Coleman interpolation theorem for cyclotomic local units ($f_u({\zeta }_n-1)=u_n,N{f}_u=f_u$)

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 $f_u(0)\equiv 1(\mathrm{mod}p)$.