no. 3
On the classgroups of imaginary abelian fields
Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 467-492

Let p be an odd prime, χ an odd, p-adic Dirichlet character and K the cyclic imaginary extension of Q associated to χ. We define a “χ-part” of the Sylow p-subgroup of the class group of K and prove a result relating its p-divisibility to that of the generalized Bernoulli number B 1,χ -1 . This uses the results of Mazur and Wiles in Iwasawa theory over Q. The more difficult case, in which p divides the order of χ is our chief concern. In this case the result is new and confirms an earlier conjecture of G. Gras.

Soit p un nombre premier impair, soit χ un caractère impair de Dirichlet p-adique et soit K l’extension cyclique imaginaire de Q associée à χ. On définit une “χ-partie” du p-sous-groupe de Sylow du groupe de classe de K et on démontre un résultat établissant un lien entre sa p-divisibilité et celle du nombre de Bernoulli généralisé B 1,χ -1 . On utilise les résultats de Mazur et Wiles de la Théorie d’Iwasawa sur Q. Nous nous intéressons principalement au cas plus difficile où p divise l’ordre de χ. Dans cette situation le résultat est nouveau et confirme une conjecture de G. Gras.

Solomon, David. On the classgroups of imaginary abelian fields. Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 467-492. doi: 10.5802/aif.1221
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