Unités cyclotomiques, unités semi-locales et ${\mathbb{Z}}_{\ell }$-extensions. II

Gillard, Roland

Annales de l'Institut Fourier, Tome 29 (1979) no. 4, pp. 1-15.

Soient $K$ un corps abélien réel, $\ell$ un nombre premier, premier à $[K:\mathbf{Q}]$ et $Y_n$ le quotient du groupe des unités semi-locales de $K\left(\sqrt[{\ell }^n]{1}\right)$ par celui des unités cyclotomiques : on donne la structure galoisienne de la limite projective des $Y_n$, généralisant un théorème d’Iwasawa, et on applique ceci à la comparaison de conjecture classique sur la limite projective des groupes de classes.

Let $K$ an abelian number field, $\ell$ a prime number, prime to $[K:\mathbf{Q}]$, $Y_n$ the quotient of the group of semi-local units in $K\left(\sqrt[{\ell }^n]{1}\right)$ by the group of cyclotomic units. By giving the Galois structure of ${\displaystyle \underleftarrow{lim}{Y}_n}$, we generalise a theorem of Iwasawa and use this result for comparing classical conjectures about projective limits of class groups.

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