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

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

Sous-modules d'unités en théorie d'Iwasawa.

We give a necessary and sufficient "Galois descent" condition to the freeness of the Iwasawa module built from Sinnott's circular units. Then we describe explicit examples for which this condition is not fulfilled

Indices isotypiques des éléments cyclotomiques.

25 pages, revised version, accepted for publication by Tokyo J. Maths.Given $F$ a real abelian field, $p$ an odd prime and $\chi$ any Dirichlet character of $F$ we give a method for computing the $\chi$-index $\displaystyle \left (H^1(G_S,\mathbb{Z}_p(r))^\chi: C^F(r)^\chi\right)$ where the Tate twist $r$ is an odd integer $r\geq 3$, the group $C^F(r)$ is the group of higher circular units, $G_S$ is the Galois group over $F$ of the maximal $S$ ramified algebraic extension of $F$, and $S$ is the set of places of $F$ dividing $p$. This $\chi$-index can now be computed in terms only of elementary arithmetic of finite fields \FM_\ell. Our work generalizes previous results by Kurihara who used the assumption that the order of $\chi$ divides $p-1$

Sur la torsion de la distribution ordinaire universelle attachée à un corps de nombres

International audienceWe study the torsion subgroup of the universal ordinary distribution related to a general number field. We describe a way to control this subgroup. We apply this method to the special case of an imaginary quadratic field, and we give examples of such fields where these torsion subgroups are non-trivial

Indices isotypiques des éléments cyclotomiques.

Given $F$ a real abelian field, $p$ an odd prime and $\chi$ any Dirichlet character of $F$ we give a method for computing the $\chi$-index $\displaystyle \left (H^1(G_S,\mathbb{Z}_p(r))^\chi: C^F(r)^\chi\right)$ where the Tate twist $r$ is an odd integer $r\geq 3$, the group $C^F(r)$ is the group of higher circular units, $G_S$ is the Galois group over $F$ of the maximal $S$ ramified algebraic extension of $F$, and $S$ is the set of places of $F$ dividing $p$. This $\chi$-index can now be computed in terms only of elementary arithmetic of finite fields \FM_\ell. Our work generalizes previous results by Kurihara who used the assumption that the order of $\chi$ divides $p-1$

On modified circular units and annihilation of real classes

For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae