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

Belliard, Jean-Robert

Canadian journal of mathematics, Tome 61 (2009) no. 3, pp. 518-533

Résumé

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 $\lambda $ and $\mu $ attached to the inverse limit ${{X}_{\infty }}=\underleftarrow{\lim }\,{{X}_{n}}$ . 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, i.e., without using Iwasawa's Main Conjecture.

Détail

DOI : 10.4153/CJM-2009-027-0

Mots-clés : 11R23

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