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

@article{AIF_1990__40_3_467_0,
     author = {Solomon, David},
     title = {On the classgroups of imaginary abelian fields},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {40},
     number = {3},
     year = {1990},
     pages = {467-492},
     doi = {10.5802/aif.1221},
     zbl = {0694.12004},
     mrnumber = {92a:11133},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1990__40_3_467_0}
}
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. http://www.numdam.org/item/AIF_1990__40_3_467_0/

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