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},
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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. http://archive.numdam.org/articles/10.5802/aif.1221/

[1] L. Federer, B. Gross, Regulators and Iwasawa Modules, Inventiones Mathematicae, 62 (1981), 443-457. | MR | Zbl

[2] B. Ferrero & R. Greenberg, On the Behavior of the p-Adic L-function at s = 0, Inventiones Mathematicae, 50 (1978), 91-102. | MR | Zbl

[3] B. Ferrero & L. Washington, The Iwasawa invariant µp vanishes for abelian number fields, Annals of Math., 109 (1979), 377-396. | MR | Zbl

[4] G. Gras, Etude d'invariants relatifs aux groupes des classes des corps abéliens, Astérisque, 41-42 (1977), 35-53. | Numdam | MR | Zbl

[5] R. Greenberg, On p-Adic L-functions and Cyclotomic Fields II, Nagoya Math. J., 67 (1977), 139-158. | MR | Zbl

[6] K. Iwasawa, Riemann-Hurwitz Formula and p-Adic Galois Representations for Number Fields, Tôhoku Math. J., 33 (1981), 263-288. | MR | Zbl

[7] B. Mazur & A. Wiles, Class Fields of Abelian Extensions of ℚ, Inventiones Mathematicae, 76 (1984), 179-330. | MR | Zbl

[8] K. Rubin, Kolyvagin's System of Gauss Sums, Preprint. | Zbl

[9] K. Rubin, The Main Conjecture. Appendix to : Cyclotomic Fields I and II, combined 2nd edition, by S. Lang. Grad. Texts in Math., 121, Springer-Verlag, New York (1990), 397-419. | Zbl

[10] D. Solomon, On Lichtenbaum's Conjecture in the Case of Number Fields, PhD. Thesis, Brown University, 1988. | MR

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