it is $D^k\log f_u$ at zero, where the formal logarithm may be normalized by replacing $f_u$ by $f_u/f_u(0)$. There is no factorial division in this definition. Uniqueness of the Coleman power series gives $f_{uv}=f_u{f}_v$, so ${\delta }_k(uv)={\delta }_k(u)+{\delta }_k(v)$.

Let ${\chi }_{\mathrm{cyc}}:\mathrm{Gal}(K_{\infty }/{\mathbb{Q}}_p)\to {\mathbb{Z}}_p^{\times }$ be the cyclotomic character, with $\sigma ({\zeta }_n)={\zeta }_n^{{\chi }_{\mathrm{cyc}}(\sigma )}$. If $b={\chi }_{\mathrm{cyc}}(\sigma )$, interpolation and uniqueness giveThe binomial series is integral for $b\in {\mathbb{Z}}_p$, and for any formal series $g$ the chain rule givesIteration, followed by evaluation at zero, therefore provesThus these additive Coates–Wiles homomorphisms have cyclotomic weight $k$.

Let $M_f$ be the matrix of multiplication by $f$ in the basis from Question 1. Define the Coleman norm operator by the norm for a finite free ring extension:The entries lie in $\varphi (R)$, and $\varphi :R\to \varphi (R)$ is an isomorphism, so this determines $Nf$ uniquely. If $f$ is a unit, $M_f$ is invertible and its determinant is a unit. Also $M_{fg}=M_f{M}_g$, giving $N(fg)=NfNg$. Splitting the conjugates, or diagonalizing multiplication after adjoining the roots of unity and inverting $p$, gives the equivalent formulaIn particular, this definition has the required target $R^{\times }$, rather than merely a ring with extra roots of unity in its coefficients.

For the requested ring congruence, observe that the reduction of $\varphi$ is the injective map $\overline{g}(T)\mapsto \overline{g}(T^p)$. If $\varphi (g)\in p^{k}R$, its reduction first gives $g=p{g}_1$. Then $\varphi (g_1)\in p^{k-1}R$; repeat to obtain $g\in p^{k}R$. Applying this to $g=f-1$ provesThis congruence reflection for Frobenius substitution is valid for all $f\in R$, without assuming $f$ is a unit.

A further property needed for the fixed points is Coleman norm contraction. If $g=1+p^{k}h$ with $k\ge 1$, expand the determinant:Here $e_j(M_h)\in \varphi (R)$ is the coefficient of degree $j$ in $\det (I+t{M}_h)$. The trace formula from Question 1 makes the linear term divisible by $p^{k+1}$; every later term is divisible by $p^{2k}$ and $2k\ge k+1$. Reflecting the resulting ring congruence through $\varphi$ givesIf $Nf=f$ and $f\in 1+pR$, iteration gives $f\in 1+p^{k}R$ for every $k$. Since ${\bigcap }_k{p}^{k}R=0$, $f=1$.

To identify all fixed points, we also need $Nf\equiv f(\mathrm{mod}pR)$ for every unit $f$. In characteristic $p$, the extension ${\mathbb{F}}_p[[T]]/{\mathbb{F}}_p[[T^p]]$ has degree $p$ and its norm for a finite free ring extension is $\overline{f}\mapsto {\overline{f}}^p$. Indeed, after passing to fraction fields it is a purely inseparable field extension of degree $p$, so multiplication has only the eigenvalue $\overline{f}$, with multiplicity $p$, after splitting. Its determinant is ${\overline{f}}^p$. Thereforeand congruence reflection for Frobenius substitution gives $Nf\equiv f(\mathrm{mod}pR)$.

Now choose any lift $f_0\in R^{\times }$ of a given $a\in A^{\times }$ and put $f_n=N^n{f}_0$. Multiplicativity givesThus $(f_n)$ is a Cauchy sequence for the finer of the topologies on integral formal power series and converges to a unit $w(a)$ reducing to $a$. The Coleman norm operator is continuous: multiplication matrices and their determinants are continuous, and the finite-free coordinate decomposition and its inverse preserve the $p$-adic filtration defining the finer of the topologies on integral formal power series. Hence $Nw(a)=w(a)$. If two fixed units reduce to $a$, their quotient is fixed and belongs to $1+pR$, so it is $1$ by the preceding argument. This proves uniqueness, independence of the initial lift, and multiplicativity of the Coleman norm-fixed lift. In particular,Every unit $f\in R$ consequently has a unique factorization $f=w(\overline{f})v$ with $v\in 1+pR$, giving the canonical Coleman decomposition of power-series units:The final assertion as printed is false. The original PDF identifies a ring with a product of unit groups. With multiplication, $A$ has a zero and is not a group. With addition, it has exponent $p$, whereas $W\times A^{\times }$ has a nonidentity element of order $2$: $-1\in W$ because $p$ is odd and $N(-1)=(-1{)}^p=-1$. Thus neither interpretation makes the printed assertion true. The two boxed isomorphisms above are the valid natural conclusions; they do not require guessing which symbols the author intended to replace.

The formal power series ring $R={\mathbb{Z}}_p[[T]]$ is complete for two useful topologies. In the $p$-adic topology, convergence means that all coefficient errors are uniformly divisible by arbitrarily large powers of $p$. In the $(p,T)$-adic topology, only finitely many coefficients are controlled at each precision. The latter topology is compact, since $R/(p,T{)}^n$ is finite and $R$ is its inverse limit. The former is finer and is not compact: the series $T^n$ converge to zero in the latter but remain pairwise separated modulo $p$ in the former. Coleman norm contraction uses the former; compact interpolation arguments use the latter.