O ct 2 00 6 Global Units modulo Circular Units : descent without Iwasawa ’ s Main Conjecture . ∗

Iwasawa’s classical asymptotical formula relates the orders of the p-parts Xn of the ideal class groups along a Zp-extension F∞/F of a number field F , to Iwasawa structural invariants λ and μ attached to the inverse limit X∞ = lim ← Xn. It relies on ”good” descent properties satisfied by Xn. If F is abelian and F∞ is cyclotomic it is known that the p-parts of the orders of the global units modulo circular units Un/Cn are asymptotically equivalent to the p-parts of the ideal class numbers. This suggests that these quotients Un/Cn, so to speak unit class groups, satisfy also good descent properties. We show this directly, i.e. without using Iwasawa’s Main Conjecture. 0 Introduction LetK be a number field and p an odd prime (p 6= 2) and letK∞/K be a Zp-extension (quite soon K∞/K will be the cyclotomic Zp-extension). Recall the usual notations : Γ = Gal(K∞/K) is the Galois group of K∞/K, Kn is the n -layer of K∞ (so that [Kn : K] = p ), Γn = Gal(K∞/Kn), and Gn = Gal(Kn/K) ∼= Γ/Γn. Let us consider a sequence (Mn)n∈N of Zp[Gn]-modules equipped with norm maps Mn −→ Mn−1 and the inverse limit of this sequence M∞ = lim ← Mn seen as a Λ = Zp[[Γ]]-module. The general philosophy of Iwasawa theory is to study the simpler Λ-structure of M∞, then to try and recollect information about the Mn’s themselves from that structure. For instance, if M∞ is Λ-torsion, one can attach two invariants λ and μ to M∞. If we assume further that (i) the Γn coinvariants (M∞)Γn are finite, ∗2000 Mathematics Subject Classification. Primary 11R23.