Published online by Cambridge University Press: 20 November 2018
Iwasawa's classical asymptotical formula relates the orders of the $p$-parts ${{X}_{n}}$ of the ideal class groups along a ${{\mathbb{Z}}_{p}}$-extension ${{F}_{\infty }}/F$ of a number field $F$ to Iwasawa structural invariants $\lambda $ and $\mu $ attached to the inverse limit ${{X}_{\infty }}=\underleftarrow{\lim }\,{{X}_{n}}$. It relies on “good” descent properties satisfied by ${{X}_{n}}$. If $F$ is abelian and ${{F}_{\infty }}$ is cyclotomic, it is known that the $p$-parts of the orders of the global units modulo circular units ${{U}_{n}}/{{C}_{n}}$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients ${{U}_{n}}/{{C}_{n}}$, so to speak unit class groups, also satisfy good descent properties. We show this directly, i.e., without using Iwasawa's Main Conjecture.