p-ramified Iwasawa module ($X=\mathrm{Gal}(M_{\infty }/F_{\infty })$)

Let $M_{\infty }$ be the maximal abelian pro-$p$ extension of $F_{\infty }$ unramified away from primes over $p$. Its Galois group is the p-ramified Iwasawa module. In the cyclotomic tower of $\mathbb{Q}({\mu }_p)$ for odd $p$, its Iwasawa-module rank is $(p-1)/2$. It is therefore quite different from the torsion unramified Iwasawa module.

Passing to norm limits in Artin reciprocity gives an exact sequence connecting closures of global units, local pro-$p$ units at primes over $p$, the p-ramified Iwasawa module and the unramified Iwasawa module. It explains arithmetic Iwasawa-module ranks by comparing local and global unit ranks. Roots of unity and the precise splitting conventions must be retained when seeking exact integral statements.

A Coleman power series encodes a norm-compatible sequence of local units in a cyclotomic tower by a single integral power series evaluated at ${\zeta }_{p^n}-1$. A logarithmic derivative and trace correction give a measure, relating the local images of cyclotomic units to p-adic L-functions.