# On the classgroups of imaginary abelian fields

Annales de l'institut Fourier (1990)

- Volume: 40, Issue: 3, page 467-492
- ISSN: 0373-0956

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topSolomon, David. "On the classgroups of imaginary abelian fields." Annales de l'institut Fourier 40.3 (1990): 467-492. <http://eudml.org/doc/74885>.

@article{Solomon1990,

abstract = {Let $p$ be an odd prime, $\chi $ an odd, $p$-adic Dirichlet character and $K$ the cyclic imaginary extension of $\{\bf Q\}$ associated to $\chi $. We define a “$\chi $-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 $B_\{1,\chi ^\{-1\}\}$. This uses the results of Mazur and Wiles in Iwasawa theory over $\{\bf Q\}$. The more difficult case, in which $p$ divides the order of $\chi $ is our chief concern. In this case the result is new and confirms an earlier conjecture of G. Gras.},

author = {Solomon, David},

journal = {Annales de l'institut Fourier},

keywords = {p-adic L-function; main conjecture; imaginary abelian fields; p-adic Dirichlet character; class group; p-divisibility; generalized Bernoulli number; Iwasawa theory},

language = {eng},

number = {3},

pages = {467-492},

publisher = {Association des Annales de l'Institut Fourier},

title = {On the classgroups of imaginary abelian fields},

url = {http://eudml.org/doc/74885},

volume = {40},

year = {1990},

}

TY - JOUR

AU - Solomon, David

TI - On the classgroups of imaginary abelian fields

JO - Annales de l'institut Fourier

PY - 1990

PB - Association des Annales de l'Institut Fourier

VL - 40

IS - 3

SP - 467

EP - 492

AB - Let $p$ be an odd prime, $\chi $ an odd, $p$-adic Dirichlet character and $K$ the cyclic imaginary extension of ${\bf Q}$ associated to $\chi $. We define a “$\chi $-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 $B_{1,\chi ^{-1}}$. This uses the results of Mazur and Wiles in Iwasawa theory over ${\bf Q}$. The more difficult case, in which $p$ divides the order of $\chi $ is our chief concern. In this case the result is new and confirms an earlier conjecture of G. Gras.

LA - eng

KW - p-adic L-function; main conjecture; imaginary abelian fields; p-adic Dirichlet character; class group; p-divisibility; generalized Bernoulli number; Iwasawa theory

UR - http://eudml.org/doc/74885

ER -

## References

top- [1] L. FEDERER, B. GROSS, Regulators and Iwasawa Modules, Inventiones Mathematicae, 62 (1981), 443-457. Zbl0468.12005MR83f:12005
- [2] B. FERRERO & R. GREENBERG, On the Behavior of the p-Adic L-function at s = 0, Inventiones Mathematicae, 50 (1978), 91-102. Zbl0441.12003MR80f:12016
- [3] B. FERRERO & L. WASHINGTON, The Iwasawa invariant µp vanishes for abelian number fields, Annals of Math., 109 (1979), 377-396. Zbl0443.12001MR81a:12005
- [4] G. GRAS, Etude d'invariants relatifs aux groupes des classes des corps abéliens, Astérisque, 41-42 (1977), 35-53. Zbl0445.12002MR56 #5489
- [5] R. GREENBERG, On p-Adic L-functions and Cyclotomic Fields II, Nagoya Math. J., 67 (1977), 139-158. Zbl0373.12007MR56 #2964
- [6] K. IWASAWA, Riemann-Hurwitz Formula and p-Adic Galois Representations for Number Fields, Tôhoku Math. J., 33 (1981), 263-288. Zbl0468.12004MR83b:12003
- [7] B. MAZUR & A. WILES, Class Fields of Abelian Extensions of ℚ, Inventiones Mathematicae, 76 (1984), 179-330. Zbl0545.12005MR85m:11069
- [8] K. RUBIN, Kolyvagin's System of Gauss Sums, Preprint. Zbl0727.11044
- [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. Zbl0704.11038
- [10] D. SOLOMON, On Lichtenbaum's Conjecture in the Case of Number Fields, PhD. Thesis, Brown University, 1988.

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