Indices isotypiques des \'el\'ements 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 $\displaystyle (H^1(G_S,\mathbb{Z}_p(r))^\chi: C^F(r)^\chi)$ 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 $p-1$.