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


Published:20090601
Printed: Jun 2009
Abstract
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.