Formules de classes pour les corps abéliens réels

Nous montrons des raffinements $p$-adique et “caractères par caractères” de la formule d’indice de Sinnott pour un corps abélien totalement réel. De tels raffinements ont aussi été obtenus par Kuz’min avec des méthodes différentes (voir les commentaires en introduction). Nous donnons des applications à la théorie d’Iwasawa des unités semi- locales et cyclotomiques.

We show $p$-adic and “character by character” refinements of Sinnott’s index formula for a totally real abelian number field. Such refinements have also been obtained by Kuz’min by different methods (but see comments in the introduction). Applications are given to Iwasawa theory of semi-local units and cyclotomic units.

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