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

TL;DR

This paper demonstrates that the unit class groups, formed by global units modulo circular units in abelian cyclotomic extensions, satisfy descent properties similar to ideal class groups without relying on Iwasawa's Main Conjecture.

Contribution

It establishes the descent properties of unit class groups directly, bypassing the need for Iwasawa's Main Conjecture in the analysis.

Findings

Unit class groups satisfy good descent properties.

Asymptotic equivalence of unit class groups and ideal class numbers.

Descent properties proven without Iwasawa's Main Conjecture.

Abstract

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a $\ZM_p$-extension $F_\infty/F$ of a number field $F$, to Iwasawa structural invariants $\la$ and $\mu$ attached to the inverse limit $X_\infty=\limpro X_n$. 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, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa's Main Conjecture.