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

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.

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.

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     title = {On the classgroups of imaginary abelian fields},
     journal = {Annales de l'Institut Fourier},
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Solomon, David. On the classgroups of imaginary abelian fields. Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 467-492. doi : 10.5802/aif.1221. https://aif.centre-mersenne.org/articles/10.5802/aif.1221/

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