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

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

2009-12-13

arxiv.org/abs/0912.2528v1

Canadian Journal of Mathematics (Journal Canadien de Math\'ematiques) 61, 3 (2009) 518--533

Mathematics

Number Theory

Scientific paper

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.

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