Global Units Modulo Circular Units: Descent Without Iwasawa's Main Conjecture

http://dx.doi.org/10.4153/CJM-2009-027-0
Canad. J. Math. 61(2009), 518-533

  Published:2009-06-01
 Printed: Jun 2009

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 $\la$ and $\mu$ attached to the inverse limit $X_\infty=\varprojlim X_n$. 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, \emph{i.e.,} without using Iwasawa's Main Conjecture.