Class groups of abelian fields, and the main conjecture

Greither, Cornelius

Annales de l'Institut Fourier, Tome 42 (1992) no. 3, pp. 449-499.

Le début de cet article est consacré à une démonstration de la Conjecture Principale en théorie d’Iwasawa, le cas $p=2$ étant inclus, par la méthode de systèmes eulériens due à Kolyvagin. Au cours de cette démonstration on obtient un résultat assez général sur le groupe quotient des unités semilocales par les unités cyclotomiques. Ensuite, on en tire des théorèmes donnant l’ordre des parties $\chi$ de certains groupes de classes pour les corps abéliens sur $\mathbb{Q}$. D’abord, on traite des groupes de classes relatives comme Solomon vient de le fait pour $p$ impair, et puis, les groupes de classes des corps abéliens réels. Ces méthodes permettent aussi une généralisation de la conjecture de Gras.

This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case $p=2$, by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of $\chi$-parts of $p$-class groups of abelian number fields: first for relative class groups of real fields (again including the case $p=2$). As a consequence, a generalization of the Gras conjecture is stated and proved.

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