Global Units modulo Circular Units : descent without Iwasawa's Main Conjecture

Iwasawa’s classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a ${\text{ZM}}_p$-extension $F_{\mathrm{\infty }}\mathrm{/}F$ of a number field $F$, to Iwasawa structural invariants $\text{la}$ and $\mu$ attached to the inverse limit $X_{\mathrm{\infty }}=\text{limpro}{X}_n$. It relies on “good” descent properties satisfied by $X_n$. If $F$ is abelian and $F_{\mathrm{\infty }}$ is cyclotomic it is known that the $p$-parts of the orders of the global units modulo circular units $U_n\mathrm{/}{C}_n$ are asymptotically equivalent to the $p$-parts of the ideal class numbers. This suggests that these quotients $U_n\mathrm{/}{C}_n$, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa’s Main Conjecture.